diff --git a/.github/workflows/continued-kappa-research.yml b/.github/workflows/continued-kappa-research.yml
new file mode 100644
index 0000000000..bcc90e948b
--- /dev/null
+++ b/.github/workflows/continued-kappa-research.yml
@@ -0,0 +1,22 @@
+name: Verify shared entrance gauge construction
+on:
+  pull_request:
+    paths:
+      - 'research/coordinated-crossover-pr200/**'
+      - '.github/workflows/continued-kappa-research.yml'
+  workflow_dispatch:
+permissions:
+  contents: read
+jobs:
+  immutable-offline:
+    runs-on: ubuntu-24.04
+    timeout-minutes: 45
+    steps:
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
+        with:
+          python-version: '3.14'
+      - run: python3 -B research/coordinated-crossover-pr200/verify.py
+      - run: git diff --exit-code
diff --git a/.github/workflows/five-stage-source527-banks.yml b/.github/workflows/five-stage-source527-banks.yml
new file mode 100644
index 0000000000..b04fbb0c09
--- /dev/null
+++ b/.github/workflows/five-stage-source527-banks.yml
@@ -0,0 +1,27 @@
+name: Verify five-stage source527 banks
+on:
+  pull_request:
+    paths:
+      - 'research/five-stage-source527-banks/**'
+      - '.github/workflows/five-stage-source527-banks.yml'
+  workflow_dispatch:
+permissions:
+  contents: read
+jobs:
+  immutable-offline:
+    strategy:
+      fail-fast: false
+      matrix:
+        python: ['3.11', '3.13']
+    runs-on: ubuntu-24.04
+    timeout-minutes: 45
+    steps:
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
+        with:
+          python-version: ${{ matrix.python }}
+      - run: python3 -m pip install -r research/five-stage-source527-banks/requirements.txt
+      - run: python3 -B research/five-stage-source527-banks/verify.py --output "$RUNNER_TEMP/five-stage-source527-verification"
+      - run: git diff --exit-code
diff --git a/certificates/five-stage-source527-banks-certificate.json b/certificates/five-stage-source527-banks-certificate.json
new file mode 100644
index 0000000000..2226306c63
--- /dev/null
+++ b/certificates/five-stage-source527-banks-certificate.json
@@ -0,0 +1,221 @@
+{
+  "assembly": {
+    "absorption_gap": "1250279579073467285345512074190197026907819906920435464620488918066153108666297669/1954512839642729925891856438544184164269186757889391397973415959412569525898931681000000000000000000",
+    "margins": {
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+      "balanced_prefix": "1387839644683050891856438544184164269186757889391397973415959412569525898931681/1954512839642729925891856438544184164269186757889391397973415959412569525898931681",
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+    },
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+      },
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+    },
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+    },
+    "m": 110,
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+    "rank_mass": 1613040
+  },
+  "finite_bridge": {
+    "proof": "PROOF.md",
+    "representation": "Exact powers with source-bound finite overcharges",
+    "rows": {
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+      "degree_gap": "23971789/25",
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+        "exponent": 30000
+      },
+      "induction_gap_lower": 1,
+      "strict_literal_gap": 1
+    }
+  },
+  "independent_base_two_pass": true,
+  "kappa": "710069340338651/1000000000000000000",
+  "kappa_decimal": "0.000710069340338651",
+  "kappa_scientific": "7.100693403386509e-04",
+  "next_bit_grid_excluded": [
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+  ],
+  "ordinary_bootstrap_chain": [
+    "384599/10000000000",
+    "44381019416110288180090970134959/62500000000000000000000000000000000",
+    "11102711843119811022038583720240445172210982052089/15625000000000000000000000000000000000000000000000000",
+    "2775679285465403142283807894652896154162821571091121589974101068319/3906250000000000000000000000000000000000000000000000000000000000000000"
+  ],
+  "schema": "source527-parity-fused-five-stage-banks/1",
+  "scope": "Conditionalfiniteconstruction; inheritedallsizecompiler,weightedselector,commonancestorchart,restoredrow,routing,prime,recovery,complexsymbolicandanalyticinterfacesremainassumptions."
+}
diff --git a/certificates/five-stage-source527-banks-publication-audit.json b/certificates/five-stage-source527-banks-publication-audit.json
new file mode 100644
index 0000000000..38f901119e
--- /dev/null
+++ b/certificates/five-stage-source527-banks-publication-audit.json
@@ -0,0 +1,86 @@
+{
+  "all_inputs_unchanged": true,
+  "certificate_sha256": "90a94f6e72999b556ecc043dc9a54efecbe96503cee9dbe5838d6726904ec161",
+  "files": 127,
+  "independent_replay": {
+    "files": 127,
+    "fresh_stages": [
+      "banks",
+      "bit",
+      "complex",
+      "finite",
+      "math",
+      "primes",
+      "raw",
+      "scalar"
+    ],
+    "inputs_unchanged": true,
+    "kappa": "710069340338651/1000000000000000000",
+    "kappa_scientific": "7.100693403386509e-04",
+    "manifest_sha256": "3851372231a7baaa86e1abd8b41407aab7450c2d76d0487f1fea4f806f70c90b",
+    "negative_controls": [
+      "omitted banks stage",
+      "omitted bit stage",
+      "omitted complex stage",
+      "omitted finite stage",
+      "omitted math stage",
+      "omitted primes stage",
+      "omitted raw stage",
+      "omitted scalar stage"
+    ],
+    "seconds": 352.48472391601535,
+    "status": "PASS_IMMUTABLE_PARITY_FUSED_SOURCE527_FIVE_STAGE_BANKED_CONSTRUCTION",
+    "timings": {
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+    }
+  },
+  "manifest_sha256": "3851372231a7baaa86e1abd8b41407aab7450c2d76d0487f1fea4f806f70c90b",
+  "published_package_byte_identical": true,
+  "root_replay": {
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+      "complex",
+      "finite",
+      "math",
+      "primes",
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+      "scalar"
+    ],
+    "inputs_unchanged": true,
+    "kappa": "710069340338651/1000000000000000000",
+    "kappa_scientific": "7.100693403386509e-04",
+    "manifest_sha256": "3851372231a7baaa86e1abd8b41407aab7450c2d76d0487f1fea4f806f70c90b",
+    "negative_controls": [
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+    }
+  },
+  "status": "PASS_TWO_IDENTICAL_IMMUTABLE_PARITY_PACKAGE_REPLAYS"
+}
diff --git a/certificates/five-stage-source527-banks-verification.json b/certificates/five-stage-source527-banks-verification.json
new file mode 100644
index 0000000000..6e39b0abfd
--- /dev/null
+++ b/certificates/five-stage-source527-banks-verification.json
@@ -0,0 +1,39 @@
+{
+  "files": 127,
+  "fresh_stages": [
+    "banks",
+    "bit",
+    "complex",
+    "finite",
+    "math",
+    "primes",
+    "raw",
+    "scalar"
+  ],
+  "inputs_unchanged": true,
+  "kappa": "710069340338651/1000000000000000000",
+  "kappa_scientific": "7.100693403386509e-04",
+  "manifest_sha256": "3851372231a7baaa86e1abd8b41407aab7450c2d76d0487f1fea4f806f70c90b",
+  "negative_controls": [
+    "omitted banks stage",
+    "omitted bit stage",
+    "omitted complex stage",
+    "omitted finite stage",
+    "omitted math stage",
+    "omitted primes stage",
+    "omitted raw stage",
+    "omitted scalar stage"
+  ],
+  "seconds": 352.77735449999454,
+  "status": "PASS_IMMUTABLE_PARITY_FUSED_SOURCE527_FIVE_STAGE_BANKED_CONSTRUCTION",
+  "timings": {
+    "banks": 48.09752395801479,
+    "bit": 37.713670458993874,
+    "complex": 31.69509837499936,
+    "finite": 0.13805145799415186,
+    "math": 0.4710108329891227,
+    "primes": 11.649015916016651,
+    "raw": 8.302082208014326,
+    "scalar": 193.73732774998643
+  }
+}
diff --git a/research/balanced-nullity-banks-163/AUDIT-TARGETS.json b/research/balanced-nullity-banks-163/AUDIT-TARGETS.json
new file mode 100644
index 0000000000..31acd84088
--- /dev/null
+++ b/research/balanced-nullity-banks-163/AUDIT-TARGETS.json
@@ -0,0 +1,54 @@
+[
+  "Mathproof.BeyondNLogN.TakeoverBit.partialSwap_apply",
+  "Mathproof.BeyondNLogN.TakeoverBit.partialSwap_zero",
+  "Mathproof.BeyondNLogN.TakeoverBit.partialSwap_comp_disjoint",
+  "Mathproof.BeyondNLogN.TakeoverBit.partialSwap_involutive",
+  "Mathproof.BeyondNLogN.TakeoverBit.partialSwap_id",
+  "Mathproof.BeyondNLogN.TakeoverBit.partialSwap_commute",
+  "Mathproof.BeyondNLogN.TakeoverBit.bankSwap_eq_sum",
+  "Mathproof.BeyondNLogN.TakeoverBit.bankSwap_full",
+  "Mathproof.BeyondNLogN.TakeoverBit.weighted_apply",
+  "Mathproof.BeyondNLogN.TakeoverBit.weighted_comp",
+  "Mathproof.BeyondNLogN.TakeoverBit.weighted_id",
+  "Mathproof.BeyondNLogN.TakeoverBit.weightedBank_eq",
+  "Mathproof.BeyondNLogN.TakeoverBit.weightedBank_full",
+  "Mathproof.BeyondNLogN.TakeoverBit.coordinateProjector_apply",
+  "Mathproof.BeyondNLogN.TakeoverBit.coordinateProjector_idempotent",
+  "Mathproof.BeyondNLogN.TakeoverBit.coordinateProjector_disjoint",
+  "Mathproof.BeyondNLogN.TakeoverBit.coordinateProjector_sum",
+  "Mathproof.BeyondNLogN.TakeoverBit.coordinateBank_full",
+  "Mathproof.BeyondNLogN.TakeoverBit.bankAddress_bijective",
+  "Mathproof.BeyondNLogN.TakeoverBit.bankAddress_roles_injective",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.selected_block_capacity",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.undeferred_block_capacity",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.selected_bank_full",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.undeferred_bank_full",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.inherited_rank_mass",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.removed_exterior_count",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.packed_rank_mass",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.packed_max_rank",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.normalized_volume",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.packed_deficit",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.actual_role_counts",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.selected_weighted_bank_full",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.undeferred_weighted_bank_full",
+  "Mathproof.BeyondNLogN.TakeoverBit.Contents.run_eq_address",
+  "Mathproof.BeyondNLogN.TakeoverBit.Contents.run_full",
+  "Mathproof.BeyondNLogN.TakeoverBit.Contents.p12_selected_contents",
+  "Mathproof.BeyondNLogN.TakeoverBit.Contents.p12_undeferred_contents",
+  "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.inherited_rank_mass",
+  "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.removed_exterior_count",
+  "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.packed_rank_mass",
+  "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.packed_max_rank",
+  "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.normalized_volume",
+  "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.packed_deficit",
+  "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.actual_role_counts",
+  "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.actual_bank_and_inventory_certificate",
+  "Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008.literal_charge_lt",
+  "Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008.finite_row_reserve",
+  "Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008.row_reserve_value",
+  "Mathproof.BeyondNLogN.TakeoverBalanced16320261009.core_slacks_positive",
+  "Mathproof.BeyondNLogN.TakeoverBalanced16320261009.all_47_slacks_positive",
+  "Mathproof.BeyondNLogN.TakeoverBalanced16320261009.strengthened_public_margin",
+  "Mathproof.BeyondNLogN.TakeoverBalanced16320261009.all_47_concrete_guards"
+]
diff --git a/research/balanced-nullity-banks-163/BankAndBalanced.lean b/research/balanced-nullity-banks-163/BankAndBalanced.lean
new file mode 100644
index 0000000000..b681bd7acf
--- /dev/null
+++ b/research/balanced-nullity-banks-163/BankAndBalanced.lean
@@ -0,0 +1,625 @@
+/- Publication snapshot: actual bank endpoints and conditional finite arithmetic only. -/
+import Mathlib.LinearAlgebra.Prod
+import Mathlib.Tactic.Abel
+import Mathlib.Algebra.BigOperators.Group.List.Defs
+import Mathlib.Data.List.Pairwise
+import Mathlib.Data.List.FinRange
+import Mathlib.Data.Fintype.Card
+import Mathlib.Data.Nat.Basic
+import Mathlib.Tactic.NormNum
+import Mathlib.Tactic.FinCases
+import Mathlib.Tactic.GCongr
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Positivity
+import Mathlib.Data.Real.Basic
+import Mathlib.Tactic.NormNum
+
+
+/-! Exact partial bit address interchanges for split-projector bank packing.
+No supplier, routing, recurrence, precision, or tape theorem is assumed here. -/
+namespace Mathproof.BeyondNLogN.TakeoverBit
+
+variable {R V : Type*} [CommRing R] [AddCommGroup V] [Module R V]
+
+/-- Exchange the P-address component between two arbitrary incoming banks. -/
+def partialSwap (P : V →ₗ[R] V) : (V × V) →ₗ[R] (V × V) where
+  toFun z := (z.1 - P z.1 + P z.2, z.2 - P z.2 + P z.1)
+  map_add' a b := by
+    apply Prod.ext <;> simp only [Prod.fst_add, Prod.snd_add, map_add] <;> abel
+  map_smul' r z := by
+    apply Prod.ext <;> simp only [Prod.smul_fst, Prod.smul_snd, map_smul,
+      RingHom.id_apply, smul_add, smul_sub]
+
+@[simp] theorem partialSwap_apply (P : V →ₗ[R] V) (z : V × V) :
+    partialSwap P z = (z.1 - P z.1 + P z.2, z.2 - P z.2 + P z.1) := rfl
+
+@[simp] theorem partialSwap_zero : partialSwap (0 : V →ₗ[R] V) = LinearMap.id := by
+  apply LinearMap.ext
+  intro z
+  simp
+
+/-- Every disjoint projector pair composes by adding its exchanged components.
+The incoming pair is arbitrary, so correlated dirty contents are included. -/
+theorem partialSwap_comp_disjoint (P Q : V →ₗ[R] V)
+    (hPQ : P.comp Q = 0) (hQP : Q.comp P = 0) :
+    (partialSwap P).comp (partialSwap Q) = partialSwap (P + Q) := by
+  have hpq (z : V) : P (Q z) = 0 := by
+    simpa using LinearMap.congr_fun hPQ z
+  have hqp (z : V) : Q (P z) = 0 := by
+    simpa using LinearMap.congr_fun hQP z
+  apply LinearMap.ext
+  intro z
+  rcases z with ⟨x,y⟩
+  apply Prod.ext <;>
+    simp only [LinearMap.comp_apply, partialSwap_apply, map_add, map_sub,
+      LinearMap.add_apply, hpq] <;> abel
+
+/-- A completed split-projector core is its own exact inverse. -/
+theorem partialSwap_involutive (P : V →ₗ[R] V) (hP : P.comp P = P) :
+    Function.Involutive (partialSwap P) := by
+  have hpp (z : V) : P (P z) = P z := by
+    simpa using LinearMap.congr_fun hP z
+  intro z
+  rcases z with ⟨x,y⟩
+  apply Prod.ext <;> simp only [partialSwap_apply, map_add, map_sub, hpp] <;> abel
+
+@[simp] theorem partialSwap_id (z : V × V) :
+    partialSwap (LinearMap.id : V →ₗ[R] V) z = (z.2,z.1) := by
+  simp
+
+/-- Split-projector factors commute at the actual address-map level. -/
+theorem partialSwap_commute (P Q : V →ₗ[R] V)
+    (hPQ : P.comp Q = 0) (hQP : Q.comp P = 0) :
+    (partialSwap P).comp (partialSwap Q) =
+      (partialSwap Q).comp (partialSwap P) := by
+  rw [partialSwap_comp_disjoint P Q hPQ hQP,
+    partialSwap_comp_disjoint Q P hQP hPQ, add_comm]
+
+
+
+
+/-- Execute completed cores in any explicit chronological bank schedule. -/
+def bankSwap : List (V →ₗ[R] V) → ((V × V) →ₗ[R] (V × V))
+  | [] => LinearMap.id
+  | P :: Ps => (partialSwap P).comp (bankSwap Ps)
+
+private theorem comp_sum_zero (P : V →ₗ[R] V) (Ps : List (V →ₗ[R] V))
+    (h : ∀ Q ∈ Ps, P.comp Q = 0) : P.comp Ps.sum = 0 := by
+  induction Ps with
+  | nil => simp
+  | cons Q Ps ih =>
+    have hQ := h Q (by simp)
+    have ht : ∀ A ∈ Ps, P.comp A = 0 := fun A hA => h A (by simp [hA])
+    simp only [List.sum_cons, LinearMap.comp_add, hQ, ih ht, zero_add]
+
+private theorem sum_comp_zero (P : V →ₗ[R] V) (Ps : List (V →ₗ[R] V))
+    (h : ∀ Q ∈ Ps, Q.comp P = 0) : Ps.sum.comp P = 0 := by
+  induction Ps with
+  | nil => simp
+  | cons Q Ps ih =>
+    have hQ := h Q (by simp)
+    have ht : ∀ A ∈ Ps, A.comp P = 0 := fun A hA => h A (by simp [hA])
+    simp only [List.sum_cons, LinearMap.add_comp, hQ, ih ht, zero_add]
+
+/-- A whole shared bank executes exactly the swap on the sum of its disjoint
+address blocks. The statement is on every input, not only clean scratch. -/
+theorem bankSwap_eq_sum (Ps : List (V →ₗ[R] V))
+    (h : Ps.Pairwise (fun P Q => P.comp Q = 0 ∧ Q.comp P = 0)) :
+    bankSwap Ps = partialSwap Ps.sum := by
+  induction Ps with
+  | nil => simp [bankSwap]
+  | cons P Ps ih =>
+    rcases List.pairwise_cons.mp h with ⟨hp, ht⟩
+    rw [bankSwap, ih ht, List.sum_cons]
+    exact partialSwap_comp_disjoint P Ps.sum
+      (comp_sum_zero P Ps (fun Q hQ => (hp Q hQ).1))
+      (sum_comp_zero P Ps (fun Q hQ => (hp Q hQ).2))
+
+/-- A complete split partition needs no exterior auxiliary child. -/
+theorem bankSwap_full (Ps : List (V →ₗ[R] V))
+    (h : Ps.Pairwise (fun P Q => P.comp Q = 0 ∧ Q.comp P = 0))
+    (hsum : Ps.sum = LinearMap.id) (z : V × V) :
+    bankSwap Ps z = (z.2,z.1) := by
+  rw [bankSwap_eq_sum Ps h, hsum]
+  exact partialSwap_id z
+
+/-- One common ancestor-dependent weighted chart, retained for the whole bank. -/
+def weighted (A : (V × V) ≃ₗ[R] (V × V))
+    (T : (V × V) →ₗ[R] (V × V)) : (V × V) →ₗ[R] (V × V) :=
+  A.toLinearMap.comp (T.comp A.symm.toLinearMap)
+
+@[simp] theorem weighted_apply (A : (V × V) ≃ₗ[R] (V × V))
+    (T : (V × V) →ₗ[R] (V × V)) (z : V × V) :
+    weighted A T z = A (T (A.symm z)) := rfl
+
+/-- Chart conjugations telescope exactly, with no restriction on dirty data. -/
+theorem weighted_comp (A : (V × V) ≃ₗ[R] (V × V))
+    (T U : (V × V) →ₗ[R] (V × V)) :
+    (weighted A T).comp (weighted A U) = weighted A (T.comp U) := by
+  apply LinearMap.ext
+  intro z
+  simp
+
+@[simp] theorem weighted_id (A : (V × V) ≃ₗ[R] (V × V)) :
+    weighted A (LinearMap.id : (V × V) →ₗ[R] (V × V)) = LinearMap.id := by
+  apply LinearMap.ext
+  intro z
+  simp
+
+def weightedBank (A : (V × V) ≃ₗ[R] (V × V)) :
+    List (V →ₗ[R] V) → ((V × V) →ₗ[R] (V × V))
+  | [] => LinearMap.id
+  | P :: Ps => (weighted A (partialSwap P)).comp (weightedBank A Ps)
+
+/-- Every paid weighted core is retained in its chronological product. -/
+theorem weightedBank_eq (A : (V × V) ≃ₗ[R] (V × V))
+    (Ps : List (V →ₗ[R] V)) : weightedBank A Ps = weighted A (bankSwap Ps) := by
+  induction Ps with
+  | nil => simp [weightedBank, bankSwap]
+  | cons P Ps ih =>
+    rw [weightedBank, ih, weighted_comp, bankSwap]
+
+/-- Heterogeneous completed-core packing has the exact full weighted endpoint. -/
+theorem weightedBank_full (A : (V × V) ≃ₗ[R] (V × V))
+    (Ps : List (V →ₗ[R] V))
+    (h : Ps.Pairwise (fun P Q => P.comp Q = 0 ∧ Q.comp P = 0))
+    (hsum : Ps.sum = LinearMap.id) (z : V × V) :
+    weightedBank A Ps z = A ((A.symm z).2,(A.symm z).1) := by
+  rw [weightedBank_eq, weighted_apply, bankSwap_full Ps h hsum]
+
+
+
+section CoordinateBanks
+variable {ι κ : Type*} [DecidableEq ι]
+
+/-- Explicit split address blocks from an actual coordinate-owner ledger. -/
+def coordinateProjector (owner : κ → ι) (s : ι) :
+    (κ → R) →ₗ[R] (κ → R) where
+  toFun v k := if owner k = s then v k else 0
+  map_add' v w := by
+    funext k
+    by_cases h : owner k = s <;> simp [h]
+  map_smul' r v := by
+    funext k
+    by_cases h : owner k = s <;> simp [h]
+
+@[simp] theorem coordinateProjector_apply (owner : κ → ι) (s : ι)
+    (v : κ → R) (k : κ) :
+    coordinateProjector (R := R) owner s v k = if owner k = s then v k else 0 := rfl
+
+theorem coordinateProjector_idempotent (owner : κ → ι) (s : ι) :
+    (coordinateProjector (R := R) owner s).comp (coordinateProjector (R := R) owner s) =
+      coordinateProjector (R := R) owner s := by
+  apply LinearMap.ext
+  intro v
+  funext k
+  by_cases h : owner k = s <;> simp [h]
+
+theorem coordinateProjector_disjoint (owner : κ → ι) (s t : ι) (hst : s ≠ t) :
+    (coordinateProjector (R := R) owner s).comp (coordinateProjector (R := R) owner t) = 0 := by
+  apply LinearMap.ext
+  intro v
+  funext k
+  by_cases h : owner k = s
+  · have ht : owner k ≠ t := by simpa [h] using hst
+    simp [h,hst]
+  · simp [h]
+
+private theorem coordinateProjector_sum_apply (owner : κ → ι)
+    (roles : List ι) (hroles : roles.Nodup) (v : κ → R) (k : κ) :
+    (roles.map (coordinateProjector (R := R) owner)).sum v k =
+      if owner k ∈ roles then v k else 0 := by
+  induction roles with
+  | nil => simp
+  | cons s roles ih =>
+    rcases List.nodup_cons.mp hroles with ⟨hs, ht⟩
+    simp only [List.map_cons, List.sum_cons, LinearMap.add_apply, Pi.add_apply,
+      coordinateProjector_apply, ih ht, List.mem_cons]
+    by_cases he : owner k = s
+    · have hn : owner k ∉ roles := by simpa [he] using hs
+      simp [he,hs]
+    · simp [he]
+
+/-- A disjoint exhaustive coordinate ledger gives an actual identity projector. -/
+theorem coordinateProjector_sum (owner : κ → ι) (roles : List ι)
+    (hroles : roles.Nodup) (hcover : ∀ k, owner k ∈ roles) :
+    (roles.map (coordinateProjector (R := R) owner)).sum = LinearMap.id := by
+  apply LinearMap.ext
+  intro v
+  funext k
+  rw [coordinateProjector_sum_apply owner roles hroles v k]
+  simp [hcover k]
+
+/-- The concrete coordinate-owner construction supplies every projector premise
+of the heterogeneous bank endpoint, including arbitrary incoming correlations. -/
+theorem coordinateBank_full (owner : κ → ι) (roles : List ι)
+    (hroles : roles.Nodup) (hcover : ∀ k, owner k ∈ roles)
+    (z : (κ → R) × (κ → R)) :
+    bankSwap (roles.map (coordinateProjector (R := R) owner)) z = (z.2,z.1) := by
+  apply bankSwap_full
+  · rw [List.pairwise_map]
+    exact (List.nodup_iff_pairwise_ne.mp hroles).imp (fun hst =>
+      ⟨coordinateProjector_disjoint owner _ _ hst,
+        coordinateProjector_disjoint owner _ _ (Ne.symm hst)⟩)
+  · exact coordinateProjector_sum owner roles hroles hcover
+
+end CoordinateBanks
+
+section Routing
+variable {G S : Type*} [Group G]
+
+/-- Actual bank address for invocation g and fixed role embedding H_s. -/
+def bankAddress (H : S → G) (g : G) (s : S) : G := g * (H s)⁻¹
+
+/-- Every role visits every physical bank once; no favorable subset is selected. -/
+theorem bankAddress_bijective (H : S → G) (s : S) :
+    Function.Bijective (fun g => bankAddress H g s) := by
+  constructor
+  · intro g g' h
+    exact mul_right_cancel h
+  · intro b
+    refine ⟨b * H s, ?_⟩
+    simp [bankAddress, mul_assoc]
+
+/-- Distinct role embeddings rule out within-invocation physical aliasing. -/
+theorem bankAddress_roles_injective (H : S → G) (hH : Function.Injective H)
+    (g : G) : Function.Injective (bankAddress H g) := by
+  intro s t h
+  apply hH
+  have hi : (H s)⁻¹ = (H t)⁻¹ := mul_left_cancel h
+  exact inv_injective hi
+
+end Routing
+end Mathproof.BeyondNLogN.TakeoverBit
+
+
+
+
+
+/-! Concrete PR155 p12 bit orbit-bank construction and complete paid ledger.
+The inherited all-input local word, stage cover and routing contracts remain
+explicit upstream dependencies; this file proves the new address endpoints,
+coordinate capacities and exact inventory, not an all-size running-time claim. -/
+namespace Mathproof.BeyondNLogN.TakeoverBit.P12
+open Mathproof.BeyondNLogN.TakeoverBit
+
+variable {R : Type*} [CommRing R]
+
+/-- The 18 actual disjoint four-coordinate blocks for retained-gauge cores. -/
+def selectedOwner (j : Fin 72) : Fin 18 := ⟨j.val / 4, by omega⟩
+
+/-- The three actual disjoint 24-coordinate blocks for undeferred cores. -/
+def undeferredOwner (j : Fin 72) : Fin 3 := ⟨j.val / 24, by omega⟩
+
+theorem selected_block_capacity (s : Fin 18) :
+    (Finset.univ.filter (fun j : Fin 72 => selectedOwner j = s)).card = 4 := by
+  fin_cases s <;> decide
+
+theorem undeferred_block_capacity (s : Fin 3) :
+    (Finset.univ.filter (fun j : Fin 72 => undeferredOwner j = s)).card = 24 := by
+  fin_cases s <;> decide
+
+/-- Eighteen completed selected cores cover the entire address module. -/
+theorem selected_bank_full (z : (Fin 72 → R) × (Fin 72 → R)) :
+    bankSwap ((List.finRange 18).map (coordinateProjector (R := R) selectedOwner)) z =
+      (z.2,z.1) := by
+  exact coordinateBank_full selectedOwner (List.finRange 18)
+    (List.nodup_finRange 18) (fun k => by simp) z
+
+/-- Three completed undeferred cores also cover the entire address module. -/
+theorem undeferred_bank_full (z : (Fin 72 → R) × (Fin 72 → R)) :
+    bankSwap ((List.finRange 3).map (coordinateProjector (R := R) undeferredOwner)) z =
+      (z.2,z.1) := by
+  exact coordinateBank_full undeferredOwner (List.finRange 3)
+    (List.nodup_finRange 3) (fun k => by simp) z
+
+/-- Every child in the immutable freshly replayed p12 profile, including rank60. -/
+def inheritedHistogram : List (Nat × Nat) :=
+  [(1,120468),(2,68080),(3,28824),(4,7200),(5,11952),(6,3312),
+   (7,6120),(8,4464),(9,1440),(10,2016),(11,2664),(12,3528),
+   (13,1944),(14,2736),(15,1152),(16,4104),(17,6192),(18,5832),
+   (19,4536),(20,7260),(21,8580),(22,15912),(60,5720)]
+
+/-- The actual coordinate-full orbit banks remove precisely auxiliary exteriors;
+all internal, source, target, center and data exterior entries remain charged. -/
+def packedHistogram : List (Nat × Nat) :=
+  inheritedHistogram.filter (fun p => p.1 != 60)
+
+def rankMass (H : List (Nat × Nat)) : Nat :=
+  (H.map (fun p => p.1*p.2)).sum
+
+theorem inherited_rank_mass : rankMass inheritedHistogram = 2096432 := by decide
+
+theorem removed_exterior_count :
+    (inheritedHistogram.filter (fun p => p.1 == 60)).map Prod.snd = [5720] := by decide
+
+theorem packed_rank_mass : rankMass packedHistogram = 1753232 := by decide
+
+theorem packed_max_rank : ∀ p ∈ packedHistogram, p.1 ≤ 22 := by decide
+
+/-- Three private stage banks for every full orbit, including fractional stock. -/
+def normalizedVolume : ℚ := 2*1760 + 3*((5720:ℚ)/18 + (19904:ℚ)/3)
+
+theorem normalized_volume : normalizedVolume = 73132/3 := by
+  norm_num [normalizedVolume]
+
+theorem packed_deficit : (72:ℚ)*normalizedVolume - rankMass packedHistogram = 1936 := by
+  rw [packed_rank_mass, normalized_volume]
+  norm_num
+
+theorem actual_role_counts :
+    5720 + 19904 = (25624:Nat) ∧ 18*4 = (72:Nat) ∧ 3*24 = (72:Nat) := by decide
+
+
+
+
+/-- The actual common weighted chart retains the selected-bank full endpoint. -/
+theorem selected_weighted_bank_full
+    (A : ((Fin 72 → R) × (Fin 72 → R)) ≃ₗ[R] ((Fin 72 → R) × (Fin 72 → R)))
+    (z : (Fin 72 → R) × (Fin 72 → R)) :
+    weightedBank A ((List.finRange 18).map (coordinateProjector (R := R) selectedOwner)) z =
+      A ((A.symm z).2,(A.symm z).1) := by
+  rw [weightedBank_eq, weighted_apply, selected_bank_full]
+
+/-- The same exact weighted endpoint holds for every undeferred bank. -/
+theorem undeferred_weighted_bank_full
+    (A : ((Fin 72 → R) × (Fin 72 → R)) ≃ₗ[R] ((Fin 72 → R) × (Fin 72 → R)))
+    (z : (Fin 72 → R) × (Fin 72 → R)) :
+    weightedBank A ((List.finRange 3).map (coordinateProjector (R := R) undeferredOwner)) z =
+      A ((A.symm z).2,(A.symm z).1) := by
+  rw [weightedBank_eq, weighted_apply, undeferred_bank_full]
+
+end Mathproof.BeyondNLogN.TakeoverBit.P12
+
+
+
+/-! Lift the checked address maps to actual arbitrary register contents.
+No clean-scratch or independence assumption occurs in these statements. -/
+namespace Mathproof.BeyondNLogN.TakeoverBit.Contents
+open Mathproof.BeyondNLogN.TakeoverBit
+
+variable {R V B : Type*} [CommRing R] [AddCommGroup V] [Module R V]
+
+/-- Chronological execution on the complete contents of a physical bank. -/
+def run : List (V →ₗ[R] V) → ((V × V) → B) → ((V × V) → B)
+  | [], contents => contents
+  | P :: Ps, contents => run Ps (fun z => contents (partialSwap P z))
+
+theorem run_eq_address (Ps : List (V →ₗ[R] V)) (contents : (V × V) → B)
+    (z : V × V) : run Ps contents z = contents (bankSwap Ps z) := by
+  induction Ps generalizing contents with
+  | nil => rfl
+  | cons P Ps ih =>
+    simp only [run, ih, bankSwap, LinearMap.comp_apply]
+
+/-- A complete split partition executes the full transform on every dirty table. -/
+theorem run_full (Ps : List (V →ₗ[R] V))
+    (h : Ps.Pairwise (fun P Q => P.comp Q = 0 ∧ Q.comp P = 0))
+    (hsum : Ps.sum = LinearMap.id) (contents : (V × V) → B) (z : V × V) :
+    run Ps contents z = contents (z.2,z.1) := by
+  rw [run_eq_address, bankSwap_full Ps h hsum]
+
+/-- The actual p12 selected orbit bank works for arbitrary full bank contents. -/
+theorem p12_selected_contents
+    (contents : ((Fin 72 → R) × (Fin 72 → R)) → B)
+    (z : (Fin 72 → R) × (Fin 72 → R)) :
+    run ((List.finRange 18).map (coordinateProjector (R := R) P12.selectedOwner)) contents z =
+      contents (z.2,z.1) := by
+  rw [run_eq_address, P12.selected_bank_full]
+
+/-- The actual p12 undeferred orbit bank has the same unconditional endpoint. -/
+theorem p12_undeferred_contents
+    (contents : ((Fin 72 → R) × (Fin 72 → R)) → B)
+    (z : (Fin 72 → R) × (Fin 72 → R)) :
+    run ((List.finRange 3).map (coordinateProjector (R := R) P12.undeferredOwner)) contents z =
+      contents (z.2,z.1) := by
+  rw [run_eq_address, P12.undeferred_bank_full]
+
+end Mathproof.BeyondNLogN.TakeoverBit.Contents
+
+
+
+/-! Immutable PR161 p12 bank specialization.  The actual coordinate-owner,
+weighted-chart and dirty-content endpoints are reused from the independently
+checked P12 construction; only the new frozen supplier inventory is specialized.
+The retained all-input local word, charts, stage cover, routing, fallback and
+sharp all-size framework are upstream contracts, not proved by this ledger. -/
+namespace Mathproof.BeyondNLogN.TakeoverBit.PR161P12
+open Mathproof.BeyondNLogN.TakeoverBit
+
+/-- Complete child inventory from immutable PR161 head
+`d14e29157bc905be1ced0776dd893d0714013f3a`, profile_p12.json. -/
+def inheritedHistogram : List (Nat × Nat) :=
+  [(1,120972),(2,73336),(3,29832),(4,8208),(5,12096),(6,4392),
+   (7,4824),(8,2736),(9,648),(10,2160),(11,2016),(12,4464),
+   (13,1080),(14,2232),(15,216),(16,2808),(17,2736),(18,3888),
+   (19,3672),(20,7260),(21,8580),(22,15912),(60,5720)]
+
+/-- Eighteen actual four-coordinate selected cores cover Fin72, so their
+auxiliary rank60 exterior is removed; all other children remain charged. -/
+def packedHistogram : List (Nat × Nat) :=
+  inheritedHistogram.filter (fun p => p.1 != 60)
+
+/-- Three stage banks, with selected roles grouped eighteen at a time and
+undeferred roles grouped three at a time. -/
+def normalizedVolume : ℚ := 2*1760 + 3*((5720:ℚ)/18 + (17648:ℚ)/3)
+
+theorem inherited_rank_mass : P12.rankMass inheritedHistogram = 1934000 := by decide
+
+theorem removed_exterior_count :
+    (inheritedHistogram.filter (fun p => p.1 == 60)).map Prod.snd = [5720] := by decide
+
+theorem packed_rank_mass : P12.rankMass packedHistogram = 1590800 := by decide
+
+theorem packed_max_rank : ∀ p ∈ packedHistogram, p.1 ≤ 22 := by decide
+
+theorem normalized_volume : normalizedVolume = 66364/3 := by
+  norm_num [normalizedVolume]
+
+theorem packed_deficit :
+    (72:ℚ)*normalizedVolume - P12.rankMass packedHistogram = 1936 := by
+  rw [packed_rank_mass, normalized_volume]
+  norm_num
+
+theorem actual_role_counts :
+    5720 + 17648 = (23368:Nat) ∧ 18*4 = (72:Nat) ∧ 3*24 = (72:Nat) := by decide
+
+/-- New source binding assembled with actual reused coordinate projectors;
+no new abstract packing gadget or independence of register contents is assumed. -/
+theorem actual_bank_and_inventory_certificate
+    {R : Type*} [CommRing R] (z : (Fin 72 → R) × (Fin 72 → R)) :
+    (∀ s : Fin 18, (Finset.univ.filter
+      (fun j : Fin 72 => P12.selectedOwner j = s)).card = 4) ∧
+    (∀ s : Fin 3, (Finset.univ.filter
+      (fun j : Fin 72 => P12.undeferredOwner j = s)).card = 24) ∧
+    bankSwap ((List.finRange 18).map
+      (coordinateProjector (R := R) P12.selectedOwner)) z = (z.2,z.1) ∧
+    bankSwap ((List.finRange 3).map
+      (coordinateProjector (R := R) P12.undeferredOwner)) z = (z.2,z.1) ∧
+    normalizedVolume = 66364/3 ∧
+    P12.rankMass packedHistogram = 1590800 ∧
+    (72:ℚ)*normalizedVolume - P12.rankMass packedHistogram = 1936 := by
+  exact ⟨P12.selected_block_capacity, P12.undeferred_block_capacity,
+    P12.selected_bank_full z, P12.undeferred_bank_full z,
+    normalized_volume, packed_rank_mass, packed_deficit⟩
+
+end Mathproof.BeyondNLogN.TakeoverBit.PR161P12
+
+
+
+/-! The finite literal guard is discharged by an elementary universal charge
+bound. No positive-guard oracle is used. The condition s ≤ mW is the ordinary
+rank-mass bound of a finite supplier, not an unknown algorithm interface. -/
+namespace Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008
+
+theorem literal_charge_lt (W m G s : ℝ)
+    (hW : 0 ≤ W) (hm : 0 ≤ m) (hG : 0 ≤ G) (hs : s ≤ m*W) :
+    2*G*W*W+8*s+4*W+4+32*m < 64*(W+m+G+1)^3 := by
+  let X := W+m+G+1
+  have hX : 1 ≤ X := by dsimp [X]; linarith
+  have hX0 : 0 ≤ X := by linarith
+  have hWX : W ≤ X := by dsimp [X]; linarith
+  have hmX : m ≤ X := by dsimp [X]; linarith
+  have hGX : G ≤ X := by dsimp [X]; linarith
+  have hcube : 0 < X^3 := by positivity
+  have hquad : X ≤ X^2 := by nlinarith
+  have hquad_cube : X^2 ≤ X^3 := by
+    have hp := mul_nonneg (show 0 ≤ X-1 by linarith) (sq_nonneg X)
+    nlinarith
+  have hfirst : 2*G*W*W ≤ 2*X*X*X := by gcongr
+  have hsecond : m*W ≤ X*X := by gcongr
+  have hlarge : 2*G*W*W+8*s+4*W+4+32*m ≤ 50*X^3 := by
+    nlinarith
+  dsimp [X] at hlarge hcube
+  linarith
+
+theorem finite_row_reserve : (0 : ℝ) < 70000-(51*12320/25) := by norm_num
+
+theorem row_reserve_value : (70000-(51*12320/25) : ℝ) = 224336/5 := by norm_num
+
+
+end Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008
+
+
+
+/- Source-bound arithmetic for PR163 paid balanced transfer and our new nullity-bank refinement.
+   This is not a formalization of all-size integer multiplication. -/
+namespace Mathproof.BeyondNLogN.TakeoverBalanced16320261009
+noncomputable def a : ℝ := (297370240363722999999999702129759636277/500000000000000000000000000000000000000000)
+noncomputable def b : ℝ := (297370240363723/500000000000000000)
+noncomputable def eta : ℝ := (1/1000000000000000000000000)
+noncomputable def beta : ℝ := (1/1000000000000000000000000)
+noncomputable def kappa : ℝ := (118877394946471/200000000000000000)
+noncomputable def tau : ℝ := 1-a
+noncomputable def sigma : ℝ := 1-b
+noncomputable def q : ℝ := a*(1-2*eta)
+noncomputable def c : ℝ := q+eta/4
+noncomputable def eps : ℝ := (1-eta)/(1+q)
+noncomputable def lp : ℝ := 1-q
+noncomputable def lam : ℝ := (tau+lp)/2
+noncomputable def g : ℝ := eps*q
+noncomputable def r : ℝ := (g+1-eps)/2
+noncomputable def delta : ℝ := eta/8
+noncomputable def internal : ℝ := tau+(1-beta)*max (sigma-tau) 0
+noncomputable def leaf : ℝ := sigma+beta*(1-sigma)
+noncomputable def coreSlacks : List ℝ := [
+  (a) /- bit_positive -/,
+  (b-a) /- complex_above_bit -/,
+  (1/32-b) /- complex_below_one_over32 -/,
+  (beta) /- beta_positive -/,
+  (1-beta) /- beta_below_one -/,
+  ((1-beta)*b-a) /- leaf_saving_above_bit -/,
+  (q) /- q_positive -/,
+  (1-internal-q) /- q_below_internal -/,
+  (1-leaf-q) /- q_below_leaf -/,
+  (c) /- c_positive -/,
+  (1-c) /- c_below_one -/,
+  (c-q) /- q_below_reservations -/,
+  (lam-tau) /- lambda_above_tau -/,
+  (lam-sigma) /- lambda_above_sigma -/,
+  (lam-internal) /- lambda_above_internal -/,
+  (lp-lam) /- lambda_prime_above_lambda -/,
+  (lp-leaf) /- compact_leaf -/,
+  (lp-(1-c)) /- compact_reservations -/,
+  (q) /- lambda_prime_below_one -/,
+  (eps) /- epsilon_positive -/,
+  (1-eps) /- epsilon_below_one -/,
+  (1-eps) /- guard_width -/,
+  (1-eps*(1+c)) /- K_geometry -/,
+  (eps*c) /- K_dominates_log -/,
+  (1-eps) /- record_suffix -/,
+  (1-eps-delta) /- phase_local -/,
+  (r-delta) /- phase_boundary -/,
+  (1-eps-r) /- gamma_sublinear -/,
+  (eps-(1-r)/2) /- cell_above_band -/,
+  (1-eps) /- prime_interval_packing -/,
+  (r) /- alpha_positive -/,
+  (1-r) /- alpha_below_one -/,
+  (1/4-r) /- alpha_below_one_fourth -/,
+  (delta) /- delta_positive -/,
+  (1/8-delta) /- delta_below_one_eighth -/,
+  (eps-a) /- short_record_fallback -/,
+  (1-eps-g) /- small_field_exposure -/,
+  (8-eps+r-delta-g) /- artificial_boundary -/,
+  ((1-eps)-kappa) /- paid_balanced_prefix_above_kappa -/,
+  ((a)-kappa) /- coordinate_movement_above_kappa -/,
+  ((g)-kappa) /- compact_phase_layer_above_kappa -/,
+  ((a)-kappa) /- bulk_exposure_above_kappa -/,
+  ((min (1-eps-delta) (r-delta))-kappa) /- Gaussian_arithmetic_above_kappa -/,
+  ((1-eps-delta)-kappa) /- scalar_work_above_kappa -/,
+  ((eps)-kappa) /- dimension_above_kappa -/
+]
+theorem core_slacks_positive : ∀ x ∈ coreSlacks, 0 < x := by
+  norm_num [coreSlacks, a, b, eta, beta, kappa, tau, sigma, q, c, eps, lp, lam, g, r, delta, internal, leaf]
+
+theorem all_47_slacks_positive (literal row : ℝ)
+    (hl : 0 < literal) (hr : 0 < row) :
+    ∀ x ∈ coreSlacks ++ [literal, row], 0 < x := by
+  intro x hx
+  rcases List.mem_append.mp hx with h | h
+  · exact core_slacks_positive x h
+  · simp only [List.mem_cons, List.not_mem_nil, or_false] at h
+    rcases h with h | h
+    · simpa [h] using hl
+    · simpa [h] using hr
+
+theorem strengthened_public_margin :
+    (148492550769873/250000000000000000 : ℝ) < kappa ∧ kappa < g := by
+  norm_num [kappa, g, eps, c, q, a, eta]
+
+/-- Complete arithmetic composition of our certified bit bank with the
+pinned PR163 paid balanced transfer physical complex supplier. All finite guards are proved, subject
+only to explicit nonnegative counts and their standard rank-mass bound.
+This theorem does not claim the retained analytic/tape interfaces. -/
+theorem all_47_concrete_guards (W m G s : ℝ)
+    (hW : 0 ≤ W) (hm : 0 ≤ m) (hG : 0 ≤ G) (hs : s ≤ m*W) :
+    ∀ x ∈ coreSlacks ++
+      [64*(W+m+G+1)^3-(2*G*W*W+8*s+4*W+4+32*m),
+       70000-(51*12320/25)], 0 < x := by
+  apply all_47_slacks_positive
+  · exact sub_pos.mpr
+      (Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008.literal_charge_lt
+        W m G s hW hm hG hs)
+  · exact Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008.finite_row_reserve
+
+end Mathproof.BeyondNLogN.TakeoverBalanced16320261009
diff --git a/research/balanced-nullity-banks-163/LICENSE b/research/balanced-nullity-banks-163/LICENSE
new file mode 100644
index 0000000000..261eeb9e9f
--- /dev/null
+++ b/research/balanced-nullity-banks-163/LICENSE
@@ -0,0 +1,201 @@
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diff --git a/research/balanced-nullity-banks-163/MANIFEST.json b/research/balanced-nullity-banks-163/MANIFEST.json
new file mode 100644
index 0000000000..6d4996ea27
--- /dev/null
+++ b/research/balanced-nullity-banks-163/MANIFEST.json
@@ -0,0 +1,14 @@
+{
+  "arithmetic-replay.log": "393570c538268eb8ef36dd89a007861a7c4e6a6c7782f5fdd0c060e99bc37df0",
+  "AUDIT-TARGETS.json": "1fb48f1e81cbc806c2e08a4ed58ea75a87f6a533138a014a13bb31e7bac0db14",
+  "BankAndBalanced.lean": "8f32bce76e47da3c6165ea94f179ddf351ed50271996ab179505f588333b5867",
+  "expected.json": "0c94f8a31170b953ca0c35f4904e80f4f38dd4a553a60249b224520a6d590a1b",
+  "lean-audit.log": "d4b24c5a28ebe4db51f41fba151500fa4c0ce4a739069e0cc1fc180b8d2117e4",
+  "LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+  "NOTICE": "2d192a690294292eaed715da43ed74951a1130de7f38a6c5084816798f4d0c72",
+  "PROOF-ORIGINS.json": "53a97d25d58611056a51a00b6cc5d2e0cefc9b16a968f297d4b8a92baaa990c8",
+  "PROOF.md": "6e6e82b2918a440b4ae556e8fef439ac641be138f8edfb2ee803342cb6e8c56e",
+  "README.md": "10e3f5a2fc7909cde95c7d25204cf0819f638f5a66202b74ef7f645da531a267",
+  "VERIFICATION.json": "991a11ba4f6efadf7927578818c26b109e71d82379c22636ad9cf5d2f507c5d3",
+  "verify.py": "37d317bcc1185db9a08163dd325e9cd95e55007d1db45b7ed5a4d060ec85978e"
+}
diff --git a/research/balanced-nullity-banks-163/NOTICE b/research/balanced-nullity-banks-163/NOTICE
new file mode 100644
index 0000000000..7f827029f3
--- /dev/null
+++ b/research/balanced-nullity-banks-163/NOTICE
@@ -0,0 +1,13 @@
+Conditional balanced transfer with completed-core nullity banks.
+Prepared with substantial OpenAI Codex assistance. No personal author name has been added by this package.
+Upstream framework: OpenAI integer multiplication manuscript; Douglas Colkitt/CrocSwap.
+PR161 annealed modules, paired word and signed merging: eumemic, with Anthropic Claude assistance.
+Paired-cube decoder, shared ledger and pricing: icekylinx/William Porter.
+Full lineage remains in the pinned PR161 NOTICE and SOURCES.json.
+PR163 fused supplier and adapter: chafreaky; paid layout lineage PR23/29 (Zhihao Chen), RaD,
+PR34 (James Chang), PR100/103 (Rohan Arun), and retained PR141 arithmetic body.
+Bit local words and completed-core sharing retain their PR161 attribution, including
+an664 PR128, eumemic PR117, and Zhihao Chen/Swapnil Jain PR97.
+This contribution claims the displayed nullity-bank refinement and composition only,
+not authorship of upstream suppliers, balanced layout, or the all-size framework.
+Apache-2.0. Consult the pinned upstream notices for full attribution.
diff --git a/research/balanced-nullity-banks-163/PROOF-ORIGINS.json b/research/balanced-nullity-banks-163/PROOF-ORIGINS.json
new file mode 100644
index 0000000000..884821366c
--- /dev/null
+++ b/research/balanced-nullity-banks-163/PROOF-ORIGINS.json
@@ -0,0 +1,32 @@
+[
+  {
+    "filename": "TakeoverBit20261008.lean",
+    "sha256": "7f5d05001824fd59c4070be55786151777a03972499092b2222c5b673f8bbe0a",
+    "change": "proof bodies unchanged; imports removed and hoisted"
+  },
+  {
+    "filename": "TakeoverBit20261008P12.lean",
+    "sha256": "52ae9bca980c634fcc206f94832a5b72ace2a47609e1d57c663424347d2944c4",
+    "change": "proof bodies unchanged; imports removed and hoisted"
+  },
+  {
+    "filename": "TakeoverBit20261008Contents.lean",
+    "sha256": "8b9e75c1ab67d210e9a6385577c9ca965ccc42c930f5279e2ab2e23e96cc2664",
+    "change": "proof bodies unchanged; imports removed and hoisted"
+  },
+  {
+    "filename": "TakeoverBit16120261008P12.lean",
+    "sha256": "2ea23c607c23a2590660f588b18eb9f3525d58f9ac6b9461a30331002ed18aae",
+    "change": "proof bodies unchanged; imports removed and hoisted"
+  },
+  {
+    "filename": "TakeoverLiteralGuard20261008.lean",
+    "sha256": "8fe9acaccb68ff70eb43d308f0a8b34f38119779ba3514a35078d3a4348d47c6",
+    "change": "imports removed and hoisted; obsolete original-layout assembly wrapper omitted from literal guard"
+  },
+  {
+    "filename": "TakeoverBalanced16320261009.lean",
+    "sha256": "ce2087d53c07e07324556e06e4a4b940905db137b4c723291a77bb2aae511fd7",
+    "change": "proof bodies unchanged; imports removed and hoisted"
+  }
+]
diff --git a/research/balanced-nullity-banks-163/PROOF.md b/research/balanced-nullity-banks-163/PROOF.md
new file mode 100644
index 0000000000..f83a8192e0
--- /dev/null
+++ b/research/balanced-nullity-banks-163/PROOF.md
@@ -0,0 +1,64 @@
+# PR163 paid-balanced review and actual combination with our bit engine
+
+Pinned source: chafreaky/integer-mult-bounds e1813796ef5c3ca38c5dd7b9e8d81b3908ff5997, research/paired-cube-balanced-161. Full finite package replay is owned by the physical/PUBLIC lane. This lane read PROOF.md, the full reference ORIGINAL-PROOF.md, balanced_shared.py, shared_bridge.py, certificate.py, complex/prove.py and the fused verifier's exact signed decoder/frame checks.
+
+## New certified arithmetic point
+
+Use the pinned PR163 fused complex supplier b=297370240363723/(5*10^17) and our previously checked PR161 p12 completed-core nullity-sharing uniform-bit supplier. The latter's exact constants are
+
+    coarse = 302618322217 / 500000000000000,
+    atom = 302446903 / 500000000000,
+    old = 384599 / 10^10,
+    A = (1-atom)*coarse + atom*old
+      = 151223451172976757106049 / 250000000000000000000000000.
+
+The paid atom inequalities A<atom<1-A hold exactly. A exceeds the new complex saving, so set a=(1-beta)*b-zeta, beta=eta=10^-24, zeta=10^-30, q=a*(1-2*eta), epsilon=(1-eta)/(1+q). The actual supported balanced margin is g=epsilon*q. The largest strictly accepted 10^-18-grid point is
+
+    kappa = 118877394946471 / 200000000000000000
+          = 0.000594386974732355.
+
+All47 strict constraints and7 margins pass the UNCHANGED paid-balanced arithmetic body. The next grid point is rejected at these same supplier certificates and parameters. An independent Fraction recomputation verifies the seven margins and all three exact identities below. The complete JSON evidence and exact replay script are expected.json and verify.py in this upload package.
+
+This exceeds public PR163's 0.000593970203079492 by 416771652863/10^18, about0.0702%. It is a real combined arithmetic gain from existing bit headroom; it is not a substantial new complex-supplier breakthrough or a claim to solve the remaining all-size physical/analytic interfaces.
+
+## Why this is a paid layout composition
+
+The PR34/RaD positional construction, retained for deferred signed words through PR100/103 and PR141, separates the extra top bit of each long FFT axis and uses common named low-coordinate groups K_j in [K,2K). The ordinary arbitrary-coordinate router pays both directions of the axis-major/group-major permutation. Entire active, spectator, scratch, phase, control and restored-row fields travel together, preserving twiddle order and forward/inverse alignment. No scalar residual is moved across another operation.
+
+The prefix becomes O(T*p*d), hence margin1-epsilon. The original geometric inequality1-epsilon*(1+c)>0 is RETAINED. This is not permission to delete the old prefix bill without providing the positional router. The existing completed ordinary supplier must expose the arbitrary-width interchange/complete-spectator/restored-row contract; the exact complex supplier must expose the completed tensor contract. Those are the explicit all-size interfaces assumed by this conditional combination.
+
+With c=q+eta/4, r=(g+1-epsilon)/2 and delta=eta/8, exact arithmetic gives
+
+    1-epsilon-g=eta,
+    1-epsilon-r=eta/2,
+    1-epsilon*(1+c)=eta-epsilon*eta/4 >0.
+
+The internal complex exponent and stopping-leaf exponent stay unchanged. Since a<(1-beta)*b, the leaf-ordering gap is strict. The tiny positive backoffs establish eventual asymptotic inequalities, not a usable practical input threshold.
+
+## Full fixed-tape / scalar / row bills retained
+
+The public fused word has scalar_R=15681, physical_R=13041, reuse_pairs=2640, c=19385, q=3157, matched=6861, M_operations=33746. The normalized complex inventory has m66,W15681,rank1033626,deficit1320,maxchild20. Compared with161, the rank mass is unchanged but the complete child histogram changes; that is why its strict complex moment improves. The scalar word and compensation counts are not replaced by the smaller physical role stock.
+
+Our replay recomputes the full |O(66,2)| group factor, complete local literal scalar bound, logical bill, finite router guard, E, B=s+E and C0=32*66*B^2. Its local scalar bill matches the exact source-bound physical scalar_bound field. The full group factor cancels only in normalized moments, never in scalar work, routing or row stock.
+
+The row validator retains both conservative external reserves9909 and252, the full complex wire-bit count2159, row coefficient12320, polynomial degree70000 and positive degree gap224336/5. Internally borrowed/restored q-adic rows are not reinterpreted as free external stock. The odd divisor remains3, scalar denominators divide6, children retain incoming odd exponents and no child boundary rounding is introduced.
+
+Replacing PR163's bit frame refinement by our already checked bit supplier does not change the complex finite bridge, complex scalar bill or physical word. It changes only the validated bit_uniform coarse/atom/ordinary fields. The strengthened bit construction and compatible weighted local-ring implementation are separate source-bound inputs; this arithmetic replay does not assert them from a numeric exponent.
+
+## Exact complex fused-broadcast phases
+
+The fused source combines signed channels feeding the SAME target before readout. For a singleton target T=(I,b0,b1,b2), the three last contributions are
+
+    P/2 + (2*b0-1)Q/2 + (2*b1-1)R/2.
+
+Repeated application of our independently verified TakeoverMerge16120261008.decoder_merge gives exactly
+
+    (P+(2*b0-1)Q+(2*b1-1)R)/2.
+
+This equality holds in every actual complex module. An actual common complex-linear frame transports it exactly by decoder_merge_common_frame; no scalar phase is discarded or inferred merely from a binary symplectic label. Signed additions are undone in literal reverse order. Conservative original-source supports are unioned and never shrunk by cancellation. The face2, edge02 and edge12 source-cube intersection classes are disjoint, so iterative disjoint_merge_support proves exact support preservation. The actual fused verifier checks every signed source/target coefficient, value span, common cap and chronological chain.
+
+The original22 complex/P18 certificates and11 merge certificates remain unchanged and reusable. P18 fourth-root Fourier/Gauss identities certify phase energy under their displayed hypotheses; they do not supply the fused histogram, paid positional router or new exponent by themselves.
+
+## Explicit conditional boundary
+
+The exact finite arithmetic point is established. General Clifford lifts and actual phase reconciliation, common generic bases, uniform weighted local-ring compilation, arbitrary-width routing, complete row/spectator restoration, positional layout applicability, prime supply, precision/recovery, uniform setup and analytic fixed-tape transfer remain inherited all-size proof contracts. No unconditional multiplication theorem, fully closed Lean multiplier root, or general optimality result is claimed here. The next-grid failure concerns only this fixed supported margin.
diff --git a/research/balanced-nullity-banks-163/README.md b/research/balanced-nullity-banks-163/README.md
new file mode 100644
index 0000000000..ee29fa8a62
--- /dev/null
+++ b/research/balanced-nullity-banks-163/README.md
@@ -0,0 +1,78 @@
+# Conditional paid-balanced transfer with nullity banks
+
+Under the explicitly retained all-size contracts, this composition supports
+`T(n) = O(n (log n)^(1-kappa))` with
+
+    kappa = 118877394946471 / 200000000000000000
+          = 0.000594386974732355.
+
+This is about 0.0702% above pinned PR163's 0.000593970203079492.
+It combines PR163's fused complex supplier and paid balanced positional transfer
+with the PR161 completed-core nullity-bank bit refinement. Gains are not added.
+The resulting supported saving is complex-limited.
+
+## What changes
+
+For PR161's bit word, eighteen explicit four-coordinate blocks and three explicit
+24-coordinate blocks each partition Fin72. Applying completed-core bank sharing
+removes exactly 5720 rank60 exterior children while retaining all other children.
+The normalized volume is 66364/3, rank mass 1590800, deficit 1936, maximum child22.
+The paid bit moment and positive atom toll yield ordinary saving
+151223451172976757106049 / 250000000000000000000000000,
+which clears the PR163 complex supplier. The full scalar, router, group and row
+bills are retained; all47 strict constraints and seven margins pass.
+
+## Reproduce exact arithmetic (Python3.11+, standard library; assertions enabled)
+
+Obtain two separate source snapshots at these immutable commits:
+
+- PR161: https://github.com/eumemic/integer-mult-bounds , commit
+  d14e29157bc905be1ced0776dd893d0714013f3a.
+- PR163: https://github.com/chafreaky/integer-mult-bounds , commit
+  e1813796ef5c3ca38c5dd7b9e8d81b3908ff5997.
+
+The verifier requires Git checkouts pinned to those exact commits, with unchanged
+tracked sources. A reviewer can acquire them using Git; the submitter can upload
+this folder using GitHub's website without using Git locally.
+
+    python3 -B research/balanced-nullity-banks-163/verify.py --pr161 /path/to/pr161 --pr163 /path/to/pr163
+
+The verifier does not download or modify sources. It reconstructs the packed
+ledger, proves the finite coordinate partitions by enumeration, recomputes the
+bit moment using two interval implementations, checks its adjacent-grid control,
+recomputes the PR163 complex moment, and replays the unchanged paid-balanced
+assembly and full finite bill against expected.json. It rejects the next final
+10^-18 grid point at these fixed certificates and parameters. This is not a claim
+of global optimality. The verifier checks arithmetic; it does not substitute for
+upstream physical-word verification.
+
+## Lean scope
+
+BankAndBalanced.lean is a single-file Mathlib-importing snapshot of existing
+proof bodies: actual split-projector address interchanges, explicit partitions,
+weighted endpoints, arbitrary dirty-content semantics, PR161 packed inventory,
+and the47 finite arithmetic guards. See PROOF-ORIGINS.json for provenance.
+It does not assert an all-size multiplication theorem. The bit inventory theorem
+is proved alongside actual bank endpoints; connecting them to the full physical
+multiplier still uses the contracts below, not merely the histogram arithmetic.
+The Lean version is4.33.0. The compatible Mathlib commit is
+`db584cd6d46c92f209a44c0f1c829460d327499d`. VERIFICATION.json records the
+independent source replay and axiom audit. In that Mathlib environment, run
+`lake env lean research/balanced-nullity-banks-163/BankAndBalanced.lean`.
+The52 audited theorem names are listed in AUDIT-TARGETS.json.
+
+## Dependency and publication scope
+
+This is an additive research package. It depends on the unmerged PR161/PR163
+snapshots above and changes no existing source. A PR to CrocSwap/main should
+explicitly state those dependencies; it does not silently import their entire
+diffs or claim they are already merged.
+
+The completed all-input word, split-frame charts, invocation/stage cover,
+weighted local-ring implementation, arbitrary-width complete-spectator routing,
+borrowed-row restoration, paid layout applicability, precision/recovery,
+prime supply, uniform setup and sharp fixed-tape analytic transfer remain
+inherited all-size contracts. PROOF.md states these boundaries. This is a
+conditional finite witness, not a fully formalized unconditional multiplier.
+
+See NOTICE and LICENSE for attribution.
diff --git a/research/balanced-nullity-banks-163/VERIFICATION.json b/research/balanced-nullity-banks-163/VERIFICATION.json
new file mode 100644
index 0000000000..26b13611da
--- /dev/null
+++ b/research/balanced-nullity-banks-163/VERIFICATION.json
@@ -0,0 +1,400 @@
+{
+  "status": "PASS",
+  "source_sha256": "8f32bce76e47da3c6165ea94f179ddf351ed50271996ab179505f588333b5867",
+  "kernel_replay_exit_code": 0,
+  "theorem_count": 52,
+  "axiom_records": [
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.partialSwap_apply",
+      "axioms": [
+        "propext",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.partialSwap_zero",
+      "axioms": [
+        "propext",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.partialSwap_comp_disjoint",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.partialSwap_involutive",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.partialSwap_id",
+      "axioms": [
+        "propext",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.partialSwap_commute",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.bankSwap_eq_sum",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.bankSwap_full",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.weighted_apply",
+      "axioms": [
+        "propext",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.weighted_comp",
+      "axioms": [
+        "propext",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.weighted_id",
+      "axioms": [
+        "propext",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.weightedBank_eq",
+      "axioms": [
+        "propext",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.weightedBank_full",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.coordinateProjector_apply",
+      "axioms": [
+        "propext",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.coordinateProjector_idempotent",
+      "axioms": [
+        "propext",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.coordinateProjector_disjoint",
+      "axioms": [
+        "propext",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.coordinateProjector_sum",
+      "axioms": [
+        "propext",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.coordinateBank_full",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.bankAddress_bijective",
+      "axioms": [
+        "propext"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.bankAddress_roles_injective",
+      "axioms": [
+        "propext"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.selected_block_capacity",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.undeferred_block_capacity",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.selected_bank_full",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.undeferred_bank_full",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.inherited_rank_mass",
+      "axioms": [
+        "propext"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.removed_exterior_count",
+      "axioms": []
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.packed_rank_mass",
+      "axioms": [
+        "propext"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.packed_max_rank",
+      "axioms": []
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.normalized_volume",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.packed_deficit",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.actual_role_counts",
+      "axioms": []
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.selected_weighted_bank_full",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.undeferred_weighted_bank_full",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.Contents.run_eq_address",
+      "axioms": [
+        "propext",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.Contents.run_full",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.Contents.p12_selected_contents",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.Contents.p12_undeferred_contents",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.inherited_rank_mass",
+      "axioms": [
+        "propext"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.removed_exterior_count",
+      "axioms": []
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.packed_rank_mass",
+      "axioms": [
+        "propext"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.packed_max_rank",
+      "axioms": []
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.normalized_volume",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.packed_deficit",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.actual_role_counts",
+      "axioms": []
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.actual_bank_and_inventory_certificate",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008.literal_charge_lt",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008.finite_row_reserve",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008.row_reserve_value",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBalanced16320261009.core_slacks_positive",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBalanced16320261009.all_47_slacks_positive",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBalanced16320261009.strengthened_public_margin",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBalanced16320261009.all_47_concrete_guards",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    }
+  ],
+  "allowed_axioms": [
+    "propext",
+    "Classical.choice",
+    "Quot.sound"
+  ],
+  "forbidden_axiom_count": 0,
+  "sorry_count": 0,
+  "imports": "Mathlib only",
+  "lean_version": "4.33.0",
+  "mathlib_commit": "db584cd6d46c92f209a44c0f1c829460d327499d",
+  "independent_replay_sha256": "4a843d015dd3fa5daa696fcea6beca654a52ef0773720e979928cdfaa605c512",
+  "log_sha256": "d4b24c5a28ebe4db51f41fba151500fa4c0ce4a739069e0cc1fc180b8d2117e4",
+  "portable_arithmetic_replay": {
+    "status": "PASS",
+    "kappa": "118877394946471/200000000000000000",
+    "decimal": "0.000594386974732355",
+    "strict_constraints": 47,
+    "margins": 7,
+    "bit_mass": 1590800,
+    "bit_volume": "66364/3",
+    "scope": "Exact conditional finite arithmetic plus ledger and coordinate partitions; full physical realization and all-size inherited contracts are separate."
+  },
+  "arithmetic_log_sha256": "393570c538268eb8ef36dd89a007861a7c4e6a6c7782f5fdd0c060e99bc37df0",
+  "scope": "Actual universal bank/address/content endpoints, explicit inventory and finite balanced arithmetic. Does not certify all-size multiplication or discharge retained implementation/analytic contracts."
+}
diff --git a/research/balanced-nullity-banks-163/arithmetic-replay.log b/research/balanced-nullity-banks-163/arithmetic-replay.log
new file mode 100644
index 0000000000..f543fe4f7a
--- /dev/null
+++ b/research/balanced-nullity-banks-163/arithmetic-replay.log
@@ -0,0 +1,10 @@
+{
+  "status": "PASS",
+  "kappa": "118877394946471/200000000000000000",
+  "decimal": "0.000594386974732355",
+  "strict_constraints": 47,
+  "margins": 7,
+  "bit_mass": 1590800,
+  "bit_volume": "66364/3",
+  "scope": "Exact conditional finite arithmetic plus ledger and coordinate partitions; full physical realization and all-size inherited contracts are separate."
+}
diff --git a/research/balanced-nullity-banks-163/expected.json b/research/balanced-nullity-banks-163/expected.json
new file mode 100644
index 0000000000..ac35a985d5
--- /dev/null
+++ b/research/balanced-nullity-banks-163/expected.json
@@ -0,0 +1,203 @@
+{
+  "public163_pin": "e1813796ef5c3ca38c5dd7b9e8d81b3908ff5997",
+  "public161_pin": "d14e29157bc905be1ced0776dd893d0714013f3a",
+  "public163_kappa": "148492550769873/250000000000000000",
+  "combo": {
+    "kappa": "118877394946471/200000000000000000",
+    "a": "297370240363722999999999702129759636277/500000000000000000000000000000000000000000",
+    "actual_bit_saving": "151223451172976757106049/250000000000000000000000000",
+    "complex_saving": "297370240363723/500000000000000000",
+    "beta": "1/1000000000000000000000000",
+    "eta": "1/1000000000000000000000000",
+    "weakening": "1/1000000000000000000000000000000",
+    "layout": "balanced",
+    "minimum_margin": "148684971496890003109996295013224259329740670260073915526993780506219493482636277/250148434971746889753110246443448196006219493780506517363723000000000000000000000000",
+    "assembly": {
+      "parameters": {
+        "a_bit": "297370240363722999999999702129759636277/500000000000000000000000000000000000000000",
+        "a_complex": "297370240363723/500000000000000000",
+        "tau": "499702629759636277000000000297870240363723/500000000000000000000000000000000000000000",
+        "sigma": "499702629759636277/500000000000000000",
+        "beta": "1/1000000000000000000000000",
+        "h": "1/1000000000000000000000000",
+        "q": "148685120181861499999999553694639454415500000000297870240363723/250000000000000000000000000000000000000000000000000000000000000000",
+        "c": "148685120181861500000062053694639454415500000000297870240363723/250000000000000000000000000000000000000000000000000000000000000000",
+        "epsilon": "249999750000249999750000000000000000000000000000000000000000/250148434971746889753110246443448196006219493780506517363723",
+        "lambda_": "499702629759636277000000000595240480727445999999999702129759636277/500000000000000000000000000000000000000000000000000000000000000000",
+        "lambda_prime": "249851314879818138500000000446305360545584499999999702129759636277/250000000000000000000000000000000000000000000000000000000000000000",
+        "alpha_squared_power": "297369942993780006220242738461420265549234450766591279249993780506219493482636277/500296869943493779506220492886896392012438987561013034727446000000000000000000000000",
+        "delta": "1/8000000000000000000000000",
+        "C0": 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+        "C1": "1",
+        "kappa": "118877394946471/200000000000000000"
+      },
+      "constraints": {
+        "a_positive": "297370240363722999999999702129759636277/500000000000000000000000000000000000000000",
+        "a_below_b": "297870240363723/500000000000000000000000000000000000000000",
+        "b_below_one_over32": "15327629759636277/500000000000000000",
+        "beta_positive": "1/1000000000000000000000000",
+        "beta_below_one": "999999999999999999999999/1000000000000000000000000",
+        "phase_leaf_above_bit": "1/1000000000000000000000000000000",
+        "q_positive": "148685120181861499999999553694639454415500000000297870240363723/250000000000000000000000000000000000000000000000000000000000000000",
+        "q_below_internal": "297370240363722999999999702129759636277/250000000000000000000000000000000000000000000000000000000000000000",
+        "q_below_leaf": "297620240363722999999999702129759636277/250000000000000000000000000000000000000000000000000000000000000000",
+        "c_positive": "148685120181861500000062053694639454415500000000297870240363723/250000000000000000000000000000000000000000000000000000000000000000",
+        "c_below_one": "249851314879818138499999937946305360545584499999999702129759636277/250000000000000000000000000000000000000000000000000000000000000000",
+        "q_below_reservations": "1/4000000000000000000000000",
+        "lambda_above_tau": "297370240363722999999999702129759636277/500000000000000000000000000000000000000000000000000000000000000000",
+        "lambda_above_sigma": "595240480727445999999999702129759636277/500000000000000000000000000000000000000000000000000000000000000000",
+        "lambda_above_internal": "297370240363722999999999702129759636277/500000000000000000000000000000000000000000000000000000000000000000",
+        "lambda_prime_above_lambda": "297370240363722999999999702129759636277/500000000000000000000000000000000000000000000000000000000000000000",
+        "compact_leaf": "297620240363722999999999702129759636277/250000000000000000000000000000000000000000000000000000000000000000",
+        "compact_reservations": "1/4000000000000000000000000",
+        "lambda_prime_below_one": "148685120181861499999999553694639454415500000000297870240363723/250000000000000000000000000000000000000000000000000000000000000000",
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+        "C1": 1,
+        "induction_gap": 245388340315854495121991495023661354133772589464405098560835281231210593354019555143204809575641856722291194489372719301193616799647826159579757887820071223106654636111050438191495751716485818695124159328484048794755630779268936969158749727291785790306894906681006422231592986767946388346377694932706235427152142784836989450028374317183063203380895257549480826760271435164826739807632685330155718123900280161432695419226532158212485722545375693195990796362804742759742725917432842044603389791467383156312396163024378938394840961078454714202388522126089198783312815501652525725420856212824101710091243502711285243089767465596146429310562531820838246195141964036386977667992333784800904065593805038690857840165112369937462991853946434986034550591529039889942641409960245899894496155913835925603690714427978783216632282652185350804110391405344267420394508689231484590201167625358071977829989612598492896437837504075800683681711004705602678862156000874995105476159542852930583132791352717479396514543009746560892105943675938040697032178998767177924243015937188965591032407979029824417117095144958500076152110001069748144440755055131143299379561950147695675291209196170790861794308920116977841826009571049029564192723661000347022780697656581844126333817888558148438858693544050730279006391032579314348383417483341726167308484102464011875039770669245289569053370315400675272769228984745966480541333705931741952201361584450779326850521243305632190114357848274877760293759481552810837844820287785429409739193492470687215042264321232635941437404752907555527467413383180997744420400148964258443628979070098673792084717606302084025121666947854676905542764446282440636655033920168714101675111829088679088416088899017520789101335421071428013989215397416644857473658623389821671427674995252610577512050322911253131859211944038596666016247376225153558207550373063492998554888580920952391708138131289219486525718977892325465230055854953215786176257899380807103644121328698903259664929947496469383326286563627861859954182187535400895764477928092562530607757748705780763587088674121305145938437603223108021159308048538225758101153578764087878011594956574876552904477637107998204434002769826088279804383770536124007804304192063468870467893053981298039318821032128707649760060073749229537610127792812276679987477985700133781138307150786679926037397394611571161387429268032865427283431018256364862938990177608641555871765353175304933281067617839664481651571704230273983452875393518335293014396311381254899480921988612253618879517438535306676833380425595251991660952995091525032647927535458234697488586825233573202703716781037985646889946321208777879673711271946767903809948726047445877686120392643104955379850171778688163274102126680878508444266761189704537482574079118130447309771519192797202081634827220328648531606261068653634762233494516255628307404776706858390508708834601264897008646546562983946360187373787346425825026502825166589159365404652363307340702487787070996971736376432277781406562795750954160208053781639250158662196318892749978189443599641836214518616936539387692422155247530353148031561963426984668637356710756857535578502681262021631597998560237428258197520890833021959495277829596803778310503074257439014215218565832359454110570594125081714805152020433763053050634665307117361238244863227269421732556357631665326777103377909512101347728315848156030582196328671462365501813019054259602691910273427111817453707511754008411283207639828440879115320662481009495070733076384053735716696980585867475945451451632411845785680475987953695363293698311225826327156039939435741826425950384128738263556609420740617600738683900894516668818578184982813685973717501435264569198072863889402431256917351853788278324748638370087025599361388713671432149463428590762196328434465289201875753262745420886261254218348598928424459889789228762368105383041556546813071637279440584703191488864477114637272334834163218052935897385189713813648366910256840069690327797675638041266996826858514819047856599720555974644532336376341915423147840463479113504223750866935902556906829947300394079605318925784823743990331563491028308230395256964522179938306014663290083982727831715151268072100089494646507958010913026276692220056723587598417289180621183577079586468288019596167019726559547363309560062771643065546613302515146937109315291412368623507609804469300980952281034100560672316181292568391370829081955158464900787242246061359338501835361109152620295946787204589278663415921390862293932802855596468826293101157606576254299895075065183725848640195038168140115238141875608609663281932121508309849213169636906548962762790229247985424467798338812598996036676288818676668020039053269385224237236006604554516626741039697049248452711713870708320270355284148316234748223145351134971979820642035510926640676212230251793685052973119933811537507199542921985886099868559230539499820772565054396424423335994260188311756682328300590138836072346894700375172449221942310825056561920528548189480019682715977200510265557444727939596334566578509841650642419995868546617797990536589793984959654750046004443272269498433270536404174860207593718627146563818414240126343169066224110289896039479249190155643300061135375135749044698752258870903455753119806914130887885547024377713477997887249790908543042453431913066622227351526052356973191952304344647002675594522000489033148689097463503118249558277146364423179459261437672925470825192600425893579520562041434304347237999198826154802593109923761892805497881437726822119855138188118058662783447363009685726520879605684932206112719489605720892676070178453033570794379935889511795906208022176982250800694161341196693332468658133233056859508089043451636005730974242647702874311303164831450289352805610181457063655032687294375885415896338259476839287814607339274526710653286349752736405778956851518821342810708582296938181310293525162934948124660222441967947885080711857144314720550984995695087455018431062631654976,
+        "fixed_odd_divisor": 3,
+        "exact_grid": "One dyadic grid times 3^(-K), K=G*(D_complex+1); completed children preserve the incoming odd exponent."
+      },
+      "rows": {
+        "coefficient": 12320,
+        "complex_coefficient": 2159,
+        "degree": 70000,
+        "suffix_slope": 280000,
+        "degree_gap": "224336/5"
+      }
+    }
+  }
+}
diff --git a/research/balanced-nullity-banks-163/lean-audit.log b/research/balanced-nullity-banks-163/lean-audit.log
new file mode 100644
index 0000000000..5943da67d0
--- /dev/null
+++ b/research/balanced-nullity-banks-163/lean-audit.log
@@ -0,0 +1,82 @@
+'Mathproof.BeyondNLogN.TakeoverBit.partialSwap_apply' depends on axioms: [propext, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.partialSwap_zero' depends on axioms: [propext, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.partialSwap_comp_disjoint' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.partialSwap_involutive' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.partialSwap_id' depends on axioms: [propext, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.partialSwap_commute' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.bankSwap_eq_sum' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.bankSwap_full' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.weighted_apply' depends on axioms: [propext, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.weighted_comp' depends on axioms: [propext, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.weighted_id' depends on axioms: [propext, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.weightedBank_eq' depends on axioms: [propext, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.weightedBank_full' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.coordinateProjector_apply' depends on axioms: [propext, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.coordinateProjector_idempotent' depends on axioms: [propext, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.coordinateProjector_disjoint' depends on axioms: [propext, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.coordinateProjector_sum' depends on axioms: [propext, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.coordinateBank_full' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.bankAddress_bijective' depends on axioms: [propext]
+'Mathproof.BeyondNLogN.TakeoverBit.bankAddress_roles_injective' depends on axioms: [propext]
+'Mathproof.BeyondNLogN.TakeoverBit.P12.selected_block_capacity' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.P12.undeferred_block_capacity' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.P12.selected_bank_full' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.P12.undeferred_bank_full' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.P12.inherited_rank_mass' depends on axioms: [propext]
+'Mathproof.BeyondNLogN.TakeoverBit.P12.removed_exterior_count' does not depend on any axioms
+'Mathproof.BeyondNLogN.TakeoverBit.P12.packed_rank_mass' depends on axioms: [propext]
+'Mathproof.BeyondNLogN.TakeoverBit.P12.packed_max_rank' does not depend on any axioms
+'Mathproof.BeyondNLogN.TakeoverBit.P12.normalized_volume' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.P12.packed_deficit' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.P12.actual_role_counts' does not depend on any axioms
+'Mathproof.BeyondNLogN.TakeoverBit.P12.selected_weighted_bank_full' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.P12.undeferred_weighted_bank_full' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.Contents.run_eq_address' depends on axioms: [propext, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.Contents.run_full' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.Contents.p12_selected_contents' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.Contents.p12_undeferred_contents' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.PR161P12.inherited_rank_mass' depends on axioms: [propext]
+'Mathproof.BeyondNLogN.TakeoverBit.PR161P12.removed_exterior_count' does not depend on any axioms
+'Mathproof.BeyondNLogN.TakeoverBit.PR161P12.packed_rank_mass' depends on axioms: [propext]
+'Mathproof.BeyondNLogN.TakeoverBit.PR161P12.packed_max_rank' does not depend on any axioms
+'Mathproof.BeyondNLogN.TakeoverBit.PR161P12.normalized_volume' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.PR161P12.packed_deficit' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.PR161P12.actual_role_counts' does not depend on any axioms
+'Mathproof.BeyondNLogN.TakeoverBit.PR161P12.actual_bank_and_inventory_certificate' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008.literal_charge_lt' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008.finite_row_reserve' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008.row_reserve_value' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBalanced16320261009.core_slacks_positive' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBalanced16320261009.all_47_slacks_positive' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBalanced16320261009.strengthened_public_margin' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBalanced16320261009.all_47_concrete_guards' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
diff --git a/research/balanced-nullity-banks-163/verify.py b/research/balanced-nullity-banks-163/verify.py
new file mode 100644
index 0000000000..6a117777c9
--- /dev/null
+++ b/research/balanced-nullity-banks-163/verify.py
@@ -0,0 +1,96 @@
+#!/usr/bin/env python3
+"""Read-only exact arithmetic replay. Full physical supplier checks are separate.
+Requires two immutable upstream checkouts; no downloads or source edits.
+"""
+from pathlib import Path
+from fractions import Fraction as Q
+from math import lcm
+import argparse,hashlib,json,subprocess,sys
+sys.dont_write_bytecode=True
+if hasattr(sys,'set_int_max_str_digits'):sys.set_int_max_str_digits(0)
+HERE=Path(__file__).resolve().parent
+
+def need(ok,msg):
+    if not ok:raise ValueError(msg)
+
+def checkout(path,pin):
+    path=path.resolve()
+    def git(*args):
+        p=subprocess.run(['git','-C',str(path),*args],capture_output=True,text=True)
+        need(p.returncode==0,'Git failed: '+p.stderr);return p.stdout.strip()
+    need(git('rev-parse','HEAD')==pin,'Incorrect pinned checkout: '+str(path))
+    need(not git('status','--porcelain','--untracked-files=no'),'Tracked source changes in '+str(path))
+    return path
+
+def main():
+    need(not sys.flags.optimize,'Assertions must be enabled; do not use -O')
+    ap=argparse.ArgumentParser(description=__doc__)
+    ap.add_argument('--pr161',type=Path,required=True)
+    ap.add_argument('--pr163',type=Path,required=True)
+    args=ap.parse_args()
+    expected=json.loads((HERE/'expected.json').read_text())
+    r161=checkout(args.pr161,expected['public161_pin'])
+    r163=checkout(args.pr163,expected['public163_pin'])
+    sys.path.insert(0,str(r161/'scripts'))
+    import paired_cube_network as p
+    from audit_community_candidate import moment
+    ar=r163/'research/paired-cube-balanced-161/arithmetic'
+    sys.path.insert(0,str(ar))
+    import certificate as c
+    from interval_moment import saving_grid
+    n=json.loads((r161/'certificates/paired-cube-network.json').read_text())
+    raw=json.loads((r161/'research/paired-cube-bit/out/profile_p12.json').read_text())
+    inherited={int(k):int(v) for k,v in raw['child_histogram'].items()}
+    original=n['bit']['counts']
+    need(inherited=={int(k):int(v) for k,v in original['child_multiplicities'].items()},'Bit sources disagree')
+    need((raw['m'],raw['h'],raw['v'],raw['R'],raw['selected_roles'])==(72,24,1760,23368,5720),'Wrong immutable word')
+    need(raw['selected_rank_histogram']=={'20':5720},'Wrong selected ranks')
+    need(sum(k*v for k,v in inherited.items())==1934000,'Inherited rank mass')
+    packed=dict(inherited);need(packed.pop(60)==5720,'Exterior child inventory')
+    W=2*1760+3*(Q(5720,18)+Q(17648,3))
+    rank=sum(k*v for k,v in packed.items())
+    need((W,rank,72*W-rank,max(packed))==(Q(66364,3),1590800,1936,22),'Packed ledger')
+    # Actual disjoint coordinate blocks, not nullity divisibility alone.
+    need(all(sum(j//4==i for j in range(72))==4 for i in range(18)),'Selected blocks')
+    need(all(sum(j//24==i for j in range(72))==24 for i in range(3)),'Undeferred blocks')
+    profile=dict(original,W_per_vertex=W,rank_per_vertex=rank,deficit_per_vertex=1936,
+                 child_multiplicities={r:Q(v) for r,v in packed.items()},edge_count=sum(packed.values()),maxchild=22)
+    coarse=Q(302618322217,500000000000000)
+    atom=Q(302446903,500000000000)
+    old=Q(384599,10**10)
+    effective=(1-atom)*coarse+atom*old
+    need(0<effective<atom<1-effective,'Paid atom toll')
+    need(p.BAD==Q(1,10**16) and p.OLD==old,'Retained constants changed')
+    def bit_ok(a):
+        try:
+            ideal=p.exact_moment(profile,a)
+            fallback=p.BAD*Q(32*72**2*profile['edge_count'],W*72)*p.exp_upper(a*p.log_upper(Q(72)))
+            return ideal['moment_upper']+fallback<1 and rank+p.BAD*32*72**2*profile['edge_count']<72*W
+        except ValueError:return False
+    need(bit_ok(coarse),'Paid bit moment does not contract')
+    need(not bit_ok(coarse+Q(1,10**15)),'Adjacent coarse grid control accepted')
+    # Independent 80-term log / 12-term exponential enclosure implementation.
+    scale=lcm(W.denominator,*(v.denominator for v in profile['child_multiplicities'].values()))
+    _,bound=moment(72,int(scale*W),[(r,int(scale*v)) for r,v in profile['child_multiplicities'].items()],coarse)
+    _,fallback=moment(72,int(scale*W),[(1,int(scale*32*72**2*profile['edge_count']))],coarse)
+    need(bound+p.BAD*fallback<1,'Independent bit moment failed')
+    upstream=json.loads((r163/'research/paired-cube-balanced-161/certificate.json').read_text())
+    row=upstream['complex']['profile']
+    complexprofile=dict(m=row['m'],W=row['W_per_vertex'],N=row['deficit_per_vertex'],L=0,
+        total_rank=row['rank_per_vertex'],maxchild=row['maxchild'],child_multiplicities=row['child_histogram'])
+    complexmoment=saving_grid(complexprofile,10**18)
+    need(complexmoment['accepted']['saving']==c.AC,'Complex moment changed')
+    c.normalized_body_check()
+    result=c.price(row,c.AC,coarse,c.BETA,c.ETA,c.WEAKENING,'balanced',atom)
+    need(c.js(result)==expected['combo'],'Exact certificate differs')
+    need(result['kappa']==Q(118877394946471,200000000000000000),'Wrong final kappa')
+    need(result['a']<effective and len(result['assembly']['constraints'])==47 and len(result['assembly']['margins'])==7,'Supported assembly')
+    try:
+        c.assembly(result['finite_bridge'],row,result['a'],result['kappa']+Q(1,10**18),beta=c.BETA,h=c.ETA,a_complex=c.AC)
+    except (ValueError,AssertionError):pass
+    else:raise ValueError('Adjacent final grid control accepted')
+    checkout(r161,expected['public161_pin']);checkout(r163,expected['public163_pin'])
+    print(json.dumps({'status':'PASS','kappa':str(result['kappa']),
+        'decimal':'0.000594386974732355','strict_constraints':47,'margins':7,
+        'bit_mass':rank,'bit_volume':str(W),'scope':'Exact conditional finite arithmetic plus ledger and coordinate partitions; full physical realization and all-size inherited contracts are separate.'},indent=2))
+if __name__=='__main__':main()
diff --git a/research/coordinated-crossover-pr200/BANK-CHART-PROOF.md b/research/coordinated-crossover-pr200/BANK-CHART-PROOF.md
new file mode 100644
index 0000000000..99b6440b80
--- /dev/null
+++ b/research/coordinated-crossover-pr200/BANK-CHART-PROOF.md
@@ -0,0 +1,51 @@
+> Historical predecessor proof. The current16-gauge construction, including mixed4/5/6/24 residual banks, is specified and replayed by JOINT-GAUGES.md and joint/.
+
+Historical predecessor note. The current joint-gauge extension is specified in JOINT-GAUGES.md and PROOF.md; current execution starts at verify.py.
+
+# Frame refinement, concrete charts and completed banks
+
+2026-10-09. This supersedes the234-frame candidate for packaging. It is a sufficient finite construction under the unchanged all-size interfaces, not an unconditional multiplication theorem.
+
+## Admitted bit word and paid inventory
+
+The new word uses immutable PR200 at a1175449f34d39ff933d9d8ab23ced1f32b290ec, with 6,416 listed operation-frame bases (274 subspaces changed beyond PR211). All source/gauge/root endpoints,1760 compensated aliases, source/target chronology and34 terminal sinks are retained. The frame lane executed the complete F2 and defining-integer dirty-column replay. The geometry lane independently reconstructed every actual modified frame and full paid ledger, audited its terminal−102 rank3/−102 rank21 histogram substitution, and recomputed all distinct used-basis Bareiss prime witnesses from their actual bases. Everyprime q>2^80 remains eligible.
+
+The entrance classes remain exactly2200 rank20 genuine physical starts; rank21 recipient gauges remain internal splices. Sources, centre-touched roles and deleted terminals are forbidden as entrance gauges, with explicit rejected controls. Remove only2200 per-vertex rank60 exterior children after the complete bank endpoint proof. Every source, target, internal, centre, compensation and fallback child stays charged.
+
+The resulting full normalized profile is m72, W56402/3, rank1351712, deficit1936, largest child22. Integer moment normalization multiplies stock, every child count and deficit by three. Nine physical replicas supply1100 selected18×4 banks and44742 undeferred3×24 banks per stage, with31680 data registers: literal stock169206. All462078 role-copy assignments across three stages are checked exactly once.
+
+Two independent rational enclosures, including the full10^-16 bad-class32*72²-per-edge fallback, certify
+
+`coarse = 136731504666219/200000000000000000`,
+
+`ordinary = 546573370825043616599485038841961823/800000000000000000000000000000000000000`,
+
+`ordinary ≈ 0.0006832167135313045`.
+
+Adjacent coarse and preceding atom-grid controls fail for this fixed profile/envelope. No independent supplier gains are added.
+
+## All actual charts and fixed normalizers
+
+The220 unchanged rank20 gauge frames have sigma=ker A with four annihilator rows a. For G=9I-J, residual basis rows15a−(sum a)1 are Gorthogonal to sigma. Put these four columns before the20 sigma columns in B. Reusing PR197's fraction-free factorization, independently verify both B·C=dI and C·B=dI and replay every elementary factor to the identity. This constructs the literal residual projector's chart, rather than an assumed normalizing gadget.
+
+Across everyactualchart, maximum factor count is186, factor numerator magnitude5, denominator/inverse denominator18, integral inverse numerator18, and basis entry magnitude2. All nonzero inverse and factor denominators are below2^80 and thus units at every inherited primeq>2^80, including local ringsZ/q^w. The chart gzip witness hash is `f81a21df89d4702b70564f9577b5506504de429ca5df253e78bd28f2685cd432`.
+
+Embed B^-1 in the corresponding24-coordinate tensor stage and use the explicit coordinate permutation Pi_b sending its first residual block into bank intervalb. Choose fixed H_b=(b+1)Pi_bdiag(B^-1,I48), with b0..17 for selected banks or b0..2 for undeferred banks. A common inputcoordinatej outside that tensorstage maps to (b+1)e_(Pi_b(j)); these columns are distinct at everyq>2^80. This supplies literal distinct normalizers without a greedy choice or same-invocation alias. Scalar multiplication leaves conjugation unchanged.
+
+Group routing sends banklabelg to coregH_b. Each role's routing is a bijection on the completeGL cover. Distinct H_b preventtwo simultaneous roles from borrowing the same physical bank. Replica execution is serial and stages are private. After everycore completes, its arbitrary-dirty residual is conjugate to a disjoint coordinate partial swap. Each bank's partition sums to I72, so the exact ring-general identity gives the full required dirty endpoint; inverse order restores the same contents. The common weighted ancestor chart telescopes and spectators remain intact.
+
+## Fully charged extra routing
+
+Allow186 chart elementary factors,71 coordinate transpositions and72 fixed-scalar coordinate factors, totalling329. Nine full replicas contain27 stage-core occurrences. A conservative out-and-back old-selector bill is
+
+`K =54*((169206−1)+17114*72*329) =21900544398 <2^40`.
+
+These are additional paid ordinary-selector calls, never silently counted as zero work or as extra recursive children. As in PR197 PROOF5, their cost is O(K V w^(1−a_old)); the inherited positive stopped-leaf atom/row toll absorbs the fixed coefficient K. They change the finite stopping constant, not the exponent or external row order. Full group stock remains charged. All-size restored weighted selector and routing guarantees remain the original conditions.
+
+## Source-bound receipts and final assembly
+
+Frame receipt SHA256 `79160698cb761b17af5667e141367160ba0d3d447e0e6121b6de784065e0fb7f`; override witness SHA256 `65b6b30748fe8db9b472c6c4bd34ba6b6ec757f4440ad0d78f8cad139cbd906b`; independent used-frame prime witness SHA256 `038867338525dcae7f9e721b809b42b93cfe2f777db02c6f6189eca25ebed6cd`. All pinned sources stayed byte-identical through the independent geometry audit.
+
+The parent owns the admitted complex supplier, finite depth-four leaf composition and unchanged47 balanced inequalities with complete scalar/group/router/layout charges. No unattained infinite-depth limit is substituted. General compiler, common weighted chart, restored rows, paid layout, prime supply, precision/recovery and analytic transfer remain inherited. No new Lean formalization is claimed.
+
+Reproduce withgeometry/pack-admitted-frames.py againstframes/pr200-fastdirections1, geometry/audit-gauge-charts.py, andgeometry/check-normalizer-routing.py. The complete bank profile, full exact moment proof and each physical assignment hash are inpacked-pr200-fastdirections1.json.
diff --git a/research/coordinated-crossover-pr200/BASELINE.json b/research/coordinated-crossover-pr200/BASELINE.json
new file mode 100644
index 0000000000..38dc26ae37
--- /dev/null
+++ b/research/coordinated-crossover-pr200/BASELINE.json
@@ -0,0 +1,34 @@
+{
+  "base_commit": "8d8d67bcf69c5ea67d3a29dbc64ba588156d6e8d",
+  "bit_source_commit": "a1175449f34d39ff933d9d8ab23ced1f32b290ec",
+  "archive_sha256": "ddb0fbdbda0a45b25ea5c5ce5fd7858687d99211f45b6c3a674d38e1ae7c378d",
+  "bytes": 51416352,
+  "parts": [
+    {
+      "file": "baseline-pr202.part01",
+      "bytes": 12000000,
+      "sha256": "f5beb67709bf1235a2b6483883aa997a150ed4be47db978ab8be6040114cf924"
+    },
+    {
+      "file": "baseline-pr202.part02",
+      "bytes": 12000000,
+      "sha256": "21e1ff2a0ddc3e8febad3ca19d210d2e83d575f0968bb5125ab62bf6d00794fe"
+    },
+    {
+      "file": "baseline-pr202.part03",
+      "bytes": 12000000,
+      "sha256": "218c880af743a75e272cd51b3bd1ce5f36583f332823c2b5606d21d5d8b72e98"
+    },
+    {
+      "file": "baseline-pr202.part04",
+      "bytes": 12000000,
+      "sha256": "e0f95fcdd581a7617eabd53f4d0ea9e7cf0551a31fcd081e3d2154750740f114"
+    },
+    {
+      "file": "baseline-pr202.part05",
+      "bytes": 3416352,
+      "sha256": "9b64c7b8b5b7893eae071733b7ca14e280fe08b3db942c1c6d00e88d6e615286"
+    }
+  ],
+  "scope": "Exact tracked PR202 LF source, plus pinned replayed aligned complex witness. PR200 bit source agrees except its SOURCE.json receipt."
+}
diff --git a/research/coordinated-crossover-pr200/BOOTSTRAP-REVIEW.md b/research/coordinated-crossover-pr200/BOOTSTRAP-REVIEW.md
new file mode 100644
index 0000000000..090b30c875
--- /dev/null
+++ b/research/coordinated-crossover-pr200/BOOTSTRAP-REVIEW.md
@@ -0,0 +1,40 @@
+Historical predecessor note. The current joint-gauge extension is specified in JOINT-GAUGES.md and PROOF.md; current execution starts at verify.py.
+
+# Finite-bootstrap proof scope review
+
+## Sufficient conditional argument
+
+Let c be the certified coarse bit saving and let a0 be the completed atom-wrapped ordinary supplier saving. Assume the inherited all-width ordinary leaf interface, complete spectator handling, restored borrowed rows, fixed finite tapes and paid routing/layout. These assumptions must be stated for the selected new packed supplier and its retained old ordinary selector.
+
+For j=1,2,3, construct Aj using the coarse supplier at atom width w=ceil(e^c), with ordinary leaves, source exchanges and routing executed only by A(j-1). This is an acyclic call graph. Within one level, rank-r children have smaller width because r<=22<72; ordinary recursion decreases j. No limit construction or self-call through a completed leaf is used.
+
+The inherited stopped-leaf estimate is
+
+Tj(e)/V = O(e^(1-c) + e^((1-c)^2+c(1-a(j-1))) + e^c log e).
+
+Therefore the achieved ordinary saving is
+
+a(j) = (1-c)c + c a(j-1) = c - c^j(c-a0).
+
+Here 0<a0<c<1/2. Hence a(j-1)<a(j)<c<1-a(j), with fixed positive rational gaps. The ordinary toll, its logarithmic factor and any fixed finite number of additional bank selectors are absorbed by the existing strict row/adapter gap. For this packaged construction the retained finite chart/router bound is K=21,900,544,398. These extra calls are paid ordinary work; they must not be silently removed or recounted as free recursive children. Internally borrowed rows are restored by each completed leaf while complete outer fields remain spectators. The retained external reserve is unused slack, not a newly available workspace.
+
+After exactly four levels, use a4 in the original paired assembly, alongside the certified complex supplier and its retained bridge. All 47 inequalities are rechecked from the actual complete bit histogram. This proves the advertised bound only conditional on the preceding interfaces.
+
+## What the current arithmetic does and does not establish
+
+The final assembly receipt checks four exact recurrence steps, their strict ordering, the two paid moments and all 47 arithmetic constraints. The expression for a4 agrees with the finite PR185/197 recurrence. No contradiction or circular infinite bootstrap was found.
+
+The arithmetic routine reads the inherited semantic/row bridge; it does not independently implement or verify its all-width completed-leaf contract. Thus wording such as "47 inequalities fully prove the new all-size compiler" would be unjustified. Wording such as "conditional finite certificate under retained completed-leaf and all-size interfaces" is accurate. A finite column replay likewise establishes literal finite endpoint and restoration identities, not those global interfaces.
+
+The public PR197 proof already makes this distinction, pays extra bank-selector routing as a fixed constant under c<1-a(j), and uses exactly three levels. Retaining that explicit conditional argument is sufficient for this package's stated conditional scope. PR197's older lack of a globally renumbered complex replay should not be copied as a present defect if PR202's complete actual replay has been successfully checked; the new package should report its own replay evidence.
+
+## Packaging checks
+
+- State the inherited completed weighted leaf and internal-row conditions explicitly.
+- Keep all new fixed bank-routing work inside the ordinary routing toll.
+- Distinguish factor-three moment normalization from nine actual replicas.
+- State the adjacent grid exclusion only for fixed suppliers and parameters.
+- Preserve source/target caps and actual donor stock in every claimed replacement.
+- Retain original license/attribution notices and do not claim new Lean certification.
+
+No expensive replay was launched during this review.
diff --git a/research/coordinated-crossover-pr200/CENTER-RETIMING.md b/research/coordinated-crossover-pr200/CENTER-RETIMING.md
new file mode 100644
index 0000000000..d4f1e52315
--- /dev/null
+++ b/research/coordinated-crossover-pr200/CENTER-RETIMING.md
@@ -0,0 +1,15 @@
+# Expose two source heads after the copied-center prefix
+
+Retained source471 construction stage, before the final target aggregation. The source/auxiliary word and paths below are unchanged. Final compensation uses 28 pivot reads in place of the 112 target reads described here; see `TARGET-AGGREGATION.md`.
+
+The final construction reuses 471 original sources. It moves 168 actual operations from the old center phase to the beginning of the following phase. The new center prefix has 17080 operations and all 440 early source-partner mixes. New source-backed rank-2 heads are role 8584 from X98 and role 9424 from X458. Their compensation reads occur immediately after all copied-center scatters; all previous gauge reads shift by 168 operation positions and retain their relative times.
+
+`newg/schedule.json` contains only the executable event permutation and operation indices. Its original discovery metadata is not trusted. The portable executor reconstructs each actual X/Z address using all 471 source aliases and all 1760 physical donor/recipient identifications. `newg/schedule_audit.py` rebuilds all 41288 operation/mix events and independently compares the original and retimed sequences. For every event, both actual input addresses have the identical preceding-write event. Every final write is identical. The complete per-address frame-dimension sequences and paid jump histograms are identical; all 64988 consecutive frame inclusions are checked exactly. Every center sees the same final producer write, and no deferred or later middle operation consumes a center address. Both new source-head addresses are untouched throughout the prefix. Every prior alias still dies before its recipient gauge.
+
+The copied-center interface is the retained lemma in `notes/copied-centers-lemma.tex` from the pinned source archives. In its notation the common target frames are D0=B⊗F_h and D1=A⊗F_h; a center occupies D_U=D0⊥(P⊗U). Every initial ungauged dirty correction leaves each target at D0. Source injection, saved partner mixing and the retimed prefix change X/Z only. All center copies therefore transform from D_U to D0 and scatter while every target remains at the common D0. The original center remains at D_U and later advances directly to D1; the charged ranks are still dim(U) for the copy and h−dim(U) for the original. No later producer consumer is bypassed.
+
+Only after all center scatters do the new two nonzero target-gauge reads occur. The 168 deferred operations follow; then all existing gauge compensation, terminal target substitutions, side roots and partner deliveries proceed in their original relative order. The scalar executor checks this actual order: every one of the 5280 center target reads occurs before any nonzero target-frame movement. It traces every subsequent target frame, checks nesting and the actual receiver cap, then reconstructs the reversed exact annihilator paths from each cap line. In the complementary orientation all those target movements reverse before the center scatter, so every target has returned to D1. The center copy transforms to D1, with the same paid ranks, before the inverse middle mixer. `newg/boundary.py` independently reconstructs both full target ledgers and every selected source/donor/recipient path.
+
+Complete F2 and both defining integer words verify all 20163 original source/target/dirty columns, every target, restoration of all 1760 original sources and 16643 remaining dirty registers. Six controls reject missing mixing/compensation, stale adjoints, early inverse mixing and moving the new reads until after the deferred operations. The retiming does not discard a prefix cost or lower the copied-center charge. The two additional source aliases remove two whole auxiliary slots; their exact local change versus source469 is {1:+2,2:+8,3:−2,19:+8,21:−8,22:−2}, of rank mass −48. All eight new compensation reads and both reflected ledgers are paid. Complete banks, selector calls, two exact moment engines and the inherited conditional assembly are recomputed.
+
+The dependency retiming, two source-head construction, integration and verification were prepared with substantial OpenAI Codex assistance. Existing contributor notices, Apache terms and OpenAI/Anthropic disclosures remain.
diff --git a/research/coordinated-crossover-pr200/CROSSOVERS.md b/research/coordinated-crossover-pr200/CROSSOVERS.md
new file mode 100644
index 0000000000..69e511ac0a
--- /dev/null
+++ b/research/coordinated-crossover-pr200/CROSSOVERS.md
@@ -0,0 +1,13 @@
+Historical predecessor description: numerical counts below describe that predecessor. The final construction has 471 source aliases, including 28 new source-backed entrances. See NEW-SOURCE-GAUGES.md for the current scalar word, target chronology and bank counts.
+
+The current package adds three genuine gauge-backed source slots and further exact frame cuts. See GAUGE-SOURCE-BORROWING.md and CUT-COMPONENTS.md.
+
+The current word additionally reuses 440 original sources for actual auxiliary branches. It recomputes adjoints and source paths, rather than deleting a histogram family from an unchanged word. See SOURCE-BORROWING.md.
+
+# Current structural refinement
+
+The current construction changes 82 actual entrance gauges, their nested compensation order, and two aliases, in addition to operation-frame refinements. It crosses the fixed-word scope of PR212, which holds gauges and reuse pairs fixed; that does not refute its stated ceiling.
+
+The new step adds 18 rank-12 and 48 rank-13 entrances in shared nonmaximal frames. Individually maximal frames can differ even when a profitable common subspace exists. Earlier isolated gauges lost after complete target costs. Shared rank-18 groups pay one target transition for multiple dirty reads. Three additional rank-19 gauges then use those existing target frames and become profitable. JOINT-GAUGES.md gives the physical mixed-bank allocation and full ledger. No internal child family is deleted from an unchanged word.
+
+The frame-only predecessor is documented in PLATEAU-REFINEMENT.md. Original notices and historical bank comparisons are retained and labeled.
diff --git a/research/coordinated-crossover-pr200/CUT-COMPONENTS.md b/research/coordinated-crossover-pr200/CUT-COMPONENTS.md
new file mode 100644
index 0000000000..f1e86aafd9
--- /dev/null
+++ b/research/coordinated-crossover-pr200/CUT-COMPONENTS.md
@@ -0,0 +1,20 @@
+# Compatible extremal frame cuts
+
+The final package uses 117 exact operation bases from 17 connected components of PR216's additional witness, at commit `34abdc58ed77329c8bf9af5634531fc41939ba3c`. PR216 develops greatest/least feasible schedules and a concave-cost minimum-cut majorant. Its original description is preserved at `search/PR216-MAJORIZATION-CUT.md`.
+
+We independently compare its additional bases with PR211, split the differences by the actual physical edge graph and retain only components whose complete boundary inclusions are valid together with the 82 shared gauges and the 440 borrowed source paths. Seven of 24 candidate components fail these checks. The other 17 lower the paid cost and are composed together. Terminal write frames are unchanged. This does not assert reproduction or arbitrary union of PR216's whole supplier.
+
+The complete pre-normalization child delta against the frozen source440 word is:
+
+```
+{2: -189, 3: -66, 4: 51, 5: 15, 6: 87,
+ 7: -75, 9: 126, 10: 9, 12: 42, 14: -102}
+```
+
+Its rank mass is zero. Stock, deficit, all actual bank assignments, gauge charts, target boundaries and selector bill are unchanged. The full recomputed moment includes factor 24. Scalar execution over F2 and the defining integers in both directions, exact frame and prime admission, all 440 original-source boundaries and reflected paths, full source/internal/target ledgers, literal banks, two paid moment engines and exact final assembly are rerun against the composed word. The source-reuse delta itself remains `{2: -1320, 22: -1320}`.
+
+The comparison witness and source hashes are in `search/PR216-COMPONENTS.json`. Original PR216 and inherited notices remain verbatim in `search/PR216-NOTICE` and `search/PR216-NOTICE-PR211`. Composition and verification were prepared by eumemic with substantial OpenAI Codex assistance. All inherited attribution and AI disclosures remain in force.
+
+## Further least-schedule refinement
+
+After the 117 compatible component changes, a new majorant cut around that word selects 73 additional operations. All 1037 operations touching borrowed source paths or terminal writes are frozen. The greatest schedule has 15239 alternatives and selects none. The least schedule has 10632 alternatives, zero degenerate freezes and chooses 73, giving shared delta {1:+48,2:-60,3:+3,4:-39,5:+24,6:+18,7:-36,9:+33,10:+3,11:+24,12:+6,14:-30}. Its rank mass is zero. A subsequent greatest/least round selects nothing. Float flow only nominates frames; complete exact geometry, prime, scalar and paid-profile admission verifies the final bases. The cut idea and its theorem are credited to Dugongue's PR216; this further application and verification used OpenAI Codex assistance. `search/EXTREMAL-CUT.json` preserves the exact selected bases and discovery ledger.
diff --git a/research/coordinated-crossover-pr200/EXTRA-SOURCE-BORROWING.md b/research/coordinated-crossover-pr200/EXTRA-SOURCE-BORROWING.md
new file mode 100644
index 0000000000..1a42e66f05
--- /dev/null
+++ b/research/coordinated-crossover-pr200/EXTRA-SOURCE-BORROWING.md
@@ -0,0 +1,15 @@
+# Twenty-two additional source entrances
+
+The conditional claim is κ = 6.95684647723759e-4. `extra/selection.json` records22 additional source/partner/role assignments beyond the retained471-source word. Their source pairs are disjoint, and every source follows the selected auxiliary head's complete later physical aliases.
+
+Nineteen new entrances use rank-two gauges; three use ordered rank-four gauges. Role14074 follows14420, role14092 follows14432, and role14073 follows14075. For each late head, the exclusive target quartet is read at its own rank-two mix before its overlapping quartet is read at rank four after the early head. Role14073 is scheduled at deferred-operation offset168, after the retained new-source gauge reads and before its first physical operation. Source90 and partner89 supply this final entrance.
+
+Every complete source path includes its original rank-one line, rank-two K mix, applicable rank-four frame, first operation, all subsequent physical aliases and final full frame. Existing K mixes and unmixes are retimed; source and partner values restore after the reverse word. `extra/boundary.py` independently rebuilds the old source and auxiliary paths, complete new source paths, both reflections and exact primal/dual prime witnesses. `targetagg/boundary.py` separately traces their literal source events alongside the complete target program.
+
+The source/auxiliary local histogram delta is C2+3,C3+2,C4+1,C5−2,C6−1,C8+1,C10+3,C12−1,C13+3,C14+3,C15+2,C16−6,C17−5,C22−22, with rank mass−528. The resulting53 source-owned gauge entrances comprise45 at rank two, five at rank four and three at rank nineteen. The2279 bankable gauge entrances retain231 exact charts.
+
+The remaining14342 ordinary width-24 roles give344208 ordinary banks per stage. Together with the unchanged mixed-width families, the72-replica bank ledger has3590136 assignments,1061490 banks and literal stock1314930. All concrete matrices, inverses, factors, normalizers and controls are replayed.
+
+The actual scalar word restores all1760 source inputs and16621 dirty registers. All208 extra-source compensation deliveries are conservatively charged before compression, with every target transformation included in the fixed-call bill.
+
+Prepared by eumemic with substantial OpenAI Codex assistance. All inherited licenses, contributor notices and assistance disclosures remain.
diff --git a/research/coordinated-crossover-pr200/FILES.json b/research/coordinated-crossover-pr200/FILES.json
new file mode 100644
index 0000000000..b78f93b29c
--- /dev/null
+++ b/research/coordinated-crossover-pr200/FILES.json
@@ -0,0 +1,116 @@
+{
+  "files": {
+    "BANK-CHART-PROOF.md": "891d42a1a515d4a3d588e1c1a585e8bf3204669fb3d21de958340597f130da82",
+    "BASELINE.json": "34d948f1f3724370d50605e1f8784407c6ee8f95f34e3135ad6afae07d7fcd0a",
+    "BOOTSTRAP-REVIEW.md": "1a28a69268f6f84bf8848c00d0493960de8c0e1636a295a39445fb5401f30693",
+    "CENTER-RETIMING.md": "c2d1da88f1901c4fef9e64200d58068ee14c8ca2a7a48b5fb913cbdb65c3297e",
+    "CROSSOVERS.md": "5aa56e18f35bbae3ee299fe1b7f745f6eb4b0c10bfeff8c77a49b5df32bc7230",
+    "CUT-COMPONENTS.md": "35978fc6d9b34597dffb264d221c317f84a8cf72a2023af044212dcee976d909",
+    "EXTRA-SOURCE-BORROWING.md": "d9fae9a9f26596ac5eaf72418e657524c290e9da2c8f226896e689028606ad54",
+    "GAUGE-SOURCE-BORROWING.md": "4c6ba79d98d780825ebdef1057f06d965982a91b05cffe8a75660c8b2e5914a5",
+    "JOINT-GAUGES.md": "ac686d1d50d221e9f949599b20604adb3de664ecf90d2886d2f14d03284ee412",
+    "LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+    "NEW-SOURCE-GAUGES.md": "6f14aa955c15d437cdec51804f8229e73298b88c72027c8dee0488c4b657d30e",
+    "NOTICE": "9d3622f97efd456e60cb41f35420a8d0b4179cf351a835b06b7f4987b3037eb8",
+    "PLATEAU-REFINEMENT.md": "c62f2224aeb1147ef91365a604e356b3b21648636dcf652b0fdaa9375b745405",
+    "PR-STATEMENT.md": "ec96509c28bd1f1372f2c25a56d873e5bcd368f5729d1a95b22fc0a91892a5da",
+    "PR205-COMPARISON.md": "c33754b0a7f551ed8a6f2446686d2ac938f8ce1822a49bd92315f9301191a902",
+    "PROOF.md": "1aa1a3147380f1181020e8450b6a5c90d0d176a0bbbbebc066c660d8514b9e0f",
+    "README.md": "b28f7be491ea354de2a57d5dbe090b6fa0131c2ab9c3be31db9d1d3fdb649da4",
+    "SOURCE-BORROWING.md": "1e45fbcbed0050c5e49846ee728b12685034e6804cc5ac129b2a3f79d7e6ac8b",
+    "TARGET-AGGREGATION.md": "debcf6f5f40320435823fee78dc06d4e819a4e752c3788be33848cf0a707261f",
+    "TARGET-ECHELON.md": "cbee505a8ae2c02f7d4361a02f04b145e2715ccce71d5324f01423b7600655fe",
+    "TARGET-RANK-COMPRESSION.md": "70aea716677b2d3530d0321b5640fa3f4deb1c6006bfca1cde7aa9eb8db8c45b",
+    "arithmetic.py": "890c575de3ea7af9379f08ea930a842f2c0ee50475fdd7507927a066e4bac7a2",
+    "baseline-pr202.part01": "f5beb67709bf1235a2b6483883aa997a150ed4be47db978ab8be6040114cf924",
+    "baseline-pr202.part02": "21e1ff2a0ddc3e8febad3ca19d210d2e83d575f0968bb5125ab62bf6d00794fe",
+    "baseline-pr202.part03": "218c880af743a75e272cd51b3bd1ce5f36583f332823c2b5606d21d5d8b72e98",
+    "baseline-pr202.part04": "e0f95fcdd581a7617eabd53f4d0ea9e7cf0551a31fcd081e3d2154750740f114",
+    "baseline-pr202.part05": "9b64c7b8b5b7893eae071733b7ca14e280fe08b3db942c1c6d00e88d6e615286",
+    "borrow/PROVENANCE.json": "3a6c0a1eb27a6c2bc17d7d5f1c8b88096d74bafd0ff9e1555b508aa8b460b86d",
+    "borrow/boundary.py": "1c13447daf83e8818b487bb5e4e492f488ad7a72d899de1e4e4143664013f0b7",
+    "borrow/derive_profile.py": "eee27a5fff481d2eaa03ce6d8d54c9f6f8144134a207d026195d4a6240e69b57",
+    "borrow/replay.py": "a9f32a2f007aba9a2bcb6983f16c817650d37e1c181cf1b98516b961fa992ea9",
+    "borrow/selection.json": "6932749407e14b199651f0a7deff02c6de74e2c1be0a1300263f1e164927fb8b",
+    "certificate.json": "801d10801e8a1ab07be04b27081c396562aca61e54d4d63a1a1dfdee25a91ad0",
+    "echelon/PROVENANCE.json": "ada9a76a6649ed4b24c63b684cad8935226f73e52f61a81cbb6a3c8f7c20251a",
+    "echelon/selection.json": "8e6ce5e257aa7656898ab46bc3421df4a619497b1e9f5956d2c546b3014050a7",
+    "extra/PROVENANCE.json": "0a28de6f0e296deb92d310313ae6353c3eef3872c41e433ba3341f44f1ff1717",
+    "extra/boundary.json": "8554bc8c44a399f00fde31830c276ffbc89b89a5c5b909dbfcbc77f7b7c0ccc7",
+    "extra/boundary.py": "6a32671203a25055a71a7acb98396124cec591ebbf680677b40bc37e4db5a7bb",
+    "extra/selection.json": "791f6ddeb19add7026ba37461256c13efe3d35bba0bccf1cb9787a01fc16f0f2",
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+    "geometry/audit-gauge-charts.py": "111710eaa3cbef146def51419e66077e103366198790e06a434d75c99a17b508",
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+    "geometry/pack-pr200.py": "ebd1765d5edc7804bc55e36ebac6c40500d8d393c40aa76cab6668954b6d4e70",
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+    "joint/joint-banks.json": "e0a2d93f594ccb797ff7fcad312b55091aa75dc9e6556acc8ea86959dc485419",
+    "joint/joint-charts.json": "e93e3258e9f381135e1bcfd4e01f26f9a79d36255d863731c583a19c2b62a43c",
+    "joint/joint-paid.json": "4f43ab78a016e8a5c9470bd7408153851547aee3f46bbd92dca3a2aab6b914fe",
+    "joint/joint-selection.json": "13f6420d6c341c22235c9d136d5f7c32896974bb165a893fecd7e583b790d133",
+    "joint/joint_word.py": "de37cbed753fc6f13a7e5dc25c8346279cfee5f1868df8c12a5fcb6ca8f631c0",
+    "joint/pack_joint.py": "13a914aaad46083acfa0a4441d088639a09bdc24df35753d8c98332c1252cefa",
+    "joint/pairs.json": "da54b863601b5cbc21d27fc5cd623bf805f974a3e47b81ed29a85aa5bad0fa05",
+    "joint/price_joint.py": "f9f07d7ed023c8a7c7882ad79d91a239630cf44cb9563c744141c74c92cf485f",
+    "joint/replay_joint.py": "5eaf12568ba9164f2e66321e14f642342f511298040d3b99cbaa9107fb671624",
+    "newg/PROVENANCE.json": "0d00855a27d956927ee3d7a7d1ba6d850dd3d83216deff2dac9aab8f6433b037",
+    "newg/boundary.py": "94c1184da0d4d0798c7e7130671bdf7026dbb422ee0a2c76196efda743150d88",
+    "newg/derive_profile.py": "d4638e20c963d5d05b0a44de3a4c3745a94f57a1f68898f1469414d425994ede",
+    "newg/replay.py": "0cbec81a11e758156e9303f63dc8c617601da69523073fdf2a77394d76acf56a",
+    "newg/schedule.json": "ab39d61fff7a2cfcb13e2c10d65f2c603e76a91013d197260d6c7b645ccb27a1",
+    "newg/schedule_audit.py": "d426102fe0faaf2c1725add5ea88db4f17316ff055a6995ae60a7da6ba881671",
+    "newg/selection.json": "d94ef4e0b8ca98a7385b364cf245816a7cad08f850a60ce8b67aab4c3a89f3f4",
+    "newg/shared-change.json": "9590fc7dcb547d3a53e03bee34da45fede81b55ca828f340e812dc53500c4771",
+    "newg/source469-candidate.json": "de81b275ad265c93340af5f6a1c3b998d490ffc558fbe7e4608bd906648af7cd",
+    "newg/source469-frame-overlay.json": "936713763e39ec09e5c373aab779fb6f82737401e9158415ad3f7183ab9e2fc8",
+    "newg/source469-receipt.json": "a201f0fad94236ef9021588f88096fde3447ab863a2ac49b9cce07eb71cf3701",
+    "rank/PROVENANCE.json": "5b7b8b4493fff577dd35a9d872c14708ad03c126a988c01ee355c4f62a2711a4",
+    "rank/selection.json": "5c97c6413f1eb56c7b8f9ba9e839329d212fae7ac3a08c29e9efdee42f5048a0",
+    "requirements.txt": "fdf15f8e56a3349c4cf3a05e4d073f03512a65deff2383767a4eaed1c445310f",
+    "search/EXTREMAL-CUT.json": "5a41f559e12a2691c2af99621cdf72f05bbf0915531cf20cf58a47ca340bd155",
+    "search/PR211-NOTICE": "6c5d974e64c2279192aabf421ed2c1f76e18105e88e3d9b3c15c7f23c991c6d2",
+    "search/PR216-COMPONENTS.json": "dd7f9a187486d5b3cd9e204fdcaab7b2e4a35c9d058c595e4160f22e714ae896",
+    "search/PR216-MAJORIZATION-CUT.md": "4f43b15d9a51e6d08d8619fa442aa6b34c27b2e666b6e724a5b09bf7ebf5492c",
+    "search/PR216-NOTICE": "c9850418622aea8d1382a230306c940947b5da818f4854b4b272ed60cf239f6c",
+    "search/PR216-NOTICE-PR211": "6c5d974e64c2279192aabf421ed2c1f76e18105e88e3d9b3c15c7f23c991c6d2",
+    "search/SEARCH-NOTE.txt": "355c066f23e166849abeaa5d596c7c55cd2e70f6eefac9cede0972e26aa2aef0",
+    "search/additional-frames.json": "a2baff847605aded1143f92032c6443949bf4f1bc5438c39c5a67d7ea78b05cc",
+    "search/discovery.json": "440e5c7216b35f3aec9a748383e211df344a432b766f8874cd433bdd0b39b9ef",
+    "search/plateau_search.py": "9e765745f08f386a9a78d591119f8bb6a580293726cfa89d26f3a99c47d37410",
+    "search/pr211-frames.json": "56b2ed0f6b867e8cafdc32ac70419cfda36d02838844572098b3fc77f8026e58",
+    "search/pr230/NOTICE": "7dfbad087ee49b74fd5075d517eaa263c5dd3d4fbd4c69bfba34ee1cb55e837f",
+    "search/pr230/PROVENANCE.json": "19957e28a8ad26281bfe8bfaf8ae16191be53e729a8870a6154671b3384959d4",
+    "search/pr230/joint-plateau-result-prdf958.json": "cd5731260af0daa62952c3b82debdb26cfb79e62a5c615688d8ccac6f7a95fcc",
+    "search/pr230/joint-plateau-search-prdf958.py": "80f3f9e8ba1972b911b701a4de4f811d6d25c9b4c0bc6282d640c2f44caafd0d",
+    "search/pr230/opframe-bases.json": "387ae1b5e69004426d094f4947b29c532d60fdf31f727f5e8830468278fe0f92",
+    "search/relaxed-diff-count.json": "d9486c23e236ef3e4e44c0152c8288862213e527b0726cb9f5a925c43fe8e8b7",
+    "search/relaxed-discovery.json": "d3a69a12e7afb000be4013b6b0e00d56a439f3c2c1ba074b9e01d8b9aa386fd1",
+    "search/relaxed_search.py": "033d1506c5967f5f1c11b5175304ffe0f108f2ebac45ca70591a67c62d277b4e",
+    "search/source467-selection-full.json": "d7b2487a6c675de7c2da4293f76e1bae4333c9e08ff629c85617f7c6113ffd16",
+    "targetagg/PROVENANCE.json": "664118f9060e54dcd9f102279d030571a4a9c8d0507cb7c3b4876259a778fba6",
+    "targetagg/boundary.json": "edcacd4267749c32a1e87aea98de2803b9ddfba3f1ecf3c71b0e27adaa939e7f",
+    "targetagg/boundary.py": "20f4c69043754901193bcf55ec89b8d0dec4c518094c7933b77f5715f163787b",
+    "targetagg/derive_profile.py": "94b849361b563d331c5e67c80363f6391e1afb53d9d34db83b7ab711d88d087b",
+    "targetagg/profile.json": "3d4cd1da64f1912da716286d1dbd84845f2699c18914f3d663e2069f102f0f01",
+    "targetagg/replay.json": "b32aea9900e5369c168014efaa4a9aab2231aa684227b4f5ef1602bc3201b1bf",
+    "targetagg/replay.py": "e4c5080aa6b59afb045122170770ad18b9465ac7535b5336fca34c7e222bcea8",
+    "targetagg/selection.json": "b7a4d5eb2a6d2f7df867f9be5af7da1d1aa99cfbd5214195967b0ff60642d77a",
+    "targetagg/setup.py": "fd7adf0b12d0769e06438b2ee4322bfcb180d8f73a1104a3eb805ab88c5cc393",
+    "targetagg/word.py": "293a0babfac7e92ffccfe1cb69971b26a49fe56f67872b1060c3fe49d759a463",
+    "verify.py": "2869e8725c20b5b7a9c5cfdcc520ca1b8ed95bf1f8b4ce8d06ba202073e351cb",
+    "verify_inner.py": "15224eb2f535f426721f0b07f86f75086aa44e62bf526ed25baa8ae8b7e14785"
+  }
+}
diff --git a/research/coordinated-crossover-pr200/GAUGE-SOURCE-BORROWING.md b/research/coordinated-crossover-pr200/GAUGE-SOURCE-BORROWING.md
new file mode 100644
index 0000000000..97db0b8a84
--- /dev/null
+++ b/research/coordinated-crossover-pr200/GAUGE-SOURCE-BORROWING.md
@@ -0,0 +1,17 @@
+Historical predecessor description: numerical counts below describe that predecessor. The final construction has 471 source aliases, including 28 new source-backed entrances. See NEW-SOURCE-GAUGES.md for the current scalar word, target chronology and bank counts.
+
+# Three source inputs occupy genuine gauge slots
+
+The selected assignments are role 14697 from X106 (partner X105), role 14696 from X138 (partner X137), and role 14699 from X346 (partner X345). Each role has a rank-19 genuine entrance gauge; its physical slot later continues as gauge recipient 17224, 17192 or 17152, respectively. `gaugeb/selection.json` is the literal selection.
+
+After the ordinary source injections, save the partner sum in each carrier. The original passive source is then used as the arbitrary initial value of its selected dirty slot. These pairs are disjoint from the 440 copy-eliding pairs. The role is untouched in phase one. Its existing compensation read and all downstream compensation remain, including reads of the later recipient through the same physical slot. The auxiliary word is unchanged. After reversing every auxiliary gate, the passive source has been restored, so the saved carrier sum can be undone and the original source injections reversed.
+
+The scalar theorem already allows arbitrary initial dirty values. The new executor verifies their correlation with the source data directly, using the same formal source column in both positions. It checks F2 and both defining integer signs, every target, and restoration of all 1760 original sources and 16671 remaining dirty registers. Controls reject omitting the new early mix or gauge compensation, as well as the three retained copy-elision mutations.
+
+The exact source path is its original line, then the rank-2 partner plane, the genuine rank-19 gauge, every actual operation/root/alias frame, and the full rank-24 frame. `gaugeb/boundary.py` verifies every inclusion, nondegeneracy, alias timing and the reflected jump multiset. The carrier keeps its original delivery frame and target anchor. Relative to the old dirty path and passive-source path, one gauge gives local jump delta {17:+1,22:-1}; its retained rank-57 child also disappears. The three-row shared delta is {17:+9,22:-9,57:-3}, raw stock falls by 3 and raw rank mass by 216, preserving deficit.
+
+`gaugeb/derive_profile.py` subtracts each complete old physical path, replaces its passive-source chain, retains the exact target chronology, and rebuilds all paid children. It derives the delta rather than assuming it.
+
+The final bank family contains no width-5 residuals. Per stage, use 864 copies of 4×11+7×4, 8464 copies of 18×4, 78 copies of 12×6, 216 copies of 6×12, and 345408 copies of 3×24. These are 355030 banks per stage. Actual role/replica assignments, charts, two-sided inverses and distinct normalizers are enumerated. Literal stock is 1318530, with 231 charts and fixed selector bill 178427035440. After normalization, W decreases by 15, H17 increases by 216 and H22 decreases by 216; rank mass decreases by 1080 and deficit remains 46464.
+
+The construction, its integration and executable verification were prepared with substantial OpenAI Codex assistance. The original source notices remain applicable.
diff --git a/research/coordinated-crossover-pr200/JOINT-GAUGES.md b/research/coordinated-crossover-pr200/JOINT-GAUGES.md
new file mode 100644
index 0000000000..a00f4693b7
--- /dev/null
+++ b/research/coordinated-crossover-pr200/JOINT-GAUGES.md
@@ -0,0 +1,54 @@
+Historical predecessor description: numerical counts below describe that predecessor. The final construction has 471 source aliases, including 28 new source-backed entrances. See NEW-SOURCE-GAUGES.md for the current scalar word, target chronology and bank counts.
+
+The source443 predecessor retains all 82 gauge reads, but three rank-19 gauges occupy original source registers. Its literal bank counts and complete reconstruction appear in GAUGE-SOURCE-BORROWING.md; counts below describe the earlier source440 predecessor.
+
+# Shared nonmaximal entrance gauges and current literal banks
+
+The new step chooses a common subspace of several roles' admissible entrance frames. Individually maximal frames can differ and force separate target transitions. A smaller shared frame lets many actual compensation reads use the same target climb.
+
+Eighteen rank-12 gauges use one common frame. Six further groups of eight rank-13 gauges use six other frames. These 66 entrances extend the verified 13 rank-18 and three rank-19 entrances, for 82 new physical entrances in total. The exact role IDs, bases and complete response supports are in `joint/joint-selection.json`. No operation frame or scalar DAG edge changes in this step.
+
+The rank-12 group's union is targets 216–223 and864–871. The six rank-13 groups use disjoint blocks 872–879,880–887,888–895,896–903,904–911,912–919. The rank-12 frame lies below the retained rank-18 frame on targets864–871. The adapter orders all new compensation reads by increasing gauge dimension, then retains the original reads and their chronology. Every resulting target path is checked exactly.
+
+## Discovery and complete paid accounting
+
+The search intersects distinct admissible entrance frames, instead of retaining only individually maximal frames. For each fixed common frame, selecting compatible roles is a maximum-weight closure problem: a selected role requires all of its response targets. At a fixed coarse exponent, the role's benefit is its saved initial increment less its literal bank-stock reduction; each newly split target interval is charged once. A min-cut proposes the profitable subset. Float flow and scores are used only for discovery. The complete actual physical word, literal banks and two independent rational moment engines determine the certified result.
+
+Every selected gauge is nondegenerate for G=9I−J, lies in its role's first operation frame and annihilates its complete target response. Every selected role is untouched by source injection and the center phase, is not a terminal role and is not a recipient/birth alias. Physical donor starts are allowed when the entire chronological chain and scalar replay pass.
+
+The two earlier donor substitutions remain: births18892/18894 use donors8292/8293 rather than7756/7757. Their rank-8 endpoints fit the original rank-21 recipient gauges and die before the unchanged reads. All 238 source aliases remain fixed. The adapter reconstructs all 1760 pairs and the actual physical map.
+
+## Scalar chronology and target paths
+
+The original ungauged word already subtracts each dirty role's complete integer adjoint response at time zero. Move that existing compensation read to the start of the side phase. The selected role has not been touched, so its value is unchanged. No response coefficient is dropped. Several reads in one frame make the target climb once and stay there. Different shared frames are ordered along actual nested target paths; every increment remains charged.
+
+For a gauged donor the physical path is its entrance gauge, donor operations, the unchanged recipient gauge, recipient operations and the full 24-dimensional frame. Exact containment is checked throughout. The recipient reads the current donor contents, including signal and dirty value. Full scalar replay checks every source, target and dirty-register column over F2 and the defining integers in both reflected signs. All sources and dirty registers are restored. Omitted compensation at ranks 12,13,18,19 is rejected, and placing the rank-19 read before its rank-18 predecessor fails exact target chronology.
+
+## Completed mixed banks
+
+The final word reuses 440 original sources for full-residual auxiliary branches; `SOURCE-BORROWING.md` gives that separately checked transformation. The entrance gauges and charts below are unchanged.
+
+The 16674 surviving physical chains have residual dimensions 4×2200,5×3,6×13,11×48,12×18,24×14392. Use 72 complete physical replicas. Allocate the following literal patterns in each tensor stage:
+
+| Residual blocks per bank | Banks per stage |
+| --- | ---: |
+| 4×5 + 13×4 | 54 |
+| 4×11 + 7×4 | 864 |
+| 18×4 | 8425 |
+| 12×6 | 78 |
+| 6×12 | 216 |
+| 3×24 | 345408 |
+
+Every bank partitions all 72 coordinates. The checker enumerates each actual role, replica, stage, bank and coordinate offset. All 3601584 role-copy-stage assignments occur exactly once. There are 355045 banks per stage and 1065135 across three stages. Adding 253440 source/target data registers gives literal stock 1318575.
+
+For each gauge annihilator row a, the residual row 15a−(sum a)1 is G-orthogonal to the gauge. Normalize by its gcd; put residual columns before gauge columns in the 24×24 chart. The 234 actual charts pass both inverse identities and elementary-factor replay. Maximum factor count is 200, numerator 41 and denominator 54. These coefficients are units under the inherited prime condition q>2^80.
+
+Each chart sends its residual projector to a literal coordinate block. Disjoint partial swaps over each completed bank compose to the full swap on arbitrary correlated dirty inputs. Reversing the block sequence restores them and all spectators. Common weighted charts cancel between successive blocks. Every pattern rejects omitted and repeated blocks.
+
+Normalizers use the actual inverse chart, an explicit coordinate permutation and distinct scalars b+1. An outside-stage identity column witnesses distinctness. Charts, permutations and scalar factors are included in the fixed extra selector bill K=178459060896<2^40. The inherited positive ordinary-leaf exponent gap absorbs that fixed restored-selector cost.
+
+## Full paid profile and conditional composition
+
+After literal bank completion remove only the actual entrance exteriors: rank 60×2200,rank 54×13,rank 57×3,rank 36×18,rank 39×48. No internal donor/recipient increment or target split is removed. Factor 24 gives m=72, W=439525, rank=31599336, deficit=46464 and maximum child 22. The 72 literal physical replicas and factor 24 arithmetic normalization have distinct roles.
+
+Both rational moment engines include the full bad-class fallback. Four fixed ordinary-leaf levels use their preceding level. The unchanged 47-constraint assembly then composes with the exactly replayed frozen complex local-flow supplier. All-size compiler, selector, routing, precision, prime and analytic hypotheses remain conditional.
diff --git a/research/coordinated-crossover-pr200/LICENSE b/research/coordinated-crossover-pr200/LICENSE
new file mode 100644
index 0000000000..261eeb9e9f
--- /dev/null
+++ b/research/coordinated-crossover-pr200/LICENSE
@@ -0,0 +1,201 @@
+                                 Apache License
+                           Version 2.0, January 2004
+                        http://www.apache.org/licenses/
+
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diff --git a/research/coordinated-crossover-pr200/NEW-SOURCE-GAUGES.md b/research/coordinated-crossover-pr200/NEW-SOURCE-GAUGES.md
new file mode 100644
index 0000000000..540e60618f
--- /dev/null
+++ b/research/coordinated-crossover-pr200/NEW-SOURCE-GAUGES.md
@@ -0,0 +1,25 @@
+# Twenty-eight source-backed entrances, including a shared rank-4 frame
+
+Retained source471 construction stage, before the final target aggregation. The source/auxiliary word and paths below are unchanged. Final compensation uses 28 pivot reads in place of the 112 target reads described here; see `TARGET-AGGREGATION.md`.
+
+The final word retains 443 source aliases and adds 28 more: 26 use rank-2 pair planes and two share a rank-4 frame. The common-frame assignments are role 14079 from X322 and role 14077 from X378. Their four targets are 24, 25, 26 and 27. The exact common basis is in `newg/selection.json` as `gauge_basis`; all other new entrances use their exact `mix_frame`.
+
+The 28 pairs are distinct and disjoint from all prior aliases. The selected roles are previously ungauged, untouched in the admitted retimed center prefix, and outside source-injection and removed terminal roles. New target sets are disjoint except the two common-frame rows, which use exactly the same frame on their common four targets. There are 112 reads on 108 distinct targets; no new target intersects a terminal substitution.
+
+Save each partner sum after the ordinary source injections. After the copied-center scatter and before the first operation on each selected role, compensate the selected role's entire dirty response at its entrance frame. Run the original physical branch on the original passive source, including its later donor/recipient continuation. Reverse every auxiliary operation, then undo the saved partner sums and source injections. The actual scalar executor uses each original source column as the correlated initial dirty input and checks complete restoration.
+
+A source path is its original line, pair plane, selected gauge frame, every actual operation/root/alias frame, and the full frame. The rank-4 paths include a genuine rank-2-to-rank-4 transition. `newg/boundary.py` checks every exact inclusion, nondegeneracy and reflected annihilator path. It reconstructs both complete target ledgers, including all old gauges, terminal pivot operations, roots and partner deliveries. The target delta is {2:+104,4:+4,17:+4,19:+104,21:-108}. Repeated compensation at the same rank-4 frame adds no second target rank jump.
+
+Operation 18751 is enlarged to dimension six to admit role 13088 from source X1634 and partner X1633, at targets 1624–1627. The frame-only local delta is {1:+1,2:+1,3:-1,5:-1,6:+1,11:+1,12:-1}, of rank mass zero. The full joint scalar/frame/prime proof is rerun with this actual operation frame before any source-profile derivation.
+
+The final schedule defers 168 actual operations past the copied-center scatters. Two newly untouched physical heads, role 8584 from X98 and role 9424 from X458, receive their rank-2 compensation immediately after those scatters. All preceding gauge reads occur after the 168 deferred operations. `CENTER-RETIMING.md` supplies the dependency and D0/D1 proof; the complete physical frame streams are unchanged.
+
+The aggregate local source/internal/target delta from the original 443-source predecessor, including that operation change, is {1:+3,2:+107,3:+4,4:+5,5:-8,8:+1,11:+2,12:+10,14:-12,15:+3,19:+104,21:-108,22:-28}. Its rank mass is -672. Raw stock falls by 28, and the ordinary shared rank mass falls by 2016, preserving deficit. `newg/derive_profile.py` reconstructs complete source/internal/target histograms from all actual paths and checks this delta against the rebuilt 443-source baseline; that baseline already includes the operation change, so the incremental source-gauge delta excludes it. A separate endpoint/target implementation independently agrees.
+
+The common rank-4 frame has cleared Gram determinant 59049=3^10. `newg/boundary.py` supplies exact cleared-Gram factor witnesses for every selected source-gauge frame, including the new rank-4 frame; all residual factors lie below the retained prime lower bound. The rebuilt whole-word certificate admits every other used frame, including the new six-dimensional operation frame at 18751, whose cleared Gram determinant is 2125764=2^2*3^12 and residual factor is one.
+
+Final banks use 14364 full-residual roles. Per stage, use 864 mixed banks 4×11+7×4, 8464 banks 18×4, 78 banks 12×6, 216 banks 6×12, and 344736 banks 3×24. This gives 354358 banks per stage and 3594888 literal assignments across all 72 replicas. All 231 charts, inverse/factor identities, complete-block controls and normalizers are checked. Literal stock is 1316514; the fixed additional selector bill is 178127441712. These source-owned entrances do not create auxiliary banks or auxiliary-bank charts.
+
+F2 and both defining integer signs check all 20163 formal columns, every target and restoration of all 1760 sources and 16643 remaining dirty registers. Six controls reject missing new compensation/mixing, stale adjoints, missing old early mixing, premature inverse mixing and placing the two new reads after their deferred operations. The final normalized profile is m=72, W=438838, rank mass 31549872, deficit 46464 and maximum child 22. Two paid moments and exact 47-constraint assembly are rerun. The conditional result is κ = 6.94782719690783e-4.
+
+The construction, common-frame integration and verification were prepared with substantial OpenAI Codex assistance. All inherited contributor notices and licenses remain.
diff --git a/research/coordinated-crossover-pr200/NOTICE b/research/coordinated-crossover-pr200/NOTICE
new file mode 100644
index 0000000000..da33b9e931
--- /dev/null
+++ b/research/coordinated-crossover-pr200/NOTICE
@@ -0,0 +1,463 @@
+A sharper exponent for integer multiplication
+Copyright 2026 Douglas Colkitt
+
+This project includes unmodified source files from OpenAI's manuscript
+"Integer multiplication below n log n" and its accompanying repository:
+https://github.com/openai/math
+Commit: adc7f1241b42e322a6451854ab7e4b4c146bf78a
+
+The original manuscript identifies OpenAI as its author. Its Apache-2.0
+license is retained at upstream/LICENSE. The exact source URLs and hashes
+are recorded in upstream/manifest.json. Proposed modifications are supplied
+as separate files under patches/; upstream source files remain unchanged.
+
+The original local research note, scripts, certificates, patches, and
+documentation were prepared by Douglas Colkitt with assistance from OpenAI
+Codex. The community release incorporates work by the contributors identified
+in the notices below; it is not solely Douglas Colkitt's work.
+This project is not an official OpenAI release or an OpenAI endorsement.
+
+Related contributions (not imported in this checkpoint):
+Zhihao Chen (jacklightChen), PR #7, supplied an earlier ternary five-subset
+motif, finite-alphabet transfer and a different compressed producer:
+https://github.com/CrocSwap/integer-mult-bounds/pull/7
+The present checkpoint claims no priority for that motif. Related compressed
+complex-network work was submitted by eumemic in PR #3:
+https://github.com/CrocSwap/integer-mult-bounds/pull/3
+The pending contributions and their stronger bounds require their own review;
+this notice does not represent adoption or verification of their claims.
+
+Additional community work, including parallel and superseded submissions, is
+acknowledged in CONTRIBUTORS.md, with pinned submission metadata in
+docs/research/contribution-snapshot.json. These acknowledgements do not imply
+joint authorship of the checkpoint, incorporation of pending code, or verification
+of submitted mathematical claims. Original source-specific notices remain in force.
+
+Community release (accepted by conditional maintainer audit):
+The following original contributor notices apply to the imported PR #39 chain.
+The preceding non-import statements describe the preserved 2^-30 checkpoint.
+
+Ternary five-subset construction and common-four-point resynthesis
+Copyright 2026 Zhihao Chen (jacklightChen)
+
+This is Zhihao Chen's first submission of this specific construction to this
+repository. Research, proof development, code and verification were prepared
+with substantial assistance from GPT-6 Astra (OpenAI Codex), as identified by
+the contributor. This is not independent review or an OpenAI endorsement.
+Future research building on this contribution should explicitly acknowledge
+Zhihao Chen (jacklightChen), cite the accompanying note and pull request, and
+retain the preceding contributors' attribution. This notice records research
+provenance; the repository's Apache-2.0 license remains unchanged.
+
+Dependencies retained from eumemic's Claude-assisted PRs #3 and #5:
+https://github.com/CrocSwap/integer-mult-bounds/pull/3
+https://github.com/CrocSwap/integer-mult-bounds/pull/5
+PR #5 commit: d3d370c34937503bd3fa63c83d505a05e8e50893
+These include complex_circuit.py, complex_network.py, fast_gaussian.py,
+their generators, construction notes and certificates. The Gaussian input
+scaling proof is corrected in this derivative; see the modification notice below.
+Aligned pairing also appears in Bortlesboat's PR #2. Retained totals and bank
+sharing appear in dleen's PR #4 and cheaper retained centers in eumemic's PR #6.
+Their results and ideas are credited separately from this new construction.
+
+Bulk recursion and controlled tensor basis, 8 October 2026
+Copyright 2026 icekylinx
+
+The new projector batching, controlled simultaneous basis, mixed-width tape
+recursions, whole-residual complex recursion, and dependency-path precision
+guard in the batched-23 proof package are contributed by icekylinx, developed
+with substantial OpenAI GPT-6 Astra assistance. Review corrections and
+repository integration were prepared with OpenAI Codex assistance. They build on the retained
+PR #7 finite producers and the preceding contributions credited above.
+The new package retains the repository's Apache-2.0 license and the original
+authors' attribution. It is not an official OpenAI release or endorsement.
+
+Modification notice, 8 October 2026 (icekylinx): the retained fast Gaussian
+proof now bounds shifted approximation error by F=2^ceil(1.14 alpha^2).
+The P=34p precision budget and asymptotic exponent are unchanged. New batching
+checks are integrated into make verify. Existing attribution and licenses
+are retained.
+
+Partial-swap extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The partial-swap framing, binary retained-total triple construction,
+dependency-preserving positive labels and carrier reuse, unequal (25,23,57)
+common basis, improved complex batching, and exact assembly are contributed
+by icekylinx with substantial OpenAI GPT-6 Astra and Codex assistance.
+This extension directly inherits PR #10, commit
+62691e395a0458ce089a1c7b5d89e74291e95e29. The portable producer adapters
+modify the retained paired-exclusion and shared-point implementations credited
+above. The original licenses, attribution, and Gaussian correction are retained.
+
+Endpoint-gauge extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The complementary auxiliary endpoint gauges, (32,30,40) controlled basis,
+general nondegenerate binary phase residuals, retained-center complex producer,
+updated path guard and exact assembly are contributed by icekylinx with
+substantial OpenAI GPT-6 Astra and Codex assistance. This extension inherits
+f2ab41aebad47861caf6316282c1793e5513845e and its PR #10 ancestry. New portable
+adapters retain the paired-exclusion and shared-point contributions credited
+above. Original licenses, notices and the Gaussian correction are retained.
+
+Structured-basis and semantic-bulk extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The selected (32,30,36) bit construction, A1/A5 block refinements, fixed
+middle-axis I+J profiles with bounded-minor CRT certificates, mixed dyadic
+complex centers, complete data-front batching and circuit-specific assembly
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex
+assistance. The immediate parent is ed90fd940279c336ebc45968631bdebfb087b505.
+
+The semantic/bulk compatibility argument, literal scalar charge, product row
+stock and parameterization adapt Zhihao Chen (jacklightChen)'s PR #23,
+commit 661ebabc1076c9f525d7ce4967c876b203fa9827. Translated complex endpoints
+are adopted from the same contributor's PR #21,
+commit 5ba6cf0bfb68f2be8d15610e7972207c50254d6a, following eumemic's source-frame
+work and the earlier nested-basis contributions.
+
+Completed-child semantic precision, arbitrary-coordinate routing, phase-cell
+Gaussian inversion and bulk tape transfer are credited to the RaD project
+(hipotures), commit 45d9b60355872041f9275f77c057514def0d45bc, exposed through
+PR #20 at 4f8d6c8272b5ff307a0da51df545ec3cd96a8b6e. Their selected proof
+sources retain their separate CC0 license and original NOTICE under
+references/semantic-bulk/rad20. The PR #23 source retains its recorded OpenAI
+Codex assistance; RaD retains its recorded Codex and GPT-6.1-Sol Ultra
+assistance. These are source attributions, not claims about this session's
+runtime model. Related contributions by Rohan Arun and Aurel Prosz remain
+acknowledged in the imported notices. No exclusive priority or additional
+license condition is asserted.
+
+Copied retained-center extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The copied retained-center schedule in both orientations, its complex-phase
+transfer, the Wm-N+L rank budget, and the selected (25,23)/(28,28) composition
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex
+assistance. This extension inherits PR #32 at
+0ef3aeb61f55cc0b321ce6a0ef00acee25cefe52 and the preceding PR #10/#18/#24 work.
+
+The two-stage topology and paid copied-stream endpoint correction are due
+to Aurel Prosz (Paureel), c82d09eb4781b68435e3fa3eb6beea72fa900fab,
+through Zhihao Chen (jacklightChen)'s PR #29 unequal-axis composition,
+9d963275075fa98f1da821e27757b238dafd6b3c. That source also credits Swapnil
+Jain's two-stage development at ae405eb474d1486b2d8aef90139869f927f4836a.
+
+The contiguous corner method, incidence factorization and exact partition
+certificates are due to Rohan Arun (PR #31), method commit
+02f68f95369c39955c5ffcfddbd75cd83e26b305. The selected (25,23) certificate
+specializes Dominik Scholz's parameterization (PR #33),
+a499a345c040aa15b68e984a0904eadc485c7a7e. Their source notices and proof
+snapshots are preserved in references/copied-centers. Historical OpenAI
+Codex, GPT-6 Astra and Anthropic Claude Opus 5.5 assistance is credited to
+the respective imported sources. Existing PR #21/#23 and RaD attribution,
+licenses and the Gaussian correction remain in force.
+
+Copied centers with reversed (23,25) data corners, 2026-10-08
+Rohan Arun with substantial OpenAI Codex assistance supplies the composition,
+exact dimension23 controls and cost/assembly integration. James Chang's PR34
+supplies the reversed boundary family; icekylinx's PR36 supplies copied retained
+centers and the selected complex graph. The balanced assembly remains attributed
+to RaD/hipotures and its PR34 specialization. Retain Dominik Scholz PR33, Rohan
+Arun PR31, Zhihao Chen PR29/21/23, Paureel/Aurel Prosz, Swapnil Jain, icekylinx,
+eumemic, Douglas Colkitt, OpenAI and all earlier notices and hypotheses.
+This is not a claim of formal verification, new foundational rank identities,
+global optimality or measured runtime speedup.
+
+Fixed middle basis with copied reversed corners (2026-10-08):
+Rohan Arun, with substantial OpenAI Codex assistance. Composition and
+simultaneous-basis proof retain icekylinx PR32/36, James Chang PR34,
+Dominik Scholz PR35/38, Zhihao Chen, Paureel, Swapnil Jain, RaD/hipotures,
+eumemic and all preceding attribution and licenses. See research/copied-fixed-reversed.
+
+Changed-graph composition with copied centers and fixed bases, 8 October 2026
+Prepared with OpenAI Codex assistance.
+The new composition uses icekylinx PR36 copied centers and PR32 fixed-I+J
+projectors/profiler, Dominik Scholz PR35 two fixed factors and CRT argument,
+Zhihao Chen (jacklightChen) PR29/23 interfaces, Aurel Prosz (Paureel) and
+Swapnil Jain's two-stage development, Rohan Arun PR25/31 corner methods,
+Dominik Scholz PR33/38 dimension/composition work; James Chang (jamesyc)
+PR34 reversed geometry and balanced transfer; Rohan Arun PR37/39/40
+reversed integration, modular data tests and both-fixed baseline;
+RaD (hipotures) PR41 changed scalar DAGs, alternating point order and
+independent role compiler; RaD balanced routing/bulk interfaces, and all
+retained predecessors.
+Complete attribution and the conditional proof boundary are recorded in
+research/copied-fixed/PROOF.md; imported originals and source pins are retained.
+Original Apache-2.0 and CC0 notices remain applicable.
+
+Community follow-up integration (PRs #49, #45 and #26), 8 October 2026:
+Rohan Arun contributes weighted carrier matching and climbed summand/leave-one-out
+orders (PRs #44/#47/#49), with Anthropic Claude assistance. The final construction
+composes Chafik Boukhalfa's changed-graph checks, exact rational data recovery and
+reordered exclusion sums (PRs #43/#46/#48, OpenAI Codex assistance) with RaD /
+hipotures's alternating pair order and independent physical role compiler (#41).
+All predecessor attribution and conditional proof interfaces remain in force.
+
+Alejandro Zarzuelo Urdiales contributes Gaussian parity and canonical arithmetic
+proofs, finite tensor execution, profile precision accounting and mixed-center
+reference checks (#45), with substantial OpenAI Codex assistance. See
+research/gaussian-parity-synthesis/PROVENANCE.md for his full attribution.
+Ryan S / princezuda contributes historical Lean certificate arithmetic, algebraic
+frame/movement/guard lemmas and independent paired-circuit checks (#26). Original
+source notices and historical snapshots are preserved. Neither formal package
+claims a full formal proof of the multiplication algorithm or a new kappa.
+
+Joint frame compilation of skip-prefix strips
+
+The local compiler and exact finite witnesses were prepared by eumemic with
+substantial OpenAI Codex assistance. Avi Eisenberg / ikeboy with Anthropic
+Claude assistance supplied the unmodified PR #53 skip-prefix producer.
+The pinned PR #48 pipeline is Chafik Boukhalfa's contribution, incorporating
+Rohan Arun, RaD / hipotures, icekylinx, Dominik Scholz, James Chang,
+Zhihao Chen, Aurel Prosz / Paureel, Swapnil Jain, eumemic, Douglas Colkitt,
+OpenAI, David Harvey, Joris van der Hoeven and the earlier contributors
+credited in the retained sources. See references/frame-compiler/pr48/NOTICE
+and references/frame-compiler/pr53/NOTICE for their original attribution.
+No authorship or review by those predecessors is claimed for this compiler.
+
+Joint dual-skip/frame-compiler composition (2026-10-08)
+---------------------------------------------------
+Rohan Gupta / gupt1156 with Anthropic Claude assistance: PR55 dual-skip
+scalar graph; original LICENSE and NOTICE are retained under
+references/frame-compiler/pr55. eumemic with substantial OpenAI Codex
+assistance: PR57 joint reversible frame compiler, serialized physical-word
+replay and profile reconstruction. Chafik Boukhalfa with OpenAI Codex
+assistance: this composition, independent integration and exact certificate.
+Avi Eisenberg / ikeboy, Rohan Arun, RaD / hipotures, icekylinx, and all
+predecessor authors retain the notices above and in pinned source files.
+No original copyright or Apache-2.0 notice is removed.
+
+Ranked reclamation after the PR58 joint-dual composition (2026-10-08)
+----------------------------------------------------------------
+Chafik Boukhalfa with OpenAI Codex assistance: prioritize retired physical
+slots by descending current frame rank, retaining slot ID as the tie-breaker;
+complete replay, exact profiles, comparison and adversarial controls.
+The dedicated joint_dual_reclaim_compiler.py is derived from eumemic's PR57
+compiler. Its original shared compiler remains byte-identical. The PR55
+producer and all preceding attribution remain unchanged. The complete PR58
+comparison certificate and its original source/proof provenance are retained
+under references/frame-compiler/pr58.
+
+Skip-prefix strip producer, 8 October 2026
+By Avi Eisenberg (ikeboy), with Anthropic Claude (Claude Opus 5.5)
+assistance. The skip-prefix leave-one-out strip layout in
+research/skip-strips/skip_graph.py is new. Everything else reuses Chafik
+Boukhalfa's PR #43/#46/#48 research/copied-fixed pipeline, its verification
+code, exact data recovery and support-order sorting, together with:
+- Rohan Arun's PR #44 weighted carrier matching optimizer;
+- RaD (hipotures) PR #41 alternating point order;
+- every predecessor credited above.
+Complete attribution and the conditional proof boundary are recorded in
+research/skip-strips/PROOF.md. Original Apache-2.0 and CC0 notices remain
+applicable.
+
+Local split-pair skip-prefix research continuation (2026-10-08)
+Prepared with substantial OpenAI Codex assistance at the user's request.
+New contribution: singleton/pair recursive partition refinement, finite summand
+order and moment search, and verified composition with inherited paid clones.
+Avi Eisenberg/ikeboy with Anthropic Claude assistance: PR53 skip-prefix strips.
+Chafik Boukhalfa with OpenAI Codex assistance: PR54 original-envelope clone
+specialization and PR43/46/48 checking interfaces. RaD/hipotures: PR51
+whole-chain/source-partition cloning and earlier producer/interfaces.
+Rohan Arun: PR44 weighted matching and earlier fixed/reversed work, with all
+stated historical AI assistance. All predecessor notices remain applicable.
+
+Original-envelope whole-chain clones on skip-prefix strips (research/skip-clones):
+Chafik Boukhalfa, prepared with OpenAI Codex assistance. The complete-chain
+source-partition clone operation is adapted from RaD / hipotures, PR #51,
+with original envelopes retained here. The scalar base is Avi Eisenberg's
+PR #53 skip-prefix layout, prepared with Anthropic Claude assistance.
+All credited PR #43/#46/#48, PR #44 and earlier code/proof notices remain
+in force. This extension is provided under the repository's Apache-2.0 terms.
+
+Community review checkpoint #50–#61, 8 October 2026
+Maintainer integration and independent arithmetic: Douglas Colkitt, with OpenAI Codex.
+Avi Eisenberg / ikeboy: skip-prefix strip framework (#53), with Anthropic Claude.
+Rohan Gupta / gupt1156: parallel order search (#50, stated Antigravity assistance)
+and dual-suffix strips (#55, Anthropic Claude assistance).
+Eumemic: equal-frame joint compilation and paid signal reclamation (#57), with Codex.
+Chafik Boukhalfa: original-envelope paid clones (#54), composition (#58) and ranked
+reclamation (#60), with Codex. Rohan Arun: order search (#52) and positive-frame
+composition (#56), with Anthropic Claude and stated Codex validation assistance.
+RaD / hipotures: enlarged frames, source partitions and paid whole-chain clones
+(#51), with Codex. Rohan Garg / rohangar1: singleton/pair recursion and paid-clone
+composition (#59), with Codex. Alejandro Zarzuelo Urdiales: exact parameter
+refinement, scoped Lean arithmetic, matrix/parity and search tools (#61), with
+his disclosed multi-tool and Codex assistance.
+Parallel #50/#52 records are reviewed and credited without replacing the earlier
+climbed-48 implementation; their exact commit history is retained. The separate
+#51 witness is retained as research; its source-graph/negative-basis data proof
+has not received the same full replay as #56's consumed compiler machinery.
+See docs/research/community-round2-review.md for precise validation boundaries.
+All earlier licenses, contributions and notices remain applicable.
+
+Interval strips and core-aware pair assembly
+
+The interval leave-one-out strips and the core-aware group-pair assembly in
+research/pair-assembly were prepared by Avi Eisenberg / ikeboy with Anthropic
+Claude assistance. They build on PR #53's skip-prefix pipeline and its
+inherited PR #48 sources. The stacked witness in research/pair-assembly/frame
+uses eumemic's PR #57 joint frame compiler unchanged; its driver and verifier
+are adaptations of PR #57's skip_frame_compiler.py, skip_frame_compose.py and
+verify_skip_frame.py. Original notices are retained.
+
+Stopped product-ring extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The opposite-bank generic projector factorization, componentwise atom-ring
+adapters, stopped recurrence and outer ordinary-interchange conversion,
+aligned pair-star producer with rational centers, exact denominator-21
+grid and selected (23,23)/(24,24) composition are contributed by icekylinx
+with substantial OpenAI GPT-6 Astra and Codex assistance. This extension
+builds on merged PR #36, 11817ccacb564bb7f98789c20dc11d3fece207e3, and is
+integrated on upstream main 56b66d58297deca1d7dd130247d720e960f77a37.
+
+The retained ordinary leaf uses the preceding projector, corner and
+copied-center proofs. The selected two-stage topology retains Aurel Prosz
+(Paureel), Swapnil Jain and Zhihao Chen's credits; the semantic assembly
+retains PR #21/#23 and RaD's attribution. Paired producers, carrier matching,
+original source notices and each imported license remain unchanged.
+
+Partial-gauge extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The pair-first cube assembly, compatible backward frame enlargement,
+partial source-gauge schedule and stopped whole-projector composition are
+contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex
+assistance, extending PR #104 at 948ce1510df750f4c18b96bdaef436a86f8bf834.
+
+The physical deferred bit word, reflected event accounting, fan-order
+clarification and signed two-stage endpoint are adopted from Zhihao Chen
+(jacklightChen), PR #97, e15350b66e4bf796153a68c240a6cc1b788abe3e, with
+Codex assistance. Its deferred readouts, V leaves, lifted frames and frozen
+witness originate in Swapnil Jain's 741e7aa078392553815df7926ee17ac5e25a8c38,
+with the original Claude disclosure retained. Cyclic interval and core-aware
+pair primitives build on Avi Eisenberg (ikeboy), PR #62,
+ad0f25ff7b23cff7f08ad237c2254e6ecf74257e, with Anthropic Claude assistance.
+Prior two-stage, copied-center, ordinary-leaf, semantic/bulk, routing and
+precision credits remain in force. Imported source notices and licenses
+are preserved under references/partial-gauge/; see SOURCES.json.
+
+Three-stage cover extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The arbitrary-subspace Clifford frame theorem, three-stage Cayley geometry,
+weighted q-adic GL cover, unit-lower weighted compiler, charged local-ring
+fallback, uniform batching, internal row borrowing and exact composition
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra assistance. This
+repository integration uses Codex assistance and extends
+PR #115, cf60442983c153a3a0f5272ea4713f99eba13c86.
+
+The selected local complex DAG and support/label replay are adopted from
+Copyright 2026 eumemic, PR #117, cbb05ce504d571546d9b7794c186a613c659c3bf,
+with its original Apache-2.0 terms and Claude assistance disclosure.
+The unchanged bit physical word retains Zhihao Chen (jacklightChen),
+PR #97, and Swapnil Jain's underlying witness and assistance notices.
+The prior opposite-bank, stopped ordinary-leaf, copied-center, topology,
+semantic/bulk, precision and analytic credits remain applicable.
+Original source notices and license bytes are preserved in
+references/three-stage-cover/pr117/ and references/partial-gauge/.
+
+Paired-cube and shared-core extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The paired-coordinate triple-cube identity, signed overlap channels,
+original-source involution, coordinate-star centers, concrete frame/carrier
+compiler, chronological source gauges, PR97 gauge-subset selection,
+orthogonal cross-stage routing and exact supplier/assembly integration
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra assistance.
+Repository integration uses Codex assistance and extends PR #130 at
+6a9970a530119174507904e23592fd59ede19a5d.
+
+The substantial completed-core workspace-sharing principle is credited to
+an664, PR #128, 530588a019b4a74f09180680c9e3961bf649ec89, with its reported
+OpenAI Codex assistance. No source code, partition or serialized witness
+from that PR is imported. The new routing and paired-motif integration
+are described separately and claim no exclusive priority for broad sharing,
+paired-coordinate, Fourier or carrier-matching ideas.
+
+Selected coarse queries restrict the retained eumemic PR #117 positive DAG;
+its original Apache-2.0 and Claude assistance notices remain unchanged.
+The bit word and original frames retain Zhihao Chen's PR #97 and Swapnil
+Jain's frozen witness, notices and historical assistance disclosures.
+Earlier generalized Clifford, three-stage, uniform weighted recursion,
+ordinary-leaf, semantic/bulk, precision and analytic credits remain in force.
+Source-specific license bytes are preserved under references/.
+
+Source-assisted common-frame extension (2026-10-09)
+Copyright 2026 icekylinx. Apache-2.0.
+Research construction and local certificates: OpenAI GPT-6 Astra assistance.
+Repository integration and independent complex scalar replay: Codex assistance.
+Substantial dependency: eumemic PR168, fd25adb7fbaa12ee761d02c733c54d1d2a7687ee;
+its byte-preserved sources, licenses and original Claude assistance disclosure
+are retained in research/source-assisted/public/pr168_fd25adb7.tar.gz.
+Additional inherited contributions are recorded in research/source-assisted/CREDITS.md.
+The new work concerns source-assisted invertible common-frame lifts, controlled
+affine cleanup and compatible zero-fresh reuse; no exclusive claim is made for
+broad register reuse, paired cubes, frame theory or ordinary linear elimination.
+
+Fourth finite bootstrap refinement (2026-10-09)
+Prepared by eumemic with substantial OpenAI Codex assistance.
+This arithmetic refinement builds on PR207 at
+cd14825023b75af4f5919a30e5e6d548b6ade5bc by adding one finite ordinary-leaf
+level. The 302 frame overrides, completed banks, scalar words and upstream
+assistance disclosures are inherited, not newly attributed to this work.
+The Apache-2.0 license and all preceding notices remain applicable.
+
+PR211 plateau frame refinement (2026-10-09)
+Prepared by eumemic with substantial OpenAI Codex assistance.
+The additional seven equal-frame component moves build directly on Rohan
+Arun's PR211 cascade witness at c63e50a5dde96fe1459d6b29e55e47f42f104347,
+prepared with Anthropic Claude assistance. Its original notice is preserved
+byte-for-byte in search/PR211-NOTICE. The offline certificate package and
+independent stage-private bank admission derive from Dugongue's PR207 at
+cd14825023b75af4f5919a30e5e6d548b6ade5bc. Original contributor notices,
+licenses, source pins and all upstream AI disclosures remain applicable.
+
+The final refinement also includes seeded bidirectional cascade search after
+the seven initial component moves. Exact canonical subspace comparison
+records 274 changed operations beyond PR211. Search, integration and replay
+were prepared with substantial OpenAI Codex assistance.
+
+Shared rank-18 gauges and two donor substitutions (2026-10-09)
+Prepared by eumemic with substantial OpenAI Codex assistance.
+The new actual gauge/read chronology, gauged-donor handoff, shared target
+transitions, rank6 residual banks and donor substitutions extend the same
+PR200/PR211 word and PR207/PR197 completed-bank framework. All original
+licenses, notices, authorship and upstream AI disclosures remain in force.
+
+Nested-gauge extension: three rank-19 physical entrances compose with thirteen shared rank-18 entrances; all target transitions and mixed4/5/6/24 banks are explicit. Prepared by eumemic with substantial OpenAI Codex assistance. All retained authorship, licenses and AI disclosures remain applicable.
+
+Target-quartet compensation aggregation, 2026-10-09
+Prepared by eumemic with substantial OpenAI Codex assistance.
+New work: reversible D0 target setup, pivot-only early source-gauge
+compensation, exact common-frame inverse timing, complete scalar/reflected
+ledgers, paid profile and conditional assembly integration.
+All prior contributor notices and Apache-2.0 terms remain applicable.
+
+Source490 and85-group extension: prepared by eumemic with substantial OpenAI Codex assistance. Retains all preceding authorship, assistance and Apache-2.0 notices.
+
+
+Source492 and integer target-response compression (2026-10-09)
+Prepared by eumemic with substantial OpenAI Codex assistance. The new ordered
+source entrances and reversible integer-response transformations retain all
+preceding contributor, assistance and Apache-2.0 notices.
+
+Equal-frame plateau refinement on PR210's 471-source word (2026-10-10)
+Prepared by sennemmi with substantial OpenAI Codex assistance. The deterministic
+frame search, 19 operation retimings, exact price update and repository
+integration extend eumemic's PR210 head
+df95878d11190518e45ef9717c9ee05011f88ace. They do not claim the inherited
+source word, shared gauges, completed banks or upstream proof framework.
+All PR200, PR197, PR202/193, PR207, PR211, PR216 and PR210 attributions,
+licenses and AI disclosures remain in force; the repository's Apache-2.0
+license is unchanged.
+
+Source493 and signed echelon target transformations (2026-10-09)
+Prepared by eumemic with substantial OpenAI Codex assistance. New work includes
+the delayed third rank-four source entrance, four determinant-one target
+transformations, signed compensation execution, integer sign controls, complete
+reflected ledgers, paid profiles and integration. All prior licenses, contributor
+notices and OpenAI/Anthropic assistance disclosures remain applicable.
diff --git a/research/coordinated-crossover-pr200/PLATEAU-REFINEMENT.md b/research/coordinated-crossover-pr200/PLATEAU-REFINEMENT.md
new file mode 100644
index 0000000000..bbf4bc79ab
--- /dev/null
+++ b/research/coordinated-crossover-pr200/PLATEAU-REFINEMENT.md
@@ -0,0 +1,45 @@
+# Exact refinement of PR211's operation frames
+
+The initial plateau step below is followed by the seeded cascade refinement described at the end of this file. The final published witness uses both steps.
+
+The input is the frozen PR211 cascade witness at c63e50a5dde96fe1459d6b29e55e47f42f104347. The search joins adjacent operations with equal frames into connected components. A candidate replaces every frame in one component by the intersection of its next frames on both physical chains. Each accepted replacement contains the old frames and hence their value spans, is G-nondegenerate, and stays nested between every fixed boundary and neighboring operation. All scalar additions, sources, root caps, gauges, aliases, partner deliveries and terminal selections remain fixed.
+
+Seven accepted two-operation components change operations 18516/18517, 18920/18921, 19324/19325, 23544/23545, 23591/23592, 23685/23686 and 23873/23874. Before integer moment normalization, the complete child change is:
+
+| Rank | Count change |
+|---|---:|
+| 2 | +12 |
+| 4 | -15 |
+| 5 | -18 |
+| 6 | +9 |
+| 7 | +24 |
+| 8 | -12 |
+| 9 | -12 |
+| 12 | +24 |
+| 15 | -12 |
+
+The weighted rank change is zero. The count changes scale by three for the integer moment profile. Terminal cancellation and bank removal are replayed and are unchanged. Both reflected signs and complete source/dirty restoration are checked on the actual scalar word.
+
+The search score uses floating arithmetic only for discovery. The published saving and final κ use exact rational interval bounds with independent enclosure engines. The four finite leaf levels each invoke their predecessor and are independent of input size.
+
+Reproduce discovery after extracting the pinned baseline (or using PR200's exact checkout):
+
+```sh
+python3 -B search/plateau_search.py --bit-package /path/to/research/paired-cube-diagonal-bit-168 --frames search/pr211-frames.json --out /tmp/plateau-reproduction
+```
+
+The search is not an optimality certificate. The following seeded cascade step contributes additional verified gain. PR211's original notice is preserved byte-for-byte in search/PR211-NOTICE. Apache-2.0; substantial OpenAI Codex assistance.
+
+## Seeded cascade refinement
+
+After the 14-operation plateau witness, deterministic seed 2116205 drives raise/lower proposals and short positive-temperature perturbations, followed by zero-temperature relaxation. A raise propagates joins forward; a lowering propagates intersections backward. Every changed operation retains its exact value span and a nondegenerate frame, and every edge stays nested between fixed source/gauge/root/alias endpoints. The final state is the best discovered exact-admissible state; search optimality is not claimed.
+
+The final witness lists 6,416 bases and changes 274 actual subspaces relative to PR211 (2,124 relative to PR200). Raw frame IDs are not mathematical change counts: registering an equivalent basis can create another ID. The canonical comparison receipt is search/relaxed-diff-count.json.
+
+Compared with the plateau profile, before the factor-three moment normalization, the further child change is {1:+12, 3:+9, 4:-24, 5:-69, 6:+9, 7:+36, 8:+39, 9:+3, 10:+9, 12:-27, 13:-18, 15:+15}. Stock and weighted rank mass are unchanged. The full verifier reconstructs this profile from the final bases and actual scalar word.
+
+```sh
+python3 -B search/relaxed_search.py --bit-package /path/to/research/paired-cube-diagonal-bit-168 --frames search/pr211-frames.json --additional-frames search/additional-frames.json --out /tmp/cascade-reproduction
+```
+
+Search traces use floating entropy only for proposals. The frozen final bases, exact frame/word replay and rational paid moments determine the public claim.
diff --git a/research/coordinated-crossover-pr200/PR-STATEMENT.md b/research/coordinated-crossover-pr200/PR-STATEMENT.md
new file mode 100644
index 0000000000..06a3133e77
--- /dev/null
+++ b/research/coordinated-crossover-pr200/PR-STATEMENT.md
@@ -0,0 +1,23 @@
+κ = 6.95684647723759e-4
+
+Conditional **κ ≈ 6.95684647723759 × 10⁻⁴** from 493 original-source aliases, 82 equal-response groups, seven integer-response groups and four signed echelon target transformations.
+
+The new target transformations cancel selected early response coefficients even when the response rows differ. Apply an explicit integer change of target coordinates at the post-center D0 frame, execute its signed transformed responses, and undo the change at the recorded common frame. Four groups on 20 targets improve the paid target paths; their matrices have determinant one and explicit integer inverses. Three quartets replace six older pairs, and an octet replaces one older integer-response group. The actual word restores arbitrary dirty target values.
+
+An additional rank-four source entrance reuses source 90 as role 14073 after the earlier source gauges have been read. This gives 22 extra entrances beyond the retained 471-source construction, including three ordered rank-four entrances. All source aliases, exclusive rank-two reads and overlapping rank-four reads remain in the complete chronological ledger.
+
+The literal scalar word passes over F₂ and both defining-integer signs on all 20141 formal columns, restoring all 1760 sources and 16621 dirty registers. Integer execution checks the defining decoder; its F₂ reduction is identity. Seventeen corruption controls reject, including a sign error tested over the integers. Independent audits rebuild all changed source paths, every target path, both reflected frame ledgers, all 5280 copied-center target reads, the exact local integer transfers and primal/dual prime witnesses. The complete banks enumerate 3590136 literal assignments and 231 exact charts.
+
+The normalized profile is m=72, W=438310, rank mass 31511856, deficit 46464 and maximum child 22. Literal stock is 1314930. The scalar bill retains all 208 uncompressed extra-source compensation deliveries, the previous 480 equal-group basis additions and 198 integer-response basis additions, then adds all 48 echelon basis additions and 56 signed compensation units. No removed operation is credited. This adds 427680 fixed calls; the complete extra bill 177892474320 remains below 2⁴⁰.
+
+The package includes sennemmi’s nineteen PR230 operation-frame improvements at fee4c05b91d4cdebc0b9d8cbb266a98069928a9a, with the original search, result, frame witness and notice preserved in `search/pr230/`. The retained complex lift/contract, two independent exact moment engines, four finite leaf levels and all 47 strict assembly inequalities pass. The adjacent final grid point is rejected; the bit supplier binds.
+
+```sh
+python3 -B research/coordinated-crossover-pr200/verify.py --temp-root /tmp
+```
+
+Python 3.11+ with assertions enabled. Five pinned archives supply dependencies. Verification extracts a temporary source tree without network or git and checks every submitted file before and after execution.
+
+The result remains conditional on inherited compiler, weighted/restored selector, source-line, routing, precision/recovery, prime, analytic-transfer and finite-bridge assumptions. The complex supplier retains its exact local-flow contract.
+
+Provenance: Dugongue's PR216 (34abdc58ed77329c8bf9af5634531fc41939ba3c), prepared with OpenAI Codex assistance, supplies compatible cuts and the majorant idea. PR211's cascade frames are Rohan Arun's work with Anthropic Claude assistance. PR207 (Dugongue) supplies offline admission; PR200 (Chafik Boukhalfa) supplies the physical word/checkers; PR202/193 supply the complex witness; PR197 (Evan McKinney) supplies completed banks. The nineteen PR230 operation-frame changes are sennemmi’s contribution with substantial OpenAI Codex assistance. Source reuse, shared gauges, further cuts, target aggregation, integer target-response compression, signed echelon target transformations and integration were prepared by eumemic with substantial OpenAI Codex assistance. All original licenses, contributor notices and upstream AI disclosures remain.
diff --git a/research/coordinated-crossover-pr200/PR205-COMPARISON.md b/research/coordinated-crossover-pr200/PR205-COMPARISON.md
new file mode 100644
index 0000000000..9dabc3d20a
--- /dev/null
+++ b/research/coordinated-crossover-pr200/PR205-COMPARISON.md
@@ -0,0 +1,17 @@
+# Historical PR207/PR205 comparison
+
+This records the predecessor package comparison. The current PR211-based refinement and exact claim are in README.md and PLATEAU-REFINEMENT.md.
+
+# PR205 read-only comparison
+
+Pinned head: `95c58706ad231d532bd7038308946a0344d08186`. Its proof and committed certificate were read; no full verifier was run in this lane.
+
+PR205 applies the same rank-4/rank-24 completed-bank idea to the unmodified PR200 internal frame word. Its paid coarse saving, `683528191056257/10^18`, exactly matches our earlier bank-only baseline. Its moment normalization has the same width 72, stock 56,402, rank mass 4,055,136, deficit 5,808 and maximum child 22. Its complex saving is also exactly the same `700918443859411/10^18`.
+
+The new result here is the paid histogram after coordinated changes to 302 operation frames, composed with completed banks. That histogram preserves total rank mass but improves its distribution and raises the coarse saving to `136731504666219/200000000000000000`. Its final conditional kappa is `1366380910073/2000000000000000`, exceeding PR205 by `129155636718/10^18`, approximately 0.01890835%.
+
+PR205 gives a different literal bank schedule: three full copies, nine role incidences across the stages, mixed banks, and a checked 9-edge-colouring. The local construction uses nine physical replicas and stage-private banks. These share the same normalized bank stock and yield no additive gain. PR205's routing constant 5,368,513,266 is smaller than our conservative 21,900,544,398, but both are fixed ordinary-selector tolls absorbed by the same strict exponent gap. Replacing one fixed routing constant cannot be credited as another exponent gain.
+
+PR205's finite three-level bootstrap has the same acyclic recurrence and discloses the same completed-core, Clifford, weighted-compilation, restored-row and fixed-tape conditions. Its initial ordinary saving is lower than the local already-admitted atom wrapper, so borrowing it does not improve this package. Its retained PR193 complex caveats are distinct from the local PR202 actual replay evidence; they should not be blindly copied.
+
+No additional independent paid gain or missing new interface was identified. Bank packing must be credited as parallel public work; the claim here should focus on the explicitly checked coordinated frame refinement and its complete composition. The precise histogram delta is preserved in PR205-COMPARISON.json.
diff --git a/research/coordinated-crossover-pr200/PROOF.md b/research/coordinated-crossover-pr200/PROOF.md
new file mode 100644
index 0000000000..654213cc80
--- /dev/null
+++ b/research/coordinated-crossover-pr200/PROOF.md
@@ -0,0 +1,12 @@
+# Executable proof map
+
+The conditional claim is κ = 6.95684647723759e-4. Inherited all-width supplier, restored-selector, source-line, routing, precision, analytic-tape and finite-bridge conditions remain assumptions.
+
+1. `joint/replay_joint.py` checks the retained scalar word, physical frames, terminal actions, dirty restoration, both defining-integer signs and used-frame prime witnesses. The final frame witness includes sennemmi’s preserved PR230 refinement.
+2. `borrow/`, `gaugeb/` and `newg/` independently replay the retained471-source stages, their center retiming and complete paid ledgers.
+3. `extra/setup.py` adds22 source entrances, with three ordered rank-four read schedules. `extra/boundary.py` independently checks every changed source and auxiliary path, alias, reflection and primal/dual prime witness.
+4. `joint/pack_joint.py` enumerates16621 physical bank roles across72 replicas:3590136 assignments,231 charts, concrete inverses/factors, normalizers and corruption controls. Literal stock is1314930; the bank selector bill is177892046640.
+5. `targetagg/word.py` and `replay.py` execute all20141 formal columns over F₂ and both defining-integer signs, with17 rejecting mutations including the signed integer control. `boundary.py` separately reconstructs full source and target chronology, both reflected histograms, local integer transfers and prime witnesses for82 equal-response, seven integer-response and four echelon groups. `derive_profile.py` rebuilds the complete paid profile and conservative427680 added-call bill.
+6. Replay the retained exact complex lift/contract, two independent rational moment engines, four finite leaf levels and all47 strict assembly inequalities. The exact certificate rejects the adjacent final grid point.
+
+`verify.py` extracts a temporary source tree, rebuilds all generated receipts before consuming them and checks every submitted byte before and after execution. Integer execution verifies the defining decoder; only its F₂ reduction is identity.
diff --git a/research/coordinated-crossover-pr200/README.md b/research/coordinated-crossover-pr200/README.md
new file mode 100644
index 0000000000..a5620d6f64
--- /dev/null
+++ b/research/coordinated-crossover-pr200/README.md
@@ -0,0 +1,27 @@
+κ = 6.95684647723759e-4
+
+# Signed target elimination with 493 reused sources
+
+Conditional **κ ≈ 6.95684647723759 × 10⁻⁴** from 493 original-source aliases, 82 equal-response groups, seven integer-response groups and four signed echelon target transformations.
+
+The new target transformations cancel selected early response coefficients even when the response rows differ. Apply an explicit integer change of target coordinates at the post-center D0 frame, execute its signed transformed responses, and undo the change at the recorded common frame. Four groups on 20 targets improve the paid target paths; their matrices have determinant one and explicit integer inverses. Three quartets replace six older pairs, and an octet replaces one older integer-response group. The actual word restores arbitrary dirty target values.
+
+An additional rank-four source entrance reuses source 90 as role 14073 after the earlier source gauges have been read. This gives 22 extra entrances beyond the retained 471-source construction, including three ordered rank-four entrances. All source aliases, exclusive rank-two reads and overlapping rank-four reads remain in the complete chronological ledger.
+
+The literal scalar word passes over F₂ and both defining-integer signs on all 20141 formal columns, restoring all 1760 sources and 16621 dirty registers. Integer execution checks the defining decoder; its F₂ reduction is identity. Seventeen corruption controls reject, including a sign error tested over the integers. Independent audits rebuild all changed source paths, every target path, both reflected frame ledgers, all 5280 copied-center target reads, the exact local integer transfers and primal/dual prime witnesses. The complete banks enumerate 3590136 literal assignments and 231 exact charts.
+
+The normalized profile is m=72, W=438310, rank mass 31511856, deficit 46464 and maximum child 22. Literal stock is 1314930. The scalar bill retains all 208 uncompressed extra-source compensation deliveries, the previous 480 equal-group basis additions and 198 integer-response basis additions, then adds all 48 echelon basis additions and 56 signed compensation units. No removed operation is credited. This adds 427680 fixed calls; the complete extra bill 177892474320 remains below 2⁴⁰.
+
+The package includes sennemmi’s nineteen PR230 operation-frame improvements at fee4c05b91d4cdebc0b9d8cbb266a98069928a9a, with the original search, result, frame witness and notice preserved in `search/pr230/`. The retained complex lift/contract, two independent exact moment engines, four finite leaf levels and all 47 strict assembly inequalities pass. The adjacent final grid point is rejected; the bit supplier binds.
+
+```sh
+python3 -B research/coordinated-crossover-pr200/verify.py --temp-root /tmp
+```
+
+Python 3.11+ with assertions enabled. Five pinned archives supply dependencies. Verification extracts a temporary source tree without network or git and checks every submitted file before and after execution.
+
+The result remains conditional on inherited compiler, weighted/restored selector, source-line, routing, precision/recovery, prime, analytic-transfer and finite-bridge assumptions. The complex supplier retains its exact local-flow contract.
+
+Provenance: Dugongue's PR216 (34abdc58ed77329c8bf9af5634531fc41939ba3c), prepared with OpenAI Codex assistance, supplies compatible cuts and the majorant idea. PR211's cascade frames are Rohan Arun's work with Anthropic Claude assistance. PR207 (Dugongue) supplies offline admission; PR200 (Chafik Boukhalfa) supplies the physical word/checkers; PR202/193 supply the complex witness; PR197 (Evan McKinney) supplies completed banks. The nineteen PR230 operation-frame changes are sennemmi’s contribution with substantial OpenAI Codex assistance. Source reuse, shared gauges, further cuts, target aggregation, integer target-response compression, signed echelon target transformations and integration were prepared by eumemic with substantial OpenAI Codex assistance. All original licenses, contributor notices and upstream AI disclosures remain.
+
+The earlier construction notes describe the retained 471-source stages. `EXTRA-SOURCE-BORROWING.md`, `TARGET-AGGREGATION.md`, `TARGET-RANK-COMPRESSION.md`, `TARGET-ECHELON.md` and `PROOF.md` document the final construction.
diff --git a/research/coordinated-crossover-pr200/SOURCE-BORROWING.md b/research/coordinated-crossover-pr200/SOURCE-BORROWING.md
new file mode 100644
index 0000000000..7e3b0b1cc3
--- /dev/null
+++ b/research/coordinated-crossover-pr200/SOURCE-BORROWING.md
@@ -0,0 +1,21 @@
+Historical predecessor description: numerical counts below describe that predecessor. The final construction has 471 source aliases, including 28 new source-backed entrances. See NEW-SOURCE-GAUGES.md for the current scalar word, target chronology and bank counts.
+
+# Original sources replace singleton-copy auxiliaries
+
+The frozen selection removes 440 physical auxiliary slots, one for each selected original passive source. It replaces a later singleton-copy branch, rather than the source's injection branch. Every selected branch has fresh value X_s after its initial copy and an unchanged-value prefix before its selected local partner mix. Its original initial copy may already use a padded rank-two frame.
+
+Identify the branch's physical storage with the original X_s register, whose initial address is the existing source line χ_s. Omit the selected first copy into that branch and omit its nonexistent initial-dirty response. Recompute every remaining integer dirty response by the actual reverse adjoint after deleting those 440 copy operations. The original source injection into the other retained source role is still performed before any source is changed.
+
+At the selected early operation, mix X_carrier ← X_carrier + X_s. The borrowed X_s has only served as a control since its initialization, so the saved value is the original partner sum. The exact rank-two partner span contains both source lines and lies on the unchanged remaining branch path. At the original target-delivery anchor, deliver the saved carrier value without mixing again. Retain every target anchor and target frame.
+
+Reverse the auxiliary word, omitting the same deleted copies. This restores every borrowed X_s. Only then undo each early partner mix. Finally perform the original retained-source-copy cleanup. The exact replay checks all source, target and arbitrary remaining dirty columns over F2, and the defining integer decoder in both reflected signs. It restores every original source and every remaining dirty input. Omitting an early mix, retaining the old adjoint, or undoing the saved mix before restoring the borrowed source is rejected.
+
+The 67 borrowed slots that were donors retain their entire physical donor-to-recipient chain, including the current-value compensation at the recipient gauge. This gauge is an internal splice, not a newly bankable entrance. Every resulting original-source path runs from χ_s to the full ambient frame; the complementary reflected path has the same paid increments. No new source endpoint or all-size primitive is assumed.
+
+The independent actual path recount removes one local rank-two child and one local rank-22 child per borrow. Across three cores, the full profile changes by H_2−1320 and H_22−1320, W−440, and rank mass−31680, with the same deficit 1936 per vertex. The 234 existing entrance charts remain valid. Remove only the 440 actual ungauged full-residual dirty chains from the literal banks; the normalized stock is then W=439525 and rank mass 31599336, with deficit 46464.
+
+Retimed partner mixes and inverse mixes have the same scalar counts. Deleting each copy and its reverse removes 880 scalar operations. The recomputed positive adjoint coefficients do not increase, and the remaining dirty inventory decreases. The previous conservative scalar charge remains valid for both reflected words.
+
+This construction uses the existing bit-word, partner-pair, restored dirty-response, source-line and weighted-rank interfaces. The inherited all-size compiler, selector, routing and analytic conditions remain conditional. The new source borrowing, exact executor and integration were prepared with substantial OpenAI Codex assistance; upstream authorship, licenses and AI disclosures remain unchanged.
+
+The final package additionally uses three original sources as genuine rank-19 gauge inputs; see GAUGE-SOURCE-BORROWING.md. The 440-copy construction and its independent path delta above remain unchanged.
diff --git a/research/coordinated-crossover-pr200/TARGET-AGGREGATION.md b/research/coordinated-crossover-pr200/TARGET-AGGREGATION.md
new file mode 100644
index 0000000000..c7b817d94b
--- /dev/null
+++ b/research/coordinated-crossover-pr200/TARGET-AGGREGATION.md
@@ -0,0 +1,13 @@
+# Eighty-two equal-response groups
+
+The conditional claim is κ = 6.95684647723759e-4. The final equal-response layer contains76 quartets and six pairs on316 targets. `targetagg/selection.json` gives every pivot, participating source role, closing frame and split source read frame. These targets are disjoint from the seven integer-response groups and four echelon groups.
+
+At the common post-center D0, subtract each pivot value from its other group coordinates. Apply each participating compensation only to the pivot, then add the pivot back at the exact common frame before the first nonparticipating read. For arbitrary dirty values and total signed response c, the dependent output is (Y_t−Y_p)+(Y_p+c)=Y_t+c. A source role can serve several groups and retain ordinary recipients outside them.
+
+The literal word closes touched groups before their first incompatible read and sorts split source reads by frame rank. The independent audit checks all inverse timestamps, nesting and target caps, local integer transfers, both reflected ledgers and prime witnesses. All5280 copied-center target reads remain at D0 forward and D1 reflected.
+
+The final layer uses468 setup/inverse unit additions and92 pivot compensation reads. The conservative bill retains the earlier88-group480-addition charge; the six replaced pairs earn no subtraction. The seven remaining integer-response groups use186 unit basis additions, while the bill retains the earlier198. Four echelon groups add48 basis units and56 compensation units. Along with208 uncompressed source deliveries, this charges990×2×3×72=427680 added calls. Together with bank bill177892046640, the total is177892474320<2⁴⁰.
+
+The complete source and target ledgers, both defining-integer signs and17 corruption controls are checked by `targetagg/replay.py` and `targetagg/boundary.py`; `derive_profile.py` reconstructs the entire paid profile.
+
+Prepared by eumemic with substantial OpenAI Codex assistance; inherited notices and Apache-2.0 terms remain.
diff --git a/research/coordinated-crossover-pr200/TARGET-ECHELON.md b/research/coordinated-crossover-pr200/TARGET-ECHELON.md
new file mode 100644
index 0000000000..691ed4529d
--- /dev/null
+++ b/research/coordinated-crossover-pr200/TARGET-ECHELON.md
@@ -0,0 +1,13 @@
+# Four signed target transformations
+
+The new groups cover0–3,40–43,72–75 and864–871. `echelon/selection.json` records an elementary integer row-operation word, the transformed responses to every prefix source, and the common inverse frame for each group. Each operation is Y_t←Y_t+cY_p with distinct t,p and integer c. The inverse reverses the operation order and negates c.
+
+Let T denote the coordinate transformation and R the original prefix-response matrix. Starting with arbitrary dirty target vector y, the literal word computes Ty, adds the signed responses TRz, and applies T⁻¹. Its result is y+Rz. The independent checker reconstructs T, proves det(T)=1, verifies an explicit integer inverse, and tests this transfer on independent target and response variables for both defining-integer signs.
+
+The first three quartets replace six equal-response pairs. For group0–3 the setup subtracts Y0 from Y1,Y2,Y3, then subtracts the updated Y2 from Y3. Their common inverse frames have rank21. Each quartet changes its target histogram by C1−1,C2−1,C3+1. The fourth group replaces the former rank-response octet864–871, uses12 elementary setup operations and closes at rank20; relative to that old octet it changes the histogram by C2−3,C6−3,C8+4,C12+1,C20−1.
+
+The four groups execute24 setup operations and24 inverse operations. Their48 nonzero signed response additions expand to56 unit additions because some coefficients have absolute value two. All these units are added to the conservative fixed-call bill, with no credits for the replaced target operations. The transformed responses are executed at their original source-gauge frames, and all groups are inverted before subsequent incompatible reads.
+
+A sign-flip corruption is checked over the integers: F₂ alone would not detect it. Seventeen whole-word controls and separate local matrix controls reject omitted setup, omitted inverse and altered signed responses. The full source and dirty registers restore, every target remains within its exact frame cap, both reflected ledgers agree, and the new frame prime witnesses have stripped residual one.
+
+Prepared by eumemic with substantial OpenAI Codex assistance; inherited attribution, licenses and assistance notices remain.
diff --git a/research/coordinated-crossover-pr200/TARGET-RANK-COMPRESSION.md b/research/coordinated-crossover-pr200/TARGET-RANK-COMPRESSION.md
new file mode 100644
index 0000000000..ad9176595c
--- /dev/null
+++ b/research/coordinated-crossover-pr200/TARGET-RANK-COMPRESSION.md
@@ -0,0 +1,11 @@
+# Seven integer-response groups
+
+Seven disjoint groups cover targets216–223 and872–919. Their exact integer dependencies are recorded in `rank/selection.json`. A response row gives one target's coefficients against its specified prefix compensation symbols; these relations do not constrain arbitrary initial target values.
+
+For a dependent row r_t=Σ_j a_tj r_j, subtract Σ_j a_tj Y_j at post-center D0, execute retained prefix reads and omit the dependent reads, then add the combination back at the recorded common rank-20 frame before the first nonprefix read. Dependencies involve only retained rows. The transformation has identity diagonal blocks and determinant one, so it restores arbitrary dirty values over the integers and F₂.
+
+The seven groups compress27 dependent coordinates and omit108 prefix reads. Setup and inverse use174 scaled additions, conservatively expanded to186 unit additions. The complete fixed-call bill retains the earlier eight-group198-unit charge without crediting the removed octet864–871, which is now handled by the fourth signed echelon transformation.
+
+The local target histogram delta relative to the same word without these seven groups is C1+28,C8−23,C9−4,C12−3,C13−24,C20+27. Stock and rank mass are unchanged. The literal scalar replay, independent arbitrary-variable integer transfer tests, both reflected ledgers and primal/dual prime witnesses check these groups together with the whole word.
+
+Prepared by eumemic with substantial OpenAI Codex assistance. Apache-2.0; inherited contributor and assistance notices remain.
diff --git a/research/coordinated-crossover-pr200/arithmetic.py b/research/coordinated-crossover-pr200/arithmetic.py
new file mode 100644
index 0000000000..df7440e640
--- /dev/null
+++ b/research/coordinated-crossover-pr200/arithmetic.py
@@ -0,0 +1,46 @@
+from pathlib import Path
+import json,importlib.util,sys
+from fractions import Fraction as Q
+sys.set_int_max_str_digits(0)
+R=Path(__file__).resolve().parent;O=R;ROOT=R.parents[1]
+def load(n,p):
+ s=importlib.util.spec_from_file_location(n,p);m=importlib.util.module_from_spec(s);s.loader.exec_module(m);return m
+def enc(x):
+ if isinstance(x,Q):return str(x)
+ if isinstance(x,dict):return {str(k):enc(v) for k,v in x.items()}
+ if isinstance(x,(list,tuple)):return [enc(v) for v in x]
+ return x
+im=load('im',ROOT/'research/paired-cube-diagonal-bit-168/arithmetic/interval_moment.py')
+assembly=load('asm',ROOT/'scripts/paired_cube_assembly.py')
+p=json.loads((R/'geometry/packed-frames.json').read_text());base=json.loads((ROOT/'research/source-assisted-v4/certificate.json').read_text())
+def profile(cp):
+ H={int(t):n for t,n in cp['child_histogram'].items()}
+ return dict(m=cp.get('m',72),W=cp['W_per_vertex'],child_multiplicities=H,N=cp['deficit_per_vertex'],L=0,total_rank=sum(t*n for t,n in H.items()),maxchild=max(H))
+actual=json.loads((R/'complex-profile-replayed.json').read_text());assert all(actual[k]==base['complex_profile'][k] for k in ['m','W_per_vertex','rank_per_vertex','deficit_per_vertex','child_histogram']);cp=profile(actual);bp=profile(p['profile']['scaled_moment_profile'])
+def paid(row,a,bit):
+ x=im.moment(row,a)
+ if bit:
+  l,u=im.log_interval(Q(row['m']));e,f=im.exp_interval(a*l,a*u);w=Q(1,10**16)*Q(32*row['m']*sum(row['child_multiplicities'].values()),row['W']);x.update(lower=x['lower']+w*e,upper=x['upper']+w*f)
+ return x
+def certify(row,bit):
+ den=10**18;lo=0;hi=den//100
+ while hi-lo>1:
+  md=(lo+hi)//2;v=paid(row,Q(md,den),bit)
+  if v['upper']<1:lo=md
+  elif v['lower']>1:hi=md
+  else:raise ValueError('inconclusive')
+ a=Q(lo,den);z=Q(hi,den);assert paid(row,a,bit)['upper']<1<paid(row,z,bit)['lower']
+ return dict(saving=a,accepted=paid(row,a,bit),next_excluded=paid(row,z,bit))
+c=certify(cp,False);b=certify(bp,True);assert b['saving']==Q(p['coarse']['coarse_saving'])
+coarse=b['saving'];chain=[Q(p['coarse']['ordinary_saving'])]
+for i in range(4):
+ a=(1-coarse)*coarse+coarse*chain[-1];assert chain[-1]<a<coarse<1-a;chain.append(a)
+bridge=base['assembly']['finite_bridge'];bridge['rows']['degree_gap']=Q(bridge['rows']['degree_gap'])
+eta=beta=Q(1,10**24);weak=Q(1,10**30);a=min(chain[-1],(1-beta)*c['saving']-weak);q=a*(1-2*eta);ceiling=(1-eta)*q/(1+q);grid=10**18;scaled=ceiling*grid;k=Q((scaled.numerator-1)//scaled.denominator,grid)
+z=assembly.assembly(a,c['saving'],bridge,k,eta=eta,beta=beta);assert len(z['strict_constraints'])==47 and all(v>0 for v in z['strict_constraints'].values())
+try:assembly.assembly(a,c['saving'],bridge,k+Q(1,grid),eta=eta,beta=beta)
+except AssertionError:pass
+else:raise AssertionError('next k accepted')
+out=dict(status='PASS_EXACT_47_CONSTRAINT_ASSEMBLY',scope='Conditional finite supplier certificate; retained all-size interfaces and analytic assumptions',bit_profile=bp,complex_profile=cp,bit=b,complex=c,ordinary_chain=chain,bit_ordinary=chain[-1],a_bit=a,a_complex=c['saving'],eta=eta,beta=beta,kappa=k,kappa_float=float(k),binding='bit',assembly=z,adjacent_kappa_rejected=True)
+(O/'assembly.json').write_text(json.dumps(enc(out),indent=2,sort_keys=True)+'\n')
+print(json.dumps(enc({k:out[k] for k in ['status','kappa','kappa_float','bit_ordinary','a_complex']}),indent=2))
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diff --git a/research/coordinated-crossover-pr200/borrow/PROVENANCE.json b/research/coordinated-crossover-pr200/borrow/PROVENANCE.json
new file mode 100644
index 0000000000..334c7f3a81
--- /dev/null
+++ b/research/coordinated-crossover-pr200/borrow/PROVENANCE.json
@@ -0,0 +1,18 @@
+{
+  "scope": "Pre-integration source packet provenance. Portable executor adapts paths and adds mandatory controls and gauge-response checks.",
+  "precursor_packet": {
+    "status": "FROZEN_EXACT_SOURCE_BORROWING_INPUTS",
+    "files": {
+      "borrow_terminal.py": "009689944a6782123ceb2595036c3f616551b67d2b7a4b4b7cea97bf50217d9d",
+      "borrow-selection.json": "6932749407e14b199651f0a7deff02c6de74e2c1be0a1300263f1e164927fb8b",
+      "borrow-terminal-replay.json": "2bbce85126d882da80cf437a29bd15df9b72566bae7eaf22e1f21cc85b213f2e",
+      "borrow-controls.json": "a6b63ee8b85ae7f43c874fb1df619407a94a94781e0d90ce35617a7c24a64521"
+    },
+    "baseline_package": "/private/tmp/joint82-package",
+    "baseline_manifest_sha256": "90617370d62e4ba82865fef5e4de85e1a0e08aba36369d7c0d70b60c4227dca0",
+    "boundary_audit": "/private/tmp/original-gauge-order/source-borrow-boundary.json",
+    "field": "F2 with exact integer defining decoder in both signs",
+    "new_all_size_primitive": false,
+    "scope": "Source \u03c7 line tofull, exactsource+dirtyrestoration; same inheritedscalar/transvection andrankcompiler interfaces."
+  }
+}
diff --git a/research/coordinated-crossover-pr200/borrow/boundary.py b/research/coordinated-crossover-pr200/borrow/boundary.py
new file mode 100644
index 0000000000..720a3e9e42
--- /dev/null
+++ b/research/coordinated-crossover-pr200/borrow/boundary.py
@@ -0,0 +1,88 @@
+#!/usr/bin/env python3
+"""Independent source-slot endpoint and paid-path audit for440 copy elisions.
+Prepared with OpenAI Codex assistance; no scalar replay duplicated here.
+"""
+import sys,json,importlib.util
+from pathlib import Path
+from collections import Counter,defaultdict
+sys.dont_write_bytecode=True
+HERE=Path(__file__).resolve().parent;p=HERE.parent/'joint/joint_word.py'
+spec=importlib.util.spec_from_file_location('boundary_seed',p);m=importlib.util.module_from_spec(spec);spec.loader.exec_module(m)
+w=m.Candidate();C=w.C;C.decoder();C.geometry()
+selection=json.loads((HERE/'selection.json').read_text())
+recipient={d:b for b,d in w.pairs};kpair={e['passive']:e for e in w.k['entries']}
+roots=defaultdict(list)
+for j,s in enumerate(w.w['rootroles']):roots[s].append((j,w.w['root_frame'][j]))
+full=w.w['full_frame'];source_roles=set(w.source.values())
+results=[];delta=Counter();alias_count=0;all_early_delta=Counter()
+def histogram(start,events):
+ prev=start;hist=Counter()
+ for tag,f in events:
+  assert prev is None or C.sub(prev,f),('nonnested',tag,prev,f)
+  d=C.dimf[f]-(C.dimf[prev] if prev is not None else 0)
+  if d:hist[d]+=1
+  prev=f
+ return hist
+for r in selection:
+ s,n,i,early=r['role'],r['source'],r['omit_copy'],r['early']
+ assert s not in w.gauge and s not in source_roles and s not in w.donor
+ assert i==w.role_ops[s][0] and w.ops[i][0]==s and w.ops[i][2]==n
+ assert i in w.phase1
+ e=kpair[n];line=w.w['source_frame'][n];mix=e['mix_frame']
+ assert e['carrier']==r['partner'] and e['passive']==n
+ physical=[s]+([recipient[s]] if s in recipient else [])
+ events=[]
+ for j,role in enumerate(physical):
+  if j:
+   alias_count+=1
+   assert w.gauge[role]['role']==role and w.phys[role]==s
+   assert w.last[s]<w.readtime[role]
+   events.append((('alias_gauge',role),w.gauge[role]['frame']))
+  for op in w.role_ops[role]:events.append((('op',op),w.opframe[op]))
+  for root,f in roots[role]:events.append((('root',root),f))
+ events.append((('full',s),full))
+ old=histogram(None,events)
+ revised=[];inserted=False
+ for tag,f in events:
+  if tag==('op',i):continue
+  if tag==('op',early):revised.append((('early_source_mix',s),mix));inserted=True
+  revised.append((tag,f))
+ assert inserted
+ # The early partner mix reads the original passive value before any write.
+ for tag,f in revised:
+  if tag==('early_source_mix',s):break
+  if tag[0]=='op':assert w.phys[w.ops[tag[1]][0]]!=s,('borrowed source written before early mix',s,tag)
+ assert C.sub(w.w['source_frame'][e['carrier']],mix)
+ new=histogram(line,revised)
+ old_passive=histogram(e['passive_chain'][0],[(('old_source',k),f) for k,f in enumerate(e['passive_chain'][1:])])
+ assert old_passive==Counter({1:1,22:1})
+ got=Counter(new)
+ got.subtract(old);got.subtract(old_passive)
+ assert sum(rank*count for rank,count in got.items())==-w.h
+ # The complementary reverse chain has the same positive jump multiset;
+ # dim(I−P)=h−dim(P) and reverse order preserves these differences exactly.
+ dims=[C.dimf[line]]+[C.dimf[f] for tag,f in revised]
+ reflected=[w.h-d for d in reversed(dims)]
+ reflected_hist=Counter(b-a for a,b in zip(reflected,reflected[1:]) if b>a)
+ assert reflected_hist==new
+ delta.update(got)
+ results.append(dict(role=s,source=n,aliased_recipient=recipient.get(s),
+    original_source_endpoint_frame=line,omitted_copy=i,early_mix=early,
+    old_aux_histogram=dict(sorted(old.items())),new_source_histogram=dict(sorted(new.items())),
+    local_histogram_delta={rank:count for rank,count in sorted(got.items()) if count},
+    reflected_histogram_equal=True))
+assert len(results)==440 and alias_count==67
+out=dict(scope='INDEPENDENT EXACT SOURCE ENDPOINT AND FRAME-PATH AUDIT; SCALAR ADMISSION OWNED BY SEPARATE EXECUTOR',
+ sources_borrowed=len(results),borrowed_physical_donor_alias_chains=alias_count,
+ all_source_start_frames_preserved=True,all_source_end_frames_full=True,
+ every_spliced_frame_inclusion_checked=True,all_reflected_path_histograms_equal=True,
+ local_histogram_delta={rank:count for rank,count in sorted(delta.items()) if count},
+ shared_histogram_delta={rank:3*count for rank,count in sorted(delta.items()) if count},
+ local_rank_mass_delta=sum(rank*count for rank,count in delta.items()),
+ shared_rank_mass_delta=3*sum(rank*count for rank,count in delta.items()),
+ physical_role_stock_delta=-len(results),shared_m=3*w.h,
+ deficit_change=-(3*w.h)*len(results)-3*sum(rank*count for rank,count in delta.items()),
+ results=results)
+assert out['deficit_change']==0
+(HERE/'boundary.json').write_text(json.dumps(out,indent=2)+'\n')
+print(json.dumps({k:v for k,v in out.items() if k!='results'},indent=2))
diff --git a/research/coordinated-crossover-pr200/borrow/derive_profile.py b/research/coordinated-crossover-pr200/borrow/derive_profile.py
new file mode 100644
index 0000000000..57d23f4a38
--- /dev/null
+++ b/research/coordinated-crossover-pr200/borrow/derive_profile.py
@@ -0,0 +1,45 @@
+"""Reconstruct the complete paid profile from actual spliced source/dirty paths."""
+from pathlib import Path
+from collections import Counter
+import json,sys
+sys.dont_write_bytecode=True
+from replay import W,selection,adj,oldadj,regs,removed
+HERE=Path(__file__).resolve().parent;PACKAGE=HERE.parent
+base=json.loads((PACKAGE/'joint/joint-replay.json').read_text());audit=json.loads((HERE/'boundary.json').read_text());scalar=json.loads((HERE/'replay.json').read_text())
+assert scalar['F2']['dirty']==len(regs)==16674
+assert len(scalar['integer'])==2 and {r['direction']for r in scalar['integer']}=={1,-1}
+assert all(r['all_source_and_dirty_restored']and r['formal_columns']==20194 for r in [scalar['F2']]+scalar['integer'])
+assert len(audit['results'])==len(selection)==440
+assert {r['role']for r in audit['results']}=={r['role']for r in selection}
+assert all(adj[s]==oldadj[s]for s in W.gauge)
+# Every actual dirty path except a borrowed/deleted path remains unchanged.
+local=Counter({int(k):n for k,n in base['profile']['physical_internal_histogram'].items()})
+_,_,source,_=W.C.chains();source=Counter(source)
+for r in audit['results']:
+ local.subtract({int(k):n for k,n in r['old_aux_histogram'].items()})
+ source.subtract({1:1,22:1})
+ source.update({int(k):n for k,n in r['new_source_histogram'].items()})
+assert all(n>=0 for n in local.values())and all(n>=0 for n in source.values())
+# The changed adjoint agrees on every gauge, all terminal actions remain,
+# and partner deliveries retain their exact original target anchors/frames.
+target=Counter({int(k):n for k,n in base['profile']['physical_target_histogram'].items()})
+child=Counter()
+for part in [local,target,source]:
+ for rank,count in part.items():
+  if rank and count:child[rank]+=3*count
+for role,z in W.gauge.items():
+ if role not in W.donor:child[3*z['dim']]+=1
+child[2]+=2*W.v
+assert all(count>0 for count in child.values())
+delta=Counter(child);delta.subtract({int(k):n for k,n in base['profile']['child_histogram'].items()});delta={k:v for k,v in delta.items()if v}
+assert delta=={int(k):v for k,v in audit['shared_histogram_delta'].items()}=={2:-1320,22:-1320}
+physical=len(regs);stock=2*W.v+physical;mass=sum(rank*count for rank,count in child.items());deficit=72*stock-mass
+assert stock==20194 and deficit==base['profile']['deficit_per_vertex']==1936
+assert sum(rank*count for rank,count in target.items())==W.v*(W.h-1)
+assert sum(rank*count for rank,count in source.items())==W.v*(W.h-1)
+def compact(x):return {str(k):n for k,n in sorted(x.items())if n}
+profile=dict(base['profile'],R=physical,active_virtual_R=base['profile']['active_virtual_R']-440,W_per_vertex=stock,rank_per_vertex=mass,child_histogram=compact(child),physical_internal_histogram=compact(local),physical_source_histogram=compact(source),physical_target_histogram=compact(target),physical_original_sources_borrowed=440)
+assert profile['maxchild']==max(child)
+out=dict(status='PASS_COMPLETE_SOURCE_BORROWED_PROFILE',profile=profile,actual_child_delta=delta,source_boundary_audit='boundary.json',scalar_replay='replay.json',all_gauge_adjoint_coefficients_retained=True,all_target_anchor_frames_retained=True,old_adjoint_replaced=True,removed_auxiliary_copy_gates=440,removed_forward_reverse_scalar_gates=880,retimed_partner_mix_count_unchanged=True,remaining_adjoint_coefficients_do_not_increase=True)
+(HERE/'profile.json').write_text(json.dumps(out,indent=2)+'\n')
+print('PASS complete spliced source/internal/target profile',dict(W=stock,rank=mass,deficit=deficit,delta=delta),flush=True)
diff --git a/research/coordinated-crossover-pr200/borrow/replay.py b/research/coordinated-crossover-pr200/borrow/replay.py
new file mode 100644
index 0000000000..3184271aa0
--- /dev/null
+++ b/research/coordinated-crossover-pr200/borrow/replay.py
@@ -0,0 +1,125 @@
+"""Complete scalar admission of original-source singleton-copy elision.
+Prepared with OpenAI Codex assistance; Apache-2.0.
+"""
+from pathlib import Path
+from collections import Counter,defaultdict
+import json,sys,importlib.util,time,gc
+sys.dont_write_bytecode=True
+HERE=Path(__file__).resolve().parent;D=HERE.parent/'joint';spec=importlib.util.spec_from_file_location('source82',D/'joint_word.py');m=importlib.util.module_from_spec(spec);spec.loader.exec_module(m)
+W=m.Candidate();C=W.C;C.decoder();C.geometry();selection=json.loads((HERE/'selection.json').read_text())
+borrow={r['role']:r['source']for r in selection};by_source={r['source']:r for r in selection};omitted={r['omit_copy']for r in selection};early={r['early']:r for r in selection};assert len(early)==len(borrow)==len(selection)==440
+adj=[{}for _ in range(W.R)]
+for r,s in zip(W.g['roots'],W.w['rootroles']):
+ for t in r['targets']:adj[s][t]=adj[s].get(t,0)+1
+for i in reversed(W.phase1+W.rest):
+ if i in omitted:continue
+ a,b,_=W.ops[i]
+ for t,c in adj[a].items():adj[b][t]=adj[b].get(t,0)+c
+oldadj=W.adjoint()
+assert all(all(0<c<=oldadj[s].get(t,0)for t,c in row.items())for s,row in enumerate(adj))
+assert all(adj[s]==oldadj[s]for s in W.gauge), 'all gauge reads and paid target supports retained exactly'
+assert len(by_source)==440 and all(r['early'] in r['windows']for r in selection)
+# The formal column layout excludes each actual borrowed dirty slot, retaining
+# distinct original X columns and every other arbitrary dirty input.
+terminal=json.loads((D/'joint-replay.json').read_text())['selected'];removed={e['role']for e in terminal};deletedroots={e['root']for e in terminal};bywrite={i:e for e in terminal for i in e['writes']};afterwrite={e['writes'][-1]:e for e in terminal};assert not(set(early)&set(bywrite)) and not(set(borrow)&removed)
+regs=sorted(set(W.phys.values())-set(borrow)-removed);idx={r:2*W.v+j for j,r in enumerate(regs)}
+assert not set(borrow)&set(W.donor)
+at=defaultdict(list)
+for b in W.order:at[W.readtime[b]].append(b)
+deliveries=defaultdict(list)
+for e in W.k['entries']:deliveries[e['deliver_after_root']].append(e)
+
+def replay(mode='F2',direction=1,bits=24,mutation=None):
+ major=mode=='bound';binary=mode=='F2';v=W.v
+ unit=(lambda i:1)if major else(lambda i:1<<i)if binary else(lambda i:1<<(bits*i))
+ def add(x,c,y):return x+abs(c)*y if major else x^y if binary and c&1 else x if binary else x+c*y
+ X0=[unit(i)for i in range(v)];Y0=[unit(v+i)for i in range(v)];Z0={r:unit(idx[r])for r in regs};x=X0[:];y=Y0[:];z=Z0.copy();top=1;done=[]
+ def track(value):
+  nonlocal top
+  if major:top=max(top,value)
+  return value
+ def value(s):
+  s=W.phys[s];return x[borrow[s]]if s in borrow else z[s]
+ def assign(s,val):
+  s=W.phys[s]
+  if s in borrow:x[borrow[s]]=track(val)
+  else:z[s]=track(val)
+ responses=W.adjoint() if mutation=='old_adjoint' else adj
+ def read(s):
+  for t,c in responses[s].items():y[t]=track(add(y[t],-direction*c,value(s)))
+ def gate(i,sign):
+  a,b,_=W.ops[i];assign(a,add(value(a),sign,value(b)))
+ for s in range(W.R):
+  if s not in W.gauge and s not in borrow and s not in removed:read(s)
+ for n,s in W.source.items():assign(s,add(value(s),1,x[n]))
+ def forward(i):
+  if i in early:
+   r=early[i];a,b=r['partner'],r['source']
+   if mutation!='omit_early_mix' or r is not selection[0]:x[a]=track(add(x[a],1,x[b]))
+  if i not in omitted:gate(i,1);done.append(i)
+ for i in W.phase1:forward(i)
+ for r,s in zip(W.g['roots'],W.w['rootroles']):
+  if r['kind']=='center':
+   for t in r['targets']:y[t]=track(add(y[t],direction,value(s)))
+ for e in terminal:
+  for t in e['targets']:
+   if t!=e['pivot']:y[t]=track(add(y[t],-1,y[e['pivot']]))
+ for j,i in enumerate(W.rest):
+  for s in at[j]:read(s)
+  if i in bywrite:
+   e=bywrite[i];a,b,n=W.ops[i];y[e['pivot']]=track(add(y[e['pivot']],direction,value(b)))
+  else:forward(i)
+  if i in afterwrite:
+   e=afterwrite[i]
+   for t in e['targets']:
+    if t!=e['pivot']:y[t]=track(add(y[t],1,y[e['pivot']]))
+ for s in at[len(W.rest)]:read(s)
+ for j,(r,s)in enumerate(zip(W.g['roots'],W.w['rootroles'])):
+  if r['kind']=='side' and j not in deletedroots:
+   for t in r['targets']:y[t]=track(add(y[t],direction,value(s)))
+  for e in deliveries[j]:
+   a,b=e['carrier'],e['passive']
+   if b not in by_source:x[a]=track(add(x[a],1,x[b]))
+   for t in e['receivers']:y[t]=track(add(y[t],direction,x[a]))
+ for e in W.k['entries']:
+  if e['passive']not in by_source:x[e['carrier']]=track(add(x[e['carrier']],-1,x[e['passive']]))
+ if mutation=='undo_before_restore':
+  for e in W.k['entries']:
+   if e['passive']in by_source:x[e['carrier']]=track(add(x[e['carrier']],-1,x[e['passive']]))
+ for i in reversed(done):gate(i,-1)
+ if mutation!='undo_before_restore':
+  for e in W.k['entries']:
+   if e['passive']in by_source:x[e['carrier']]=track(add(x[e['carrier']],-1,x[e['passive']]))
+ for n,s in W.source.items():assign(s,add(value(s),-1,x[n]))
+ if binary:want=[Y0[t]^X0[t]for t in range(v)]
+ else:
+  want=Y0[:];cache={}
+  for root in W.g['roots']:
+   n=root['node']
+   if n not in cache:
+    mask=C.sup[n];val=0
+    while mask:low=mask&-mask;mask-=low;val=add(val,1,X0[low.bit_length()-1])
+    cache[n]=val
+   for t in root['targets']:want[t]=add(want[t],direction,cache[n])
+  for e in W.k['entries']:
+   val=add(X0[e['carrier']],1,X0[e['passive']])
+   for t in e['receivers']:want[t]=add(want[t],direction,val)
+ if major:return max(top,max(a+b for a,b in zip(y,want)),max(z[r]+Z0[r]for r in regs),max(a+b for a,b in zip(x,X0)))
+ assert y==want,('wrong targets',mode,[t for t in range(v)if y[t]!=want[t]][:10])
+ assert x==X0,('source restoration',mode,[s for s in range(v)if x[s]!=X0[s]][:10])
+ assert z==Z0,('dirty restoration',mode)
+ return dict(mode=mode,direction=direction,formal_columns=2*v+len(regs),sources=v,targets=v,dirty=len(regs),all_targets=True,all_source_and_dirty_restored=True)
+if __name__=='__main__':
+ start=time.monotonic();out=dict(status='PASS_FULL_SOURCE_BORROWED_WORD',selection=selection)
+ out['F2']=replay();print('PASS F2',out['F2'],'seconds',time.monotonic()-start,flush=True)
+ bound=replay('bound');bits=8*((bound.bit_length()+2+7)//8);out['bound']=bound;out['bits']=bits;out['terminal_sinks']=len(terminal);print('bound',bound,'bits',bits,flush=True)
+ out['integer']=[]
+ for direction in(1,-1):out['integer'].append(replay('Z',direction,bits));gc.collect();print('PASS integer',direction,'seconds',time.monotonic()-start,flush=True)
+ out['controls']={}
+ for mutation in ['omit_early_mix','old_adjoint','undo_before_restore']:
+  try:replay(mutation=mutation)
+  except AssertionError as error:out['controls'][mutation]=dict(status='REJECTED',error=str(error))
+  else:raise AssertionError('Invalid mutation accepted: '+mutation)
+ assert (1<<bits)>2*bound
+ (HERE/'replay.json').write_text(json.dumps(out,indent=2)+'\n')
+ print('PASS borrowed source word, both integer signs, all columns and controls',flush=True)
diff --git a/research/coordinated-crossover-pr200/borrow/selection.json b/research/coordinated-crossover-pr200/borrow/selection.json
new file mode 100644
index 0000000000..80d0a76822
--- /dev/null
+++ b/research/coordinated-crossover-pr200/borrow/selection.json
@@ -0,0 +1,5282 @@
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+    "role": 13,
+    "omit_copy": 9,
+    "partner": 0,
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+    "aliased_to": 17404,
+    "early": 9252
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diff --git a/research/coordinated-crossover-pr200/certificate.json b/research/coordinated-crossover-pr200/certificate.json
new file mode 100644
index 0000000000..ec75d0ed09
--- /dev/null
+++ b/research/coordinated-crossover-pr200/certificate.json
@@ -0,0 +1,200 @@
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+  ],
+  "scope": "Conditional finite supplier certificate; retained all-size interfaces and analytic assumptions",
+  "status": "PASS_EXACT_47_CONSTRAINT_ASSEMBLY"
+}
diff --git a/research/coordinated-crossover-pr200/echelon/PROVENANCE.json b/research/coordinated-crossover-pr200/echelon/PROVENANCE.json
new file mode 100644
index 0000000000..fc5e4928a4
--- /dev/null
+++ b/research/coordinated-crossover-pr200/echelon/PROVENANCE.json
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+{
+  "scope": "Four exact determinant-one integer coordinate transformations on20targets,48setup/inverseunits and56compensationunits; actualsignedZcontrol.",
+  "source_selection_sha256": "791f6ddeb19add7026ba37461256c13efe3d35bba0bccf1cb9787a01fc16f0f2",
+  "equal_response_selection_sha256": "b7a4d5eb2a6d2f7df867f9be5af7da1d1aa99cfbd5214195967b0ff60642d77a",
+  "integer_response_selection_sha256": "5c97c6413f1eb56c7b8f9ba9e839329d212fae7ac3a08c29e9efdee42f5048a0",
+  "echelon_selection_sha256": "8e6ce5e257aa7656898ab46bc3421df4a619497b1e9f5956d2c546b3014050a7",
+  "assistance": "Prepared by eumemic with substantial OpenAI Codex assistance. Apache-2.0; all inherited notices retained.",
+  "admission": "Literal F2 and both defining-integer signs,17 whole-word controls including integer sign corruption, independent complete source and target paths in both reflections, local integer transfer matrices, prime witnesses, concrete banks and exact paid assembly."
+}
diff --git a/research/coordinated-crossover-pr200/echelon/selection.json b/research/coordinated-crossover-pr200/echelon/selection.json
new file mode 100644
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diff --git a/research/coordinated-crossover-pr200/extra/PROVENANCE.json b/research/coordinated-crossover-pr200/extra/PROVENANCE.json
new file mode 100644
index 0000000000..df735d4149
--- /dev/null
+++ b/research/coordinated-crossover-pr200/extra/PROVENANCE.json
@@ -0,0 +1,9 @@
+{
+  "scope": "22 additional source entrances and all aliases; total493sourceheads; third split rank4 head14073 follows14075.",
+  "source_selection_sha256": "791f6ddeb19add7026ba37461256c13efe3d35bba0bccf1cb9787a01fc16f0f2",
+  "equal_response_selection_sha256": "b7a4d5eb2a6d2f7df867f9be5af7da1d1aa99cfbd5214195967b0ff60642d77a",
+  "integer_response_selection_sha256": "5c97c6413f1eb56c7b8f9ba9e839329d212fae7ac3a08c29e9efdee42f5048a0",
+  "echelon_selection_sha256": "8e6ce5e257aa7656898ab46bc3421df4a619497b1e9f5956d2c546b3014050a7",
+  "assistance": "Prepared by eumemic with substantial OpenAI Codex assistance. Apache-2.0; all inherited notices retained.",
+  "admission": "Literal F2 and both defining-integer signs,17 whole-word controls including integer sign corruption, independent complete source and target paths in both reflections, local integer transfer matrices, prime witnesses, concrete banks and exact paid assembly."
+}
diff --git a/research/coordinated-crossover-pr200/extra/boundary.json b/research/coordinated-crossover-pr200/extra/boundary.json
new file mode 100644
index 0000000000..9ee84dfb2b
--- /dev/null
+++ b/research/coordinated-crossover-pr200/extra/boundary.json
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+        "cleared_determinant": 324,
+        "small_prime_powers": {
+          "2": 2,
+          "3": 4,
+          "5": 0,
+          "7": 0,
+          "11": 0,
+          "13": 0,
+          "17": 0,
+          "19": 0,
+          "23": 0,
+          "29": 0,
+          "31": 0
+        },
+        "remaining_factor": 1
+      },
+      "dual": {
+        "dimension": 22,
+        "cleared_determinant": -179563863424301147460937500,
+        "small_prime_powers": {
+          "2": 2,
+          "3": 23,
+          "5": 21,
+          "7": 0,
+          "11": 0,
+          "13": 0,
+          "17": 0,
+          "19": 0,
+          "23": 0,
+          "29": 0,
+          "31": 0
+        },
+        "remaining_factor": 1
+      }
+    }
+  ],
+  "maximum_prime_residual": 1,
+  "all_source_paths_reflected": true,
+  "scope": "Independent complete changed source/auxiliary paths and exact paid histograms; target ledger, literal scalar word and assembly separately checked.",
+  "input_pins": {
+    "extra/selection.json": "791f6ddeb19add7026ba37461256c13efe3d35bba0bccf1cb9787a01fc16f0f2",
+    "extra/setup.py": "a7789d13eda75e78108cee84206692c913f63aafdb9e3e633fddd14640e4678a",
+    "extra/boundary.py": "6a32671203a25055a71a7acb98396124cec591ebbf680677b40bc37e4db5a7bb",
+    "newg/replay.py": "0cbec81a11e758156e9303f63dc8c617601da69523073fdf2a77394d76acf56a"
+  }
+}
diff --git a/research/coordinated-crossover-pr200/extra/boundary.py b/research/coordinated-crossover-pr200/extra/boundary.py
new file mode 100644
index 0000000000..17f993e3ab
--- /dev/null
+++ b/research/coordinated-crossover-pr200/extra/boundary.py
@@ -0,0 +1,90 @@
+"""Independent22-source paths, reflected ledgers, paid deltas and exact primes.
+Prepared with substantial OpenAI Codex assistance; Apache-2.0.
+Whole scalar word and complete assembly are checked separately.
+"""
+from pathlib import Path
+from collections import Counter,defaultdict
+import importlib.util,json,hashlib,sys
+sys.dont_write_bytecode=True
+D=Path(__file__).resolve().parent;P=D.parent
+sp=importlib.util.spec_from_file_location('source493_boundary_setup',D/'setup.py');setup=importlib.util.module_from_spec(sp);sp.loader.exec_module(setup)
+m=setup.m;W=m.W;C=m.C;selected=setup.extra
+compact=lambda H:{str(k):v for k,v in sorted(H.items())if v}
+oldsel=sum((json.loads((P/f'{side}/selection.json').read_text())for side in('borrow','gaugeb','newg')),[])
+assert len(oldsel)==471 and len(selected)==22 and len(m.borrow)==493
+usedroles={r['role']for r in oldsel};usedsource={r[k]for r in oldsel for k in('source','partner')};assert len(usedsource)==942
+recipient={d:b for b,d in W.pairs};roots=defaultdict(list)
+for j,r in enumerate(W.w['rootroles']):roots[r].append(W.w['root_frame'][j])
+touched={s for i in W.phase1 for s in W.ops[i][:2]};positions={i:j for j,i in enumerate(W.phase1+W.rest)}
+# Rebuild actual positive adjoint directly from literal retimed operations.
+adj=[{}for _ in range(W.R)]
+for root,role in zip(W.g['roots'],W.w['rootroles']):
+ for t in root['targets']:adj[role][t]=adj[role].get(t,0)+1
+for i in reversed(W.phase1+W.rest):
+ if i in m.omitted:continue
+ a,b,_=W.ops[i]
+ for t,c in adj[a].items():adj[b][t]=adj[b].get(t,0)+c
+assert adj==m.adj
+
+def histogram(chain,start):
+ H=Counter();prev=None;d=start
+ for f in chain:
+  assert C.nondeg(f)and(prev is None or C.sub(prev,f));n=C.dimf[f];assert n>=d
+  if n>d:H[n-d]+=1
+  prev=f;d=n
+ return H
+
+def determinant(A):
+ A=[list(r)for r in A];n=len(A)
+ if not n:return 1
+ den=1;sign=1
+ for k in range(n-1):
+  i=next((j for j in range(k,n)if A[j][k]),None)
+  if i is None:return 0
+  if i!=k:A[k],A[i]=A[i],A[k];sign=-sign
+  pivot=A[k][k]
+  for i in range(k+1,n):
+   for j in range(k+1,n):
+    x=A[i][j]*pivot-A[i][k]*A[k][j];assert x%den==0;A[i][j]=x//den
+   A[i][k]=0
+  den=pivot
+ return sign*A[-1][-1]
+
+def prime_record(B,dual=False):
+ sums=list(map(sum,B));scale=9-W.h if dual else 9;sign=1 if dual else -1
+ gram=[[scale*W.module.dot(x,y)+sign*sums[i]*sums[j]for j,y in enumerate(B)]for i,x in enumerate(B)]
+ det=determinant(gram);assert det;rest=abs(det);powers={}
+ for p in(2,3,5,7,11,13,17,19,23,29,31):
+  exponent=0
+  while rest%p==0:rest//=p;exponent+=1
+  powers[str(p)]=exponent
+ assert 0<rest<2**80
+ check=rest
+ for p,e in powers.items():check*=int(p)**e
+ assert check==abs(det)
+ return dict(dimension=len(B),cleared_determinant=det,small_prime_powers=powers,remaining_factor=rest)
+rows=[];delta=Counter();newroles=set();newsource=set();frames=set()
+for r in selected:
+ role,s,p=r['role'],r['source'],r['partner'];e=m.kpair[s];M=e['mix_frame'];G=W.register(r.get('split_gauge_basis',r['gauge_basis']));frames.add(G)
+ assert role not in usedroles|newroles|touched|m.removed|set(W.source.values())|set(W.donor)
+ assert not({s,p}&(usedsource|newsource));newsource.update((s,p));newroles.add(role)
+ assert p==e['carrier']and C.dimf[M]==2 and C.dimf[G]in(2,4)and C.sub(M,G)
+ assert adj[role]==m.adj[role]and sorted(adj[role])==r['targets'];ops=W.role_ops[role]
+ assert ops and ops[0]==r['first_operation']and C.sub(G,W.opframe[ops[0]])
+ aux=[W.opframe[i]for i in ops]+roots[role]
+ if role in recipient:
+  rec=recipient[role];assert positions[ops[-1]]<positions[W.role_ops[rec][0]]
+  aux+=[W.gauge[rec]['frame']]+[W.opframe[i]for i in W.role_ops[rec]]+roots[rec]
+ aux.append(W.w['full_frame']);chain=[W.w['source_frame'][s],M,G]+aux
+ oldaux=histogram(aux,0);oldsource=histogram(e['passive_chain'][1:],1);newhist=histogram(chain[1:],1)
+ reflection=Counter()
+ for a,b in zip(reversed(chain),list(reversed(chain))[1:]):
+  assert all(W.module.dot(x,y)==0 for x in C.A[a]for y in C.B[b]);d=len(C.A[b])-len(C.A[a]);assert d>=0
+  if d:reflection[d]+=1
+ assert reflection==newhist
+ rowdelta=Counter(newhist);rowdelta.subtract(oldaux);rowdelta.subtract(oldsource);assert sum(k*v for k,v in rowdelta.items())==-24
+ delta.update(rowdelta);rows.append(dict(role=role,source=s,partner=p,alias_recipient=recipient.get(role),source_gauge=C.dimf[G],source_path=chain,old_aux_histogram=compact(oldaux),old_source_histogram=compact(oldsource),complete_source_histogram=compact(newhist),source_delta=compact(rowdelta),reflected_equal=True))
+assert len(newroles)==22 and len(newsource)==44 and sum(k*v for k,v in delta.items())==-528
+primes=[dict(frame=f,basis=C.B[f],annihilator=C.A[f],primal=prime_record(C.B[f]),dual=prime_record(C.A[f],True))for f in sorted(frames)]
+result=dict(status='PASS_SOURCE493_INDEPENDENT_SOURCE_AUX_PATHS_BOTH_REFLECTIONS_AND_PRIMES',selected=22,source_local_delta=compact(delta),local_rank_delta=-528,shared_rank_delta=-1584,W_delta=-22,deficit_delta=0,endpoint_rows=rows,source_gauge_prime_witnesses=primes,maximum_prime_residual=max(r[k]['remaining_factor']for r in primes for k in('primal','dual')),all_source_paths_reflected=True,scope='Independent complete changed source/auxiliary paths and exact paid histograms; target ledger, literal scalar word and assembly separately checked.',input_pins={str(p.relative_to(P)):hashlib.sha256(p.read_bytes()).hexdigest()for p in(D/'selection.json',D/'setup.py',Path(__file__),P/'newg/replay.py')})
+(D/'boundary.json').write_text(json.dumps(result,indent=2)+'\n');print('PASS22 source/auxiliary paths, reflected ledgers,44 primal/dual prime witnesses',compact(delta),flush=True)
diff --git a/research/coordinated-crossover-pr200/extra/selection.json b/research/coordinated-crossover-pr200/extra/selection.json
new file mode 100644
index 0000000000..de96cef13a
--- /dev/null
+++ b/research/coordinated-crossover-pr200/extra/selection.json
@@ -0,0 +1,30433 @@
+[
+  {
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+    "source": 170,
+    "partner": 169,
+    "targets": [
+      232,
+      233,
+      234,
+      235,
+      280,
+      281,
+      282,
+      283
+    ],
+    "first_operation": 17514,
+    "first_dimension": 5,
+    "before_target_event": 0,
+    "next_event": "newg",
+    "gauge_rank": 2,
+    "gauge_basis": [
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+    ],
+    "mix_frame": 125969,
+    "score": 0.047997717412499696,
+    "local_delta": {
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+      "22": -1,
+      "3": 1,
+      "2": 8,
+      "19": 8,
+      "21": -8
+    },
+    "aggregate_score": -0.03476921211436945,
+    "aggregate_local_delta": {
+      "5": -1,
+      "22": -1,
+      "3": 1,
+      "2": 2,
+      "19": 2,
+      "21": -2
+    },
+    "new_target_groups": [
+      {
+        "targets": [
+          232,
+          233,
+          234,
+          235
+        ],
+        "pivot": 232,
+        "undo_rank": 21,
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+          [
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+          ],
+          [
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+            1,
+            1,
+            1
+          ]
+        ]
+      }
+    ],
+    "early_role": 14432,
+    "late_role": 14092,
+    "split_gauge_basis": [
+      [
+        0,
+        0,
+        1,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
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+      ],
+      [
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+        -1,
+        1,
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+        0,
+        0,
+        0
+      ],
+      [
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+        0,
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+        0,
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+        0,
+        0,
+        0,
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+        -1,
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+        0,
+        0,
+        0,
+        0
+      ],
+      [
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+        0,
+        0,
+        0
+      ]
+    ]
+  },
+  {
+    "role": 14073,
+    "source": 90,
+    "partner": 89,
+    "targets": [
+      48,
+      49,
+      50,
+      51,
+      56,
+      57,
+      58,
+      59
+    ],
+    "first_operation": 24428,
+    "first_dimension": 14,
+    "before_target_event": 4,
+    "next_event": "newg",
+    "gauge_rank": 4,
+    "gauge_basis": [
+      [
+        1,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        0,
+        1,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ],
+      [
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+        0,
+        0,
+        1,
+        -1,
+        0,
+        0,
+        -1,
+        1,
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+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
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+        0,
+        0,
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+      ],
+      [
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+        0,
+        1,
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+      [
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+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ]
+    ],
+    "mix_frame": 126149,
+    "score": 0.09272955383185888,
+    "local_delta": {
+      "14": -1,
+      "22": -1,
+      "2": 5,
+      "10": 1,
+      "17": 8,
+      "19": -4,
+      "4": 4,
+      "21": -4
+    },
+    "split_gauge_basis": [
+      [
+        1,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        0,
+        1,
+        0,
+        0,
+        0,
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+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ],
+      [
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+        0,
+        1,
+        -1,
+        0,
+        0,
+        -1,
+        1,
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+        0,
+        0,
+        0,
+        0,
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+        0,
+        0,
+        0,
+        0
+      ],
+      [
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+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        -1,
+        0,
+        0,
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+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ],
+      [
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+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        -1,
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+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ]
+    ],
+    "early_role": 14075,
+    "late_role": 14073,
+    "new_target_groups": [
+      {
+        "targets": [
+          56,
+          57,
+          58,
+          59
+        ],
+        "pivot": 56
+      },
+      {
+        "targets": [
+          48,
+          49,
+          50,
+          51
+        ],
+        "pivot": 48
+      }
+    ],
+    "derive_closing_frames": true
+  }
+]
diff --git a/research/coordinated-crossover-pr200/extra/setup.py b/research/coordinated-crossover-pr200/extra/setup.py
new file mode 100644
index 0000000000..2edf47302d
--- /dev/null
+++ b/research/coordinated-crossover-pr200/extra/setup.py
@@ -0,0 +1,28 @@
+"""Source493 role setup; actual word executed by targetagg/replay.py.
+Prepared with substantial OpenAI Codex assistance; Apache-2.0.
+The22 extra source mixes retime existing K mixes and unmixes.
+"""
+from pathlib import Path
+from collections import defaultdict
+import importlib.util,json,sys
+sys.dont_write_bytecode=True
+D=Path(__file__).resolve().parent;P=D.parent/'newg'
+sp=importlib.util.spec_from_file_location('source471base',P/'replay.py');m=importlib.util.module_from_spec(sp);sp.loader.exec_module(m)
+extra=json.loads((D/'selection.json').read_text());oldborrow=m.borrow.copy();used={s for r in m.selection+m.gauge_selection for s in[r['source'],r['partner']]};assert len(oldborrow)==471
+for r in extra:
+ role,n=r['role'],r['source'];e=m.kpair[n];assert e['carrier']==r['partner']and e['mix_frame']==r['mix_frame'];assert not({n,r['partner']}&used);used.update([n,r['partner']])
+ assert role not in m.borrow and role not in m.W.gauge and role not in m.W.donor and role not in m.removed and role not in m.W.source.values()
+ assert set(m.adj[role])==set(r['targets'])and set(m.adj[role].values())=={1}
+ assert role not in {s for i in m.W.phase1 for s in m.W.ops[i][:2]};f=m.W.register(r['split_gauge_basis'])if'split_gauge_basis'in r else e['mix_frame'];assert m.C.dimf[f]in(2,4) and m.C.nondeg(f)and m.C.sub(f,m.W.opframe[m.W.role_ops[role][0]])
+ assert all(all(m.W.module.dot(m.C.cov[t],b)==0 for b in m.C.B[f])for t in r['targets'])
+ z=dict(role=role,frame=f,dim=m.C.dimf[f],targets=r['targets']);m.W.gauge[role]=z;m.W.w['gauges'].append(z);m.W.readtime[role]=0;m.W.w['reads'][str(role)]=len(m.W.phase1)
+ m.borrow[role]=n;m.by_source[n]=r;m.gauge_borrow[role]=n;m.gauge_selection.append(r)
+m.W.order=[r['role']for r in extra]+m.W.order
+# The third ordered rank4 source entrance follows all retained new-source gauges.
+m.W.readtime[14073]=168;m.W.w['reads']['14073']=len(m.W.phase1)+168
+m.W.order.remove(14073);at_index=max(m.W.order.index(r['role'])for r in m.new_gauge_selection)+1;m.W.order.insert(at_index,14073)
+m.at=defaultdict(list)
+for role in m.W.order:m.at[m.W.readtime[role]].append(role)
+m.regs=sorted(set(m.W.phys.values())-set(m.borrow)-m.removed);m.idx={r:2*m.W.v+j for j,r in enumerate(m.regs)};assert len(m.borrow)==493 and len(m.regs)==16621
+assert all(0<=m.W.readtime[s]<=m.W.first[s]for s in m.W.gauge);m.W.exact_frames()
+m.extra_roles={r['role']for r in extra};m.extra_selection=extra
diff --git a/research/coordinated-crossover-pr200/frames/opframe-bases.json b/research/coordinated-crossover-pr200/frames/opframe-bases.json
new file mode 100644
index 0000000000..1c2c64e3ef
--- /dev/null
+++ b/research/coordinated-crossover-pr200/frames/opframe-bases.json
@@ -0,0 +1 @@
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diff --git a/research/coordinated-crossover-pr200/frames/verify-bit-frames.py b/research/coordinated-crossover-pr200/frames/verify-bit-frames.py
new file mode 100644
index 0000000000..7093a3b805
--- /dev/null
+++ b/research/coordinated-crossover-pr200/frames/verify-bit-frames.py
@@ -0,0 +1,37 @@
+import sys,json,gzip,hashlib,argparse,types,signal,contextlib,time
+from pathlib import Path
+from fractions import Fraction
+sys.set_int_max_str_digits(0);sys.dont_write_bytecode=True
+ap=argparse.ArgumentParser();ap.add_argument('--package',required=True);ap.add_argument('--frames',required=True);ap.add_argument('--out',required=True);ap.add_argument('--full',action='store_true');a=ap.parse_args();P=Path(a.package).resolve();O=Path(a.out);O.mkdir(parents=True,exist_ok=True)
+resource=types.ModuleType('resource');resource.RLIMIT_CPU=0;resource.RUSAGE_SELF=0;resource.setrlimit=lambda *xs:None;resource.getrusage=lambda *xs:types.SimpleNamespace(ru_maxrss=0);sys.modules.setdefault('resource',resource)
+if not hasattr(signal,'alarm'):signal.alarm=lambda *xs:None
+sys.path.insert(0,str(P/'bit'));sys.path.insert(0,str(P/'arithmetic'))
+from word import Candidate
+from terminal import prove
+from prime_witnesses import certificate as primes
+from prove import certify
+W=Candidate();C=W.C;B=json.loads(Path(a.frames).read_text());pins={str(p):hashlib.sha256(p.read_bytes()).hexdigest() for p in W.input_paths+[Path(a.frames),P/'selected/bit/sinks.json',P/'bit/word.py',P/'bit/base_word.py',P/'bit/terminal.py',P/'bit/prime_witnesses.py',Path(W.module.__file__)]}
+public=W.opframe[:]
+for i,rows in B:W.opframe[i]=W.register(rows)
+W.changed_frames=[i for i,(x,y) in enumerate(zip(W.original_opframe,W.opframe)) if x!=y];W.endframe={s:W.opframe[xs[-1]] for s,xs in W.role_ops.items()}
+W.exact_frames();baseline=W.row();selection=json.loads((P/'selected/bit/sinks.json').read_text())
+print('PASS full exact rational value, geometry and alias checks; changes',len(B),flush=True)
+if a.full:
+ with contextlib.redirect_stdout(sys.stderr):record=prove(W,selection)
+ record.pop('seconds');record.pop('maxrss');row=record['profile'];prime=primes(W)
+ (O/'prime-witnesses.json.gz').write_bytes(gzip.compress((json.dumps(prime,sort_keys=True,separators=(',',':'))+'\n').encode(),mtime=0));prime={k:v for k,v in prime.items() if k!='frame_witnesses'}
+ row['child_histogram']={int(k):v for k,v in row['child_histogram'].items()}
+ co=certify(row)
+else:
+ # Exact integer terminal ledger follows the original cancellation once complete
+ # target trajectories and exact all-column replay are admitted by the full mode.
+ row=baseline;record=None;prime=None;co=None
+
+def js(x):
+ if isinstance(x,Fraction):return str(x)
+ if isinstance(x,dict):return {str(k):js(v) for k,v in x.items()}
+ if isinstance(x,(tuple,list)):return [js(v) for v in x]
+ return x
+assert pins=={p:hashlib.sha256(Path(p).read_bytes()).hexdigest() for p in pins}
+R=js(dict(status='FULL_FINITE_REPLAY_PASS' if a.full else 'EXACT_FRAME_ONLY_NOT_FULL_REPLAY',input_pins=pins,changed_public_ops=len(B),baseline_profile=baseline,profile=row,terminal=record,coarse=co,prime_witnesses=prime,word_scalar_ops_unchanged=True,gauge_endpoint_and_aliases_unchanged=True))
+(O/'verification.json').write_text(json.dumps(R,indent=1)+'\n');print('PASS',R['status'],R['coarse']['coarse_saving'] if co else '',flush=True)
\ No newline at end of file
diff --git a/research/coordinated-crossover-pr200/gaugeb/PROVENANCE.json b/research/coordinated-crossover-pr200/gaugeb/PROVENANCE.json
new file mode 100644
index 0000000000..32623df31a
--- /dev/null
+++ b/research/coordinated-crossover-pr200/gaugeb/PROVENANCE.json
@@ -0,0 +1,8 @@
+{
+  "selection_sha256": "b0f9f0e67f94a3c21069f393ea27f9483d315bff2f53f77270163065a6345d1c",
+  "source440_manifest": "17fb2b198f633a8ca6b61004970c979bf9ec1861c9d02d800c6be1632c7a3c22",
+  "source440_cut117_manifest": "2e17de69bea7d0b2a0f95ffabcb5bb3a271f6c4ad18764f67c5bf6e6e453b681",
+  "scalar_prototype_sha256": "a85e28ccce6e22ed912e4cefc85a4babe1de9243bb3a0f2710304d3a88a2c2e7",
+  "geometry_screen_sha256": "bf0a8efd28e86551ce7724c04b745b846d02e0e7bcab8bf21c7d9590fceb1744",
+  "scope": "Original scalar prototype adapted to relative paths and mandatory complete integer admission; complete physical ledgers and literal banks reconstructed separately. Prepared with OpenAI Codex assistance."
+}
diff --git a/research/coordinated-crossover-pr200/gaugeb/boundary.py b/research/coordinated-crossover-pr200/gaugeb/boundary.py
new file mode 100644
index 0000000000..033d983fa4
--- /dev/null
+++ b/research/coordinated-crossover-pr200/gaugeb/boundary.py
@@ -0,0 +1,42 @@
+"""Exact original-source paths through genuine gauge donor/recipient chains.
+Prepared with OpenAI Codex assistance; Apache-2.0.
+"""
+from pathlib import Path
+from collections import Counter,defaultdict
+import json,sys,importlib.util
+sys.dont_write_bytecode=True
+HERE=Path(__file__).resolve().parent;P=HERE.parent
+spec=importlib.util.spec_from_file_location('gauge_source_boundary_word',P/'joint/joint_word.py');m=importlib.util.module_from_spec(spec);spec.loader.exec_module(m)
+W=m.Candidate();C=W.C;C.decoder();C.geometry()
+selection=json.loads((HERE/'selection.json').read_text());old=json.loads((P/'borrow/selection.json').read_text());oldroles={r['role']for r in old};oldpair={r['source']for r in old}|{r['partner']for r in old}
+recipient={d:b for b,d in W.pairs};kpair={e['passive']:e for e in W.k['entries']};roots=defaultdict(list)
+for j,s in enumerate(W.w['rootroles']):roots[s].append(W.w['root_frame'][j])
+sources=set(W.source.values());touched={s for i in W.phase1 for s in W.ops[i][:2]};full=W.w['full_frame']
+def histogram(chain):
+ assert all(C.sub(a,b)for a,b in zip(chain,chain[1:]));assert all(C.nondeg(f)for f in set(chain))
+ h=Counter(C.dimf[b]-C.dimf[a]for a,b in zip(chain,chain[1:])if C.dimf[b]>C.dimf[a])
+ dims=[C.dimf[f]for f in chain];refl=[W.h-d for d in reversed(dims)]
+ assert h==Counter(b-a for a,b in zip(refl,refl[1:])if b>a)
+ return h
+results=[];delta=Counter();rawdelta=Counter();used=set(oldpair)
+for r in selection:
+ s,n,c=r['role'],r['source'],r['partner'];assert s in W.gauge and s not in W.donor and s not in oldroles and s not in sources and s not in touched
+ assert n not in used and c not in used;used.update([n,c]);e=kpair[n];assert e['carrier']==c
+ z=W.gauge[s];g=z['frame'];p=e['mix_frame'];assert W.readtime[s]==0 and C.sub(p,g)
+ assert C.sub(W.w['source_frame'][c],p)and C.sub(p,e['deliver_frame'])
+ chain=[g]+[W.opframe[i]for i in W.role_ops[s]]+roots[s]
+ if s in recipient:
+  b=recipient[s];assert W.last[s]<W.readtime[b]
+  chain += [W.gauge[b]['frame']]+[W.opframe[i]for i in W.role_ops[b]]+roots[b]
+ chain += [full]
+ oldaux=histogram(chain);oldpassive=histogram(e['passive_chain']);newchain=[W.w['source_frame'][n],p]+chain;newhist=histogram(newchain)
+ got=Counter(newhist);got.subtract(oldaux);got.subtract(oldpassive);got={k:v for k,v in got.items()if v}
+ assert got=={z['dim']-2:1,22:-1}
+ delta.update(got)
+ for k,v in got.items():rawdelta[k]+=3*v
+ rawdelta[3*z['dim']]-=1
+ results.append(dict(role=s,source=n,partner=c,gauge_dimension=z['dim'],aliased_recipient=recipient.get(s),old_aux_histogram=dict(oldaux),old_source_histogram=dict(oldpassive),new_source_histogram=dict(newhist),new_source_frame_dimensions=[C.dimf[f]for f in newchain],reflected_histogram_equal=True,local_delta=got))
+assert len(results)==3 and Counter(r['gauge_dimension']for r in results)=={19:3}
+assert dict(delta)=={17:3,22:-3};assert dict(rawdelta)=={17:9,22:-9,57:-3};assert sum(k*v for k,v in rawdelta.items())==-216
+out=dict(status='PASS_EXACT_GAUGE_SOURCE_AND_REFLECTED_BOUNDARIES',results=results,local_delta=dict(delta),raw_child_delta=dict(rawdelta),physical_stock_delta=-3,rank_mass_delta=-216,deficit_delta=0,all_original_source_start_frames_preserved=True,all_source_end_frames_full=True,all_carrier_target_anchors_preserved=True,all_borrowed_gauge_reads_after_original_source_injections_and_partner_mix=True)
+(HERE/'boundary.json').write_text(json.dumps(out,indent=2)+'\n');print('PASS3 actual gauge/source/reflected paths',dict(rawdelta),flush=True)
diff --git a/research/coordinated-crossover-pr200/gaugeb/derive_profile.py b/research/coordinated-crossover-pr200/gaugeb/derive_profile.py
new file mode 100644
index 0000000000..d213c4813e
--- /dev/null
+++ b/research/coordinated-crossover-pr200/gaugeb/derive_profile.py
@@ -0,0 +1,32 @@
+"""Complete paid internal/source/target ledger reconstruction after three gauges become sources."""
+from pathlib import Path
+from collections import Counter
+import importlib.util,json,sys
+sys.dont_write_bytecode=True
+HERE=Path(__file__).resolve().parent;P=HERE.parent
+spec=importlib.util.spec_from_file_location('gauge_source_profile_word',P/'joint/joint_word.py');m=importlib.util.module_from_spec(spec);spec.loader.exec_module(m)
+W=m.Candidate();base=json.loads((P/'borrow/profile.json').read_text());audit=json.loads((HERE/'boundary.json').read_text());scalar=json.loads((HERE/'replay.json').read_text());r=base['profile'];selection=json.loads((HERE/'selection.json').read_text());selected={e['role']for e in selection}
+assert len(selected)==3;assert len(scalar['integer'])==2 and {s['direction']for s in scalar['integer']}=={-1,1}
+assert all(s['dirty']==16671 and s['formal_columns']==20191 and s['all_targets']and s['all_source_and_dirty_restored']for s in [scalar['F2']]+scalar['integer'])
+assert set(scalar['controls'])=={'omit_early_mix','old_adjoint','undo_before_restore','omit_gauge_mix','omit_gauge_compensation'}
+assert all(s['status']=='REJECTED'for s in scalar['controls'].values())
+local=Counter({int(k):n for k,n in r['physical_internal_histogram'].items()});source=Counter({int(k):n for k,n in r['physical_source_histogram'].items()});target=Counter({int(k):n for k,n in r['physical_target_histogram'].items()})
+for e in audit['results']:
+ local.subtract({int(k):n for k,n in e['old_aux_histogram'].items()});source.subtract({int(k):n for k,n in e['old_source_histogram'].items()});source.update({int(k):n for k,n in e['new_source_histogram'].items()})
+assert all(n>=0 for h in (local,source,target)for n in h.values())
+child=Counter()
+for h in (local,source,target):
+ for rank,n in h.items():
+  if rank and n:child[rank]+=3*n
+for s,z in W.gauge.items():
+ if s not in W.donor and s not in selected:child[3*z['dim']]+=1
+child[2]+=2*W.v
+mass=sum(rank*n for rank,n in child.items());stock=r['W_per_vertex']-3;physical=r['R']-3;deficit=72*stock-mass
+assert stock==20191 and physical==16671 and deficit==r['deficit_per_vertex']==1936
+assert sum(k*v for k,v in source.items())==W.v*(W.h-1)
+assert sum(k*v for k,v in target.items())==W.v*(W.h-1)
+delta=Counter(child);delta.subtract({int(k):n for k,n in r['child_histogram'].items()});delta={k:v for k,v in delta.items()if v};assert delta=={17:9,22:-9,57:-3}
+def compact(h):return {str(k):n for k,n in sorted(h.items())if n}
+profile=dict(r,R=physical,active_virtual_R=r['active_virtual_R']-3,W_per_vertex=stock,rank_per_vertex=mass,child_histogram=compact(child),physical_internal_histogram=compact(local),physical_source_histogram=compact(source),physical_target_histogram=compact(target),physical_original_sources_borrowed=443)
+out=dict(status='PASS_COMPLETE_443_SOURCE_PROFILE',profile=profile,actual_child_delta=delta,all_gauge_adjoint_coefficients_retained=True,all_target_anchor_frames_retained=True,source_boundary_audit='boundary.json',scalar_replay='replay.json')
+(HERE/'profile.json').write_text(json.dumps(out,indent=2)+'\n');print('PASS443 complete ledgers',dict(W=stock,R=physical,rank=mass,deficit=deficit,delta=delta),flush=True)
diff --git a/research/coordinated-crossover-pr200/gaugeb/replay.py b/research/coordinated-crossover-pr200/gaugeb/replay.py
new file mode 100644
index 0000000000..a7ab2077fa
--- /dev/null
+++ b/research/coordinated-crossover-pr200/gaugeb/replay.py
@@ -0,0 +1,141 @@
+"""Scalar prototype: original sources replace genuine gauged dirty slots.
+Prepared with OpenAI Codex assistance; Apache-2.0.
+"""
+from pathlib import Path
+from collections import Counter,defaultdict
+import json,sys,importlib.util,time,gc,hashlib
+sys.dont_write_bytecode=True
+HERE=Path(__file__).resolve().parent;D=HERE.parent/'joint';spec=importlib.util.spec_from_file_location('source82',D/'joint_word.py');m=importlib.util.module_from_spec(spec);spec.loader.exec_module(m)
+W=m.Candidate();C=W.C;C.decoder();C.geometry();selection=json.loads((HERE.parent/'borrow/selection.json').read_text());gauge_selection=json.loads((HERE/'selection.json').read_text());gauge_selection=gauge_selection['selection']if isinstance(gauge_selection,dict)else gauge_selection
+borrow={r['role']:r['source']for r in selection};by_source={r['source']:r for r in selection};omitted={r['omit_copy']for r in selection};early={r['early']:r for r in selection};assert len(early)==len(borrow)==len(selection)==440
+gauge_borrow={r['role']:r['source']for r in gauge_selection};kpair={e['passive']:e for e in W.k['entries']};used_sources=set(by_source)|{r['partner']for r in selection}
+assert len(gauge_borrow)==len(gauge_selection) and len(set(gauge_borrow.values()))==len(gauge_selection)
+for r in gauge_selection:
+ assert r['source']in kpair and r['partner']==kpair[r['source']]['carrier']
+ assert r['role']in W.gauge and r['role']not in W.donor and r['role']not in borrow
+ assert r['source']not in used_sources and r['partner']not in used_sources
+ used_sources.update([r['source'],r['partner']])
+borrow.update(gauge_borrow);by_source.update({r['source']:r for r in gauge_selection})
+input_paths=[*W.input_paths,W.framepath,D/'joint_word.py',D/'joint-selection.json',D/'pairs.json',Path(W.module.__file__),D/'joint-replay.json',(HERE/'selection.json'),Path(__file__),(HERE.parent/'borrow/selection.json')]
+input_pins={str(p):hashlib.sha256(p.read_bytes()).hexdigest()for p in input_paths}
+adj=[{}for _ in range(W.R)]
+for r,s in zip(W.g['roots'],W.w['rootroles']):
+ for t in r['targets']:adj[s][t]=adj[s].get(t,0)+1
+for i in reversed(W.phase1+W.rest):
+ if i in omitted:continue
+ a,b,_=W.ops[i]
+ for t,c in adj[a].items():adj[b][t]=adj[b].get(t,0)+c
+oldadj=W.adjoint()
+assert all(all(0<c<=oldadj[s].get(t,0)for t,c in row.items())for s,row in enumerate(adj))
+assert all(adj[s]==oldadj[s]for s in W.gauge), 'all gauge reads and paid target supports retained exactly'
+assert len(by_source)==440+len(gauge_selection) and all(r['early'] in r['windows']for r in selection)
+# The formal column layout excludes each actual borrowed dirty slot, retaining
+# distinct original X columns and every other arbitrary dirty input.
+terminal=json.loads((D/'joint-replay.json').read_text())['selected'];removed={e['role']for e in terminal};deletedroots={e['root']for e in terminal};bywrite={i:e for e in terminal for i in e['writes']};afterwrite={e['writes'][-1]:e for e in terminal};assert not(set(early)&set(bywrite)) and not(set(borrow)&removed)
+regs=sorted(set(W.phys.values())-set(borrow)-removed);idx={r:2*W.v+j for j,r in enumerate(regs)}
+assert not set(borrow)&set(W.donor)
+at=defaultdict(list)
+for b in W.order:at[W.readtime[b]].append(b)
+deliveries=defaultdict(list)
+for e in W.k['entries']:deliveries[e['deliver_after_root']].append(e)
+
+def replay(mode='F2',direction=1,bits=24,mutation=None):
+ major=mode=='bound';binary=mode=='F2';v=W.v
+ unit=(lambda i:1)if major else(lambda i:1<<i)if binary else(lambda i:1<<(bits*i))
+ def add(x,c,y):return x+abs(c)*y if major else x^y if binary and c&1 else x if binary else x+c*y
+ X0=[unit(i)for i in range(v)];Y0=[unit(v+i)for i in range(v)];Z0={r:unit(idx[r])for r in regs};x=X0[:];y=Y0[:];z=Z0.copy();top=1;done=[]
+ def track(value):
+  nonlocal top
+  if major:top=max(top,value)
+  return value
+ def value(s):
+  s=W.phys[s];return x[borrow[s]]if s in borrow else z[s]
+ def assign(s,val):
+  s=W.phys[s]
+  if s in borrow:x[borrow[s]]=track(val)
+  else:z[s]=track(val)
+ responses=W.adjoint() if mutation=='old_adjoint' else adj
+ def read(s):
+  if mutation=='omit_gauge_compensation' and s==gauge_selection[0]['role']:return
+  for t,c in responses[s].items():y[t]=track(add(y[t],-direction*c,value(s)))
+ def gate(i,sign):
+  a,b,_=W.ops[i];assign(a,add(value(a),sign,value(b)))
+ for s in range(W.R):
+  if s not in W.gauge and s not in borrow and s not in removed:read(s)
+ for n,s in W.source.items():assign(s,add(value(s),1,x[n]))
+ for r in gauge_selection:
+  if mutation!='omit_gauge_mix' or r is not gauge_selection[0]:x[r['partner']]=track(add(x[r['partner']],1,x[r['source']]))
+ def forward(i):
+  if i in early:
+   r=early[i];a,b=r['partner'],r['source']
+   if mutation!='omit_early_mix' or r is not selection[0]:x[a]=track(add(x[a],1,x[b]))
+  if i not in omitted:gate(i,1);done.append(i)
+ for i in W.phase1:forward(i)
+ for r,s in zip(W.g['roots'],W.w['rootroles']):
+  if r['kind']=='center':
+   for t in r['targets']:y[t]=track(add(y[t],direction,value(s)))
+ for e in terminal:
+  for t in e['targets']:
+   if t!=e['pivot']:y[t]=track(add(y[t],-1,y[e['pivot']]))
+ for j,i in enumerate(W.rest):
+  for s in at[j]:read(s)
+  if i in bywrite:
+   e=bywrite[i];a,b,n=W.ops[i];y[e['pivot']]=track(add(y[e['pivot']],direction,value(b)))
+  else:forward(i)
+  if i in afterwrite:
+   e=afterwrite[i]
+   for t in e['targets']:
+    if t!=e['pivot']:y[t]=track(add(y[t],1,y[e['pivot']]))
+ for s in at[len(W.rest)]:read(s)
+ for j,(r,s)in enumerate(zip(W.g['roots'],W.w['rootroles'])):
+  if r['kind']=='side' and j not in deletedroots:
+   for t in r['targets']:y[t]=track(add(y[t],direction,value(s)))
+  for e in deliveries[j]:
+   a,b=e['carrier'],e['passive']
+   if b not in by_source:x[a]=track(add(x[a],1,x[b]))
+   for t in e['receivers']:y[t]=track(add(y[t],direction,x[a]))
+ for e in W.k['entries']:
+  if e['passive']not in by_source:x[e['carrier']]=track(add(x[e['carrier']],-1,x[e['passive']]))
+ if mutation=='undo_before_restore':
+  for e in W.k['entries']:
+   if e['passive']in by_source:x[e['carrier']]=track(add(x[e['carrier']],-1,x[e['passive']]))
+ for i in reversed(done):gate(i,-1)
+ if mutation!='undo_before_restore':
+  for e in W.k['entries']:
+   if e['passive']in by_source:x[e['carrier']]=track(add(x[e['carrier']],-1,x[e['passive']]))
+ for n,s in W.source.items():assign(s,add(value(s),-1,x[n]))
+ if binary:want=[Y0[t]^X0[t]for t in range(v)]
+ else:
+  want=Y0[:];cache={}
+  for root in W.g['roots']:
+   n=root['node']
+   if n not in cache:
+    mask=C.sup[n];val=0
+    while mask:low=mask&-mask;mask-=low;val=add(val,1,X0[low.bit_length()-1])
+    cache[n]=val
+   for t in root['targets']:want[t]=add(want[t],direction,cache[n])
+  for e in W.k['entries']:
+   val=add(X0[e['carrier']],1,X0[e['passive']])
+   for t in e['receivers']:want[t]=add(want[t],direction,val)
+ if major:return max(top,max(a+b for a,b in zip(y,want)),max(z[r]+Z0[r]for r in regs),max(a+b for a,b in zip(x,X0)))
+ assert y==want,('wrong targets',mode,[t for t in range(v)if y[t]!=want[t]][:10])
+ assert x==X0,('source restoration',mode,[s for s in range(v)if x[s]!=X0[s]][:10])
+ assert z==Z0,('dirty restoration',mode)
+ return dict(mode=mode,direction=direction,formal_columns=2*v+len(regs),sources=v,targets=v,dirty=len(regs),all_targets=True,all_source_and_dirty_restored=True)
+if __name__=='__main__':
+ start=time.monotonic();out=dict(status='PASS_COMPLETE_GAUGE_SOURCE_CORRELATION',selection=selection,gauge_selection=gauge_selection,geometry_and_paid_admission_owned_by_separate_reconstructor=True)
+ out['F2']=replay();print('PASS F2',out['F2'],'seconds',time.monotonic()-start,flush=True)
+ bound=replay('bound');bits=8*((bound.bit_length()+2+7)//8);out['bound']=bound;out['bits']=bits;out['terminal_sinks']=len(terminal);print('bound',bound,'bits',bits,flush=True)
+ out['integer']=[]
+ if True:
+  for direction in(1,-1):out['integer'].append(replay('Z',direction,bits));gc.collect();print('PASS integer',direction,'seconds',time.monotonic()-start,flush=True)
+ out['controls']={}
+ for mutation in ['omit_early_mix','old_adjoint','undo_before_restore','omit_gauge_mix','omit_gauge_compensation']:
+  try:replay(mutation=mutation)
+  except AssertionError as error:out['controls'][mutation]=dict(status='REJECTED',error=str(error))
+  else:raise AssertionError('Invalid mutation accepted: '+mutation)
+ assert (1<<bits)>2*bound
+ assert input_pins=={str(p):hashlib.sha256(p.read_bytes()).hexdigest()for p in input_paths}
+ out['input_pins']=input_pins
+ (HERE/'replay.json').write_text(json.dumps(out,indent=2)+'\n')
+ print('PASS complete gauged source word, both integer signs, all columns and five controls',flush=True)
diff --git a/research/coordinated-crossover-pr200/gaugeb/selection.json b/research/coordinated-crossover-pr200/gaugeb/selection.json
new file mode 100644
index 0000000000..8906d43a1c
--- /dev/null
+++ b/research/coordinated-crossover-pr200/gaugeb/selection.json
@@ -0,0 +1,17 @@
+[
+  {
+    "role": 14697,
+    "source": 106,
+    "partner": 105
+  },
+  {
+    "role": 14696,
+    "source": 138,
+    "partner": 137
+  },
+  {
+    "role": 14699,
+    "source": 346,
+    "partner": 345
+  }
+]
diff --git a/research/coordinated-crossover-pr200/geometry/BANK-PROOF-PR200.md b/research/coordinated-crossover-pr200/geometry/BANK-PROOF-PR200.md
new file mode 100644
index 0000000000..a9516fdb77
--- /dev/null
+++ b/research/coordinated-crossover-pr200/geometry/BANK-PROOF-PR200.md
@@ -0,0 +1,61 @@
+> Historical predecessor proof. The current16-gauge construction, including mixed4/5/6/24 residual banks, is specified and replayed by JOINT-GAUGES.md and joint/.
+
+Historical predecessor bank construction. The current 4/6/24 extension, 36 replicas and gauged-donor proof are in ../JOINT-GAUGES.md; current execution starts at ../verify.py.
+
+# Completed entrance banks on the PR200 physical bit word
+
+2026-10-09. A sufficient finite construction, retaining the upstream all-size interfaces. This is not an unconditional multiplication theorem and is not a new Lean certificate.
+
+## Exact unchanged input
+
+PR200 commit `a1175449f34d39ff933d9d8ab23ced1f32b290ec`, package `research/paired-cube-diagonal-bit-168`. Its graph, scalar word, fixed source-pair chronology, physical frame bases, compensated handoffs, terminal sinks and prime witnesses are unchanged. The source includes 18,908 logical auxiliary roles, 1,760 compensated aliases and 34 terminal sinks, leaving 17,114 physical auxiliary chains. There are 2,200 surviving genuine rank-20 entrance gauges. The 1,760 rank-21 recipient gauges are internal splices and receive no separate bank allocation. Deleted terminals are neither gauges nor alias endpoints. The remaining 14,914 physical chains are retained completely.
+
+The original data channels are 1,760 sources and 1,760 targets. The original complete paid profile is width72, stock20,634, rank mass1,483,712, deficit1,936, with2,200 rank60 exterior children. Source births are not treated as dirty entrance gauges.
+
+## Universal endpoint and explicit allocation
+
+Reuse our PR186 completed-core endpoint and paid group routing. For a coordinate projector P and arbitrary incoming addresses x,y over the eligible local ring, the literal map is
+
+`S_P(x,y) = (x - P*x + P*y, y - P*y + P*x)`.
+
+On an included coordinate this exchanges x and y; on every other coordinate it preserves both. Thus disjoint coordinate blocks partitioning all72 coordinates compose exactly to the full swap, including arbitrary correlated dirty contents and spectator fields. The reverse block schedule is the literal inverse. This is a ring-general identity; it does not replace the odd-prime address space with F2 just because the payload decoder is binary.
+
+The established common weighted chart conjugates every block by the same Q_A, so intermediate Q_A^-1 Q_A cancels. For each role, the inherited fixed normalizer sends its true residual projector to its assigned block. Group-indexed routing assigns a role invocation g to bank gH^-1, so each occurrence visits every bank exactly once. Normalizers inside a bank can be made distinct by the inherited finite scalar choice without changing conjugation. Replicas execute serially and stages own different banks. No uninitialized bank or unfinished core is reused.
+
+Use nine physical replicas of the full unchanged core. In each stage, allocate the9*2,200 selected role-copy occurrences into1,100 banks with18 four-coordinate blocks. Allocate the9*14,914 other occurrences into44,742 banks with three24-coordinate blocks. Both partitions fill every bank exactly; no positive-rank completion child remains. The generator in `pack-pr200.py` maps sorted actual surviving role IDs and replica IDs to explicit bank/block numbers and hashes every assignment. It checks all462,078 role-copy uses across three stages exactly once. Nine replicas are physical, whereas factor three merely clears the rational moment stock.
+
+There are45,842 banks per stage,137,526 across all stages, and31,680 data registers in nine replicas. The literal persistent stock is169,206, exactly9*(56,402/3).
+
+## Fully charged child inventory and exact price
+
+Retain every internal, source, target, center, compensating and terminal-substitution child. Remove exactly the2,200 rank60 exterior children from the original per-vertex histogram. No other child count changes. Then
+
+`W = 2*1760 + 3*(2200/18 +14914/3) =56402/3`,
+
+`rank =1483712 -2200*60 =1351712`,
+
+`72*W -rank =1936`.
+
+The largest retained child is22. On the integer normalization used by the moment verifier, stock is56,402, rank mass4,055,136, deficit5,808, and every child count is multiplied by three. This normalization removes6,600 rank60 exteriors, corresponding to the2,200 per-vertex children above; it must not be confused with the nine-replica literal realization.
+
+The original worst-case bad fraction10^-16 and fallback32*72^2 per actual retained child are charged. Independent outward interval calculations based on4/3 and2 certify
+
+`a0=683528191056257/10^18`.
+
+They reject the adjacent coarse grid point for this fixed profile and worst-case envelope. The established atom wrapper uses a_old=384599/10^10 and the first10^-24 atom above a0/(1+a0-a_old), giving the paid ordinary saving
+
+`A_B=170771888233960718836822306974273386891/(250*10^39)`
+
+`A_B≈0.0006830875529358429`.
+
+Both strict adapter/borrowed-row tolls remain charged. The parent must compose this bit saving with an admitted complex supplier, its full scalar/group/router/layout bills and the unchanged47-constraint balanced body. Component gains are never added. No final kappa is claimed in this lane until that assembly runs.
+
+## Other crossover ideas actually assessed
+
+The current new word still has exactly the4/24 residual families and both tile72 exactly. The rational heterogeneous configuration theorem instantiates the same volume-filling construction; there is no new LP improvement to add. It is ready for future genuinely different residual types. The optional nonorthogonal parity cover cannot be substituted into this odd-prime address model. Even in its proper binary-address model, its overlapping local cores must stay charged. The Parseval phase identity similarly repairs a binary Gaussian endpoint, but no unchanged PR200 local child is a new arbitrary Gram-resolving coding core. These identities are retained as design tools and are not hidden supplier oracles.
+
+## Verification and retained scope
+
+`python -B geometry/pack-pr200.py --full` executes the immutable PR200 bit admission, its exact rational frame/source/gauge/partner chronology, all original and terminal-modified F2 and defining integer columns, adverse controls and all24,401 frame prime witnesses. It then checks literal bank coordinate columns, inverse columns, omitted/duplicate blocks, exact assignments and both rational moment engines. On Windows the owned wrapper supplies only the removed ru_maxrss diagnostic when POSIX resource is unavailable; no mathematical checker changes.
+
+General Clifford/tensor realization, uniform weighted local-ring compilation, common chart selection, restored rows, paid routing/layout, prime supply, precision/recovery and all-size analytic transfer remain the original stated conditions. This construction changes a finite word's dirty-bank allocation under those same conditions. Credit PR186's original bank proof, Chafik Boukhalfa's PR200 physical word, eumemic's PR168 lineage, James Chang's terminal substitution, and every retained upstream attribution.
\ No newline at end of file
diff --git a/research/coordinated-crossover-pr200/geometry/audit-gauge-charts.py b/research/coordinated-crossover-pr200/geometry/audit-gauge-charts.py
new file mode 100644
index 0000000000..d0b050bc99
--- /dev/null
+++ b/research/coordinated-crossover-pr200/geometry/audit-gauge-charts.py
@@ -0,0 +1,51 @@
+"""Reuse PR197's exact gauge chart factors on unchanged PR200 entrances.
+Checks both inverse matrix identities, literal elementary factor replay,
+all factor/unit denominators, and a conservative fully charged routing bound.
+"""
+from pathlib import Path
+from collections import Counter
+from fractions import Fraction as Q
+from math import gcd
+import sys,json,hashlib,importlib.util
+sys.set_int_max_str_digits(0);sys.dont_write_bytecode=True
+HERE=Path(__file__).resolve().parent
+spec=importlib.util.spec_from_file_location('own_bank',HERE/'pack-pr200.py');G=importlib.util.module_from_spec(spec);spec.loader.exec_module(G)
+packing=HERE/'retained-packing.py'
+sp=importlib.util.spec_from_file_location('retained_exact_chart_factorizer',packing);P=importlib.util.module_from_spec(sp);sp.loader.exec_module(P)
+word=G.proof.Candidate();C=word.C
+selected={s:z for s,z in word.gauge.items() if s not in word.donor};inventory=Counter(z['frame'] for z in selected.values())
+G.need(len(selected)==2200 and len(inventory)==220,'actual unchanged220gauge frames')
+records=[];maxops=maxnum=maxden=maxC=maxB=0
+for f,count in sorted(inventory.items()):
+ A=C.A[f];sigma=C.B[f];G.need(len(A)==4 and len(sigma)==20,'actual rank4 residual')
+ residual=[]
+ for a in A:
+  r=[15*x-sum(a) for x in a];d0=gcd(*r);residual.append([x//d0 for x in r])
+ G.need(all(9*sum(x*y for x,y in zip(a,b))-sum(a)*sum(b)==0 for a in residual for b in sigma),'literal Gorthogonal residual basis')
+ B=list(map(list,zip(*(residual+sigma))));inv,d,ops=P.inverse_integer(B)
+ for left,right in [(B,inv),(inv,B)]:
+  G.need(all(sum(x*y for x,y in zip(row,col))==d*int(i==j) for i,row in enumerate(left) for j,col in enumerate(zip(*right))),'two exact inverse identities')
+ replay=[list(map(Q,r)) for r in B]
+ for op,i,j,n,den in ops:
+  if op=='swap':replay[i],replay[j]=replay[j],replay[i]
+  elif op=='scale':replay[i]=[Q(n,den)*x for x in replay[i]]
+  else:
+   G.need(op=='add','retained elementary factor type');replay[i]=[x+Q(n,den)*y for x,y in zip(replay[i],replay[j])]
+ G.need(all(x==int(i==j) for i,row in enumerate(replay) for j,x in enumerate(row)),'independent factor replay')
+ maxops=max(maxops,len(ops));maxnum=max(maxnum,max(abs(o[3]) for o in ops));maxden=max(maxden,d,max(o[4] for o in ops));maxC=max(maxC,max(abs(x) for r in inv for x in r));maxB=max(maxB,max(abs(x) for r in B for x in r))
+ records.append(dict(frame=f,count=count,basis=B,inverse_numerators=inv,inverse_denominator=d,factors=ops))
+G.need(max(maxnum,maxden,maxC,maxB,18)<2**80,'all literal inverse and chart units below inherited prime bound')
+# t in1..18 supplies18 distinct fixed scalar choices per selected bank. At most
+# one scalar per earlier normalizer is forbidden; conjugation stays unchanged.
+# Rational distinctness survives q>2^80 by this bound on cleared differences.
+cleared_difference_bound=2*18*max(maxC,1)*max(maxden,1)
+G.need(cleared_difference_bound<2**80,'distinct normalizers remain distinct modulo every allowed prime')
+factor_bound=maxops+71+72
+physicalstock=169206;physicalchains=17114;full_core_stage_occurrences=27
+calls=2*full_core_stage_occurrences*((physicalstock-1)+physicalchains*72*factor_bound)
+G.need(calls<2**40,'fully literal replicated extra selector bound')
+raw=json.dumps(records,sort_keys=True,separators=(',',':')).encode()
+import gzip
+(HERE/'pr200-gauge-charts.json.gz').write_bytes(gzip.compress(raw,mtime=0))
+out=dict(status='PASS_ALL_ACTUAL_GAUGE_CHARTS_AND_ROUTING_BILL',source_pin='a1175449f34d39ff933d9d8ab23ced1f32b290ec',retained_factorizer_sha256=hashlib.sha256(packing.read_bytes()).hexdigest(),gauges=2200,distinct_gauge_frames=220,max_chart_factors=maxops,max_factor_numerator=maxnum,max_denominator_or_inverse_denominator=maxden,max_inverse_numerator=maxC,max_basis_numerator=maxB,chart_witness_sha256=hashlib.sha256((HERE/'pr200-gauge-charts.json.gz').read_bytes()).hexdigest(),greedy_distinct_scalars=list(range(1,19)),cleared_normalizer_difference_bound=cleared_difference_bound,permutation_transpositions_bound=71,extra_distinct_scalar_coordinate_scales_bound=72,combined_chart_factor_bound=factor_bound,nine_full_replicas=True,total_stage_core_occurrences=27,literal_persistent_stock=physicalstock,conservative_extra_selector_calls=calls,selector_calls_below_2pow40=True,prime_guard='Every nonzero factor/inverse denominator and fixed scalar is smaller than2^80; all actual changes are invertible for every inherited primeq>2^80. Distinct normalizers remain distinct by cleared entry bounds.',routing_payment='These finite extra calls use inherited completed old ordinary selector; they are O(K V w^(1-a_old)), absorbed by strict finite-leaf atom/row toll as in PR197 PROOF5. No extra positive-rank children and no new external row reserve.',scope='Exact finite address chart, inverse, unit and replicated routing bill on same220gauge frames; allsize weighted/restored selector guarantee stays inherited.')
+(HERE/'PR200-GAUGE-CHART-AUDIT.json').write_text(json.dumps(out,indent=2,sort_keys=True));print(json.dumps(out,indent=2))
\ No newline at end of file
diff --git a/research/coordinated-crossover-pr200/geometry/base_two_moment.py b/research/coordinated-crossover-pr200/geometry/base_two_moment.py
new file mode 100644
index 0000000000..14a90fa0e7
--- /dev/null
+++ b/research/coordinated-crossover-pr200/geometry/base_two_moment.py
@@ -0,0 +1,63 @@
+#!/usr/bin/env python3
+"""Maintainer arithmetic cross-check of the pinned PR39 certificate.
+
+Uses no producer/checker imports. Counts are inputs, not proved by this script.
+80-term rational logarithms and 12-term exponentials independently enclose
+the two moments. The accompanying written review addresses their realization.
+"""
+import argparse
+from fractions import Fraction as Q
+from hashlib import sha256
+import json
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parents[1]
+GRID = 10**30
+
+
+def require(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def log_unit(x):
+    require(1 <= x <= 2, "logarithm range")
+    z = (x-1)/(x+1)
+    lo = 2*sum((z**(2*j+1)/(2*j+1) for j in range(80)), Q(0))
+    return lo, lo+2*z**161/(161*(1-z*z))
+
+
+def log_bounds(x):
+    k = 0
+    while x > 2:
+        x /= 2
+        k += 1
+    lo, hi = log_unit(x)
+    l2, u2 = log_unit(Q(2))
+    lo, hi = lo+k*l2, hi+k*u2
+    # Outward rational rounding keeps later certificates compact.
+    return Q((lo*GRID).__floor__(), GRID), Q((hi*GRID).__ceil__(), GRID)
+
+
+def exp_bounds(x):
+    require(0 <= x < 1, "exponential range")
+    term = total = Q(1)
+    for j in range(1, 13):
+        term *= x/j
+        total += term
+    tail = term*x/13/(1-x/14)
+    return total, total+tail
+
+
+def moment(m, W, entries, saving):
+    lo = hi = Q(0)
+    for t, count in entries:
+        t, count = int(t), int(count)
+        require(0 < t < m and count > 0, "improper child")
+        l, u = log_bounds(Q(m, t))
+        lower = exp_bounds(saving*l)[0]
+        upper = exp_bounds(saving*u)[1]
+        lo += Q(t*count, m*W)*lower
+        hi += Q(t*count, m*W)*upper
+    return Q((lo*GRID).__floor__(), GRID), Q((hi*GRID).__ceil__(), GRID)
+
diff --git a/research/coordinated-crossover-pr200/geometry/check-normalizer-routing.py b/research/coordinated-crossover-pr200/geometry/check-normalizer-routing.py
new file mode 100644
index 0000000000..2bbe2d3bcc
--- /dev/null
+++ b/research/coordinated-crossover-pr200/geometry/check-normalizer-routing.py
@@ -0,0 +1,23 @@
+"""Literal fixed scalars distinguish all same-invocation bank normalizers.
+No greedy unknown chart is assumed: H_b=(b+1)*Pi_b*diag(B^-1,I_48).
+All gauges share the same unchanged24coordinate stage; outside-coordinate
+unit columns distinguish H_b for b=0..17 at every primeq>2^80.
+"""
+from pathlib import Path
+import json
+H=Path(__file__).resolve().parent;records=[]
+for stage in range(3):
+ local=set(range(24*stage,24*(stage+1)));j=next(i for i in range(72) if i not in local)
+ for width,blocks in [(4,18),(24,3)]:
+  original=list(range(24*stage,24*stage+width));witness=[]
+  for block in range(blocks):
+   desired=list(range(block*width,(block+1)*width));src=original+[i for i in range(72) if i not in original];dst=desired+[i for i in range(72) if i not in desired]
+   pi=dict(zip(src,dst));assert set(pi)==set(pi.values())==set(range(72))
+   assert [pi[k] for k in original]==desired
+   # Embedded B^-1 fixes e_j exactly, so H_b e_j is this literal unit column.
+   witness.append(dict(block=block,scalar=block+1,unit_column=j,result_row=pi[j],result_coefficient=block+1))
+  for a,left in enumerate(witness):
+   for right in witness[a+1:]:assert (left['result_row'],left['result_coefficient'])!=(right['result_row'],right['result_coefficient'])
+  records.append(dict(stage=stage,width=width,blocks=blocks,outside_stage_column=j,witnesses=witness))
+out=dict(status='PASS_LITERAL_DISTINCT_NORMALIZERS',records=records,prime_guard='All1..18 are different nonzero units at every primeq>2^80, and remain units/distinct in Z/q^w. Each normalizer scales a literal outside-stage unit column differently. Scaling doesnotchange projectorconjugation.',per_role_scalar_coordinate_factors=72,already_charged_in_chart_bound329=True,no_same_invocation_bank_alias=True)
+(H/'LITERAL-NORMALIZER-ROUTING.json').write_text(json.dumps(out,indent=2,sort_keys=True));print(out['status'])
\ No newline at end of file
diff --git a/research/coordinated-crossover-pr200/geometry/pack-admitted-frames.py b/research/coordinated-crossover-pr200/geometry/pack-admitted-frames.py
new file mode 100644
index 0000000000..c4e8ffe01f
--- /dev/null
+++ b/research/coordinated-crossover-pr200/geometry/pack-admitted-frames.py
@@ -0,0 +1,38 @@
+"""Compose admitted frame-only changes with concrete stage-private entrance banks.
+Reconstructs every modified frame/physical paid chain, audits all actual prime
+witnesses, and independently recomputes two exact paid bank moment engines.
+The referenced frame receipt provides the full dirty-column terminal replay.
+"""
+from pathlib import Path
+from fractions import Fraction as Q
+from collections import Counter
+import sys,json,gzip,hashlib,argparse,importlib.util
+sys.set_int_max_str_digits(0);sys.dont_write_bytecode=True
+HERE=Path(__file__).resolve().parent
+spec=importlib.util.spec_from_file_location('geometry_bank_pr200',HERE/'pack-pr200.py');G=importlib.util.module_from_spec(spec);spec.loader.exec_module(G)
+need=G.need
+ap=argparse.ArgumentParser();ap.add_argument('--candidate',required=True);a=ap.parse_args()
+D=Path(a.candidate);record=json.loads((D/'checked/verification.json').read_text());frames=D/'opframe-bases.json';primefile=D/'checked/prime-witnesses.json.gz'
+need(record['status']=='FULL_FINITE_REPLAY_PASS','complete independent frame dirty-word admission')
+for path,want in record['input_pins'].items():need(hashlib.sha256(Path(path).read_bytes()).hexdigest()==want,'actual immutable input pin '+path)
+need(record['word_scalar_ops_unchanged'] and record['gauge_endpoint_and_aliases_unchanged'],'unchanged whole-word endpoint classes')
+word=G.proof.Candidate();before={k:word.w[k] for k in ['gauges','pairs','rootroles','root_frame','source_frame','sources','reads']};changes=json.loads(frames.read_text())
+for index,B in changes:word.opframe[index]=word.register(B)
+word.changed_frames=[i for i,(x,y) in enumerate(zip(word.original_opframe,word.opframe)) if x!=y];word.endframe={s:word.opframe[xs[-1]] for s,xs in word.role_ops.items()}
+word.exact_frames();row=word.row()
+need(G.encode(row)==record['baseline_profile'],'independent complete modified physical baseline ledger')
+need(before=={k:word.w[k] for k in before},'bank source/gauge/endpoints truly unchanged')
+need(Counter({int(k):v for k,v in record['terminal']['child_delta'].items()})==Counter({3:-102,21:-102}),'paid34terminal child substitution unchanged')
+H={int(k):v for k,v in row['child_histogram'].items()};H[3]-=102;H[21]-=102
+need(H=={int(k):v for k,v in record['profile']['child_histogram'].items()},'independent terminal-paid complete histogram')
+prime=G.raw_prime_certificate(word);need(prime==json.loads(gzip.decompress(primefile.read_bytes())),'independent allused modifiedframe prime witness audit')
+packed=G.packed(record['profile'],word.w);coarse=G.proof.certify(packed['scaled_moment_profile'])
+from base_two_moment import moment as alternate
+q=packed['scaled_moment_profile'];hist=list(q['child_histogram'].items());W=q['W_per_vertex'];a0=coarse['coarse_saving'];fallback=32*72**2*sum(n for r,n in hist)
+_,upper=alternate(72,W,hist,a0);_,bad=alternate(72,W,[(1,fallback)],a0);need(upper+Q(1,10**16)*bad<1,'independent paid moment accepted')
+lower,_=alternate(72,W,hist,a0+Q(1,10**18));badlower,_=alternate(72,W,[(1,fallback)],a0+Q(1,10**18));need(lower+Q(1,10**16)*badlower>1,'independent adjacent paid moment rejected')
+allocation=G.literal_allocation(word.w);endpoints=G.endpoint_checks()
+for path,want in record['input_pins'].items():need(hashlib.sha256(Path(path).read_bytes()).hexdigest()==want,'inputs remain identical')
+out=dict(status='PASS_ADMITTED_FRAME_BANK_COMPOSITION',frame_receipt_sha256=hashlib.sha256((D/'checked/verification.json').read_bytes()).hexdigest(),opframe_override_sha256=hashlib.sha256(frames.read_bytes()).hexdigest(),opframe_changes=len(changes),profile=packed,coarse=coarse,literal_allocation=allocation,endpoints=endpoints,prime_witness_sha256=hashlib.sha256(primefile.read_bytes()).hexdigest(),all_used_frame_prime_audit={k:v for k,v in prime.items() if k!='frame_witnesses'},whole_word_receipt=record['status'],unchanged_entrance_alias_and_terminal_classes=True,scope='Sufficient finite dirty-bank composition of independently admitted full frame word. Original allsize supplier interfaces retained; parent owns finalbalanced assembly.')
+name='packed-'+D.name+'.json';(HERE/name).write_text(json.dumps(G.encode(out),indent=2,sort_keys=True))
+print(json.dumps(G.encode(dict(status=out['status'],output=str(HERE/name),changed_ops=len(changes),stock=packed['W_per_vertex'],rank=packed['rank_per_vertex'],coarse=coarse['coarse_saving'],ordinary=coarse['ordinary_saving'],ordinary_float=float(coarse['ordinary_saving']),frames=prime['total_used_frames'])),indent=2))
\ No newline at end of file
diff --git a/research/coordinated-crossover-pr200/geometry/pack-pr200.py b/research/coordinated-crossover-pr200/geometry/pack-pr200.py
new file mode 100644
index 0000000000..1407b17750
--- /dev/null
+++ b/research/coordinated-crossover-pr200/geometry/pack-pr200.py
@@ -0,0 +1,121 @@
+"""Apply established completed entrance banks to the immutable PR200 physical word.
+Light mode is candidate pricing, never complete word admission. --full replays all
+source/target/dirty columns and exact frame/prime/terminal checks through PR200.
+"""
+from pathlib import Path
+from fractions import Fraction as Q
+from collections import Counter
+import json,sys,importlib.util,hashlib,contextlib,types
+# Windows shim supplies only the removed resource diagnostic, never proof data.
+try: import resource
+except ImportError:
+ resource=types.ModuleType("resource");resource.RUSAGE_SELF=0
+ resource.getrusage=lambda _:types.SimpleNamespace(ru_maxrss=0)
+ sys.modules["resource"]=resource
+sys.set_int_max_str_digits(0);sys.dont_write_bytecode=True
+HERE=Path(__file__).resolve().parent
+PKG=HERE.parents[2]/'research/paired-cube-diagonal-bit-168'
+sys.path.insert(0,str(PKG/'bit'));sys.path.insert(0,str(PKG/'arithmetic'))
+spec=importlib.util.spec_from_file_location('pr200_paid_proof',PKG/'bit/prove.py')
+proof=importlib.util.module_from_spec(spec);spec.loader.exec_module(proof)
+# Match Linux provenance spelling only; determinants and factor identities untouched.
+raw_prime_certificate=proof.prime_certificate
+def portable_prime_certificate(word):
+ record=raw_prime_certificate(word)
+ record['input_sha256']={k.replace(chr(92),'/'):v for k,v in record['input_sha256'].items()}
+ return record
+proof.prime_certificate=portable_prime_certificate
+def need(ok,why):
+ if not ok:raise ValueError(why)
+def encode(z):
+ if isinstance(z,Q):return str(z)
+ if isinstance(z,dict):return {str(k):encode(v) for k,v in z.items()}
+ if isinstance(z,(tuple,list)):return [encode(v) for v in z]
+ return z
+
+def endpoint_checks():
+ records=[]
+ for width,copies in [(4,18),(24,3)]:
+  for ring in [0,2,3,5]:
+   def run(column,blocks):
+    x=[int(j==column) for j in range(72)];y=[int(72+j==column) for j in range(72)]
+    for block in blocks:
+     for j in range(block*width,(block+1)*width):x[j],y[j]=y[j],x[j]
+    return x+y
+   for column in range(144):
+    want=[int(j==(column+72)%144) for j in range(144)]
+    need(run(column,range(copies))==want,'all dirty address columns full swap')
+    need(run(column,list(range(copies))+list(reversed(range(copies))))==[int(j==column) for j in range(144)],'all inverse columns')
+   for blocks in [list(range(copies-1)),list(range(copies))+[copies-1]]:
+    need(run(72-width,blocks)!=[int(j==144-width) for j in range(144)],'adverse omitted/duplicate block')
+   records.append(dict(width=width,blocks=copies,address_ring=ring,forward_columns=144,inverse_columns=144,negative_controls=2))
+ return records
+
+def literal_allocation(word_data):
+ pairs=word_data['pairs'];donor={b:a for a,b in pairs}
+ selected=sorted(z['role'] for z in word_data['gauges'] if z['role'] not in donor)
+ sinks=json.loads((PKG/'selected/bit/sinks.json').read_text())
+ removed={word_data['rootroles'][e['root']] for e in sinks}
+ need(len(removed)==34 and not removed.intersection(selected),'terminal roles excluded from selected entrance banks')
+ need(not set(selected).intersection(word_data['sources'].values()),'source births are not entrance gauges')
+ allroles={donor.get(s,s) for s in range(18908)}-removed
+ rest=sorted(allroles-set(selected))
+ need(len(allroles)==17114 and len(selected)==2200 and len(rest)==14914,'every actual retained physical chain exactly once')
+ records=[]
+ for stage in range(3):
+  totalbanks=0;assigned=set();digest=hashlib.sha256()
+  for family,(roles,width,blocks) in enumerate([(selected,4,18),(rest,24,3)]):
+   occurrences=[(replica,role) for replica in range(9) for role in roles]
+   need(len(occurrences)%blocks==0,'exact full banks, no omitted residual child')
+   for offset in range(0,len(occurrences),blocks):
+    group=occurrences[offset:offset+blocks]
+    need(len(group)==blocks and blocks*width==72,'each literal bank fills 72 coordinates')
+    for block,label in enumerate(group):
+     need(label not in assigned,'each role-copy receives one bank slot per stage');assigned.add(label)
+     digest.update(f'{stage}:{family}:{offset//blocks}:{block}:{label[0]}:{label[1]}\n'.encode())
+   totalbanks+=len(occurrences)//blocks
+  need(len(assigned)==9*17114 and totalbanks==45842,'complete stage-private physical stock')
+  records.append(dict(stage=stage,assigned_role_copies=len(assigned),selected_banks=1100,undeferred_banks=44742,total_banks=totalbanks,allocation_sha256=digest.hexdigest()))
+ return dict(stages=records,data_registers=2*9*1760,total_persistent_registers=3*45842+2*9*1760,arbitrary_dirty_contents=True,normalization='Common ancestor weighted chart; literal disjoint coordinate partialSwap blocks; role normalizers chosen distinctly inside each bank by inherited fixed conjugation/routing construction.')
+
+def packed(row,word_data):
+ pairs=word_data['pairs'];recipients={b for a,b in pairs};gauges=word_data['gauges']
+ starts={z['role']:z['dim'] for z in gauges if z['role'] not in recipients}
+ need(Counter(starts.values())=={20:2200},'genuine surviving entrance gauges')
+ need(not set(starts).intersection(word_data['sources'].values()),'source births are not bank entrance gauges')
+ touched={s for i in word_data['phase1'] for s in word_data['ops'][i][:2]}
+ need(not set(starts).intersection(touched),'entrance gauges untouched throughout centre phase')
+ need(Counter(z['dim'] for z in gauges if z['role'] in recipients)=={21:1760},'spliced recipients not bank roles')
+ need(row['R']==17114 and row['terminal_sinks']==34,'retained exact physical terminals')
+ rest=row['R']-len(starts)
+ H={int(k):v for k,v in row['child_histogram'].items()};need(H.pop(60)==2200,'remove exactly actual entrance exteriors')
+ W=Q(2*row['v'])+3*(Q(len(starts),18)+Q(rest,3))
+ mass=sum(k*v for k,v in H.items());need(72*W-mass==1936,'full telescoping deficit')
+ t=9
+ need(t*2200%18==0 and t*rest%3==0,'integral stage-private configuration')
+ scaled=dict(row,W_per_vertex=int(3*W),rank_per_vertex=3*mass,deficit_per_vertex=3*1936,maxchild=max(H),child_histogram={k:3*v for k,v in H.items()})
+ return dict(m=72,W_per_vertex=W,rank_per_vertex=mass,deficit_per_vertex=1936,maxchild=max(H),child_histogram=H,selected_entrances=2200,undeferred_physical_chains=rest,physical_replicas=t,selected_banks_per_stage=t*2200//18,undeferred_banks_per_stage=t*rest//3,scaled_moment_profile=scaled)
+
+def main():
+ import gzip
+ cert=json.loads((PKG/'certificate.json').read_text());row=cert['bit']['profile']
+ data=json.loads(gzip.decompress((PKG/'selected/bit/word_p12.json.gz').read_bytes()))
+ full='--full' in sys.argv
+ if full:
+  import io
+  sink=io.StringIO()
+  with contextlib.redirect_stdout(sink):proof.main()
+  actual=json.loads(sink.getvalue());need(actual==cert['bit'],'complete word admission reproduces frozen bit certificate')
+  row=actual['profile']
+ p=packed(row,data)
+ coarse=proof.certify(p['scaled_moment_profile'])
+ from base_two_moment import moment as alternate_moment
+ q=p['scaled_moment_profile'];H=list(q['child_histogram'].items());a=coarse['coarse_saving'];W=q['W_per_vertex'];edgecount=sum(n for r,n in H)
+ _,upper=alternate_moment(72,W,H,a);_,bad=alternate_moment(72,W,[(1,32*72**2*edgecount)],a)
+ need(upper+Q(1,10**16)*bad<1,'independent base-two paid acceptance')
+ lower,_=alternate_moment(72,W,H,a+Q(1,10**18));badlow,_=alternate_moment(72,W,[(1,32*72**2*edgecount)],a+Q(1,10**18))
+ need(lower+Q(1,10**16)*badlow>1,'independent adjacent paid exclusion')
+ out=dict(status='PASS_FULL_BIT_ADMISSION' if full else 'PRICED_CANDIDATE_NOT_ADMITTED',base_pin='a1175449f34d39ff933d9d8ab23ced1f32b290ec',base_certificate_sha256=hashlib.sha256((PKG/'certificate.json').read_bytes()).hexdigest(),profile=p,coarse=coarse,endpoints=endpoint_checks(),literal_allocation=literal_allocation(data),universal_proof='Established ring-general disjoint partialSwap endpoint, arbitrary dirty contents, nine physical replicas, same weighted chart and charged routing. No source-birth bank gauge or characteristic-two odd-field substitution.',scope='Bit endpoint, complete paid child profile and exact moments only. Full root balanced assembly remains primary-owned.')
+ (HERE/('packed-pr200-full.json' if full else 'packed-pr200-priced.json')).write_text(json.dumps(encode(out),sort_keys=True,indent=2))
+ print(json.dumps(encode(dict(status=out['status'],W=p['W_per_vertex'],rank=p['rank_per_vertex'],coarse=coarse['coarse_saving'],effective=coarse['ordinary_saving'],effective_float=float(coarse['ordinary_saving']),selected=2200,rest=p['undeferred_physical_chains'])),indent=2))
+if __name__=='__main__':main()
\ No newline at end of file
diff --git a/research/coordinated-crossover-pr200/geometry/retained-packing.py b/research/coordinated-crossover-pr200/geometry/retained-packing.py
new file mode 100644
index 0000000000..b7e31dc08a
--- /dev/null
+++ b/research/coordinated-crossover-pr200/geometry/retained-packing.py
@@ -0,0 +1,186 @@
+"""Exact gauge charts and completed-residual bank allocation.
+
+Apache-2.0. Prepared with substantial OpenAI Codex assistance.
+"""
+from array import array
+from collections import Counter
+from fractions import Fraction as Q
+from math import gcd
+from pathlib import Path
+import hashlib,json,sys
+
+def inverse_integer(a):
+    """Fraction-free Gauss-Jordan; return integral C,d with a C = d I."""
+    n=len(a)
+    m=[list(row)+[int(i==j) for j in range(n)] for i,row in enumerate(a)]
+    previous=1
+    factors=[]
+    for k in range(n):
+        pivot=next((i for i in range(k,n) if m[i][k]),None)
+        if pivot is None:
+            raise ValueError('Singular basis')
+        m[k],m[pivot]=m[pivot],m[k]
+        if pivot != k:factors.append(('swap',k,pivot,1,1))
+        p=m[k][k]
+        if p != previous:
+            ratio=Q(previous,p)
+            factors.append(('scale',k,k,ratio.numerator,ratio.denominator))
+        for i in range(n):
+            if i==k:continue
+            u=m[i][k]
+            if u:
+                ratio=Q(-u,previous)
+                factors.append(('add',i,k,ratio.numerator,ratio.denominator))
+            for j in range(2*n):
+                if j==k:continue
+                value=p*m[i][j]-u*m[k][j]
+                assert value % previous==0
+                m[i][j]=value//previous
+            m[i][k]=0
+        previous=p
+    d=m[0][0]
+    assert d and all(m[i][j]==d*int(i==j) for i in range(n) for j in range(n))
+    c=[row[n:] for row in m]
+    common=abs(d)
+    for row in c:
+        for x in row:common=gcd(common,x)
+    d//=common
+    c=[[x//common for x in row] for row in c]
+    if d<0:d=-d;c=[[-x for x in row] for row in c]
+    return c,d,factors
+
+def color_incidence(left,right,nleft,nright,colors):
+    """Incremental two-color swaps; parallel edges are identified separately."""
+    assert len(left)==len(right)
+    full=(1<<colors)-1
+    lf=[full]*nleft;rf=[full]*nright
+    lc=[array('i',[-1])*colors for _ in range(nleft)]
+    rc=[{} for _ in range(nright)]
+    labels=array('i',[-1])*len(left)
+    swaps=longest=0
+    def remove(e):
+        u,v,c=left[e],right[e],labels[e]
+        assert lc[u][c]==e and rc[v][c]==e
+        lc[u][c]=-1;del rc[v][c]
+        lf[u]|=1<<c;rf[v]|=1<<c
+    def add(e,c):
+        u,v=left[e],right[e]
+        assert lf[u]&(1<<c) and rf[v]&(1<<c)
+        labels[e]=c;lc[u][c]=e;rc[v][c]=e
+        lf[u]&=~(1<<c);rf[v]&=~(1<<c)
+    for e,(u,v) in enumerate(zip(left,right)):
+        common=lf[u]&rf[v]
+        if common:
+            add(e,(common&-common).bit_length()-1)
+            continue
+        assert lf[u] and rf[v]
+        a=(lf[u]&-lf[u]).bit_length()-1
+        b=(rf[v]&-rf[v]).bit_length()-1
+        path=[];r=u
+        while True:
+            edge=lc[r][b]
+            if edge<0:break
+            path.append(edge);bank=right[edge]
+            assert bank!=v
+            edge=rc[bank].get(a,-1)
+            if edge<0:break
+            path.append(edge);r=left[edge]
+        changes=[(edge,b if labels[edge]==a else a) for edge in path]
+        for edge,_ in changes:remove(edge)
+        for edge,c in changes:add(edge,c)
+        add(e,b)
+        swaps+=1;longest=max(longest,len(path))
+    return labels,{'swaps':swaps,'longest_swapped_path':longest}
+
+def check_coloring(left,right,labels,nleft,nright,colors,full_left=False):
+    assert len(left)==len(right)==len(labels)
+    lm=[0]*nleft;rm=[0]*nright
+    for u,v,c in zip(left,right,labels):
+        assert 0<=u<nleft and 0<=v<nright and 0<=c<colors
+        bit=1<<c
+        assert not lm[u]&bit and not rm[v]&bit
+        lm[u]|=bit;rm[v]|=bit
+    if full_left:assert all(mask==(1<<colors)-1 for mask in lm)
+
+def build(BIT):
+    sys.path.insert(0,str(BIT/'scripts'))
+    import paired_cube_bit_physical as P
+    chk=P.loaded();chk.frames()
+    wordpath=BIT/'build/source-assisted-bit-bootstrap/emitted-word.json'
+    w=json.loads(wordpath.read_text())
+    receipt=json.loads((BIT/'build/source-assisted-bit-bootstrap/emitted-physical.json').read_text())
+    assert hashlib.sha256(wordpath.read_bytes()).hexdigest()==receipt['word_sha256']
+    R=w['R'];donors={a for a,b in w['pairs']};recipients={b for a,b in w['pairs']};gauges={z['role']:z for z in w['gauges']}
+    assert len(donors)==len(recipients)==len(w['pairs'])==1760
+    assert not donors&recipients and not donors&gauges.keys() and recipients<=gauges.keys()
+    physical=sorted(set(range(R))-recipients);ren={r:i for i,r in enumerate(physical)}
+    assert len(physical)==receipt['physical_R']==17588
+    selected=set(gauges)-recipients
+    assert len(selected)==2200 and {gauges[r]['dim'] for r in selected}=={20}
+    assert {gauges[r]['dim'] for r in recipients}=={21}
+    # Interior rank-21 recipient starts are already paid splices, not entrance residuals.
+    inventory=Counter(gauges[r]['frame'] for r in selected)
+    digest=hashlib.sha256();maxops=maxnum=maxden=0
+    for f,count in sorted(inventory.items()):
+     A=chk.A[f];source=chk.B[f];assert len(A)==4 and len(source)==20
+     residual=[]
+     for a in A:
+      row=[15*x-sum(a) for x in a];d=gcd(*row);residual.append([x//d for x in row])
+     assert all(9*sum(x*y for x,y in zip(a,b))-sum(a)*sum(b)==0 for a in residual for b in source)
+     B=list(map(list,zip(*(residual+source))))
+     C,d,ops=inverse_integer(B)
+     for left,right in ((B,C),(C,B)):
+      assert all(sum(x*y for x,y in zip(row,col))==d*int(i==j) for i,row in enumerate(left) for j,col in enumerate(zip(*right)))
+     replay=[list(map(Q,row)) for row in B]
+     for op,i,j,num,den in ops:
+      if op=='swap':replay[i],replay[j]=replay[j],replay[i]
+      elif op=='scale':replay[i]=[Q(num,den)*x for x in replay[i]]
+      else:
+       assert op=='add';replay[i]=[x+Q(num,den)*y for x,y in zip(replay[i],replay[j])]
+     assert all(x==int(i==j) for i,row in enumerate(replay) for j,x in enumerate(row))
+     maxops=max(maxops,len(ops));maxnum=max(maxnum,max(abs(op[3]) for op in ops));maxden=max(maxden,d,max(op[4] for op in ops))
+     digest.update(json.dumps([f,count,B,C,d,ops],separators=(',',':')).encode())
+    assert max(maxnum,maxden)<2**80
+    families={4:sorted(selected),24:sorted(set(physical)-selected)}
+    degree=9;copies=3;small=degree*len(selected)//6
+    assert 6*small==degree*len(selected)
+    large=(degree*len(families[24])-2*small)//3
+    assert 3*large+2*small==degree*len(families[24])
+    left=[];right=[];offsets=[];counts=Counter();bank=0
+    for pattern,n in (({4:6,24:2},small),({24:3},large)):
+     for _ in range(n):
+      offset=0
+      for rank,mult in sorted(pattern.items()):
+       for _ in range(mult):
+        role=families[rank][counts[rank]//degree]
+        left.append(ren[role]);right.append(bank);offsets.append(offset);counts[rank]+=1;offset+=rank
+      assert offset==72
+      bank+=1
+    assert all(counts[r]==degree*len(rs) for r,rs in families.items())
+    labels,stats=color_incidence(left,right,len(physical),bank,degree)
+    check_coloring(left,right,labels,len(physical),bank,degree,full_left=True)
+    first={};rejected=False
+    for e,r in enumerate(left):
+     if r not in first:first[r]=e;continue
+     old=labels[e];labels[e]=labels[first[r]]
+     try:check_coloring(left,right,labels,len(physical),bank,degree,full_left=True)
+     except AssertionError:rejected=True
+     finally:labels[e]=old
+     break
+    assert rejected
+    incidence=hashlib.sha256()
+    for r,b,c,o in zip(left,right,labels,offsets):incidence.update(f'{r},{b},{c},{o}\n'.encode())
+    H=Counter({int(r):copies*n for r,n in receipt['child_histogram'].items()})
+    assert H.pop(60)==copies*len(selected)
+    assert not any(r>22 for r in H)
+    W=2*w['v']*copies+bank;mass=sum(r*n for r,n in H.items())
+    assert 72*W-mass==copies*receipt['deficit_per_vertex']==5808
+    K=18*((W-1)+len(physical)*72*(maxops+71))
+    record=dict(status='PASS exact actual-chain chart and incidence checks',source_commit='201737a1ec4f936e166e2481fb9e88104cb2ccc7',
+     word_sha256=receipt['word_sha256'],physical_roles=len(physical),interior_splices=len(recipients),gauge_roles=len(selected),
+     distinct_gauges=len(inventory),copies=copies,banks=bank,patterns=[dict(rank4=6,rank24=2,count=small),dict(rank4=0,rank24=3,count=large)],
+     incidences=len(left),W=W,m=72,rank_mass=mass,deficit=5808,maxchild=max(H),child_histogram=dict(sorted(H.items())),
+     chart_sha256=digest.hexdigest(),incidence_sha256=incidence.hexdigest(),max_chart_factors=maxops,max_num=maxnum,max_den=maxden,
+     conservative_selector_calls=K,color_stats=stats,conflicting_assignment_rejected=rejected)
+
+    return record
diff --git a/research/coordinated-crossover-pr200/joint/REMATCH-NOTE.txt b/research/coordinated-crossover-pr200/joint/REMATCH-NOTE.txt
new file mode 100644
index 0000000000..89f04b0591
--- /dev/null
+++ b/research/coordinated-crossover-pr200/joint/REMATCH-NOTE.txt
@@ -0,0 +1,15 @@
+Bit donor rematching experiment
+
+Prepared with OpenAI Codex assistance; Apache-2.0. Uses PR200 scalar word/checkers, PR211 bases, and our frozen plateau plus relaxed frame refinements.
+
+Keep the 238 original source-injection donor pairs verbatim. For each of the other 1,522 rank-21 births, enumerate nongauge, nonroot, nonsource donors whose last use is before the unchanged birth read, and whose exact endpoint frame is contained in the birth gauge. All donors are single-use in the bipartite matching. Sort donors by descending endpoint rank and augment the matching; endpoint rank orders every relevant concave paid-moment saving because all recipient gauges have the same rank 21. This is weighted greedy for a transversal matroid, with weights depending on donors only.
+
+The maximum-rank matching changes 355 pair assignments, but its symmetric difference with the original matching has 124 connected alternating components. Components with identical donor-rank multisets have zero paid effect and were reverted. Only two strictly improving components remain, each a single recipient:
+- birth 18892: old donor 7756 (rank 7), new donor 8292 (rank 8).
+- birth 18894: old donor 7757 (rank 7), new donor 8293 (rank 8).
+
+The full proposed pair list is pairs.json in [birth, donor] orientation. All 1,760 pairs remain; no new source donor is introduced and all original source aliases stay fixed. The exact frame/handoff/row checks on the full maximum-rank matching passed. The two-component reduction has the identical paid delta; the parent verifier must independently replay the actual minimal word.
+
+Paid child delta before bank replica scaling: rank13 +6; rank14 -6; rank16 -6; rank17 +6. Weighted rank delta and stock delta are both zero.
+
+Only literal finite-word/paid-moment/assembly replay can certify kappa; this search itself makes no new kappa claim.
diff --git a/research/coordinated-crossover-pr200/joint/arithmetic.py b/research/coordinated-crossover-pr200/joint/arithmetic.py
new file mode 100644
index 0000000000..bd3941d39e
--- /dev/null
+++ b/research/coordinated-crossover-pr200/joint/arithmetic.py
@@ -0,0 +1,46 @@
+from pathlib import Path
+import json,importlib.util,sys
+from fractions import Fraction as Q
+sys.set_int_max_str_digits(0)
+O=Path(__file__).resolve().parent;R=O.parent;ROOT=R.parents[1]
+def load(n,p):
+ s=importlib.util.spec_from_file_location(n,p);m=importlib.util.module_from_spec(s);s.loader.exec_module(m);return m
+def enc(x):
+ if isinstance(x,Q):return str(x)
+ if isinstance(x,dict):return {str(k):enc(v) for k,v in x.items()}
+ if isinstance(x,(list,tuple)):return [enc(v) for v in x]
+ return x
+im=load('im',ROOT/'research/paired-cube-diagonal-bit-168/arithmetic/interval_moment.py')
+assembly=load('asm',ROOT/'scripts/paired_cube_assembly.py')
+saved=json.loads((O/'joint-paid.json').read_text());p=dict(profile=dict(scaled_moment_profile=saved['profile']),coarse=saved['coarse']);base=json.loads((ROOT/'research/source-assisted-v4/certificate.json').read_text())
+def profile(cp):
+ H={int(t):n for t,n in cp['child_histogram'].items()}
+ return dict(m=cp.get('m',72),W=cp['W_per_vertex'],child_multiplicities=H,N=cp['deficit_per_vertex'],L=0,total_rank=sum(t*n for t,n in H.items()),maxchild=max(H))
+actual=json.loads((R/'complex-profile-replayed.json').read_text());assert all(actual[k]==base['complex_profile'][k] for k in ['m','W_per_vertex','rank_per_vertex','deficit_per_vertex','child_histogram']);cp=profile(actual);bp=profile(p['profile']['scaled_moment_profile'])
+def paid(row,a,bit):
+ x=im.moment(row,a)
+ if bit:
+  l,u=im.log_interval(Q(row['m']));e,f=im.exp_interval(a*l,a*u);w=Q(1,10**16)*Q(32*row['m']*sum(row['child_multiplicities'].values()),row['W']);x.update(lower=x['lower']+w*e,upper=x['upper']+w*f)
+ return x
+def certify(row,bit):
+ den=10**18;lo=0;hi=den//100
+ while hi-lo>1:
+  md=(lo+hi)//2;v=paid(row,Q(md,den),bit)
+  if v['upper']<1:lo=md
+  elif v['lower']>1:hi=md
+  else:raise ValueError('inconclusive')
+ a=Q(lo,den);z=Q(hi,den);assert paid(row,a,bit)['upper']<1<paid(row,z,bit)['lower']
+ return dict(saving=a,accepted=paid(row,a,bit),next_excluded=paid(row,z,bit))
+c=certify(cp,False);b=certify(bp,True);assert b['saving']==Q(p['coarse']['coarse_saving'])
+coarse=b['saving'];chain=[Q(p['coarse']['ordinary_saving'])]
+for i in range(4):
+ a=(1-coarse)*coarse+coarse*chain[-1];assert chain[-1]<a<coarse<1-a;chain.append(a)
+bridge=base['assembly']['finite_bridge'];bridge['rows']['degree_gap']=Q(bridge['rows']['degree_gap'])
+eta=beta=Q(1,10**24);weak=Q(1,10**30);a=min(chain[-1],(1-beta)*c['saving']-weak);q=a*(1-2*eta);ceiling=(1-eta)*q/(1+q);grid=10**18;scaled=ceiling*grid;k=Q((scaled.numerator-1)//scaled.denominator,grid)
+z=assembly.assembly(a,c['saving'],bridge,k,eta=eta,beta=beta);assert len(z['strict_constraints'])==47 and all(v>0 for v in z['strict_constraints'].values())
+try:assembly.assembly(a,c['saving'],bridge,k+Q(1,grid),eta=eta,beta=beta)
+except AssertionError:pass
+else:raise AssertionError('next k accepted')
+out=dict(status='PASS_EXACT_47_CONSTRAINT_ASSEMBLY',scope='Conditional finite supplier certificate; retained all-size interfaces and analytic assumptions',bit_profile=bp,complex_profile=cp,bit=b,complex=c,ordinary_chain=chain,bit_ordinary=chain[-1],a_bit=a,a_complex=c['saving'],eta=eta,beta=beta,kappa=k,kappa_float=float(k),binding='bit',assembly=z,adjacent_kappa_rejected=True)
+(O/'assembly.json').write_text(json.dumps(enc(out),indent=2,sort_keys=True)+'\n')
+print(json.dumps(enc({k:out[k] for k in ['status','kappa','kappa_float','bit_ordinary','a_complex']}),indent=2))
diff --git a/research/coordinated-crossover-pr200/joint/assembly.json b/research/coordinated-crossover-pr200/joint/assembly.json
new file mode 100644
index 0000000000..ec75d0ed09
--- /dev/null
+++ b/research/coordinated-crossover-pr200/joint/assembly.json
@@ -0,0 +1,200 @@
+{
+  "a_bit": "21755280055709184141297133175833154864464800141095339760920300439829072853269246792290854835584685812259264151/31250000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+  "a_complex": "700918443859411/1000000000000000000",
+  "adjacent_kappa_rejected": true,
+  "assembly": {
+    "absorption_gap": "14404304083223483917555372739613008511168702806910361478002009707003601906224959463556118962371200768072323598588374864887381363029693293289/16304356246118722202367725281368755497104325608731401975023258900683122414318347093301193939091469749360670323028827074468523657706711000000000000000000000000",
+    "margins": {
+      "Gaussian_arithmetic": "18148304530310059763344145833762676644718606244961791281717136767387131507380326023009619423937280629979215652184227679888491660414571172925531476342293289/26086969993789955523788360450190008795366920973970243160037214241092995862909355349281910302546351598977072516846123319149637852330737600000000000000000000000",
+      "balanced_prefix": "11342690331443787352100281368755497104325608731401975023258900683122414318347093301193939091469749360670323028827074468523657706711/16304356246118722202367725281368755497104325608731401975023258900683122414318347093301193939091469749360670323028827074468523657706711",
+      "bulk_exposure": "21755280055709184141297133175833154864464800141095339760920300439829072853269246792290854835584685812259264151/31250000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "compact_phase_layer": "11342690331443787352083977012509378382123241006120606267761796357513682916372070042293255969055431013577021834887982998774297036387971172925531476342293289/16304356246118722202367725281368755497104325608731401975023258900683122414318347093301193939091469749360670323028827074468523657706711000000000000000000000000",
+      "coordinate_movement": "21755280055709184141297133175833154864464800141095339760920300439829072853269246792290854835584685812259264151/31250000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "dimension": "16293013555787278415015625000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000/16304356246118722202367725281368755497104325608731401975023258900683122414318347093301193939091469749360670323028827074468523657706711",
+      "scalar_work": "90741522651550298816785946593797858112402502125934431430574101139370583144801723150650829609343676538269283036677504278439900983364971172925531476342293289/130434849968949777618941802250950043976834604869851215800186071205464979314546776746409551512731757994885362584230616595748189261653688000000000000000000000000"
+    },
+    "minimum_margin": "11342690331443787352083977012509378382123241006120606267761796357513682916372070042293255969055431013577021834887982998774297036387971172925531476342293289/16304356246118722202367725281368755497104325608731401975023258900683122414318347093301193939091469749360670323028827074468523657706711000000000000000000000000",
+    "parameters": {
+      "C0": {
+        "base": 2,
+        "exponent": 210000
+      },
+      "C1": 1,
+      "a_bit": "21755280055709184141297133175833154864464800141095339760920300439829072853269246792290854835584685812259264151/31250000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "a_complex": "700918443859411/1000000000000000000",
+      "alpha_squared_power": "22685380662887574704184258381264875486448849737522581291020697040636097234719163343487195060525180374247344863715057467297954743098971172925531476342293289/32608712492237444404735450562737510994208651217462803950046517801366244828636694186602387878182939498721340646057654148937047315413422000000000000000000000000",
+      "beta": "1/1000000000000000000000000",
+      "c": "10877640027854592070652451082636521723048258773414494047305285755114395331294862475844987588719489636882839784645164415314187740735849/15625000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "delta": "1/8000000000000000000000000",
+      "epsilon": "16293013555787278415015625000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000/16304356246118722202367725281368755497104325608731401975023258900683122414318347093301193939091469749360670323028827074468523657706711",
+      "eta": "1/1000000000000000000000000",
+      "kappa": "695684647723759/1000000000000000000",
+      "lambda_": "31228244719944290815858702888579446900844719341156037836072234564024971068242070514128009584993488167456987528139854835584685812259264151/31250000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "lambda_prime": "15614122359972145407929351455167363478276951741226585505952694714244885604668705137524155012411280510363117160215354835584685812259264151/15625000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "q": "10877640027854592070648544832636521723048258773414494047305285755114395331294862475844987588719489636882839784645164415314187740735849/15625000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
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+}
diff --git a/research/coordinated-crossover-pr200/joint/joint-banks.json b/research/coordinated-crossover-pr200/joint/joint-banks.json
new file mode 100644
index 0000000000..b0583c17dc
--- /dev/null
+++ b/research/coordinated-crossover-pr200/joint/joint-banks.json
@@ -0,0 +1,1161 @@
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diff --git a/research/coordinated-crossover-pr200/joint/joint-charts.json b/research/coordinated-crossover-pr200/joint/joint-charts.json
new file mode 100644
index 0000000000..9d0d318fb9
--- /dev/null
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diff --git a/research/coordinated-crossover-pr200/joint/joint-paid.json b/research/coordinated-crossover-pr200/joint/joint-paid.json
new file mode 100644
index 0000000000..12a915bbdc
--- /dev/null
+++ b/research/coordinated-crossover-pr200/joint/joint-paid.json
@@ -0,0 +1,528 @@
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+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        -1
+      ]
+    ],
+    "role_targets": {
+      "14545": [
+        904,
+        906,
+        908,
+        910
+      ],
+      "14551": [
+        905,
+        907,
+        909,
+        911
+      ],
+      "14620": [
+        905,
+        907,
+        908,
+        910
+      ],
+      "14621": [
+        904,
+        906,
+        909,
+        911
+      ],
+      "14800": [
+        904,
+        906,
+        908,
+        910
+      ],
+      "14805": [
+        905,
+        907,
+        909,
+        911
+      ],
+      "14902": [
+        905,
+        906,
+        909,
+        910
+      ],
+      "14903": [
+        904,
+        907,
+        908,
+        911
+      ]
+    }
+  },
+  {
+    "roles": [
+      14557,
+      14563,
+      14622,
+      14623,
+      14810,
+      14815,
+      14904,
+      14905
+    ],
+    "gauge_basis": [
+      [
+        1,
+        1,
+        1,
+        1,
+        1,
+        1,
+        1,
+        1,
+        1,
+        1,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        4,
+        4
+      ],
+      [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        1
+      ],
+      [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        1
+      ],
+      [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        1
+      ],
+      [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        1
+      ],
+      [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        1
+      ],
+      [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        1
+      ],
+      [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        1
+      ],
+      [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        0,
+        0,
+        0,
+        1,
+        1
+      ],
+      [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        0,
+        0,
+        1,
+        1
+      ],
+      [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        0,
+        1,
+        1
+      ],
+      [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        1,
+        1
+      ],
+      [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        1,
+        1
+      ]
+    ],
+    "role_targets": {
+      "14557": [
+        912,
+        914,
+        916,
+        918
+      ],
+      "14563": [
+        913,
+        915,
+        917,
+        919
+      ],
+      "14622": [
+        913,
+        915,
+        916,
+        918
+      ],
+      "14623": [
+        912,
+        914,
+        917,
+        919
+      ],
+      "14810": [
+        912,
+        914,
+        916,
+        918
+      ],
+      "14815": [
+        913,
+        915,
+        917,
+        919
+      ],
+      "14904": [
+        913,
+        914,
+        917,
+        918
+      ],
+      "14905": [
+        912,
+        915,
+        916,
+        919
+      ]
+    }
+  }
+]
diff --git a/research/coordinated-crossover-pr200/joint/joint_word.py b/research/coordinated-crossover-pr200/joint/joint_word.py
new file mode 100644
index 0000000000..52be474a77
--- /dev/null
+++ b/research/coordinated-crossover-pr200/joint/joint_word.py
@@ -0,0 +1,50 @@
+"""Actual joint entrance gauges on the frozen relaxed PR211 frame word.
+Experimental adapter, prepared with OpenAI Codex assistance; Apache-2.0.
+Pinned supplier files are never modified.
+"""
+from pathlib import Path
+from copy import deepcopy
+from collections import defaultdict
+import sys,json,hashlib
+sys.dont_write_bytecode=True
+D=Path(__file__).resolve().parent
+PACKAGE=D.parent;ROOT=PACKAGE.parents[1]
+P=ROOT/'research/paired-cube-diagonal-bit-168'
+sys.path.insert(0,str(P/'bit'))
+from word import Candidate as Original
+class Candidate(Original):
+ def __init__(self):
+  super().__init__()
+  self.original_w=deepcopy(self.w)
+  pairs=json.loads((D/'pairs.json').read_text())
+  assert len(pairs)==1760 and set(b for b,d in pairs)==set(self.donor)
+  assert len(set(d for b,d in pairs))==1760
+  old=dict(self.pairs);changed=[(b,d) for b,d in pairs if old[b]!=d]
+  assert changed==[(18892,8292),(18894,8293)]
+  assert all(d not in self.source.values() and d not in self.gauge and d not in self.rootroles for b,d in changed)
+  assert all(self.last[d]<self.readtime[b] for b,d in pairs)
+  self.pairs=pairs;self.donor=dict(pairs);self.w['pairs']=[[d,b] for b,d in pairs]
+  self.phys={s:self.donor.get(s,s) for s in range(self.R)}
+  assert all(self.phys[a]!=self.phys[b] for a,b,n in self.ops)
+  self.framepath=PACKAGE/'frames/opframe-bases.json'
+  for i,rows in json.loads(self.framepath.read_text()):self.opframe[i]=self.register(rows)
+  self.changed_frames=[i for i,(a,b) in enumerate(zip(self.original_opframe,self.opframe))if a!=b]
+  self.endframe={s:self.opframe[xs[-1]]for s,xs in self.role_ops.items()}
+  groups=json.loads((D/'joint-selection.json').read_text())
+  oldorder=self.order[:];touched={s for i in self.phase1 for s in self.ops[i][:2]};sources=set(self.source.values())
+  co=[set()for _ in range(self.R)]
+  for r,s in zip(self.g['roots'],self.w['rootroles']):co[s].update(r['targets'])
+  for a,b,n in reversed(self.ops):co[b].update(co[a])
+  newroles=[]
+  for group in groups:
+   f=self.register(group['gauge_basis']);assert self.C.dimf[f] in (12,13,18,19) and self.C.nondeg(f)
+   for s in group['roles']:
+    assert s not in self.gauge and s not in sources and s not in touched and s not in self.donor
+    assert self.C.sub(f,self.opframe[self.role_ops[s][0]])
+    targets=sorted(co[s]);assert targets==group['role_targets'][str(s)]
+    z=dict(role=s,frame=f,dim=self.C.dimf[f],targets=targets)
+    self.gauge[s]=z;self.w['gauges'].append(z);self.w['reads'][str(s)]=len(self.phase1);self.readtime[s]=0;newroles.append(s)
+  self.order=sorted(newroles,key=lambda s:self.gauge[s]['dim'])+oldorder
+  assert len(newroles)==82 and len(set(newroles))==82
+  self.newroles=newroles
+  assert self.original_w==json.loads(__import__('gzip').decompress(self.input_paths[2].read_bytes()))
diff --git a/research/coordinated-crossover-pr200/joint/pack_joint.py b/research/coordinated-crossover-pr200/joint/pack_joint.py
new file mode 100644
index 0000000000..94f8d09f08
--- /dev/null
+++ b/research/coordinated-crossover-pr200/joint/pack_joint.py
@@ -0,0 +1,93 @@
+"""Literal4/5/6/11/12/24 banks and charts after493 original-source aliases.
+Source493 integration prepared with OpenAI Codex assistance; Apache-2.0.
+"""
+from pathlib import Path
+from collections import Counter
+from fractions import Fraction as Q
+from math import gcd
+import importlib.util,json,hashlib,sys
+sys.dont_write_bytecode=True
+from joint_word import Candidate,D,P
+PACKAGE=D.parent;ROOT=PACKAGE.parents[1]
+spec=importlib.util.spec_from_file_location('retained_chart',PACKAGE/'geometry/retained-packing.py');chart=importlib.util.module_from_spec(spec);spec.loader.exec_module(chart)
+W=Candidate();C=W.C
+removed={W.w['rootroles'][e['root']]for e in json.loads((P/'selected/bit/sinks.json').read_text())}
+borrowed={r['role']for r in json.loads((PACKAGE/'borrow/selection.json').read_text())}
+assert len(borrowed)==440 and not borrowed&(set(W.gauge)|set(W.source.values())|set(W.donor)|removed)
+gauge_borrowed={r['role']for r in json.loads((PACKAGE/'gaugeb/selection.json').read_text())}
+assert len(gauge_borrowed)==3 and gauge_borrowed<=set(W.gauge) and not gauge_borrowed&(borrowed|set(W.donor)|removed)
+new_gauge_sources={r['role']for r in json.loads((PACKAGE/'newg/selection.json').read_text())}
+assert len(new_gauge_sources)==28 and not new_gauge_sources&(set(W.gauge)|borrowed|gauge_borrowed|set(W.donor)|removed)
+extra_rows=json.loads((PACKAGE/'extra/selection.json').read_text())
+extra_source_roles={r['role']for r in extra_rows}
+assert len(extra_rows)==len(extra_source_roles)==22
+# Every newly reused register had a full-width24 physical bank in source471.
+assert not extra_source_roles&(set(W.gauge)|set(W.source.values())|set(W.donor)|removed|borrowed|gauge_borrowed|new_gauge_sources)
+assert len({r['source']for r in extra_rows})==len({r['partner']for r in extra_rows})==22
+assert len({s for r in extra_rows for s in (r['source'],r['partner'])})==44
+physical=set(range(W.R))-set(W.donor)-removed-borrowed-gauge_borrowed-new_gauge_sources-extra_source_roles
+selected={s for s in W.gauge if s in physical}
+assert Counter(W.gauge[s]['dim']for s in selected)=={20:2200,18:13,12:18,13:48}
+families={r:sorted(s for s in selected if 24-W.gauge[s]['dim']==r)for r in [4,5,6,11,12]}
+families[24]=sorted(physical-selected)
+assert {r:len(xs)for r,xs in families.items()}=={4:2200,6:13,5:0,11:48,12:18,24:14342}
+assert set().union(*map(set,families.values()))==physical and len(physical)==16621
+# Every actual gauge chart and its two-sided inverse.
+charts=[];maxops=maxnum=maxden=0
+for f,count in sorted(Counter(W.gauge[s]['frame']for s in selected).items()):
+ residual=[]
+ for a in C.A[f]:
+  row=[15*x-sum(a)for x in a];d=gcd(*row);residual.append([x//d for x in row])
+ source=C.B[f]
+ assert len(residual)+len(source)==24
+ assert all(9*sum(x*y for x,y in zip(a,b))-sum(a)*sum(b)==0 for a in residual for b in source)
+ B=list(map(list,zip(*(residual+list(source)))));inverse,den,ops=chart.inverse_integer(B)
+ for left,right in ((B,inverse),(inverse,B)):
+  assert all(sum(x*y for x,y in zip(row,col))==den*int(i==j)for i,row in enumerate(left)for j,col in enumerate(zip(*right)))
+ replay=[list(map(Q,row))for row in B]
+ for op,i,j,num,d in ops:
+  if op=='swap':replay[i],replay[j]=replay[j],replay[i]
+  elif op=='scale':replay[i]=[Q(num,d)*x for x in replay[i]]
+  else:replay[i]=[x+Q(num,d)*y for x,y in zip(replay[i],replay[j])]
+ assert all(x==int(i==j)for i,row in enumerate(replay)for j,x in enumerate(row))
+ maxops=max(maxops,len(ops));maxnum=max(maxnum,max(abs(o[3])for o in ops));maxden=max(maxden,den,max(o[4]for o in ops))
+ charts.append(dict(frame=f,count=count,residual_rank=len(residual),basis=B,inverse_numerator=inverse,inverse_denominator=den,factors=ops))
+assert max(maxnum,maxden)<2**80
+print('PASS charts',len(charts),'max factors',maxops,flush=True)
+
+T=72;patterns=[([11]*4+[4]*7,864),([4]*18,8464),([6]*12,78),([12]*6,216),([24]*3,344208)]
+assert all(sum(blocks)==72 for blocks,_ in patterns)
+incidence=hashlib.sha256();assignments=0;normalizers=[];controls=[]
+for stage in range(3):
+ used=Counter();bank=0
+ for widths,count in patterns:
+  j=next(i for i in range(72)if not 24*stage<=i<24*(stage+1));witness=[];offset=0
+  for block,rank in enumerate(widths):
+   original=list(range(24*stage,24*stage+rank));target=list(range(offset,offset+rank));src=original+[i for i in range(72)if i not in original];dst=target+[i for i in range(72)if i not in target];pi=dict(zip(src,dst));assert set(pi)==set(pi.values())==set(range(72));witness.append((pi[j],block+1));offset+=rank
+  assert len(set(witness))==len(widths)
+  normalizers.append(dict(stage=stage,widths=widths,outside_stage_column=j,scaled_column_witness=witness))
+  for _ in range(count):
+   offset=0
+   for block,rank in enumerate(widths):
+    q=used[rank];role=families[rank][q//T];replica=q%T;used[rank]+=1
+    incidence.update(f'{stage},{rank},{role},{replica},{bank},{offset}\n'.encode());assignments+=1;offset+=rank
+   assert offset==72;bank+=1
+ assert bank==353830 and used=={r:T*len(rs)for r,rs in families.items()if rs}
+ if stage==0:
+  for widths,_ in patterns:
+   def endpoint(indices):
+    state=list(range(144));offsets=[sum(widths[:b])for b in range(len(widths))]
+    for b in indices:
+     for j in range(offsets[b],offsets[b]+widths[b]):state[j],state[72+j]=state[72+j],state[j]
+    return state
+   blocks=len(widths);full=list(range(72,144))+list(range(72))
+   assert endpoint(range(blocks))==full
+   assert endpoint(list(range(blocks))+list(reversed(range(blocks))))==list(range(144))
+   assert endpoint(range(blocks-1))!=full;controls.append(dict(widths=widths,mutation='omitted last block',status='REJECTED'))
+   assert endpoint(list(range(blocks))+[0])!=full;controls.append(dict(widths=widths,mutation='repeated first block',status='REJECTED'))
+assert assignments==3*T*16621==3590136
+banks=3*353830;literal_stock=banks+T*2*W.v;assert literal_stock==1314930
+K=2*3*T*((literal_stock-1)+16621*72*(maxops+71+72));assert 0<K<2**40
+out=dict(extra_source_roles=sorted(extra_source_roles),status='PASS_ACTUAL_JOINT_GAUGE_CHARTS_AND_COMPLETE_BANKS',families={r:len(rs)for r,rs in families.items()},physical_replicas=T,bank_patterns=[dict(widths=widths,banks_per_stage=n)for widths,n in patterns],banks_total=banks,assignments=assignments,literal_stock=literal_stock,integer_normalization=24,W=438310,deficit=46464,m=72,charts=len(charts),max_chart_factors=maxops,max_factor_numerator=maxnum,max_denominator=maxden,incidence_sha256=incidence.hexdigest(),normalizers=normalizers,controls=controls,conservative_extra_selector_calls=K,chart_sha256=hashlib.sha256(json.dumps(charts,sort_keys=True,separators=(',',':')).encode()).hexdigest())
+(D/'joint-charts.json').write_text(json.dumps(charts,separators=(',',':'))+'\n');(D/'joint-banks.json').write_text(json.dumps(out,indent=2)+'\n')
+print('PASS complete banks',patterns,'stock',literal_stock,'routing',K,flush=True)
diff --git a/research/coordinated-crossover-pr200/joint/pairs.json b/research/coordinated-crossover-pr200/joint/pairs.json
new file mode 100644
index 0000000000..33b56aa01d
--- /dev/null
+++ b/research/coordinated-crossover-pr200/joint/pairs.json
@@ -0,0 +1 @@
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12891], [18328, 11454], [18329, 11121], [18330, 11456], [18331, 11123], [18332, 12847], [18333, 14456], [18334, 12884], [18335, 14462], [18336, 11103], [18337, 8832], [18338, 11105], [18339, 8834], [18340, 328], [18341, 337], [18342, 332], [18343, 341], [18344, 14457], [18345, 5152], [18346, 14463], [18347, 5168], [18348, 8030], [18349, 7116], [18350, 8031], [18351, 7117], [18352, 14134], [18353, 11363], [18354, 14143], [18355, 11365], [18356, 49], [18357, 10302], [18358, 53], [18359, 10304], [18360, 11343], [18361, 4608], [18362, 11346], [18363, 4624], [18364, 4365], [18365, 4358], [18366, 4380], [18367, 4378], [18368, 12835], [18369, 11400], [18370, 12872], [18371, 11402], [18372, 11419], [18373, 4355], [18374, 11421], [18375, 4371], [18376, 13933], [18377, 8775], [18378, 13941], [18379, 8777], [18380, 8833], [18381, 12857], [18382, 8835], [18383, 12894], [18384, 8852], [18385, 4704], [18386, 8854], [18387, 4720], [18388, 11211], [18389, 11306], [18390, 11214], [18391, 11308], [18392, 11455], [18393, 11437], [18394, 11457], [18395, 11439], [18396, 12855], [18397, 12852], [18398, 12892], [18399, 12889], [18400, 11286], [18401, 8794], [18402, 11289], [18403, 8796], [18404, 7120], [18405, 7136], [18406, 7121], [18407, 7137], [18408, 12866], [18409, 10322], [18410, 12903], [18411, 10324], [18412, 11509], [18413, 11248], [18414, 11511], [18415, 11251], [18416, 7304], [18417, 7144], [18418, 7305], [18419, 7145], [18420, 4384], [18421, 9392], [18422, 4400], [18423, 9395], [18424, 11491], [18425, 8813], [18426, 11493], [18427, 8815], [18428, 11012], [18429, 7128], [18430, 11014], [18431, 7129], [18432, 14126], [18433, 12843], [18434, 14135], [18435, 12880], [18436, 12832], [18437, 11249], [18438, 12869], [18439, 11252], [18440, 13934], [18441, 11344], [18442, 13942], [18443, 11347], [18444, 7448], [18445, 4992], [18446, 7449], [18447, 5008], [18448, 11342], [18449, 9578], [18450, 11345], [18451, 9580], [18452, 4672], [18453, 4768], [18454, 4688], [18455, 4784], [18456, 13660], [18457, 11268], [18458, 13665], [18459, 11270], [18460, 12938], [18461, 10490], [18462, 12975], [18463, 10492], [18464, 11053], [18465, 10288], [18466, 11055], [18467, 10291], [18468, 7488], [18469, 12932], [18470, 7489], [18471, 12969], [18472, 11034], [18473, 8724], [18474, 11036], [18475, 8726], [18476, 625], [18477, 12921], [18478, 629], [18479, 12958], [18480, 7608], [18481, 5504], [18482, 7609], [18483, 5520], [18484, 12910], [18485, 13946], [18486, 12947], [18487, 13954], [18488, 11071], [18489, 8925], [18490, 11073], [18491, 8927], [18492, 616], [18493, 12928], [18494, 620], [18495, 12965], [18496, 11850], [18497, 11125], [18498, 11852], [18499, 11127], [18500, 11830], [18501, 14468], [18502, 11832], [18503, 14474], [18504, 11107], [18505, 8960], [18506, 11109], [18507, 8962], [18508, 369], [18509, 360], [18510, 373], [18511, 364], [18512, 14469], [18513, 5824], [18514, 14475], [18515, 5840], [18516, 81], [18517, 72], [18518, 85], [18519, 76], [18520, 14152], [18521, 11755], [18522, 14161], [18523, 11757], [18524, 8032], [18525, 11774], [18526, 8033], [18527, 11777], [18528, 11735], [18529, 5280], [18530, 11738], [18531, 5296], [18532, 7942], [18533, 10454], [18534, 7943], [18535, 10456], [18536, 12909], [18537, 11794], [18538, 12946], [18539, 11796], [18540, 9412], [18541, 7148], [18542, 9415], [18543, 7149], [18544, 13949], [18545, 8907], [18546, 13957], [18547, 8909], [18548, 11911], [18549, 12931], [18550, 11913], [18551, 12968], [18552, 8978], [18553, 5376], [18554, 8980], [18555, 5392], [18556, 11599], [18557, 11696], [18558, 11602], [18559, 11698], [18560, 11851], [18561, 11831], [18562, 11853], [18563, 11833], [18564, 5344], [18565, 12926], [18566, 5360], [18567, 12963], [18568, 11676], [18569, 8924], [18570, 11679], [18571, 8926], [18572, 7958], [18573, 7164], [18574, 7959], [18575, 7165], [18576, 12940], [18577, 10472], [18578, 12977], [18579, 10474], [18580, 8961], [18581, 11638], [18582, 8963], [18583, 11641], [18584, 7332], [18585, 7172], [18586, 7333], [18587, 7173], [18588, 7476], [18589, 12929], [18590, 7477], [18591, 12966], [18592, 11891], [18593, 8943], [18594, 11893], [18595, 8945], [18596, 11016], [18597, 7156], [18598, 11018], [18599, 7157], [18600, 14144], [18601, 12917], [18602, 14153], [18603, 12954], [18604, 12906], [18605, 11639], [18606, 12943], [18607, 11642], [18608, 13950], [18609, 11736], [18610, 13958], [18611, 11739], [18612, 7588], [18613, 5664], [18614, 7589], [18615, 5680], [18616, 11734], [18617, 9582], [18618, 11737], [18619, 9584], [18620, 11813], [18621, 5440], [18622, 11815], [18623, 5456], [18624, 13670], [18625, 11658], [18626, 13675], [18627, 11660], [18628, 401], [18629, 392], [18630, 405], [18631, 396], [18632, 14481], [18633, 6304], [18634, 14487], [18635, 6320], [18636, 113], [18637, 104], [18638, 117], [18639, 108], [18640, 14170], [18641, 12057], [18642, 14179], [18643, 12059], [18644, 9071], [18645, 11605], [18646, 9073], [18647, 11608], [18648, 12037], [18649, 7700], [18650, 12040], [18651, 7701], [18652, 7960], [18653, 10458], [18654, 7961], [18655, 10460], [18656, 12983], [18657, 12096], [18658, 13020], [18659, 12098], [18660, 7974], [18661, 7708], [18662, 7975], [18663, 7709], [18664, 13965], [18665, 9015], [18666, 13973], [18667, 9017], [18668, 8034], [18669, 13005], [18670, 8035], [18671, 13042], [18672, 9088], [18673, 7712], [18674, 9090], [18675, 7713], [18676, 12213], [18677, 11996], [18678, 12215], [18679, 11998], [18680, 12157], [18681, 12137], [18682, 12159], [18683, 12139], [18684, 7944], [18685, 13000], [18686, 7945], [18687, 13037], [18688, 11972], [18689, 9032], [18690, 11975], [18691, 9034], [18692, 9436], [18693, 7188], [18694, 9439], [18695, 7189], [18696, 13014], [18697, 10586], [18698, 13051], [18699, 10588], [18700, 12078], [18701, 11928], [18702, 12081], [18703, 11931], [18704, 7356], [18705, 7196], [18706, 7357], [18707, 7197], [18708, 13003], [18709, 7500], [18710, 13040], [18711, 7501], [18712, 12195], [18713, 9051], [18714, 12197], [18715, 9053], [18716, 7180], [18717, 11020], [18718, 7181], [18719, 11022], [18720, 14162], [18721, 12991], [18722, 14171], [18723, 13028], [18724, 12980], [18725, 11929], [18726, 13017], [18727, 11932], [18728, 13966], [18729, 12038], [18730, 13974], [18731, 12041], [18732, 7612], [18733, 6144], [18734, 7613], [18735, 6160], [18736, 12036], [18737, 9602], [18738, 12039], [18739, 9604], [18740, 12115], [18741, 5920], [18742, 12117], [18743, 5936], [18744, 13680], [18745, 11952], [18746, 13685], [18747, 11954], [18748, 8036], [18749, 13079], [18750, 8037], [18751, 13116], [18752, 9182], [18753, 7732], [18754, 9184], [18755, 7733], [18756, 12417], [18757, 12000], [18758, 12419], [18759, 12002], [18760, 12357], [18761, 12337], [18762, 12359], [18763, 12339], [18764, 13077], [18765, 13074], [18766, 13114], [18767, 13111], [18768, 11978], [18769, 9126], [18770, 11981], [18771, 9128], [18772, 7728], [18773, 7208], [18774, 7729], [18775, 7209], [18776, 13088], [18777, 10590], [18778, 13125], [18779, 10592], [18780, 9165], [18781, 12276], [18782, 9167], [18783, 12279], [18784, 8078], [18785, 7216], [18786, 8079], [18787, 7217], [18788, 7200], [18789, 8150], [18790, 7201], [18791, 8151], [18792, 12397], [18793, 9147], [18794, 12399], [18795, 9149], [18796, 7962], [18797, 11038], [18798, 7963], [18799, 11040], [18800, 14180], [18801, 13065], [18802, 14189], [18803, 13102], [18804, 13054], [18805, 11935], [18806, 13091], [18807, 11938], [18808, 13982], [18809, 12232], [18810, 13990], [18811, 12235], [18812, 9456], [18813, 8206], [18814, 9459], [18815, 8207], [18816, 12230], [18817, 9620], [18818, 12233], [18819, 9622], [18820, 12319], [18821, 7780], [18822, 12321], [18823, 7781], [18824, 13690], [18825, 11956], [18826, 13695], [18827, 11958], [18828, 12282], [18829, 9241], [18830, 12285], [18831, 9243], [18832, 8080], [18833, 7232], [18834, 8081], [18835, 7233], [18836, 8152], [18837, 13151], [18838, 8153], [18839, 13188], [18840, 12507], [18841, 9221], [18842, 12509], [18843, 9223], [18844, 11056], [18845, 9476], [18846, 11058], [18847, 9479], [18848, 14198], [18849, 13139], [18850, 14207], [18851, 13176], [18852, 13128], [18853, 11941], [18854, 13165], [18855, 11944], [18856, 13998], [18857, 12238], [18858, 14006], [18859, 12241], [18860, 7964], [18861, 8208], [18862, 7965], [18863, 8209], [18864, 12236], [18865, 9638], [18866, 12239], [18867, 9640], [18868, 8290], [18869, 12435], [18870, 8291], [18871, 12437], [18872, 13700], [18873, 12042], [18874, 13705], [18875, 12044], [18876, 13202], [18877, 11961], [18878, 13239], [18879, 11964], [18880, 14014], [18881, 12244], [18882, 14022], [18883, 12247], [18884, 7966], [18885, 8210], [18886, 7967], [18887, 8211], [18888, 12242], [18889, 9658], [18890, 12245], [18891, 9660], [18892, 8292], [18893, 12439], [18894, 8293], [18895, 12441], [18896, 13710], [18897, 12046], [18898, 13715], [18899, 12048], [18900, 8212], [18901, 12443], [18902, 8213], [18903, 12445], [18904, 13720], [18905, 12066], [18906, 13725], [18907, 12068]]
diff --git a/research/coordinated-crossover-pr200/joint/price_joint.py b/research/coordinated-crossover-pr200/joint/price_joint.py
new file mode 100644
index 0000000000..1fc0156aef
--- /dev/null
+++ b/research/coordinated-crossover-pr200/joint/price_joint.py
@@ -0,0 +1,21 @@
+from pathlib import Path
+from fractions import Fraction as Q
+import json,sys,importlib.util
+sys.dont_write_bytecode=True;sys.set_int_max_str_digits(0)
+D=Path(__file__).resolve().parent;PACKAGE=D.parent;ROOT=PACKAGE.parents[1];P=ROOT/'research/paired-cube-diagonal-bit-168'
+sys.path.insert(0,str(P/'bit'));sys.path.insert(0,str(P/'arithmetic'))
+from prove import certify
+record=json.loads((PACKAGE/'targetagg/profile.json').read_text());banks=json.loads((D/'joint-banks.json').read_text());r=record['profile'];H={int(k):n for k,n in r['child_histogram'].items()};assert H.pop(60)==2200 and H.pop(54)==13 and H.pop(57,0)==0 and H.pop(36)==18 and H.pop(39)==48
+H={t:24*n for t,n in H.items()};mass=sum(t*n for t,n in H.items());assert 72*banks['W']-mass==banks['deficit']
+row=dict(m=72,W_per_vertex=banks['W'],child_histogram=H,rank_per_vertex=mass,deficit_per_vertex=banks['deficit'],maxchild=max(H));coarse=certify(row)
+spec=importlib.util.spec_from_file_location('base_two_moment',ROOT/'research/coordinated-crossover-pr200/geometry/base_two_moment.py');alt=importlib.util.module_from_spec(spec);spec.loader.exec_module(alt)
+a=coarse['coarse_saving'];fb=32*72**2*sum(H.values());_,upper=alt.moment(72,banks['W'],list(H.items()),a);_,bad=alt.moment(72,banks['W'],[(1,fb)],a);assert upper+Q(1,10**16)*bad<1
+lower,_=alt.moment(72,banks['W'],list(H.items()),a+Q(1,10**18));badlower,_=alt.moment(72,banks['W'],[(1,fb)],a+Q(1,10**18));assert lower+Q(1,10**16)*badlower>1
+def enc(x):
+ if isinstance(x,Q):return str(x)
+ if isinstance(x,dict):return {str(k):enc(v)for k,v in x.items()}
+ if isinstance(x,(list,tuple)):return [enc(v)for v in x]
+ return x
+(D/'joint-paid.json').write_text(json.dumps(enc(dict(profile=row,coarse=coarse,independent_base2_pass=True)),indent=2)+'\n')
+import subprocess
+subprocess.run([sys.executable,'-B',str(D/'arithmetic.py')],check=True)
diff --git a/research/coordinated-crossover-pr200/joint/replay_joint.py b/research/coordinated-crossover-pr200/joint/replay_joint.py
new file mode 100644
index 0000000000..40bf4efb39
--- /dev/null
+++ b/research/coordinated-crossover-pr200/joint/replay_joint.py
@@ -0,0 +1,47 @@
+from pathlib import Path
+import json,hashlib,gzip,sys,time
+sys.dont_write_bytecode=True
+from joint_word import Candidate,P,D
+from terminal import prove
+from prime_witnesses import certificate
+W=Candidate();pins={str(p):hashlib.sha256(p.read_bytes()).hexdigest()for p in [*W.input_paths,W.framepath,D/'joint_word.py',D/'joint-selection.json',D/'pairs.json',P/'selected/bit/sinks.json',P/'bit/base_word.py',P/'bit/word.py',P/'bit/terminal.py',P/'bit/prime_witnesses.py',Path(W.module.__file__)]}
+W.exact_frames();print('PASS exact frames and alias endpoints',flush=True)
+record=prove(W,json.loads((P/'selected/bit/sinks.json').read_text()));record.pop('seconds');record.pop('maxrss');print('PASS full reflected supplier replay',flush=True)
+primes=certificate(W);(D/'joint-primes.json.gz').write_bytes(gzip.compress((json.dumps(primes,sort_keys=True,separators=(',',':'))+'\n').encode(),mtime=0))
+record['input_pins']=pins;record['prime_summary']={k:v for k,v in primes.items()if k!='frame_witnesses'}
+assert pins=={p:hashlib.sha256(Path(p).read_bytes()).hexdigest()for p in pins}
+(D/'joint-replay.json').write_text(json.dumps(record,indent=2)+'\n')
+print('PASS joint gauge whole-word and prime admission',record['profile']['child_histogram'],flush=True)
+
+control=Candidate()
+omitted=control.newroles[0]
+control.order.remove(omitted)
+try:control.formal(2)
+except ValueError as error:
+ assert 'defining decoder' in str(error),str(error)
+ print('PASS omitted new compensation read rejected',flush=True)
+else:raise AssertionError('Omitted new gauge compensation was accepted')
+
+control=Candidate()
+role=next(s for s in control.newroles if control.gauge[s]['dim']==19)
+control.order.remove(role);control.order.insert(0,role)
+control.exact_frames()
+try:control.row()
+except (ValueError,AssertionError) as error:
+ assert 'target frame chronology' in str(error),str(error)
+ print('PASS reversed nested gauge read order rejected',flush=True)
+else:raise AssertionError('Rank19 read before rank18 was accepted')
+control=Candidate();control.order.remove(role)
+try:control.formal(2)
+except ValueError as error:
+ assert 'defining decoder' in str(error),str(error)
+ print('PASS omitted nested rank19 compensation rejected',flush=True)
+else:raise AssertionError('Omitted nested compensation was accepted')
+
+for dim in (12,13):
+ control=Candidate();role=next(s for s in control.newroles if control.gauge[s]['dim']==dim);control.order.remove(role)
+ try:control.formal(2)
+ except ValueError as error:
+  assert 'defining decoder' in str(error),str(error)
+  print('PASS omitted rank'+str(dim)+' shared gauge compensation rejected',flush=True)
+ else:raise AssertionError('Shared compensation omission accepted')
diff --git a/research/coordinated-crossover-pr200/newg/PROVENANCE.json b/research/coordinated-crossover-pr200/newg/PROVENANCE.json
new file mode 100644
index 0000000000..eaa6e19b8b
--- /dev/null
+++ b/research/coordinated-crossover-pr200/newg/PROVENANCE.json
@@ -0,0 +1,8 @@
+{
+  "source469_final_manifest_sha256": "9bfe66dde122a7408dbc18b3f87f12d153b7c2a6191c9864db8692c7a0344d0d",
+  "source471_selection_sha256": "d94ef4e0b8ca98a7385b364cf245816a7cad08f850a60ce8b67aab4c3a89f3f4",
+  "discovery_schedule_sha256": "5230a1503cdf1f0eefb4f09b1b9d6dedd8afab9378d30be463bb3597513d166e",
+  "discovery_schedule_audit_sha256": "de4668f5d0c509fd18a722ad6050c65b5f7ad4b14ad3e9142f7bc9a95dc1d527",
+  "independent_reflected_ledger_sha256": "ef582523844f66baa356d8916c6aaff5437151804f7cbb516de69cfe0993b37f",
+  "scope": "The executable schedule is normalized to indices only and every actual address/frame rebuilt. Two retimed source heads extend source469; all source/target/frame/center/paid proofs are rerun. Prepared with OpenAI Codex assistance."
+}
diff --git a/research/coordinated-crossover-pr200/newg/boundary.py b/research/coordinated-crossover-pr200/newg/boundary.py
new file mode 100644
index 0000000000..dde771921f
--- /dev/null
+++ b/research/coordinated-crossover-pr200/newg/boundary.py
@@ -0,0 +1,111 @@
+#!/usr/bin/env python3
+"""Independent exact full-path and target-ledger audit of28 new source gauges.
+No scalar claim. Prepared with OpenAI Codex assistance.
+"""
+from pathlib import Path
+from collections import defaultdict,Counter
+import importlib.util,json,sys
+sys.dont_write_bytecode=True
+D=Path(__file__).resolve().parent;P=D.parent/'joint';sp=importlib.util.spec_from_file_location('ind_newgauges',P/'joint_word.py');m=importlib.util.module_from_spec(sp);sp.loader.exec_module(m)
+w=m.Candidate();C=w.C;w.exact_frames();base=w.row();selected=json.loads((D/'selection.json').read_text());co=w.adjoint();recipient={d:b for b,d in w.pairs};kpair={e['passive']:e for e in w.k['entries']};roots=defaultdict(list)
+schedule=json.loads((D/'schedule.json').read_text());original_rest=w.rest[:];w.phase1=schedule['new_phase1'];w.rest=schedule['deferred_ops']+original_rest
+actual_order=w.phase1+w.rest;w.role_ops=defaultdict(list)
+for i in actual_order:
+ for s in w.ops[i][:2]:w.role_ops[s].append(i)
+w.readtime={s:t+168 for s,t in w.readtime.items()}
+touched={s for i in w.phase1 for s in w.ops[i][:2]}
+new_phase_roles={8584,9424}
+for j,s in enumerate(w.w['rootroles']):roots[s].append((j,w.w['root_frame'][j]))
+term=json.loads((P/'joint-replay.json').read_text())['selected'];removed={r['role']for r in term};deleted={r['root']for r in term};bywrite={i:e for e in term for i in e['writes']};after={e['writes'][-1]:e for e in term};tt={t for e in term for t in e['targets']};source_roles=set(w.source.values())
+old=json.loads((D.parent/'borrow/selection.json').read_text())+json.loads((D.parent/'gaugeb/selection.json').read_text());old_sources={r['source']for r in old};old_roles={r['role']for r in old};old_members=old_sources|{r['partner']for r in old}
+assert not old_members&({r['source']for r in selected}|{r['partner']for r in selected})
+assert len({r['source']for r in selected})==len(selected)==28
+assert not(old_sources&{r['source']for r in selected})and not(old_roles&{r['role']for r in selected})
+assert len({r['role']for r in selected})==28
+assert sum(len(r['targets'])for r in selected)==112 and len({t for r in selected for t in r['targets']})==108
+local_delta=Counter();endpoint_rows=[];gframes={};prime_records=[]
+from prime_witnesses import det,factor_witness,validate_factor
+def H(chain,start_dim):
+ out=Counter();previous=None;d=start_dim
+ for f in chain:
+  assert previous is None or C.sub(previous,f)
+  n=C.dimf[f];assert n>=d
+  if n>d:out[n-d]+=1
+  previous=f;d=n
+ return out
+for r in selected:
+ s,n=r['role'],r['source'];e=kpair[n];mix=e['mix_frame'];f=w.register(r['gauge_basis'])if 'gauge_basis'in r else mix;gframes[s]=f;assert mix==r['mix_frame']and e['carrier']==r['partner']and C.sub(mix,f)
+ assert s not in touched|source_roles|removed|set(w.donor)|set(w.gauge)
+ assert sorted(co[s])==r['targets']and not(set(r['targets'])&tt)
+ chain=[w.opframe[i]for i in w.role_ops[s]]+[g for j,g in roots[s]]
+ if s in recipient:
+  b=recipient[s];chain += [w.gauge[b]['frame']]+[w.opframe[i]for i in w.role_ops[b]]+[g for j,g in roots[b]]
+ chain.append(w.w['full_frame']);assert C.sub(f,chain[0])
+ new=[w.w['source_frame'][n],mix,f]+chain;assert all(C.nondeg(g)for g in set(new))and all(C.sub(a,b)for a,b in zip(new,new[1:]))
+ old_aux=H(chain,0);old_source=H([mix,w.w['full_frame']],1);new_source=H(new[1:],1)
+ delta=Counter(new_source);delta.subtract(old_aux);delta.subtract(old_source);delta={d:c for d,c in delta.items()if c}
+ assert sum(d*c for d,c in delta.items())==-24
+ # Reflected annihilator path has precisely reversed nested frame inclusions.
+ for a,b in zip(new,new[1:]):assert C.sub(a,b)and all(w.module.dot(x,y)==0 for x in C.A[b]for y in C.B[a])
+ reflected=Counter(C.dimf[b]-C.dimf[a]for a,b in zip(new,new[1:])if C.dimf[b]>C.dimf[a]);assert reflected==new_source
+ local_delta.update(delta);endpoint_rows.append(dict(role=s,source=n,alias_recipient=recipient.get(s),source_line=C.dimf[new[0]],K_mix=C.dimf[mix],source_gauge=C.dimf[f],final_dimension=C.dimf[new[-1]],auxiliary_delta=delta,old_aux_histogram=dict(old_aux),old_source_histogram=dict(old_source),complete_source_histogram=dict(new_source),reflected_equal=True))
+assert not(set(schedule['deferred_ops'])&set(bywrite))
+# Complete actual target chronology, including all terminal substitutions and K.
+at=defaultdict(list)
+for s in w.order:at[w.readtime[s]].append(s)
+deliv=defaultdict(list)
+for e in w.k['entries']:deliv[e['deliver_after_root']].append(e)
+def target_hist(extra):
+ current=[None]*w.v;hist=Counter();events=0;paths=[[]for _ in range(w.v)]
+ def move(t,f):
+  nonlocal events
+  previous=current[t];assert previous is None or C.sub(previous,f),('target nesting',t,previous,f)
+  assert all(w.module.dot(C.cov[t],x)==0 for x in C.B[f]),('target cap',t,f)
+  d=C.dimf[f]-(0 if previous is None else C.dimf[previous]);assert d>=0
+  if d:hist[d]+=1
+  current[t]=f;paths[t].append(f);events+=1
+ for j,i in enumerate(w.rest):
+  if extra:
+   for r in selected:
+    if j==(0 if r['role']in new_phase_roles else 168):
+     for t in r['targets']:move(t,gframes[r['role']])
+  for s in at[j]:
+   for t in w.gauge[s]['targets']:move(t,w.gauge[s]['frame'])
+  if i in bywrite:move(bywrite[i]['pivot'],w.opframe[i])
+  if i in after:
+   for t in after[i]['targets']:move(t,after[i]['root_frame'])
+ for s in at[len(w.rest)]:
+  for t in w.gauge[s]['targets']:move(t,w.gauge[s]['frame'])
+ for j,r in enumerate(w.g['roots']):
+  if r['kind']=='side'and j not in deleted:
+   for t in r['targets']:move(t,w.w['root_frame'][j])
+  for e in deliv[j]:
+   for t in e['receivers']:move(t,e['deliver_frame'])
+ for t,f in enumerate(current):
+  d=w.h-1-C.dimf[f]
+  if d:hist[d]+=1
+ assert sum(d*c for d,c in hist.items())==w.v*(w.h-1)
+ reflected=Counter()
+ for t,chain in enumerate(paths):
+  # Reverse the exact annihilator bases. The cap annihilator is cov[t].
+  last=[C.cov[t]];previous_dim=1
+  for f in reversed(chain):
+   basis=C.A[f];assert len(basis)==w.h-C.dimf[f]
+   assert all(w.module.dot(a,b)==0 for a in last for b in C.B[f])
+   step=len(basis)-previous_dim;assert step>=0
+   if step:reflected[step]+=1
+   last=basis;previous_dim=len(basis)
+  step=w.h-previous_dim
+  if step:reflected[step]+=1
+ assert reflected==hist,('independent reflected target histogram',reflected,hist)
+ return hist,events
+Y0,e0=target_hist(False);Y1,e1=target_hist(True);ydelta=Counter(Y1);ydelta.subtract(Y0);local_delta.update(ydelta)
+compact=lambda c:{str(d):n for d,n in sorted(c.items())if n}
+assert sum(d*c for d,c in local_delta.items())==-28*24
+for f in sorted(set(gframes.values())):
+ B=C.B[f];ss=list(map(sum,B));gram=[[9*sum(a*b for a,b in zip(x,y))-ss[i]*ss[j]for j,y in enumerate(B)]for i,x in enumerate(B)]
+ determinant=det(gram);powers,residual=factor_witness(determinant);validate_factor(determinant,powers,residual)
+ prime_records.append(dict(frame=f,dimension=len(B),basis=B,cleared_gram_determinant=determinant,small_prime_powers=powers,remaining_factor=residual))
+assert any(r['dimension']==4 for r in prime_records)
+res=dict(status='PASS_EXACT_28_SHARED_SOURCE_GAUGE_PATHS_TARGETS_AND_PRIMES',scope='Complete physical source paths, donor-recipient chains, reflected annihilator nesting, complete terminal/gauge/root/K target chronology, all exact paid local deltas. Whole scalar replay and global assembly owned separately.',selected=28,touched_targets=108,explicit_reflected_target_histograms_equal=True,additional_target_events=e1-e0,target_delta=compact(ydelta),old_complete_target_histogram=compact(Y0),new_complete_target_histogram=compact(Y1),local_delta=compact(local_delta),local_rank_delta=sum(d*c for d,c in local_delta.items()),W_delta=-28,shared_rank_delta=3*sum(d*c for d,c in local_delta.items()),deficit_delta=-28*72-3*sum(d*c for d,c in local_delta.items()),endpoint_rows=endpoint_rows,source_gauge_prime_witnesses=prime_records)
+(D/'boundary.json').write_text(json.dumps(res,indent=2)+'\n');print(json.dumps({k:v for k,v in res.items()if k not in ('endpoint_rows','source_gauge_prime_witnesses')},indent=2),flush=True)
diff --git a/research/coordinated-crossover-pr200/newg/derive_profile.py b/research/coordinated-crossover-pr200/newg/derive_profile.py
new file mode 100644
index 0000000000..fc5595d597
--- /dev/null
+++ b/research/coordinated-crossover-pr200/newg/derive_profile.py
@@ -0,0 +1,62 @@
+"""Reconstruct all actual source/internal/target paths after 28 new source-backed gauges."""
+from pathlib import Path
+from collections import Counter
+import json,sys,importlib.util
+sys.dont_write_bytecode=True
+HERE=Path(__file__).resolve().parent;P=HERE.parent
+spec=importlib.util.spec_from_file_location('newg_profile_word',P/'joint/joint_word.py');m=importlib.util.module_from_spec(spec);spec.loader.exec_module(m)
+W=m.Candidate();r=json.loads((P/'gaugeb/profile.json').read_text())['profile'];audit=json.loads((HERE/'boundary.json').read_text());scalar=json.loads((HERE/'replay.json').read_text());sel=json.loads((HERE/'selection.json').read_text());oldg={e['role']for e in json.loads((P/'gaugeb/selection.json').read_text())}
+assert len(sel)==28 and len(scalar['integer'])==2 and {x['direction']for x in scalar['integer']}=={-1,1}
+assert all(x['formal_columns']==20163 and x['dirty']==16643 and x['all_targets']and x['all_source_and_dirty_restored']for x in [scalar['F2']]+scalar['integer'])
+assert len(scalar['controls'])==6 and all(x['status']=='REJECTED'for x in scalar['controls'].values())
+schedule=json.loads((HERE/'schedule-audit.json').read_text())
+assert schedule['source_heads']==471 and schedule['deferred_operations']==168 and schedule['center_prefix_operations']==17080
+assert schedule['all_exact_address_subspace_streams_preserved'] and schedule['all_actual_address_frame_dimension_streams_preserved'] and schedule['all_address_jump_histograms_preserved'] and schedule['no_later_center_middle_consumer']
+assert all(x['all_forward_centers_at_D0']and x['all_reflected_centers_at_D1']and x['actual_target_histogram']==audit['new_complete_target_histogram']==x['actual_reflected_target_histogram']for x in [scalar['F2']]+scalar['integer'])
+def counter(h):return Counter({int(k):n for k,n in h.items()})
+local=counter(r['physical_internal_histogram']);source=counter(r['physical_source_histogram']);oldtarget=counter(r['physical_target_histogram'])
+assert oldtarget==counter(audit['old_complete_target_histogram']);target=counter(audit['new_complete_target_histogram'])
+assert {e['role']for e in audit['endpoint_rows']}=={e['role']for e in sel}
+for e in audit['endpoint_rows']:
+ local.subtract(counter(e['old_aux_histogram']));source.subtract(counter(e['old_source_histogram']));source.update(counter(e['complete_source_histogram']))
+assert all(n>=0 for h in (local,source,target)for n in h.values())
+child=Counter()
+for h in(local,source,target):
+ for k,n in h.items():
+  if k and n:child[k]+=3*n
+for s,z in W.gauge.items():
+ if s not in W.donor and s not in oldg:child[3*z['dim']]+=1
+child[2]+=2*W.v
+stock=r['W_per_vertex']-28;physical=r['R']-28;mass=sum(k*n for k,n in child.items());deficit=72*stock-mass
+assert stock==20163 and physical==16643 and mass==1449800 and deficit==1936
+assert sum(k*n for k,n in source.items())==sum(k*n for k,n in target.items())==W.v*(W.h-1)
+delta=Counter(child);delta.subtract(counter(r['child_histogram']));delta={k:v for k,v in delta.items()if v};assert delta=={int(k):3*n for k,n in audit['local_delta'].items()}
+def compact(h):return {str(k):n for k,n in sorted(h.items())if n}
+profile=dict(r,R=physical,active_virtual_R=physical+r['reused_registers'],W_per_vertex=stock,rank_per_vertex=mass,child_histogram=compact(child),physical_internal_histogram=compact(local),physical_source_histogram=compact(source),physical_target_histogram=compact(target),physical_original_sources_borrowed=471,source_backed_new_rank2_gauges=26,source_backed_shared_rank4_gauges=2,center_prefix_operations=17080,deferred_center_independent_operations=168,source_heads_exposed_by_retiming=2,copied_center_frame_chronology="D0 forward and D1 complementary reverse")
+assert profile['active_virtual_R']==18403
+# Legacy selected_* fields consistently describe actual auxiliary-bank entrances.
+# Data/source-owned entrances are recorded separately and never banked twice.
+bank_ranks=Counter(z['dim']for role,z in W.gauge.items()if role not in W.donor and role not in oldg)
+source_ranks=Counter(W.gauge[role]['dim']for role in oldg)
+source_ranks.update(e['source_gauge']for e in audit['endpoint_rows'])
+all_ranks=bank_ranks+source_ranks
+profile['retained_selected_roles']=r['selected_roles']
+profile['retained_selected_rank_histogram']=r['selected_rank_histogram']
+profile['selected_rank_histogram']=compact(bank_ranks)
+profile['selected_roles']=sum(bank_ranks.values())
+profile['selected_roles_scope']='Actual remaining bankable auxiliary entrances; source-owned data entrances are counted separately.'
+profile['bankable_auxiliary_entrances']=sum(bank_ranks.values())
+profile['source_owned_gauge_entrances']=sum(source_ranks.values())
+profile['source_owned_gauge_rank_histogram']=compact(source_ranks)
+profile['all_gauge_entrances']=sum(all_ranks.values())
+profile['all_gauge_rank_histogram']=compact(all_ranks)
+assert profile['bankable_auxiliary_entrances']==2279
+assert profile['source_owned_gauge_entrances']==31
+assert profile['all_gauge_entrances']==2310
+assert bank_ranks=={20:2200,18:13,12:18,13:48}
+assert source_ranks=={19:3,2:26,4:2}
+assert profile['R']+profile['reused_registers']==profile['active_virtual_R']==18403
+assert profile['W_per_vertex']==2*profile['v']+profile['R']
+assert profile['maxchild']==max(child)
+out=dict(status='PASS_COMPLETE_471_SOURCE_PROFILE',profile=profile,actual_child_delta=delta,all_original_gauge_adjoint_coefficients_retained=True,all_original_target_anchors_retained=True,additional_source_gauge_target_reads=112,all_new_target_chronology_reconstructed=True)
+(HERE/'profile.json').write_text(json.dumps(out,indent=2)+'\n');print('PASS471 complete ledgers',dict(W=stock,R=physical,rank=mass,deficit=deficit,delta=delta),flush=True)
diff --git a/research/coordinated-crossover-pr200/newg/replay.py b/research/coordinated-crossover-pr200/newg/replay.py
new file mode 100644
index 0000000000..a0af83c245
--- /dev/null
+++ b/research/coordinated-crossover-pr200/newg/replay.py
@@ -0,0 +1,208 @@
+"""Scalar prototype: source471 with168 frame-preserving commuting operations retimed after centers.
+Prepared with OpenAI Codex assistance; Apache-2.0.
+"""
+from pathlib import Path
+from collections import Counter,defaultdict
+import json,sys,importlib.util,time,gc,hashlib
+sys.dont_write_bytecode=True
+HERE=Path(__file__).resolve().parent;D=HERE.parent/'joint';spec=importlib.util.spec_from_file_location('source82',D/'joint_word.py');m=importlib.util.module_from_spec(spec);spec.loader.exec_module(m)
+W=m.Candidate();C=W.C
+W.changed_frames=[i for i,(a,b)in enumerate(zip(W.original_opframe,W.opframe))if a!=b]
+W.endframe={s:W.opframe[xs[-1]]for s,xs in W.role_ops.items()}
+C.decoder();C.geometry();selection=json.loads((HERE.parent/'borrow/selection.json').read_text());new_gauge_selection=json.loads((HERE/'selection.json').read_text());new_gauge_selection=new_gauge_selection['selection']if isinstance(new_gauge_selection,dict)else new_gauge_selection;gauge_selection=json.loads((HERE.parent/'gaugeb/selection.json').read_text())+new_gauge_selection
+schedule=json.loads((HERE/'schedule.json').read_text())
+original_phase1=W.phase1[:];original_rest=W.rest[:];omitted_for_schedule={r['omit_copy']for r in selection};deferred=schedule['deferred_ops'];assert len(deferred)==168
+W.phase1=schedule['new_phase1'];W.rest=deferred+original_rest
+assert set(W.phase1)|set(W.rest)|omitted_for_schedule==set(range(len(W.ops)))
+assert not(set(W.phase1)&set(W.rest))and not(set(W.phase1+W.rest)&omitted_for_schedule)
+W.role_ops=defaultdict(list)
+for i in W.phase1+W.rest:
+ for role in W.ops[i][:2]:W.role_ops[role].append(i)
+pos={i:j for j,i in enumerate(W.rest)};W.first={role:pos.get(ii[0],-1)for role,ii in W.role_ops.items()};W.last={role:pos.get(ii[-1],-1)for role,ii in W.role_ops.items()};W.position={i:j for j,i in enumerate(W.phase1+W.rest)}
+W.endframe={role:W.opframe[ii[-1]]for role,ii in W.role_ops.items()}
+W.readtime={role:t+168 for role,t in W.readtime.items()};new_phase_roles={8584,9424}
+assert new_phase_roles<=set(schedule['new_untouched_roles'])
+for role,t in W.readtime.items():W.w['reads'][str(role)]=len(W.phase1)+t
+knew={e['passive']:e for e in W.k['entries']};raw_response=W.adjoint();newroles=[];target_frames={};touched={s for i in W.phase1 for s in W.ops[i][:2]}
+for r in new_gauge_selection:
+ role,source=r['role'],r['source'];mix=knew[source]['mix_frame'];f=W.register(r['gauge_basis'])if 'gauge_basis'in r else mix;targets=sorted(raw_response[role]);assert targets==r['targets'] and mix==r['mix_frame'] and C.sub(mix,f)
+ assert role not in W.gauge and role not in W.donor and role not in W.source.values() and role not in touched
+ assert C.dimf[f]in(2,4) and C.nondeg(f) and C.sub(f,W.opframe[W.role_ops[role][0]])
+ assert all(t not in target_frames or C.sub(target_frames[t],f) and C.sub(f,target_frames[t])for t in targets);target_frames.update({t:f for t in targets})
+ assert all(all(W.module.dot(C.cov[t],b)==0 for b in C.B[f])for t in targets)
+ z=dict(role=role,frame=f,dim=C.dimf[f],targets=targets);W.gauge[role]=z;W.w['gauges'].append(z);W.readtime[role]=0 if role in new_phase_roles else 168;W.w['reads'][str(role)]=len(W.phase1)+W.readtime[role];newroles.append(role)
+W.order=newroles+W.order
+W.exact_frames()
+assert all(0<=W.readtime[s]<=W.first[s]for s in W.gauge)
+assert all(W.last[d]<W.readtime[b]and C.sub(W.endframe[d],W.gauge[b]['frame'])for b,d in W.pairs)
+assert all(not(set(W.gauge)&set(W.ops[i][:2]))for i in W.phase1)
+print('PASS scalar schedule timing, all gauge/alias readtimes, new source prefix inclusions; independent target/reflected ledger required',flush=True)
+borrow={r['role']:r['source']for r in selection};by_source={r['source']:r for r in selection};omitted={r['omit_copy']for r in selection};early={r['early']:r for r in selection};assert len(early)==len(borrow)==len(selection)==440
+gauge_borrow={r['role']:r['source']for r in gauge_selection};kpair={e['passive']:e for e in W.k['entries']};used_sources=set(by_source)|{r['partner']for r in selection}
+assert len(gauge_borrow)==len(gauge_selection) and len(set(gauge_borrow.values()))==len(gauge_selection)
+for r in gauge_selection:
+ assert r['source']in kpair and r['partner']==kpair[r['source']]['carrier']
+ assert r['role']in W.gauge and r['role']not in W.donor and r['role']not in borrow
+ assert r['source']not in used_sources and r['partner']not in used_sources
+ used_sources.update([r['source'],r['partner']])
+borrow.update(gauge_borrow);by_source.update({r['source']:r for r in gauge_selection})
+input_paths=[*W.input_paths,W.framepath,D/'joint_word.py',D/'joint-selection.json',D/'pairs.json',Path(W.module.__file__),D/'joint-replay.json',(HERE/'selection.json'),(HERE.parent/'gaugeb/selection.json'),Path(__file__),HERE/'schedule.json',(HERE.parent/'borrow/selection.json')]
+input_pins={str(p):hashlib.sha256(p.read_bytes()).hexdigest()for p in input_paths}
+adj=[{}for _ in range(W.R)]
+for r,s in zip(W.g['roots'],W.w['rootroles']):
+ for t in r['targets']:adj[s][t]=adj[s].get(t,0)+1
+for i in reversed(W.phase1+W.rest):
+ if i in omitted:continue
+ a,b,_=W.ops[i]
+ for t,c in adj[a].items():adj[b][t]=adj[b].get(t,0)+c
+oldadj=W.adjoint()
+assert all(all(0<c<=oldadj[s].get(t,0)for t,c in row.items())for s,row in enumerate(adj))
+assert all(adj[s]==oldadj[s]for s in W.gauge), 'all gauge reads and paid target supports retained exactly'
+assert len(by_source)==440+len(gauge_selection) and all(r['early'] in r['windows']for r in selection)
+# The formal column layout excludes each actual borrowed dirty slot, retaining
+# distinct original X columns and every other arbitrary dirty input.
+terminal=json.loads((D/'joint-replay.json').read_text())['selected'];removed={e['role']for e in terminal};deletedroots={e['root']for e in terminal};bywrite={i:e for e in terminal for i in e['writes']};afterwrite={e['writes'][-1]:e for e in terminal};assert not(set(early)&set(bywrite)) and not(set(borrow)&removed)
+regs=sorted(set(W.phys.values())-set(borrow)-removed);idx={r:2*W.v+j for j,r in enumerate(regs)}
+assert not set(borrow)&set(W.donor)
+at=defaultdict(list)
+for b in W.order:at[W.readtime[b]].append(b)
+deliveries=defaultdict(list)
+for e in W.k['entries']:deliveries[e['deliver_after_root']].append(e)
+
+def replay(mode='F2',direction=1,bits=24,mutation=None):
+ major=mode=='bound';binary=mode=='F2';v=W.v
+ unit=(lambda i:1)if major else(lambda i:1<<i)if binary else(lambda i:1<<(bits*i))
+ def add(x,c,y):return x+abs(c)*y if major else x^y if binary and c&1 else x if binary else x+c*y
+ target_frames=[None]*v;target_paths=[[]for _ in range(v)];center_reads=[];target_hist=Counter()
+ def target_move(t,f):
+  previous=target_frames[t];assert previous is None or C.sub(previous,f),('actual target chronology',t,previous,f)
+  assert all(W.module.dot(C.cov[t],b)==0 for b in C.B[f]),('actual target cap',t,f)
+  d=C.dimf[f]-(0 if previous is None else C.dimf[previous]);assert d>=0
+  if d:target_hist[d]+=1
+  target_frames[t]=f;target_paths[t].append(f)
+ X0=[unit(i)for i in range(v)];Y0=[unit(v+i)for i in range(v)];Z0={r:unit(idx[r])for r in regs};x=X0[:];y=Y0[:];z=Z0.copy();top=1;done=[]
+ def track(value):
+  nonlocal top
+  if major:top=max(top,value)
+  return value
+ def value(s):
+  s=W.phys[s];return x[borrow[s]]if s in borrow else z[s]
+ def assign(s,val):
+  s=W.phys[s]
+  if s in borrow:x[borrow[s]]=track(val)
+  else:z[s]=track(val)
+ responses=W.adjoint() if mutation=='old_adjoint' else adj
+ def read(s):
+  if mutation=='omit_gauge_compensation' and s==new_gauge_selection[-1]['role']:return
+  for t,c in responses[s].items():
+   if s in W.gauge:target_move(t,W.gauge[s]['frame'])
+   else:assert target_frames[t]is None,'initial correction leaves target atD0'
+   y[t]=track(add(y[t],-direction*c,value(s)))
+ def gate(i,sign):
+  a,b,_=W.ops[i];assign(a,add(value(a),sign,value(b)))
+ for s in range(W.R):
+  if s not in W.gauge and s not in borrow and s not in removed:read(s)
+ for n,s in W.source.items():assign(s,add(value(s),1,x[n]))
+ for r in gauge_selection:
+  if mutation!='omit_gauge_mix' or r is not new_gauge_selection[-1]:x[r['partner']]=track(add(x[r['partner']],1,x[r['source']]))
+ def forward(i):
+  if i not in omitted:gate(i,1);done.append(i)
+ for k in schedule['center_prefix_events']:
+  event=schedule['events'][k];i=event['op']
+  if event['kind']=='early_mix':
+   r=early[i];a,b=r['partner'],r['source']
+   if mutation!='omit_early_mix' or r is not selection[0]:x[a]=track(add(x[a],1,x[b]))
+  else:forward(i)
+ for r,s in zip(W.g['roots'],W.w['rootroles']):
+  if r['kind']=='center':
+   for t in r['targets']:
+    assert all(f is None for f in target_frames),'copied centers scatter before any nonzero target movement'
+    center_reads.append((s,t));y[t]=track(add(y[t],direction,value(s)))
+ for e in terminal:
+  assert all(target_frames[t]is None for t in e['targets']),'terminal setup afterD0center scatter'
+  for t in e['targets']:
+   if t!=e['pivot']:y[t]=track(add(y[t],-1,y[e['pivot']]))
+ for j,i in enumerate(W.rest):
+  for s in at[j]:
+   if mutation!='late_phase1_gauge' or s not in new_phase_roles:read(s)
+  if mutation=='late_phase1_gauge'and j==168:
+   for s in sorted(new_phase_roles):read(s)
+  if i in bywrite:
+   e=bywrite[i];a,b,n=W.ops[i];target_move(e['pivot'],W.opframe[i]);y[e['pivot']]=track(add(y[e['pivot']],direction,value(b)))
+  else:forward(i)
+  if i in afterwrite:
+   e=afterwrite[i]
+   for t in e['targets']:
+    target_move(t,e['root_frame'])
+    if t!=e['pivot']:y[t]=track(add(y[t],1,y[e['pivot']]))
+ for s in at[len(W.rest)]:read(s)
+ for j,(r,s)in enumerate(zip(W.g['roots'],W.w['rootroles'])):
+  if r['kind']=='side' and j not in deletedroots:
+   for t in r['targets']:target_move(t,W.w['root_frame'][j]);y[t]=track(add(y[t],direction,value(s)))
+  for e in deliveries[j]:
+   a,b=e['carrier'],e['passive']
+   if b not in by_source:x[a]=track(add(x[a],1,x[b]))
+   for t in e['receivers']:target_move(t,e['deliver_frame']);y[t]=track(add(y[t],direction,x[a]))
+ for e in W.k['entries']:
+  if e['passive']not in by_source:x[e['carrier']]=track(add(x[e['carrier']],-1,x[e['passive']]))
+ if mutation=='undo_before_restore':
+  for e in W.k['entries']:
+   if e['passive']in by_source:x[e['carrier']]=track(add(x[e['carrier']],-1,x[e['passive']]))
+ for i in reversed(done):gate(i,-1)
+ if mutation!='undo_before_restore':
+  for e in W.k['entries']:
+   if e['passive']in by_source:x[e['carrier']]=track(add(x[e['carrier']],-1,x[e['passive']]))
+ for n,s in W.source.items():assign(s,add(value(s),-1,x[n]))
+ if binary:want=[Y0[t]^X0[t]for t in range(v)]
+ else:
+  want=Y0[:];cache={}
+  for root in W.g['roots']:
+   n=root['node']
+   if n not in cache:
+    mask=C.sup[n];val=0
+    while mask:low=mask&-mask;mask-=low;val=add(val,1,X0[low.bit_length()-1])
+    cache[n]=val
+   for t in root['targets']:want[t]=add(want[t],direction,cache[n])
+  for e in W.k['entries']:
+   val=add(X0[e['carrier']],1,X0[e['passive']])
+   for t in e['receivers']:want[t]=add(want[t],direction,val)
+ # Finish every local target at its codimension-one cap, then reverse the exact
+ # annihilator stream. All copied-center scatters lie before these movements,
+ # hence atD0 forward and atD1 after the complementary movements reverse.
+ reflected=Counter()
+ for t,chain in enumerate(target_paths):
+  assert chain
+  d=W.h-1-C.dimf[chain[-1]];assert d>=0
+  if d:target_hist[d]+=1
+  previous=[C.cov[t]];dimension=1
+  for f in reversed(chain):
+   assert all(W.module.dot(a,b)==0 for a in previous for b in C.B[f])
+   basis=C.A[f];step=len(basis)-dimension;assert step>=0
+   if step:reflected[step]+=1
+   previous=basis;dimension=len(basis)
+  if W.h>dimension:reflected[W.h-dimension]+=1
+ assert reflected==target_hist
+ assert len(center_reads)==sum(len(r['targets'])for r in W.g['roots']if r['kind']=='center')
+ assert sum(k*n for k,n in target_hist.items())==W.v*(W.h-1)
+ if major:return max(top,max(a+b for a,b in zip(y,want)),max(z[r]+Z0[r]for r in regs),max(a+b for a,b in zip(x,X0)))
+ assert y==want,('wrong targets',mode,[t for t in range(v)if y[t]!=want[t]][:10])
+ assert x==X0,('source restoration',mode,[s for s in range(v)if x[s]!=X0[s]][:10])
+ assert z==Z0,('dirty restoration',mode)
+ return dict(mode=mode,direction=direction,formal_columns=2*v+len(regs),sources=v,targets=v,dirty=len(regs),all_targets=True,all_source_and_dirty_restored=True,copied_center_target_reads=len(center_reads),all_forward_centers_at_D0=True,all_reflected_centers_at_D1=True,actual_target_histogram={str(k):n for k,n in sorted(target_hist.items())},actual_reflected_target_histogram={str(k):n for k,n in sorted(reflected.items())})
+if __name__=='__main__':
+ start=time.monotonic();out=dict(status='PASS_SCALAR_SOURCE471_RETIMED_PHASE1_GAUGES',selection=selection,gauge_selection=gauge_selection,new_gauge_selection=new_gauge_selection,geometry_and_paid_admission=False)
+ out['F2']=replay();print('PASS F2',out['F2'],'seconds',time.monotonic()-start,flush=True)
+ bound=replay('bound');bits=8*((bound.bit_length()+2+7)//8);out['bound']=bound;out['bits']=bits;out['terminal_sinks']=len(terminal);print('bound',bound,'bits',bits,flush=True)
+ out['integer']=[]
+ if True:
+  for direction in(1,-1):out['integer'].append(replay('Z',direction,bits));gc.collect();print('PASS integer',direction,'seconds',time.monotonic()-start,flush=True)
+ out['controls']={}
+ for mutation in ['omit_early_mix','old_adjoint','undo_before_restore','omit_gauge_mix','omit_gauge_compensation','late_phase1_gauge']:
+  try:replay(mutation=mutation)
+  except AssertionError as error:out['controls'][mutation]=dict(status='REJECTED',error=str(error))
+  else:raise AssertionError('Invalid mutation accepted: '+mutation)
+ assert (1<<bits)>2*bound
+ assert input_pins=={str(p):hashlib.sha256(p.read_bytes()).hexdigest()for p in input_paths}
+ out['input_pins']=input_pins
+ (HERE/'replay.json').write_text(json.dumps(out,indent=2)+'\n')
+ print('PASS actual471 source word, both integer signs, all columns and six controls',flush=True)
diff --git a/research/coordinated-crossover-pr200/newg/schedule.json b/research/coordinated-crossover-pr200/newg/schedule.json
new file mode 100644
index 0000000000..51c2c8df5a
--- /dev/null
+++ b/research/coordinated-crossover-pr200/newg/schedule.json
@@ -0,0 +1 @@
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diff --git a/research/coordinated-crossover-pr200/newg/schedule_audit.py b/research/coordinated-crossover-pr200/newg/schedule_audit.py
new file mode 100644
index 0000000000..c94e006726
--- /dev/null
+++ b/research/coordinated-crossover-pr200/newg/schedule_audit.py
@@ -0,0 +1,66 @@
+"""Independent event/value and address chronology audit of the actual471 retiming."""
+from pathlib import Path
+from collections import Counter,defaultdict
+import importlib.util,json,sys,hashlib
+sys.dont_write_bytecode=True
+HERE=Path(__file__).resolve().parent
+spec=importlib.util.spec_from_file_location('scalar471',HERE/'replay.py');m=importlib.util.module_from_spec(spec);spec.loader.exec_module(m);W=m.W;C=m.C
+schedule=m.schedule;events=[]
+def port(s):
+ p=W.phys[s];return('x',m.borrow[p])if p in m.borrow else('z',p)
+for e in schedule['events']:
+ i=e['op']
+ if e['kind']=='early_mix':
+  r=m.early[i];events.append(dict(op=i,kind='early_mix',dest=('x',r['partner']),source=('x',r['source']),frame=m.kpair[r['source']]['mix_frame']))
+ else:
+  a,b,n=W.ops[i];events.append(dict(op=i,kind='operation',dest=port(a),source=port(b),frame=W.opframe[i]))
+prefix=schedule['center_prefix_events'];deferred=schedule['deferred_events'];rest=schedule['remaining_events'];neworder=prefix+deferred+rest
+assert [events[k]['op']for k in prefix if events[k]['kind']=='operation']==W.phase1
+assert [events[k]['op']for k in deferred+rest if events[k]['kind']=='operation']==W.rest
+assert len(neworder)==len(set(neworder))==len(events)
+def trace(order):
+ writes={};inputs={};frames={};edges=0;paths=defaultdict(list)
+ for k in order:
+  e=events[k];a,b,f=e['dest'],e['source'],e['frame'];inputs[k]=(writes.get(a,-1),writes.get(b,-1))
+  for p in[a,b]:
+   old=frames.get(p)
+   assert old is None or C.sub(old,f),('address descent',k,p,old,f)
+   if old is not None:edges+=1
+   frames[p]=f;paths[p].append(f)
+  writes[a]=k
+ return inputs,writes,frames,edges,paths
+oldinput,oldwrite,oldframes,oldedges,oldpaths=trace(range(len(events)));newinput,newwrite,newframes,newedges,newpaths=trace(neworder)
+assert oldinput==newinput and oldwrite==newwrite and oldframes==newframes
+assert set(oldpaths)==set(newpaths)
+def jumps(path):
+ previous=0;out=Counter()
+ for f in path:
+  d=C.dimf[f];assert d>=previous
+  if d>previous:out[d-previous]+=1
+  previous=d
+ return out
+assert all(jumps(oldpaths[p])==jumps(newpaths[p])for p in oldpaths)
+assert all([C.dimf[f]for f in oldpaths[p]]==[C.dimf[f]for f in newpaths[p]]for p in oldpaths), 'complete actual address-frame dimension stream changed' 
+assert all(all(C.sub(a,b)and C.sub(b,a)for a,b in zip(oldpaths[p],newpaths[p]))for p in oldpaths), 'complete exact physical subspace stream changed'
+_,prefixwrite,prefixframes,_,_=trace(prefix)
+for r,s in zip(W.g['roots'],W.w['rootroles']):
+ if r['kind']=='center':assert prefixwrite[port(s)]==oldwrite[port(s)]
+centerports={port(s)for r,s in zip(W.g['roots'],W.w['rootroles'])if r['kind']=='center'}
+assert all(not(centerports&{events[k]['dest'],events[k]['source']})for k in deferred+rest),'no later middle consumer bypassed atcopied center'
+center_ranks=Counter()
+for j,(r,s)in enumerate(zip(W.g['roots'],W.w['rootroles'])):
+ if r['kind']=='center':
+  f=W.w['root_frame'][j];assert C.sub(prefixframes[port(s)],f)and C.sub(f,prefixframes[port(s)])
+  center_ranks[C.dimf[f]]+=1
+newports={port(s)for s in m.new_phase_roles}
+assert all(not(newports&{events[k]['dest'],events[k]['source']})for k in prefix)
+assert {events[k]['op']for k in prefix if events[k]['kind']=='early_mix'}==set(m.early)
+assert all(events[k]['kind']=='operation'for k in deferred+rest)
+assert all(W.readtime[s]==0 for s in m.new_phase_roles)
+assert all(W.readtime[s]>=168 for s in W.gauge if s not in m.new_phase_roles)
+assert all(W.readtime[s]<=W.first[s]for s in W.gauge)
+assert all(W.last[d]<W.readtime[b]for b,d in W.pairs)
+# The only pre-center target reads remain original ungauged corrections at0.
+# New source/Kmix movement affects X only; all gauge corrections followcenter.
+receipt=dict(status='PASS_SOURCE471_EXACT_EVENT_AND_ADDRESS_RETIMING',events=len(events),center_prefix_events=len(prefix),center_prefix_operations=len(W.phase1),deferred_operations=len(deferred),early_source_mixes_in_prefix=len(m.early),source_heads=len(m.borrow),formal_columns=2*W.v+len(m.regs),every_actual_input_preceding_write_preserved=True,every_final_write_preserved=True,all_physical_frame_paths_nested=True,all_actual_address_frame_dimension_streams_preserved=True,all_exact_address_subspace_streams_preserved=True,all_address_jump_histograms_preserved=True,physical_frame_inclusions_checked=newedges,all_center_reads_see_identical_last_write=True,no_later_center_middle_consumer=True,copied_center_rank_histogram=dict(center_ranks),copied_center_paid_copy_rank_mass=sum(d*n for d,n in center_ranks.items()),new_source_heads_physically_untouched_before_centers=True,all_target_gauge_reads_after_center_scatter=True,all_new_gauges_before_deferred_ops=True,all_original_gauge_reads_shifted_by168=True,all_alias_lifetimes_retained=True,input_pins={p.name:hashlib.sha256(p.read_bytes()).hexdigest()for p in[HERE/'replay.py',HERE/'selection.json',HERE/'schedule.json',Path(__file__)]})
+(HERE/'schedule-audit.json').write_text(json.dumps(receipt,indent=2)+'\n');print(json.dumps(receipt,indent=2),flush=True)
diff --git a/research/coordinated-crossover-pr200/newg/selection.json b/research/coordinated-crossover-pr200/newg/selection.json
new file mode 100644
index 0000000000..3a0ad4bcfb
--- /dev/null
+++ b/research/coordinated-crossover-pr200/newg/selection.json
@@ -0,0 +1,550 @@
+[
+  {
+    "role": 12627,
+    "source": 114,
+    "partner": 113,
+    "mix_frame": 126095,
+    "targets": [
+      32,
+      33,
+      34,
+      35
+    ]
+  },
+  {
+    "role": 14075,
+    "source": 82,
+    "partner": 81,
+    "mix_frame": 126167,
+    "targets": [
+      48,
+      49,
+      50,
+      51
+    ]
+  },
+  {
+    "role": 14424,
+    "source": 26,
+    "partner": 25,
+    "mix_frame": 126293,
+    "targets": [
+      120,
+      121,
+      122,
+      123
+    ]
+  },
+  {
+    "role": 14422,
+    "source": 18,
+    "partner": 17,
+    "mix_frame": 126311,
+    "targets": [
+      128,
+      129,
+      130,
+      131
+    ]
+  },
+  {
+    "role": 14080,
+    "source": 10,
+    "partner": 9,
+    "mix_frame": 126329,
+    "targets": [
+      328,
+      329,
+      330,
+      331
+    ]
+  },
+  {
+    "role": 12644,
+    "source": 330,
+    "partner": 329,
+    "mix_frame": 125609,
+    "targets": [
+      384,
+      385,
+      386,
+      387
+    ]
+  },
+  {
+    "role": 14072,
+    "source": 2,
+    "partner": 1,
+    "mix_frame": 126347,
+    "targets": [
+      408,
+      409,
+      410,
+      411
+    ]
+  },
+  {
+    "role": 14436,
+    "source": 586,
+    "partner": 585,
+    "mix_frame": 125033,
+    "targets": [
+      480,
+      481,
+      482,
+      483
+    ]
+  },
+  {
+    "role": 14434,
+    "source": 578,
+    "partner": 577,
+    "mix_frame": 125051,
+    "targets": [
+      488,
+      489,
+      490,
+      491
+    ]
+  },
+  {
+    "role": 14098,
+    "source": 474,
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index 0000000000..afaf38f79c
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new file mode 100644
index 0000000000..5a6c8165c9
--- /dev/null
+++ b/research/coordinated-crossover-pr200/newg/source469-receipt.json
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diff --git a/research/coordinated-crossover-pr200/rank/PROVENANCE.json b/research/coordinated-crossover-pr200/rank/PROVENANCE.json
new file mode 100644
index 0000000000..d43dbcc2e9
--- /dev/null
+++ b/research/coordinated-crossover-pr200/rank/PROVENANCE.json
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+  "assistance": "Prepared by eumemic with substantial OpenAI Codex assistance. Apache-2.0; all inherited notices retained.",
+  "admission": "Literal F2 and both defining-integer signs,17 whole-word controls including integer sign corruption, independent complete source and target paths in both reflections, local integer transfer matrices, prime witnesses, concrete banks and exact paid assembly."
+}
diff --git a/research/coordinated-crossover-pr200/rank/selection.json b/research/coordinated-crossover-pr200/rank/selection.json
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diff --git a/research/coordinated-crossover-pr200/requirements.txt b/research/coordinated-crossover-pr200/requirements.txt
new file mode 100644
index 0000000000..784141bdc2
--- /dev/null
+++ b/research/coordinated-crossover-pr200/requirements.txt
@@ -0,0 +1,2 @@
+numpy==2.3.5
+scipy==1.17.0
diff --git a/research/coordinated-crossover-pr200/search/EXTREMAL-CUT.json b/research/coordinated-crossover-pr200/search/EXTREMAL-CUT.json
new file mode 100644
index 0000000000..0d897b948d
--- /dev/null
+++ b/research/coordinated-crossover-pr200/search/EXTREMAL-CUT.json
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+  "elapsed": 55.79363829101203,
+  "baseline_manifest_sha256": "2e17de69bea7d0b2a0f95ffabcb5bb3a271f6c4ad18764f67c5bf6e6e453b681",
+  "theorem_source": "Dugongue PR216, 34abdc58ed77329c8bf9af5634531fc41939ba3c; OpenAI Codex assistance"
+}
diff --git a/research/coordinated-crossover-pr200/search/PR211-NOTICE b/research/coordinated-crossover-pr200/search/PR211-NOTICE
new file mode 100644
index 0000000000..9edc257237
--- /dev/null
+++ b/research/coordinated-crossover-pr200/search/PR211-NOTICE
@@ -0,0 +1,18 @@
+Packed diagonal bit supplier. Apache-2.0.
+Prepared by Rohan Arun (rohanarun) with Anthropic Claude assistance.
+
+Reused, with credit:
+- Evan McKinney, PR197 (OpenAI Codex assistance): completed-bank packing, exact
+  gauge charts, nine-colour bank incidence, routing bound, arithmetic, audit
+  and endpoint controls. banks.py and controls.py are byte-identical extracts;
+  packing.py, arithmetic.py, audit.py and verify.py are adaptations to PR200.
+- Chafik Boukhalfa, PR200 at a1175449f34d39ff933d9d8ab23ced1f32b290ec (Anthropic
+  Claude and OpenAI Codex assistance): the fixed-coordinate/face-diagonal bit
+  word, physical layer, terminal sinks, prime witnesses and package verifier.
+- Avi Eisenberg, PR193 at 187e1010ac8b259af8e9b5166f68b64bc27b4b47: the v4
+  source-assisted complex supplier and balanced assembly bridge; icekylinx for
+  the source-assisted method (PR184) and the paired-cube framework (PR144).
+- maxime-fleury, PR204: the model of this lever and its predicted value.
+- eumemic (PR168 lineage), jamesyc (PR166 terminal lemma), PR185/187 finite
+  ordinary-leaf composition, and every contributor credited by those packages.
+Original notices remain in the pinned checkouts.
diff --git a/research/coordinated-crossover-pr200/search/PR216-COMPONENTS.json b/research/coordinated-crossover-pr200/search/PR216-COMPONENTS.json
new file mode 100644
index 0000000000..348f67ced4
--- /dev/null
+++ b/research/coordinated-crossover-pr200/search/PR216-COMPONENTS.json
@@ -0,0 +1,35376 @@
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+  },
+  "fixed_coarse_numerator_delta": -0.014457314550327283,
+  "elapsed": 14.928294375014957,
+  "source_files": {
+    "witnesses/additional-bases.json": "f065829ab1d04dc9d7370b4585d9807f1969cfadb6cf67ced67495f78f10d28b",
+    "research/MAJORIZATION-CUT.md": "4f43b15d9a51e6d08d8619fa442aa6b34c27b2e666b6e724a5b09bf7ebf5492c"
+  },
+  "source_url": "https://github.com/CrocSwap/integer-mult-bounds/pull/216"
+}
diff --git a/research/coordinated-crossover-pr200/search/PR216-MAJORIZATION-CUT.md b/research/coordinated-crossover-pr200/search/PR216-MAJORIZATION-CUT.md
new file mode 100644
index 0000000000..df375e80da
--- /dev/null
+++ b/research/coordinated-crossover-pr200/search/PR216-MAJORIZATION-CUT.md
@@ -0,0 +1,44 @@
+# Global extremal-frame majorization cuts
+
+2026-10-09. Pure mathematics and checked-integer C++ discovery, no Lean certificate, no all-column/prime closure, no new published kappa.
+
+## Sufficient theorem
+
+Let a legal finite physical compiler word have one frame F_i at each operation, fixed source/gauge/root/alias/target boundaries, and paid directed chain edges i -> j. Every edge has F_i subset F_j. Let f be increasing and concave on permitted nonnegative rank gaps, with f(0)=0. Construct ANY second legal schedule M_i with F_i subset M_i for every mutable i and identical fixed boundaries. Assume every M_i is nondegenerate and contains the required exact value span.
+
+There is a linear-size explicit minimum-cut majorant for selecting F_i or M_i independently at each operation. A strict reduction of its objective is a sufficient certificate of a strict reduction of the actual complete paid rank objective. All stock, endpoints, source/target legality, chronological aliases and total rank mass remain unchanged. This is a legal-neighborhood optimization theorem; it does not claim global optimality or an all-size multiplication theorem.
+
+For an edge with both endpoints mutable, write d_i=dim(M_i)-dim(F_i), d_j=dim(M_j)-dim(F_j), r=dim(F_j)-dim(F_i). The 00,01,11 cases always nest, and have costs E00=f(r), E01=f(r+d_j), E11=f(r+d_j-d_i). The 10 case nests precisely when M_i subset F_j; if legal, E10=f(r-d_i).
+
+Define unary contributions u_i=E11-E01 and u_j=E01-E00. The affine objective E00+u_i*x_i+u_j*x_j agrees with every 00,01,11 cost. If 10 is illegal add a hard arc i -> j, prohibiting x_i=1,x_j=0. If 10 is legal its affine price is E00+E11-E01. Concavity implies E00+E11 >= E01+E10: these are two inner points versus two outer points with equal sum. Thus the affine price majorizes the true 10 cost. Both implications follow without requiring equal subspaces or a common join direction. For fixed boundaries retain the exact one-variable cost; no hard arc is needed where an endpoint contains both alternatives.
+
+Sum over ALL actual physical and target-chain edges, including terminal pivot movements. Positive unary u becomes i -> sink; negative unary becomes source -> i with the usual constant offset. Use a hard capacity strictly greater than the sum of absolute unary capacities. The unchanged all-zero assignment is feasible and has the baseline objective. Consequently a minimizing cut with negative surrogate delta has actual delta at most that negative number. This proves sufficiency. Concavity defects are explicitly bounded, so the surrogate can also quantify how far it is from the best schedule in its binary neighborhood; this is useful for dropping an unproductive route, not a research result by itself.
+
+A contraction schedule L_i subset F_i reduces to this theorem by replacing each frame with its ordinary annihilator, reversing every paid edge and exchanging F,L accordingly. Dimensions change from k to h-k, so every paid rank gap is exactly preserved. Nondegeneracy and required-value containment are verified on the original physical frames, not presumed from the annihilator representation.
+
+## Concrete second schedules
+
+The greatest feasible candidate is reconstructed backwards: M_i is the intersection of every successor frame, including all literal endpoint caps. The required values and current F_i stay contained because the initial word is legal. If a candidate frame is degenerate, freeze that operation at its current frame and restart backwards propagation. Since each failed round freezes at least one operation, the process terminates in at most N rounds with a legal schedule; in the observed PR200 input no freeze was necessary.
+
+The least candidate is reconstructed forwards: L_i is the span of its exact required source-label value vectors and every predecessor frame, including literal source/gauge/target boundaries. This is a subspace of the current legal F_i. Freeze a degenerate candidate at F_i and continue forwards; successors still remain within their current frames. Again the observed PR200 input needed no freeze. Both candidates are explicit finite constructions rather than assumed transformation gadgets.
+
+## C++ implementation and actual results
+
+- `greatest.cpp` is the complete loader, exact fraction-free integer basis/kernel operations, checked signed-128 multiplication/subtraction, canonical interning, full physical/target graph reconstruction, nondegeneracy predicate for G=I-J/9, Dinic nomination and literal histogram recount.
+- Input is the unchanged PR200 a1175449f34d39ff933d9d8ab23ced1f32b290ec word plus the already admitted 302-frame checkpoint. Staged immutable JSON and executable are on F:/proof/research/beyond-nlogn-20261009/threshold/frames. No Python was executed for this lane.
+- Current graph: 41,288 operation nodes, 126,520 paid edges, rank mass 405,456. The stage factor3 is not part of this graph-level histogram.
+- Complete greatest schedule changes14,487 operations and worsens entropy867200.2563234223 -> 898639.1065801512. Rejected as a whole.
+- Its majorant cut chooses48 operations and improves entropy to867140.3594551166. Exact histogram delta {0:-24,2:+96,4:-72,10:-48,12:+48}. Surrogate delta -45.67759759837121, actual entropy delta -59.8968683057. This is a candidate beyond prior common-direction cuts, not a threshold-scale improvement.
+- On that checkpoint the full least schedule worsens entropy to897647.8189150952. Rejected as a whole.
+- Its majorant cut chooses1410 operations and reaches entropy866438.7403496904. Final cumulative1458 operation changes relative to the 302 checkpoint. Rank mass still405456. Total graph entropy improvement761.5159737319. There are zero degenerate freezes.
+- `greatest-summary.json` contains every full exact histogram and surrogate decision ledger. `greatest-bases.json` contains the explicit integer rational bases for all1458 additional operations relative to the 302 checkpoint; it is NOT a standalone public-relative replacement file. Compose with the original302 witness before any full replay.
+
+Every changed basis is independently reduced and every literal physical/target edge is rechecked for containment; all source/gauge/root/alias/read/terminal identities remain fixed. This does NOT replace the untouched original all-formal-column and all-used-frame prime certificate. New bases need that admission before publication or kappa pricing as a record. No such closure is claimed in this lane.
+
+## Threshold assessment and route selection
+
+The new graph entropy reduction is too small for the requested kappa >= .0007. At baseline normalized delta1936, approximately48 additional useful rank-deficit units would be threshold-scale at approximately unchanged entropy. The copied-center loss528 is therefore a higher-leverage target than continuing tiny frame cuts. Parent is pursuing a concrete hybrid264 missing-port and center dirty-completion route. This lane stops expensive frame cuts and preserves all exact witnesses and the generic sufficient construction for reuse.
+
+## Retrieval and literature provenance
+
+Prior exact cut contracts and proofs were read from the whole-subspace mincut notes, the rank-moment-to-exponent notes, PR200 frame continuation and actual public integer geometry checker. The canonical tree search found Mathlib `Analysis/SpecialFunctions/Log/NegMulLog.lean` (`strictConcaveOn_negMulLog`) and existing Problem18ProductAwareCost/Calculus majorization mechanisms; these are source retrieval, no new imports or Lean certification. The ordinary directed minimum-cut method and concavity/majorization are classical. This particular variable-frame majorant applies those methods to the compiler's complete physical/target ledger. Publication novelty is not established.
\ No newline at end of file
diff --git a/research/coordinated-crossover-pr200/search/PR216-NOTICE b/research/coordinated-crossover-pr200/search/PR216-NOTICE
new file mode 100644
index 0000000000..ba5ea80dd7
--- /dev/null
+++ b/research/coordinated-crossover-pr200/search/PR216-NOTICE
@@ -0,0 +1,399 @@
+A sharper exponent for integer multiplication
+Copyright 2026 Douglas Colkitt
+
+This project includes unmodified source files from OpenAI's manuscript
+"Integer multiplication below n log n" and its accompanying repository:
+https://github.com/openai/math
+Commit: adc7f1241b42e322a6451854ab7e4b4c146bf78a
+
+The original manuscript identifies OpenAI as its author. Its Apache-2.0
+license is retained at upstream/LICENSE. The exact source URLs and hashes
+are recorded in upstream/manifest.json. Proposed modifications are supplied
+as separate files under patches/; upstream source files remain unchanged.
+
+The original local research note, scripts, certificates, patches, and
+documentation were prepared by Douglas Colkitt with assistance from OpenAI
+Codex. The community release incorporates work by the contributors identified
+in the notices below; it is not solely Douglas Colkitt's work.
+This project is not an official OpenAI release or an OpenAI endorsement.
+
+Related contributions (not imported in this checkpoint):
+Zhihao Chen (jacklightChen), PR #7, supplied an earlier ternary five-subset
+motif, finite-alphabet transfer and a different compressed producer:
+https://github.com/CrocSwap/integer-mult-bounds/pull/7
+The present checkpoint claims no priority for that motif. Related compressed
+complex-network work was submitted by eumemic in PR #3:
+https://github.com/CrocSwap/integer-mult-bounds/pull/3
+The pending contributions and their stronger bounds require their own review;
+this notice does not represent adoption or verification of their claims.
+
+Additional community work, including parallel and superseded submissions, is
+acknowledged in CONTRIBUTORS.md, with pinned submission metadata in
+docs/research/contribution-snapshot.json. These acknowledgements do not imply
+joint authorship of the checkpoint, incorporation of pending code, or verification
+of submitted mathematical claims. Original source-specific notices remain in force.
+
+Community release (accepted by conditional maintainer audit):
+The following original contributor notices apply to the imported PR #39 chain.
+The preceding non-import statements describe the preserved 2^-30 checkpoint.
+
+Ternary five-subset construction and common-four-point resynthesis
+Copyright 2026 Zhihao Chen (jacklightChen)
+
+This is Zhihao Chen's first submission of this specific construction to this
+repository. Research, proof development, code and verification were prepared
+with substantial assistance from GPT-6 Astra (OpenAI Codex), as identified by
+the contributor. This is not independent review or an OpenAI endorsement.
+Future research building on this contribution should explicitly acknowledge
+Zhihao Chen (jacklightChen), cite the accompanying note and pull request, and
+retain the preceding contributors' attribution. This notice records research
+provenance; the repository's Apache-2.0 license remains unchanged.
+
+Dependencies retained from eumemic's Claude-assisted PRs #3 and #5:
+https://github.com/CrocSwap/integer-mult-bounds/pull/3
+https://github.com/CrocSwap/integer-mult-bounds/pull/5
+PR #5 commit: d3d370c34937503bd3fa63c83d505a05e8e50893
+These include complex_circuit.py, complex_network.py, fast_gaussian.py,
+their generators, construction notes and certificates. The Gaussian input
+scaling proof is corrected in this derivative; see the modification notice below.
+Aligned pairing also appears in Bortlesboat's PR #2. Retained totals and bank
+sharing appear in dleen's PR #4 and cheaper retained centers in eumemic's PR #6.
+Their results and ideas are credited separately from this new construction.
+
+Bulk recursion and controlled tensor basis, 8 October 2026
+Copyright 2026 icekylinx
+
+The new projector batching, controlled simultaneous basis, mixed-width tape
+recursions, whole-residual complex recursion, and dependency-path precision
+guard in the batched-23 proof package are contributed by icekylinx, developed
+with substantial OpenAI GPT-6 Astra assistance. Review corrections and
+repository integration were prepared with OpenAI Codex assistance. They build on the retained
+PR #7 finite producers and the preceding contributions credited above.
+The new package retains the repository's Apache-2.0 license and the original
+authors' attribution. It is not an official OpenAI release or endorsement.
+
+Modification notice, 8 October 2026 (icekylinx): the retained fast Gaussian
+proof now bounds shifted approximation error by F=2^ceil(1.14 alpha^2).
+The P=34p precision budget and asymptotic exponent are unchanged. New batching
+checks are integrated into make verify. Existing attribution and licenses
+are retained.
+
+Partial-swap extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The partial-swap framing, binary retained-total triple construction,
+dependency-preserving positive labels and carrier reuse, unequal (25,23,57)
+common basis, improved complex batching, and exact assembly are contributed
+by icekylinx with substantial OpenAI GPT-6 Astra and Codex assistance.
+This extension directly inherits PR #10, commit
+62691e395a0458ce089a1c7b5d89e74291e95e29. The portable producer adapters
+modify the retained paired-exclusion and shared-point implementations credited
+above. The original licenses, attribution, and Gaussian correction are retained.
+
+Endpoint-gauge extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The complementary auxiliary endpoint gauges, (32,30,40) controlled basis,
+general nondegenerate binary phase residuals, retained-center complex producer,
+updated path guard and exact assembly are contributed by icekylinx with
+substantial OpenAI GPT-6 Astra and Codex assistance. This extension inherits
+f2ab41aebad47861caf6316282c1793e5513845e and its PR #10 ancestry. New portable
+adapters retain the paired-exclusion and shared-point contributions credited
+above. Original licenses, notices and the Gaussian correction are retained.
+
+Structured-basis and semantic-bulk extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The selected (32,30,36) bit construction, A1/A5 block refinements, fixed
+middle-axis I+J profiles with bounded-minor CRT certificates, mixed dyadic
+complex centers, complete data-front batching and circuit-specific assembly
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex
+assistance. The immediate parent is ed90fd940279c336ebc45968631bdebfb087b505.
+
+The semantic/bulk compatibility argument, literal scalar charge, product row
+stock and parameterization adapt Zhihao Chen (jacklightChen)'s PR #23,
+commit 661ebabc1076c9f525d7ce4967c876b203fa9827. Translated complex endpoints
+are adopted from the same contributor's PR #21,
+commit 5ba6cf0bfb68f2be8d15610e7972207c50254d6a, following eumemic's source-frame
+work and the earlier nested-basis contributions.
+
+Completed-child semantic precision, arbitrary-coordinate routing, phase-cell
+Gaussian inversion and bulk tape transfer are credited to the RaD project
+(hipotures), commit 45d9b60355872041f9275f77c057514def0d45bc, exposed through
+PR #20 at 4f8d6c8272b5ff307a0da51df545ec3cd96a8b6e. Their selected proof
+sources retain their separate CC0 license and original NOTICE under
+references/semantic-bulk/rad20. The PR #23 source retains its recorded OpenAI
+Codex assistance; RaD retains its recorded Codex and GPT-6.1-Sol Ultra
+assistance. These are source attributions, not claims about this session's
+runtime model. Related contributions by Rohan Arun and Aurel Prosz remain
+acknowledged in the imported notices. No exclusive priority or additional
+license condition is asserted.
+
+Copied retained-center extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The copied retained-center schedule in both orientations, its complex-phase
+transfer, the Wm-N+L rank budget, and the selected (25,23)/(28,28) composition
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex
+assistance. This extension inherits PR #32 at
+0ef3aeb61f55cc0b321ce6a0ef00acee25cefe52 and the preceding PR #10/#18/#24 work.
+
+The two-stage topology and paid copied-stream endpoint correction are due
+to Aurel Prosz (Paureel), c82d09eb4781b68435e3fa3eb6beea72fa900fab,
+through Zhihao Chen (jacklightChen)'s PR #29 unequal-axis composition,
+9d963275075fa98f1da821e27757b238dafd6b3c. That source also credits Swapnil
+Jain's two-stage development at ae405eb474d1486b2d8aef90139869f927f4836a.
+
+The contiguous corner method, incidence factorization and exact partition
+certificates are due to Rohan Arun (PR #31), method commit
+02f68f95369c39955c5ffcfddbd75cd83e26b305. The selected (25,23) certificate
+specializes Dominik Scholz's parameterization (PR #33),
+a499a345c040aa15b68e984a0904eadc485c7a7e. Their source notices and proof
+snapshots are preserved in references/copied-centers. Historical OpenAI
+Codex, GPT-6 Astra and Anthropic Claude Opus 5.5 assistance is credited to
+the respective imported sources. Existing PR #21/#23 and RaD attribution,
+licenses and the Gaussian correction remain in force.
+
+Copied centers with reversed (23,25) data corners, 2026-10-08
+Rohan Arun with substantial OpenAI Codex assistance supplies the composition,
+exact dimension23 controls and cost/assembly integration. James Chang's PR34
+supplies the reversed boundary family; icekylinx's PR36 supplies copied retained
+centers and the selected complex graph. The balanced assembly remains attributed
+to RaD/hipotures and its PR34 specialization. Retain Dominik Scholz PR33, Rohan
+Arun PR31, Zhihao Chen PR29/21/23, Paureel/Aurel Prosz, Swapnil Jain, icekylinx,
+eumemic, Douglas Colkitt, OpenAI and all earlier notices and hypotheses.
+This is not a claim of formal verification, new foundational rank identities,
+global optimality or measured runtime speedup.
+
+Fixed middle basis with copied reversed corners (2026-10-08):
+Rohan Arun, with substantial OpenAI Codex assistance. Composition and
+simultaneous-basis proof retain icekylinx PR32/36, James Chang PR34,
+Dominik Scholz PR35/38, Zhihao Chen, Paureel, Swapnil Jain, RaD/hipotures,
+eumemic and all preceding attribution and licenses. See research/copied-fixed-reversed.
+
+Changed-graph composition with copied centers and fixed bases, 8 October 2026
+Prepared with OpenAI Codex assistance.
+The new composition uses icekylinx PR36 copied centers and PR32 fixed-I+J
+projectors/profiler, Dominik Scholz PR35 two fixed factors and CRT argument,
+Zhihao Chen (jacklightChen) PR29/23 interfaces, Aurel Prosz (Paureel) and
+Swapnil Jain's two-stage development, Rohan Arun PR25/31 corner methods,
+Dominik Scholz PR33/38 dimension/composition work; James Chang (jamesyc)
+PR34 reversed geometry and balanced transfer; Rohan Arun PR37/39/40
+reversed integration, modular data tests and both-fixed baseline;
+RaD (hipotures) PR41 changed scalar DAGs, alternating point order and
+independent role compiler; RaD balanced routing/bulk interfaces, and all
+retained predecessors.
+Complete attribution and the conditional proof boundary are recorded in
+research/copied-fixed/PROOF.md; imported originals and source pins are retained.
+Original Apache-2.0 and CC0 notices remain applicable.
+
+Community follow-up integration (PRs #49, #45 and #26), 8 October 2026:
+Rohan Arun contributes weighted carrier matching and climbed summand/leave-one-out
+orders (PRs #44/#47/#49), with Anthropic Claude assistance. The final construction
+composes Chafik Boukhalfa's changed-graph checks, exact rational data recovery and
+reordered exclusion sums (PRs #43/#46/#48, OpenAI Codex assistance) with RaD /
+hipotures's alternating pair order and independent physical role compiler (#41).
+All predecessor attribution and conditional proof interfaces remain in force.
+
+Alejandro Zarzuelo Urdiales contributes Gaussian parity and canonical arithmetic
+proofs, finite tensor execution, profile precision accounting and mixed-center
+reference checks (#45), with substantial OpenAI Codex assistance. See
+research/gaussian-parity-synthesis/PROVENANCE.md for his full attribution.
+Ryan S / princezuda contributes historical Lean certificate arithmetic, algebraic
+frame/movement/guard lemmas and independent paired-circuit checks (#26). Original
+source notices and historical snapshots are preserved. Neither formal package
+claims a full formal proof of the multiplication algorithm or a new kappa.
+
+Joint frame compilation of skip-prefix strips
+
+The local compiler and exact finite witnesses were prepared by eumemic with
+substantial OpenAI Codex assistance. Avi Eisenberg / ikeboy with Anthropic
+Claude assistance supplied the unmodified PR #53 skip-prefix producer.
+The pinned PR #48 pipeline is Chafik Boukhalfa's contribution, incorporating
+Rohan Arun, RaD / hipotures, icekylinx, Dominik Scholz, James Chang,
+Zhihao Chen, Aurel Prosz / Paureel, Swapnil Jain, eumemic, Douglas Colkitt,
+OpenAI, David Harvey, Joris van der Hoeven and the earlier contributors
+credited in the retained sources. See references/frame-compiler/pr48/NOTICE
+and references/frame-compiler/pr53/NOTICE for their original attribution.
+No authorship or review by those predecessors is claimed for this compiler.
+
+Joint dual-skip/frame-compiler composition (2026-10-08)
+---------------------------------------------------
+Rohan Gupta / gupt1156 with Anthropic Claude assistance: PR55 dual-skip
+scalar graph; original LICENSE and NOTICE are retained under
+references/frame-compiler/pr55. eumemic with substantial OpenAI Codex
+assistance: PR57 joint reversible frame compiler, serialized physical-word
+replay and profile reconstruction. Chafik Boukhalfa with OpenAI Codex
+assistance: this composition, independent integration and exact certificate.
+Avi Eisenberg / ikeboy, Rohan Arun, RaD / hipotures, icekylinx, and all
+predecessor authors retain the notices above and in pinned source files.
+No original copyright or Apache-2.0 notice is removed.
+
+Ranked reclamation after the PR58 joint-dual composition (2026-10-08)
+----------------------------------------------------------------
+Chafik Boukhalfa with OpenAI Codex assistance: prioritize retired physical
+slots by descending current frame rank, retaining slot ID as the tie-breaker;
+complete replay, exact profiles, comparison and adversarial controls.
+The dedicated joint_dual_reclaim_compiler.py is derived from eumemic's PR57
+compiler. Its original shared compiler remains byte-identical. The PR55
+producer and all preceding attribution remain unchanged. The complete PR58
+comparison certificate and its original source/proof provenance are retained
+under references/frame-compiler/pr58.
+
+Skip-prefix strip producer, 8 October 2026
+By Avi Eisenberg (ikeboy), with Anthropic Claude (Claude Opus 5.5)
+assistance. The skip-prefix leave-one-out strip layout in
+research/skip-strips/skip_graph.py is new. Everything else reuses Chafik
+Boukhalfa's PR #43/#46/#48 research/copied-fixed pipeline, its verification
+code, exact data recovery and support-order sorting, together with:
+- Rohan Arun's PR #44 weighted carrier matching optimizer;
+- RaD (hipotures) PR #41 alternating point order;
+- every predecessor credited above.
+Complete attribution and the conditional proof boundary are recorded in
+research/skip-strips/PROOF.md. Original Apache-2.0 and CC0 notices remain
+applicable.
+
+Local split-pair skip-prefix research continuation (2026-10-08)
+Prepared with substantial OpenAI Codex assistance at the user's request.
+New contribution: singleton/pair recursive partition refinement, finite summand
+order and moment search, and verified composition with inherited paid clones.
+Avi Eisenberg/ikeboy with Anthropic Claude assistance: PR53 skip-prefix strips.
+Chafik Boukhalfa with OpenAI Codex assistance: PR54 original-envelope clone
+specialization and PR43/46/48 checking interfaces. RaD/hipotures: PR51
+whole-chain/source-partition cloning and earlier producer/interfaces.
+Rohan Arun: PR44 weighted matching and earlier fixed/reversed work, with all
+stated historical AI assistance. All predecessor notices remain applicable.
+
+Original-envelope whole-chain clones on skip-prefix strips (research/skip-clones):
+Chafik Boukhalfa, prepared with OpenAI Codex assistance. The complete-chain
+source-partition clone operation is adapted from RaD / hipotures, PR #51,
+with original envelopes retained here. The scalar base is Avi Eisenberg's
+PR #53 skip-prefix layout, prepared with Anthropic Claude assistance.
+All credited PR #43/#46/#48, PR #44 and earlier code/proof notices remain
+in force. This extension is provided under the repository's Apache-2.0 terms.
+
+Community review checkpoint #50–#61, 8 October 2026
+Maintainer integration and independent arithmetic: Douglas Colkitt, with OpenAI Codex.
+Avi Eisenberg / ikeboy: skip-prefix strip framework (#53), with Anthropic Claude.
+Rohan Gupta / gupt1156: parallel order search (#50, stated Antigravity assistance)
+and dual-suffix strips (#55, Anthropic Claude assistance).
+Eumemic: equal-frame joint compilation and paid signal reclamation (#57), with Codex.
+Chafik Boukhalfa: original-envelope paid clones (#54), composition (#58) and ranked
+reclamation (#60), with Codex. Rohan Arun: order search (#52) and positive-frame
+composition (#56), with Anthropic Claude and stated Codex validation assistance.
+RaD / hipotures: enlarged frames, source partitions and paid whole-chain clones
+(#51), with Codex. Rohan Garg / rohangar1: singleton/pair recursion and paid-clone
+composition (#59), with Codex. Alejandro Zarzuelo Urdiales: exact parameter
+refinement, scoped Lean arithmetic, matrix/parity and search tools (#61), with
+his disclosed multi-tool and Codex assistance.
+Parallel #50/#52 records are reviewed and credited without replacing the earlier
+climbed-48 implementation; their exact commit history is retained. The separate
+#51 witness is retained as research; its source-graph/negative-basis data proof
+has not received the same full replay as #56's consumed compiler machinery.
+See docs/research/community-round2-review.md for precise validation boundaries.
+All earlier licenses, contributions and notices remain applicable.
+
+Interval strips and core-aware pair assembly
+
+The interval leave-one-out strips and the core-aware group-pair assembly in
+research/pair-assembly were prepared by Avi Eisenberg / ikeboy with Anthropic
+Claude assistance. They build on PR #53's skip-prefix pipeline and its
+inherited PR #48 sources. The stacked witness in research/pair-assembly/frame
+uses eumemic's PR #57 joint frame compiler unchanged; its driver and verifier
+are adaptations of PR #57's skip_frame_compiler.py, skip_frame_compose.py and
+verify_skip_frame.py. Original notices are retained.
+
+Stopped product-ring extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The opposite-bank generic projector factorization, componentwise atom-ring
+adapters, stopped recurrence and outer ordinary-interchange conversion,
+aligned pair-star producer with rational centers, exact denominator-21
+grid and selected (23,23)/(24,24) composition are contributed by icekylinx
+with substantial OpenAI GPT-6 Astra and Codex assistance. This extension
+builds on merged PR #36, 11817ccacb564bb7f98789c20dc11d3fece207e3, and is
+integrated on upstream main 56b66d58297deca1d7dd130247d720e960f77a37.
+
+The retained ordinary leaf uses the preceding projector, corner and
+copied-center proofs. The selected two-stage topology retains Aurel Prosz
+(Paureel), Swapnil Jain and Zhihao Chen's credits; the semantic assembly
+retains PR #21/#23 and RaD's attribution. Paired producers, carrier matching,
+original source notices and each imported license remain unchanged.
+
+Partial-gauge extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The pair-first cube assembly, compatible backward frame enlargement,
+partial source-gauge schedule and stopped whole-projector composition are
+contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex
+assistance, extending PR #104 at 948ce1510df750f4c18b96bdaef436a86f8bf834.
+
+The physical deferred bit word, reflected event accounting, fan-order
+clarification and signed two-stage endpoint are adopted from Zhihao Chen
+(jacklightChen), PR #97, e15350b66e4bf796153a68c240a6cc1b788abe3e, with
+Codex assistance. Its deferred readouts, V leaves, lifted frames and frozen
+witness originate in Swapnil Jain's 741e7aa078392553815df7926ee17ac5e25a8c38,
+with the original Claude disclosure retained. Cyclic interval and core-aware
+pair primitives build on Avi Eisenberg (ikeboy), PR #62,
+ad0f25ff7b23cff7f08ad237c2254e6ecf74257e, with Anthropic Claude assistance.
+Prior two-stage, copied-center, ordinary-leaf, semantic/bulk, routing and
+precision credits remain in force. Imported source notices and licenses
+are preserved under references/partial-gauge/; see SOURCES.json.
+
+Three-stage cover extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The arbitrary-subspace Clifford frame theorem, three-stage Cayley geometry,
+weighted q-adic GL cover, unit-lower weighted compiler, charged local-ring
+fallback, uniform batching, internal row borrowing and exact composition
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra assistance. This
+repository integration uses Codex assistance and extends
+PR #115, cf60442983c153a3a0f5272ea4713f99eba13c86.
+
+The selected local complex DAG and support/label replay are adopted from
+Copyright 2026 eumemic, PR #117, cbb05ce504d571546d9b7794c186a613c659c3bf,
+with its original Apache-2.0 terms and Claude assistance disclosure.
+The unchanged bit physical word retains Zhihao Chen (jacklightChen),
+PR #97, and Swapnil Jain's underlying witness and assistance notices.
+The prior opposite-bank, stopped ordinary-leaf, copied-center, topology,
+semantic/bulk, precision and analytic credits remain applicable.
+Original source notices and license bytes are preserved in
+references/three-stage-cover/pr117/ and references/partial-gauge/.
+
+Paired-cube and shared-core extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The paired-coordinate triple-cube identity, signed overlap channels,
+original-source involution, coordinate-star centers, concrete frame/carrier
+compiler, chronological source gauges, PR97 gauge-subset selection,
+orthogonal cross-stage routing and exact supplier/assembly integration
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra assistance.
+Repository integration uses Codex assistance and extends PR #130 at
+6a9970a530119174507904e23592fd59ede19a5d.
+
+The substantial completed-core workspace-sharing principle is credited to
+an664, PR #128, 530588a019b4a74f09180680c9e3961bf649ec89, with its reported
+OpenAI Codex assistance. No source code, partition or serialized witness
+from that PR is imported. The new routing and paired-motif integration
+are described separately and claim no exclusive priority for broad sharing,
+paired-coordinate, Fourier or carrier-matching ideas.
+
+Selected coarse queries restrict the retained eumemic PR #117 positive DAG;
+its original Apache-2.0 and Claude assistance notices remain unchanged.
+The bit word and original frames retain Zhihao Chen's PR #97 and Swapnil
+Jain's frozen witness, notices and historical assistance disclosures.
+Earlier generalized Clifford, three-stage, uniform weighted recursion,
+ordinary-leaf, semantic/bulk, precision and analytic credits remain in force.
+Source-specific license bytes are preserved under references/.
+
+Source-assisted common-frame extension (2026-10-09)
+Copyright 2026 icekylinx. Apache-2.0.
+Research construction and local certificates: OpenAI GPT-6 Astra assistance.
+Repository integration and independent complex scalar replay: Codex assistance.
+Substantial dependency: eumemic PR168, fd25adb7fbaa12ee761d02c733c54d1d2a7687ee;
+its byte-preserved sources, licenses and original Claude assistance disclosure
+are retained in research/source-assisted/public/pr168_fd25adb7.tar.gz.
+Additional inherited contributions are recorded in research/source-assisted/CREDITS.md.
+The new work concerns source-assisted invertible common-frame lifts, controlled
+affine cleanup and compatible zero-fresh reuse; no exclusive claim is made for
+broad register reuse, paired cubes, frame theory or ordinary linear elimination.
diff --git a/research/coordinated-crossover-pr200/search/PR216-NOTICE-PR211 b/research/coordinated-crossover-pr200/search/PR216-NOTICE-PR211
new file mode 100644
index 0000000000..9edc257237
--- /dev/null
+++ b/research/coordinated-crossover-pr200/search/PR216-NOTICE-PR211
@@ -0,0 +1,18 @@
+Packed diagonal bit supplier. Apache-2.0.
+Prepared by Rohan Arun (rohanarun) with Anthropic Claude assistance.
+
+Reused, with credit:
+- Evan McKinney, PR197 (OpenAI Codex assistance): completed-bank packing, exact
+  gauge charts, nine-colour bank incidence, routing bound, arithmetic, audit
+  and endpoint controls. banks.py and controls.py are byte-identical extracts;
+  packing.py, arithmetic.py, audit.py and verify.py are adaptations to PR200.
+- Chafik Boukhalfa, PR200 at a1175449f34d39ff933d9d8ab23ced1f32b290ec (Anthropic
+  Claude and OpenAI Codex assistance): the fixed-coordinate/face-diagonal bit
+  word, physical layer, terminal sinks, prime witnesses and package verifier.
+- Avi Eisenberg, PR193 at 187e1010ac8b259af8e9b5166f68b64bc27b4b47: the v4
+  source-assisted complex supplier and balanced assembly bridge; icekylinx for
+  the source-assisted method (PR184) and the paired-cube framework (PR144).
+- maxime-fleury, PR204: the model of this lever and its predicted value.
+- eumemic (PR168 lineage), jamesyc (PR166 terminal lemma), PR185/187 finite
+  ordinary-leaf composition, and every contributor credited by those packages.
+Original notices remain in the pinned checkouts.
diff --git a/research/coordinated-crossover-pr200/search/SEARCH-NOTE.txt b/research/coordinated-crossover-pr200/search/SEARCH-NOTE.txt
new file mode 100644
index 0000000000..df3e13ab9a
--- /dev/null
+++ b/research/coordinated-crossover-pr200/search/SEARCH-NOTE.txt
@@ -0,0 +1,15 @@
+PR211 residual plateau enlargement experiment
+
+Input: PR211 frozen 6,191 operation-frame bases at c63e50a5dde96fe1459d6b29e55e47f42f104347, applied to the PR200 physical word.
+
+Result: 7 accepted equal-frame component moves change 14 operation frames. Exact rational frame/value-span/nondegeneracy and complete physical-chain predicates pass. Scalar operations, source and gauge boundaries, root caps, alias pairs and partner deliveries remain fixed. The 34 terminal deletions have an operation-frame-independent cancellation: their internal-path change is reproduced on the terminal pivot target, leaving the same rank-3/rank-21 correction. The parent verifier must still replay complete scalar columns and certify the final paid moment before publication.
+
+Additional child histogram delta (before the three-copy scaling) is:
+rank 2: +12; rank 4: -15; rank 5: -18; rank 6: +9; rank 7: +24; rank 8: -12; rank 9: -12; rank 12: +24; rank 15: -12.
+Weighted rank delta is zero. Stock and endpoint gauges are unchanged.
+Floating search objective improvement is 1.1846038379180897 in sum r log r over the internal chains. This is a heuristic score, not the certified kappa.
+
+Reproduce:
+python3 -B plateau_search.py --bit-package /path/to/research/paired-cube-diagonal-bit-168 --frames /path/to/research/cascade-frames-bit/opframe-bases.json --out /path/to/results
+
+Provenance: PR200 physical word and checkers by Chafik Boukhalfa; PR211 cascade frame witness by Rohan Arun with Anthropic Claude assistance. This residual search was prepared with OpenAI Codex assistance. Apache-2.0.
diff --git a/research/coordinated-crossover-pr200/search/additional-frames.json b/research/coordinated-crossover-pr200/search/additional-frames.json
new file mode 100644
index 0000000000..60e8a5650f
--- /dev/null
+++ b/research/coordinated-crossover-pr200/search/additional-frames.json
@@ -0,0 +1 @@
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index 0000000000..1aab6d6d34
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+      "ops": [
+        18516,
+        18517
+      ],
+      "old_dim": 5,
+      "new_dim": 6,
+      "gain": 0.201355135506887
+    },
+    {
+      "ops": [
+        18920,
+        18921
+      ],
+      "old_dim": 5,
+      "new_dim": 6,
+      "gain": 0.201355135506887
+    },
+    {
+      "ops": [
+        19324,
+        19325
+      ],
+      "old_dim": 5,
+      "new_dim": 6,
+      "gain": 0.201355135506887
+    },
+    {
+      "ops": [
+        23544,
+        23545
+      ],
+      "old_dim": 9,
+      "new_dim": 12,
+      "gain": 0.14513460784935717
+    },
+    {
+      "ops": [
+        23591,
+        23592
+      ],
+      "old_dim": 9,
+      "new_dim": 12,
+      "gain": 0.14513460784935717
+    },
+    {
+      "ops": [
+        23685,
+        23686
+      ],
+      "old_dim": 9,
+      "new_dim": 12,
+      "gain": 0.14513460784935717
+    },
+    {
+      "ops": [
+        23873,
+        23874
+      ],
+      "old_dim": 9,
+      "new_dim": 12,
+      "gain": 0.14513460784935717
+    }
+  ],
+  "entropy_gain": 1.1846038379180897,
+  "elapsed": 28.685559957986698
+}
diff --git a/research/coordinated-crossover-pr200/search/plateau_search.py b/research/coordinated-crossover-pr200/search/plateau_search.py
new file mode 100644
index 0000000000..1144c784bf
--- /dev/null
+++ b/research/coordinated-crossover-pr200/search/plateau_search.py
@@ -0,0 +1,104 @@
+"""Exact admissibility search on equal-frame operation plateaux.
+
+Searches frame enlargements on connected equal-frame operation components.
+The floating entropy objective proposes moves only; paid-moment and scalar-word
+admission belong to the parent verifier. Every output candidate is checked with
+the inherited exact rational frame and physical-chain predicates.
+
+Prepared with OpenAI Codex assistance. Apache-2.0.
+Retains PR200's physical word and exact integer checker (Chafik Boukhalfa),
+and consumes PR211's frozen cascade frame input (Rohan Arun, Claude assisted).
+"""
+from pathlib import Path
+from collections import Counter,defaultdict
+import sys,json,time,math,argparse
+sys.dont_write_bytecode=True
+ap=argparse.ArgumentParser(description=__doc__)
+ap.add_argument('--bit-package',required=True,type=Path)
+ap.add_argument('--frames',required=True,type=Path)
+ap.add_argument('--out',required=True,type=Path)
+args=ap.parse_args();P=args.bit_package.resolve();D=args.out.resolve();D.mkdir(parents=True,exist_ok=True)
+sys.path.insert(0,str(P/'bit'))
+from word import Candidate
+started=time.monotonic();W=Candidate();C=W.C
+for i,rows in json.loads((args.frames).read_text()):W.opframe[i]=W.register(rows)
+base=W.opframe[:]
+W.changed_frames=[i for i,(a,b) in enumerate(zip(W.original_opframe,W.opframe))if a!=b]
+W.endframe={s:W.opframe[xs[-1]]for s,xs in W.role_ops.items()}
+W.exact_frames();baseline=W.row();print('initialized',len(W.ops),len(C.B),'frames',time.monotonic()-started,flush=True)
+# Fixed boundary refs are encoded as -frame-1, while zero frame gets its own id.
+zero=W.register([]);full=W.w['full_frame'];before=[[]for _ in W.ops];after=[[]for _ in W.ops]
+starts={b:W.w['source_frame'][x]for x,b in W.source.items()};starts.update({b:z['frame']for b,z in W.gauge.items()})
+ends={s:W.w['root_frame'][j]for j,s in enumerate(W.w['rootroles'])}
+ends.update({d:W.gauge[b]['frame']for b,d in W.pairs})
+for s,xs in W.role_ops.items():
+ chain=[-starts.get(s,zero)-1]+xs+[-ends.get(s,full)-1]
+ for j,i in enumerate(xs,1):before[i].append(chain[j-1]);after[i].append(chain[j+1])
+assert all(len(x)==2 for x in before+after)
+def frame(ref):return W.opframe[ref]if ref>=0 else-ref-1
+F=[0]+[r*math.log(r)for r in range(1,W.h+1)]
+def merit(d,prev,nxt):return sum(F[d-C.dimf[p]]for p in prev)+sum(F[C.dimf[n]-d]for n in nxt)
+nondeg={};intersections={};changes=[];tot=0
+
+def good(f):
+ if f not in nondeg:nondeg[f]=C.nondeg(f)
+ return nondeg[f]
+def intersection(a,b):
+ if C.sub(a,b):return a
+ if C.sub(b,a):return b
+ k=tuple(sorted((a,b)))
+ if k not in intersections:
+  rows,_=W.module.kernel(C.A[a]+C.A[b],W.h)
+  intersections[k]=W.register(rows)
+ return intersections[k]
+# Move connected equal-frame plateaux as units.
+parent=list(range(len(W.ops)))
+def find(i):
+ while parent[i]!=i:parent[i]=parent[parent[i]];i=parent[i]
+ return i
+def union(i,j):
+ i,j=find(i),find(j)
+ if i!=j:parent[j]=i
+for i,ns in enumerate(after):
+ for j in ns:
+  if j>=0 and C.dimf[W.opframe[i]]==C.dimf[W.opframe[j]] and C.sub(W.opframe[i],W.opframe[j]):union(i,j)
+groups=defaultdict(list)
+for i in range(len(W.ops)):groups[find(i)].append(i)
+print('groups',len(groups),'largest',sorted(map(len,groups.values()),reverse=True)[:20],flush=True)
+stats=Counter();gains=[]
+for sweep in range(8):
+ num=0;gain=0
+ for xs in groups.values():
+  members=set(xs);old=W.opframe[xs[0]];d=C.dimf[old]
+  if any(not(C.dimf[W.opframe[i]]==d and C.sub(old,W.opframe[i]))for i in xs):continue
+  prev=[frame(j)for i in xs for j in before[i]if j not in members]
+  nxt=[frame(j)for i in xs for j in after[i]if j not in members]
+  stats['groups']+=1
+  if min(C.dimf[f]for f in nxt)<=d:continue
+  stats['dimension_room']+=1;f=nxt[0]
+  for n in nxt[1:]:
+   f=intersection(f,n)
+   if C.dimf[f]<=d:break
+  dd=C.dimf[f]
+  if dd<=d:continue
+  stats['intersection_room']+=1
+  if not good(f):stats['degenerate']+=1;continue
+  assert C.sub(old,f)
+  improvement=merit(dd,prev,nxt)-merit(d,prev,nxt)
+  stats['positive'if improvement>1e-10 else 'nonpositive']+=1
+  gains.append([improvement,len(xs),d,dd])
+  if improvement<=1e-10:continue
+  for i in xs:W.opframe[i]=f
+  num+=len(xs);gain+=improvement;changes.append(dict(ops=xs,old_dim=d,new_dim=dd,gain=improvement))
+ print('sweep',sweep,'changes',num,'entropy_gain',gain,'stats',dict(stats),'seconds',time.monotonic()-started,flush=True)
+ tot+=gain
+ if not num:break
+print('best gains',sorted(gains,reverse=True)[:20],flush=True)
+W.changed_frames=[i for i,(a,b)in enumerate(zip(W.original_opframe,W.opframe))if a!=b]
+W.endframe={s:W.opframe[xs[-1]]for s,xs in W.role_ops.items()}
+W.exact_frames();row=W.row();delta=Counter({int(k):n for k,n in row['child_histogram'].items()});delta.subtract({int(k):n for k,n in baseline['child_histogram'].items()})
+frames=[[i,C.B[f]]for i,f in enumerate(W.opframe)if f!=base[i]]
+(D/'plateau-additional-frames.json').write_text(json.dumps(frames,separators=(',',':'))+'\n')
+(D/'plateau-all-frames.json').write_text(json.dumps([[i,C.B[f]]for i,f in enumerate(W.opframe)if f!=W.w['op_frame'][i]],separators=(',',':'))+'\n')
+out=dict(status='EXACT_FRAMES_AND_NESTED_CHAINS_PASS_NOT_FULL_REPLAY',baseline=baseline,profile=row,child_delta={str(k):v for k,v in sorted(delta.items())if v},changes=changes,entropy_gain=tot,elapsed=time.monotonic()-started)
+(D/'plateau-result.json').write_text(json.dumps(out,indent=2)+'\n');print('DONE additional',len(frames),'delta',out['child_delta'],'gain',tot,flush=True)
diff --git a/research/coordinated-crossover-pr200/search/pr211-frames.json b/research/coordinated-crossover-pr200/search/pr211-frames.json
new file mode 100644
index 0000000000..8433d7724c
--- /dev/null
+++ b/research/coordinated-crossover-pr200/search/pr211-frames.json
@@ -0,0 +1 @@
+[[9404, [[0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]], [9405, [[0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]], [9415, [[1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]], [9451, [[1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]], [9477, [[0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0]]], [9487, [[1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]], [9515, [[1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0]]], [9516, [[1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0]]], [9549, [[0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0]]], [9585, [[0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1]]], [9615, [[1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]], [9616, [[1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]], [9617, [[1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]], [9621, [[0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]], [9622, [[0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]], [9623, [[0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]], [9636, [[1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]], [9637, [[1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]], [9710, [[1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]], [9711, [[1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]], [9856, [[1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]], [9857, [[1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 1, 0, 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0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1]]]]
\ No newline at end of file
diff --git a/research/coordinated-crossover-pr200/search/pr230/NOTICE b/research/coordinated-crossover-pr200/search/pr230/NOTICE
new file mode 100644
index 0000000000..bc1cfc52a1
--- /dev/null
+++ b/research/coordinated-crossover-pr200/search/pr230/NOTICE
@@ -0,0 +1,441 @@
+A sharper exponent for integer multiplication
+Copyright 2026 Douglas Colkitt
+
+This project includes unmodified source files from OpenAI's manuscript
+"Integer multiplication below n log n" and its accompanying repository:
+https://github.com/openai/math
+Commit: adc7f1241b42e322a6451854ab7e4b4c146bf78a
+
+The original manuscript identifies OpenAI as its author. Its Apache-2.0
+license is retained at upstream/LICENSE. The exact source URLs and hashes
+are recorded in upstream/manifest.json. Proposed modifications are supplied
+as separate files under patches/; upstream source files remain unchanged.
+
+The original local research note, scripts, certificates, patches, and
+documentation were prepared by Douglas Colkitt with assistance from OpenAI
+Codex. The community release incorporates work by the contributors identified
+in the notices below; it is not solely Douglas Colkitt's work.
+This project is not an official OpenAI release or an OpenAI endorsement.
+
+Related contributions (not imported in this checkpoint):
+Zhihao Chen (jacklightChen), PR #7, supplied an earlier ternary five-subset
+motif, finite-alphabet transfer and a different compressed producer:
+https://github.com/CrocSwap/integer-mult-bounds/pull/7
+The present checkpoint claims no priority for that motif. Related compressed
+complex-network work was submitted by eumemic in PR #3:
+https://github.com/CrocSwap/integer-mult-bounds/pull/3
+The pending contributions and their stronger bounds require their own review;
+this notice does not represent adoption or verification of their claims.
+
+Additional community work, including parallel and superseded submissions, is
+acknowledged in CONTRIBUTORS.md, with pinned submission metadata in
+docs/research/contribution-snapshot.json. These acknowledgements do not imply
+joint authorship of the checkpoint, incorporation of pending code, or verification
+of submitted mathematical claims. Original source-specific notices remain in force.
+
+Community release (accepted by conditional maintainer audit):
+The following original contributor notices apply to the imported PR #39 chain.
+The preceding non-import statements describe the preserved 2^-30 checkpoint.
+
+Ternary five-subset construction and common-four-point resynthesis
+Copyright 2026 Zhihao Chen (jacklightChen)
+
+This is Zhihao Chen's first submission of this specific construction to this
+repository. Research, proof development, code and verification were prepared
+with substantial assistance from GPT-6 Astra (OpenAI Codex), as identified by
+the contributor. This is not independent review or an OpenAI endorsement.
+Future research building on this contribution should explicitly acknowledge
+Zhihao Chen (jacklightChen), cite the accompanying note and pull request, and
+retain the preceding contributors' attribution. This notice records research
+provenance; the repository's Apache-2.0 license remains unchanged.
+
+Dependencies retained from eumemic's Claude-assisted PRs #3 and #5:
+https://github.com/CrocSwap/integer-mult-bounds/pull/3
+https://github.com/CrocSwap/integer-mult-bounds/pull/5
+PR #5 commit: d3d370c34937503bd3fa63c83d505a05e8e50893
+These include complex_circuit.py, complex_network.py, fast_gaussian.py,
+their generators, construction notes and certificates. The Gaussian input
+scaling proof is corrected in this derivative; see the modification notice below.
+Aligned pairing also appears in Bortlesboat's PR #2. Retained totals and bank
+sharing appear in dleen's PR #4 and cheaper retained centers in eumemic's PR #6.
+Their results and ideas are credited separately from this new construction.
+
+Bulk recursion and controlled tensor basis, 8 October 2026
+Copyright 2026 icekylinx
+
+The new projector batching, controlled simultaneous basis, mixed-width tape
+recursions, whole-residual complex recursion, and dependency-path precision
+guard in the batched-23 proof package are contributed by icekylinx, developed
+with substantial OpenAI GPT-6 Astra assistance. Review corrections and
+repository integration were prepared with OpenAI Codex assistance. They build on the retained
+PR #7 finite producers and the preceding contributions credited above.
+The new package retains the repository's Apache-2.0 license and the original
+authors' attribution. It is not an official OpenAI release or endorsement.
+
+Modification notice, 8 October 2026 (icekylinx): the retained fast Gaussian
+proof now bounds shifted approximation error by F=2^ceil(1.14 alpha^2).
+The P=34p precision budget and asymptotic exponent are unchanged. New batching
+checks are integrated into make verify. Existing attribution and licenses
+are retained.
+
+Partial-swap extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The partial-swap framing, binary retained-total triple construction,
+dependency-preserving positive labels and carrier reuse, unequal (25,23,57)
+common basis, improved complex batching, and exact assembly are contributed
+by icekylinx with substantial OpenAI GPT-6 Astra and Codex assistance.
+This extension directly inherits PR #10, commit
+62691e395a0458ce089a1c7b5d89e74291e95e29. The portable producer adapters
+modify the retained paired-exclusion and shared-point implementations credited
+above. The original licenses, attribution, and Gaussian correction are retained.
+
+Endpoint-gauge extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The complementary auxiliary endpoint gauges, (32,30,40) controlled basis,
+general nondegenerate binary phase residuals, retained-center complex producer,
+updated path guard and exact assembly are contributed by icekylinx with
+substantial OpenAI GPT-6 Astra and Codex assistance. This extension inherits
+f2ab41aebad47861caf6316282c1793e5513845e and its PR #10 ancestry. New portable
+adapters retain the paired-exclusion and shared-point contributions credited
+above. Original licenses, notices and the Gaussian correction are retained.
+
+Structured-basis and semantic-bulk extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The selected (32,30,36) bit construction, A1/A5 block refinements, fixed
+middle-axis I+J profiles with bounded-minor CRT certificates, mixed dyadic
+complex centers, complete data-front batching and circuit-specific assembly
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex
+assistance. The immediate parent is ed90fd940279c336ebc45968631bdebfb087b505.
+
+The semantic/bulk compatibility argument, literal scalar charge, product row
+stock and parameterization adapt Zhihao Chen (jacklightChen)'s PR #23,
+commit 661ebabc1076c9f525d7ce4967c876b203fa9827. Translated complex endpoints
+are adopted from the same contributor's PR #21,
+commit 5ba6cf0bfb68f2be8d15610e7972207c50254d6a, following eumemic's source-frame
+work and the earlier nested-basis contributions.
+
+Completed-child semantic precision, arbitrary-coordinate routing, phase-cell
+Gaussian inversion and bulk tape transfer are credited to the RaD project
+(hipotures), commit 45d9b60355872041f9275f77c057514def0d45bc, exposed through
+PR #20 at 4f8d6c8272b5ff307a0da51df545ec3cd96a8b6e. Their selected proof
+sources retain their separate CC0 license and original NOTICE under
+references/semantic-bulk/rad20. The PR #23 source retains its recorded OpenAI
+Codex assistance; RaD retains its recorded Codex and GPT-6.1-Sol Ultra
+assistance. These are source attributions, not claims about this session's
+runtime model. Related contributions by Rohan Arun and Aurel Prosz remain
+acknowledged in the imported notices. No exclusive priority or additional
+license condition is asserted.
+
+Copied retained-center extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The copied retained-center schedule in both orientations, its complex-phase
+transfer, the Wm-N+L rank budget, and the selected (25,23)/(28,28) composition
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex
+assistance. This extension inherits PR #32 at
+0ef3aeb61f55cc0b321ce6a0ef00acee25cefe52 and the preceding PR #10/#18/#24 work.
+
+The two-stage topology and paid copied-stream endpoint correction are due
+to Aurel Prosz (Paureel), c82d09eb4781b68435e3fa3eb6beea72fa900fab,
+through Zhihao Chen (jacklightChen)'s PR #29 unequal-axis composition,
+9d963275075fa98f1da821e27757b238dafd6b3c. That source also credits Swapnil
+Jain's two-stage development at ae405eb474d1486b2d8aef90139869f927f4836a.
+
+The contiguous corner method, incidence factorization and exact partition
+certificates are due to Rohan Arun (PR #31), method commit
+02f68f95369c39955c5ffcfddbd75cd83e26b305. The selected (25,23) certificate
+specializes Dominik Scholz's parameterization (PR #33),
+a499a345c040aa15b68e984a0904eadc485c7a7e. Their source notices and proof
+snapshots are preserved in references/copied-centers. Historical OpenAI
+Codex, GPT-6 Astra and Anthropic Claude Opus 5.5 assistance is credited to
+the respective imported sources. Existing PR #21/#23 and RaD attribution,
+licenses and the Gaussian correction remain in force.
+
+Copied centers with reversed (23,25) data corners, 2026-10-08
+Rohan Arun with substantial OpenAI Codex assistance supplies the composition,
+exact dimension23 controls and cost/assembly integration. James Chang's PR34
+supplies the reversed boundary family; icekylinx's PR36 supplies copied retained
+centers and the selected complex graph. The balanced assembly remains attributed
+to RaD/hipotures and its PR34 specialization. Retain Dominik Scholz PR33, Rohan
+Arun PR31, Zhihao Chen PR29/21/23, Paureel/Aurel Prosz, Swapnil Jain, icekylinx,
+eumemic, Douglas Colkitt, OpenAI and all earlier notices and hypotheses.
+This is not a claim of formal verification, new foundational rank identities,
+global optimality or measured runtime speedup.
+
+Fixed middle basis with copied reversed corners (2026-10-08):
+Rohan Arun, with substantial OpenAI Codex assistance. Composition and
+simultaneous-basis proof retain icekylinx PR32/36, James Chang PR34,
+Dominik Scholz PR35/38, Zhihao Chen, Paureel, Swapnil Jain, RaD/hipotures,
+eumemic and all preceding attribution and licenses. See research/copied-fixed-reversed.
+
+Changed-graph composition with copied centers and fixed bases, 8 October 2026
+Prepared with OpenAI Codex assistance.
+The new composition uses icekylinx PR36 copied centers and PR32 fixed-I+J
+projectors/profiler, Dominik Scholz PR35 two fixed factors and CRT argument,
+Zhihao Chen (jacklightChen) PR29/23 interfaces, Aurel Prosz (Paureel) and
+Swapnil Jain's two-stage development, Rohan Arun PR25/31 corner methods,
+Dominik Scholz PR33/38 dimension/composition work; James Chang (jamesyc)
+PR34 reversed geometry and balanced transfer; Rohan Arun PR37/39/40
+reversed integration, modular data tests and both-fixed baseline;
+RaD (hipotures) PR41 changed scalar DAGs, alternating point order and
+independent role compiler; RaD balanced routing/bulk interfaces, and all
+retained predecessors.
+Complete attribution and the conditional proof boundary are recorded in
+research/copied-fixed/PROOF.md; imported originals and source pins are retained.
+Original Apache-2.0 and CC0 notices remain applicable.
+
+Community follow-up integration (PRs #49, #45 and #26), 8 October 2026:
+Rohan Arun contributes weighted carrier matching and climbed summand/leave-one-out
+orders (PRs #44/#47/#49), with Anthropic Claude assistance. The final construction
+composes Chafik Boukhalfa's changed-graph checks, exact rational data recovery and
+reordered exclusion sums (PRs #43/#46/#48, OpenAI Codex assistance) with RaD /
+hipotures's alternating pair order and independent physical role compiler (#41).
+All predecessor attribution and conditional proof interfaces remain in force.
+
+Alejandro Zarzuelo Urdiales contributes Gaussian parity and canonical arithmetic
+proofs, finite tensor execution, profile precision accounting and mixed-center
+reference checks (#45), with substantial OpenAI Codex assistance. See
+research/gaussian-parity-synthesis/PROVENANCE.md for his full attribution.
+Ryan S / princezuda contributes historical Lean certificate arithmetic, algebraic
+frame/movement/guard lemmas and independent paired-circuit checks (#26). Original
+source notices and historical snapshots are preserved. Neither formal package
+claims a full formal proof of the multiplication algorithm or a new kappa.
+
+Joint frame compilation of skip-prefix strips
+
+The local compiler and exact finite witnesses were prepared by eumemic with
+substantial OpenAI Codex assistance. Avi Eisenberg / ikeboy with Anthropic
+Claude assistance supplied the unmodified PR #53 skip-prefix producer.
+The pinned PR #48 pipeline is Chafik Boukhalfa's contribution, incorporating
+Rohan Arun, RaD / hipotures, icekylinx, Dominik Scholz, James Chang,
+Zhihao Chen, Aurel Prosz / Paureel, Swapnil Jain, eumemic, Douglas Colkitt,
+OpenAI, David Harvey, Joris van der Hoeven and the earlier contributors
+credited in the retained sources. See references/frame-compiler/pr48/NOTICE
+and references/frame-compiler/pr53/NOTICE for their original attribution.
+No authorship or review by those predecessors is claimed for this compiler.
+
+Joint dual-skip/frame-compiler composition (2026-10-08)
+---------------------------------------------------
+Rohan Gupta / gupt1156 with Anthropic Claude assistance: PR55 dual-skip
+scalar graph; original LICENSE and NOTICE are retained under
+references/frame-compiler/pr55. eumemic with substantial OpenAI Codex
+assistance: PR57 joint reversible frame compiler, serialized physical-word
+replay and profile reconstruction. Chafik Boukhalfa with OpenAI Codex
+assistance: this composition, independent integration and exact certificate.
+Avi Eisenberg / ikeboy, Rohan Arun, RaD / hipotures, icekylinx, and all
+predecessor authors retain the notices above and in pinned source files.
+No original copyright or Apache-2.0 notice is removed.
+
+Ranked reclamation after the PR58 joint-dual composition (2026-10-08)
+----------------------------------------------------------------
+Chafik Boukhalfa with OpenAI Codex assistance: prioritize retired physical
+slots by descending current frame rank, retaining slot ID as the tie-breaker;
+complete replay, exact profiles, comparison and adversarial controls.
+The dedicated joint_dual_reclaim_compiler.py is derived from eumemic's PR57
+compiler. Its original shared compiler remains byte-identical. The PR55
+producer and all preceding attribution remain unchanged. The complete PR58
+comparison certificate and its original source/proof provenance are retained
+under references/frame-compiler/pr58.
+
+Skip-prefix strip producer, 8 October 2026
+By Avi Eisenberg (ikeboy), with Anthropic Claude (Claude Opus 5.5)
+assistance. The skip-prefix leave-one-out strip layout in
+research/skip-strips/skip_graph.py is new. Everything else reuses Chafik
+Boukhalfa's PR #43/#46/#48 research/copied-fixed pipeline, its verification
+code, exact data recovery and support-order sorting, together with:
+- Rohan Arun's PR #44 weighted carrier matching optimizer;
+- RaD (hipotures) PR #41 alternating point order;
+- every predecessor credited above.
+Complete attribution and the conditional proof boundary are recorded in
+research/skip-strips/PROOF.md. Original Apache-2.0 and CC0 notices remain
+applicable.
+
+Local split-pair skip-prefix research continuation (2026-10-08)
+Prepared with substantial OpenAI Codex assistance at the user's request.
+New contribution: singleton/pair recursive partition refinement, finite summand
+order and moment search, and verified composition with inherited paid clones.
+Avi Eisenberg/ikeboy with Anthropic Claude assistance: PR53 skip-prefix strips.
+Chafik Boukhalfa with OpenAI Codex assistance: PR54 original-envelope clone
+specialization and PR43/46/48 checking interfaces. RaD/hipotures: PR51
+whole-chain/source-partition cloning and earlier producer/interfaces.
+Rohan Arun: PR44 weighted matching and earlier fixed/reversed work, with all
+stated historical AI assistance. All predecessor notices remain applicable.
+
+Original-envelope whole-chain clones on skip-prefix strips (research/skip-clones):
+Chafik Boukhalfa, prepared with OpenAI Codex assistance. The complete-chain
+source-partition clone operation is adapted from RaD / hipotures, PR #51,
+with original envelopes retained here. The scalar base is Avi Eisenberg's
+PR #53 skip-prefix layout, prepared with Anthropic Claude assistance.
+All credited PR #43/#46/#48, PR #44 and earlier code/proof notices remain
+in force. This extension is provided under the repository's Apache-2.0 terms.
+
+Community review checkpoint #50–#61, 8 October 2026
+Maintainer integration and independent arithmetic: Douglas Colkitt, with OpenAI Codex.
+Avi Eisenberg / ikeboy: skip-prefix strip framework (#53), with Anthropic Claude.
+Rohan Gupta / gupt1156: parallel order search (#50, stated Antigravity assistance)
+and dual-suffix strips (#55, Anthropic Claude assistance).
+Eumemic: equal-frame joint compilation and paid signal reclamation (#57), with Codex.
+Chafik Boukhalfa: original-envelope paid clones (#54), composition (#58) and ranked
+reclamation (#60), with Codex. Rohan Arun: order search (#52) and positive-frame
+composition (#56), with Anthropic Claude and stated Codex validation assistance.
+RaD / hipotures: enlarged frames, source partitions and paid whole-chain clones
+(#51), with Codex. Rohan Garg / rohangar1: singleton/pair recursion and paid-clone
+composition (#59), with Codex. Alejandro Zarzuelo Urdiales: exact parameter
+refinement, scoped Lean arithmetic, matrix/parity and search tools (#61), with
+his disclosed multi-tool and Codex assistance.
+Parallel #50/#52 records are reviewed and credited without replacing the earlier
+climbed-48 implementation; their exact commit history is retained. The separate
+#51 witness is retained as research; its source-graph/negative-basis data proof
+has not received the same full replay as #56's consumed compiler machinery.
+See docs/research/community-round2-review.md for precise validation boundaries.
+All earlier licenses, contributions and notices remain applicable.
+
+Interval strips and core-aware pair assembly
+
+The interval leave-one-out strips and the core-aware group-pair assembly in
+research/pair-assembly were prepared by Avi Eisenberg / ikeboy with Anthropic
+Claude assistance. They build on PR #53's skip-prefix pipeline and its
+inherited PR #48 sources. The stacked witness in research/pair-assembly/frame
+uses eumemic's PR #57 joint frame compiler unchanged; its driver and verifier
+are adaptations of PR #57's skip_frame_compiler.py, skip_frame_compose.py and
+verify_skip_frame.py. Original notices are retained.
+
+Stopped product-ring extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The opposite-bank generic projector factorization, componentwise atom-ring
+adapters, stopped recurrence and outer ordinary-interchange conversion,
+aligned pair-star producer with rational centers, exact denominator-21
+grid and selected (23,23)/(24,24) composition are contributed by icekylinx
+with substantial OpenAI GPT-6 Astra and Codex assistance. This extension
+builds on merged PR #36, 11817ccacb564bb7f98789c20dc11d3fece207e3, and is
+integrated on upstream main 56b66d58297deca1d7dd130247d720e960f77a37.
+
+The retained ordinary leaf uses the preceding projector, corner and
+copied-center proofs. The selected two-stage topology retains Aurel Prosz
+(Paureel), Swapnil Jain and Zhihao Chen's credits; the semantic assembly
+retains PR #21/#23 and RaD's attribution. Paired producers, carrier matching,
+original source notices and each imported license remain unchanged.
+
+Partial-gauge extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The pair-first cube assembly, compatible backward frame enlargement,
+partial source-gauge schedule and stopped whole-projector composition are
+contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex
+assistance, extending PR #104 at 948ce1510df750f4c18b96bdaef436a86f8bf834.
+
+The physical deferred bit word, reflected event accounting, fan-order
+clarification and signed two-stage endpoint are adopted from Zhihao Chen
+(jacklightChen), PR #97, e15350b66e4bf796153a68c240a6cc1b788abe3e, with
+Codex assistance. Its deferred readouts, V leaves, lifted frames and frozen
+witness originate in Swapnil Jain's 741e7aa078392553815df7926ee17ac5e25a8c38,
+with the original Claude disclosure retained. Cyclic interval and core-aware
+pair primitives build on Avi Eisenberg (ikeboy), PR #62,
+ad0f25ff7b23cff7f08ad237c2254e6ecf74257e, with Anthropic Claude assistance.
+Prior two-stage, copied-center, ordinary-leaf, semantic/bulk, routing and
+precision credits remain in force. Imported source notices and licenses
+are preserved under references/partial-gauge/; see SOURCES.json.
+
+Three-stage cover extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The arbitrary-subspace Clifford frame theorem, three-stage Cayley geometry,
+weighted q-adic GL cover, unit-lower weighted compiler, charged local-ring
+fallback, uniform batching, internal row borrowing and exact composition
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra assistance. This
+repository integration uses Codex assistance and extends
+PR #115, cf60442983c153a3a0f5272ea4713f99eba13c86.
+
+The selected local complex DAG and support/label replay are adopted from
+Copyright 2026 eumemic, PR #117, cbb05ce504d571546d9b7794c186a613c659c3bf,
+with its original Apache-2.0 terms and Claude assistance disclosure.
+The unchanged bit physical word retains Zhihao Chen (jacklightChen),
+PR #97, and Swapnil Jain's underlying witness and assistance notices.
+The prior opposite-bank, stopped ordinary-leaf, copied-center, topology,
+semantic/bulk, precision and analytic credits remain applicable.
+Original source notices and license bytes are preserved in
+references/three-stage-cover/pr117/ and references/partial-gauge/.
+
+Paired-cube and shared-core extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The paired-coordinate triple-cube identity, signed overlap channels,
+original-source involution, coordinate-star centers, concrete frame/carrier
+compiler, chronological source gauges, PR97 gauge-subset selection,
+orthogonal cross-stage routing and exact supplier/assembly integration
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra assistance.
+Repository integration uses Codex assistance and extends PR #130 at
+6a9970a530119174507904e23592fd59ede19a5d.
+
+The substantial completed-core workspace-sharing principle is credited to
+an664, PR #128, 530588a019b4a74f09180680c9e3961bf649ec89, with its reported
+OpenAI Codex assistance. No source code, partition or serialized witness
+from that PR is imported. The new routing and paired-motif integration
+are described separately and claim no exclusive priority for broad sharing,
+paired-coordinate, Fourier or carrier-matching ideas.
+
+Selected coarse queries restrict the retained eumemic PR #117 positive DAG;
+its original Apache-2.0 and Claude assistance notices remain unchanged.
+The bit word and original frames retain Zhihao Chen's PR #97 and Swapnil
+Jain's frozen witness, notices and historical assistance disclosures.
+Earlier generalized Clifford, three-stage, uniform weighted recursion,
+ordinary-leaf, semantic/bulk, precision and analytic credits remain in force.
+Source-specific license bytes are preserved under references/.
+
+Source-assisted common-frame extension (2026-10-09)
+Copyright 2026 icekylinx. Apache-2.0.
+Research construction and local certificates: OpenAI GPT-6 Astra assistance.
+Repository integration and independent complex scalar replay: Codex assistance.
+Substantial dependency: eumemic PR168, fd25adb7fbaa12ee761d02c733c54d1d2a7687ee;
+its byte-preserved sources, licenses and original Claude assistance disclosure
+are retained in research/source-assisted/public/pr168_fd25adb7.tar.gz.
+Additional inherited contributions are recorded in research/source-assisted/CREDITS.md.
+The new work concerns source-assisted invertible common-frame lifts, controlled
+affine cleanup and compatible zero-fresh reuse; no exclusive claim is made for
+broad register reuse, paired cubes, frame theory or ordinary linear elimination.
+
+Fourth finite bootstrap refinement (2026-10-09)
+Prepared by eumemic with substantial OpenAI Codex assistance.
+This arithmetic refinement builds on PR207 at
+cd14825023b75af4f5919a30e5e6d548b6ade5bc by adding one finite ordinary-leaf
+level. The 302 frame overrides, completed banks, scalar words and upstream
+assistance disclosures are inherited, not newly attributed to this work.
+The Apache-2.0 license and all preceding notices remain applicable.
+
+PR211 plateau frame refinement (2026-10-09)
+Prepared by eumemic with substantial OpenAI Codex assistance.
+The additional seven equal-frame component moves build directly on Rohan
+Arun's PR211 cascade witness at c63e50a5dde96fe1459d6b29e55e47f42f104347,
+prepared with Anthropic Claude assistance. Its original notice is preserved
+byte-for-byte in search/PR211-NOTICE. The offline certificate package and
+independent stage-private bank admission derive from Dugongue's PR207 at
+cd14825023b75af4f5919a30e5e6d548b6ade5bc. Original contributor notices,
+licenses, source pins and all upstream AI disclosures remain applicable.
+
+The final refinement also includes seeded bidirectional cascade search after
+the seven initial component moves. Exact canonical subspace comparison
+records 274 changed operations beyond PR211. Search, integration and replay
+were prepared with substantial OpenAI Codex assistance.
+
+Shared rank-18 gauges and two donor substitutions (2026-10-09)
+Prepared by eumemic with substantial OpenAI Codex assistance.
+The new actual gauge/read chronology, gauged-donor handoff, shared target
+transitions, rank6 residual banks and donor substitutions extend the same
+PR200/PR211 word and PR207/PR197 completed-bank framework. All original
+licenses, notices, authorship and upstream AI disclosures remain in force.
+
+Nested-gauge extension: three rank-19 physical entrances compose with thirteen shared rank-18 entrances; all target transitions and mixed4/5/6/24 banks are explicit. Prepared by eumemic with substantial OpenAI Codex assistance. All retained authorship, licenses and AI disclosures remain applicable.
+
+Equal-frame plateau refinement on PR210's 471-source word (2026-10-10)
+Prepared by sennemmi with substantial OpenAI Codex assistance. The deterministic
+frame search, 19 operation retimings, exact price update and repository
+integration extend eumemic's PR210 head
+df95878d11190518e45ef9717c9ee05011f88ace. They do not claim the inherited
+source word, shared gauges, completed banks or upstream proof framework.
+All PR200, PR197, PR202/193, PR207, PR211, PR216 and PR210 attributions,
+licenses and AI disclosures remain in force; the repository's Apache-2.0
+license is unchanged.
diff --git a/research/coordinated-crossover-pr200/search/pr230/PROVENANCE.json b/research/coordinated-crossover-pr200/search/pr230/PROVENANCE.json
new file mode 100644
index 0000000000..ff39579387
--- /dev/null
+++ b/research/coordinated-crossover-pr200/search/pr230/PROVENANCE.json
@@ -0,0 +1,27 @@
+{
+  "pull_request": "https://github.com/CrocSwap/integer-mult-bounds/pull/230",
+  "author": "sennemmi",
+  "assistance": "Substantial OpenAI Codex assistance, as disclosed by the original contributor.",
+  "imported_head": "fee4c05b91d4cdebc0b9d8cbb266a98069928a9a",
+  "scope": "Nineteen operation-frame changes inherited from the independently verified492-source composition and replayed with493sourcealiases,82equal-response groups,seven integer-response groups andfour signed echelon groups. Original search,result,framewitness andnotice remain byte-preserved.",
+  "original_files": {
+    "NOTICE": "7dfbad087ee49b74fd5075d517eaa263c5dd3d4fbd4c69bfba34ee1cb55e837f",
+    "joint-plateau-result-prdf958.json": "cd5731260af0daa62952c3b82debdb26cfb79e62a5c615688d8ccac6f7a95fcc",
+    "joint-plateau-search-prdf958.py": "80f3f9e8ba1972b911b701a4de4f811d6d25c9b4c0bc6282d640c2f44caafd0d",
+    "opframe-bases.json": "387ae1b5e69004426d094f4947b29c532d60fdf31f727f5e8830468278fe0f92"
+  },
+  "final_frame_sha256": "9a2bba5c72aebc9ee4f21e59b10ce810b2da98a1fa1748fbc44c50a080e584ff",
+  "raw_child_delta": {
+    "1": -27,
+    "2": -36,
+    "3": 51,
+    "4": -30,
+    "7": 30,
+    "8": -15,
+    "9": -15,
+    "10": 24,
+    "11": -24,
+    "12": 30,
+    "15": -15
+  }
+}
diff --git a/research/coordinated-crossover-pr200/search/pr230/joint-plateau-result-prdf958.json b/research/coordinated-crossover-pr200/search/pr230/joint-plateau-result-prdf958.json
new file mode 100644
index 0000000000..b2b61f1227
--- /dev/null
+++ b/research/coordinated-crossover-pr200/search/pr230/joint-plateau-result-prdf958.json
@@ -0,0 +1 @@
+{"accepted_moves":[{"new_dim":6,"old_dim":5,"ops":[18750],"score_gain":1.909542504884438},{"new_dim":12,"old_dim":9,"ops":[23544,23545],"score_gain":0.145134607849343},{"new_dim":12,"old_dim":9,"ops":[23591,23592],"score_gain":0.145134607849343},{"new_dim":12,"old_dim":9,"ops":[23638,23639],"score_gain":0.145134607849343},{"new_dim":12,"old_dim":9,"ops":[23685,23686],"score_gain":0.145134607849343},{"new_dim":12,"old_dim":9,"ops":[23732,23733],"score_gain":0.145134607849343},{"new_dim":14,"old_dim":13,"ops":[27900],"score_gain":0.468087938927262},{"new_dim":14,"old_dim":13,"ops":[27904],"score_gain":0.468087938927262},{"new_dim":14,"old_dim":13,"ops":[27908],"score_gain":0.468087938927262},{"new_dim":14,"old_dim":13,"ops":[27912],"score_gain":0.468087938927262},{"new_dim":14,"old_dim":13,"ops":[28052],"score_gain":0.468087938927262},{"new_dim":14,"old_dim":13,"ops":[28056],"score_gain":0.468087938927262},{"new_dim":14,"old_dim":13,"ops":[28060],"score_gain":0.468087938927262},{"new_dim":14,"old_dim":13,"ops":[28064],"score_gain":0.468087938927262}],"baseline_profile":{"R":17148,"W_per_vertex":20668,"changed_operation_frames":11904,"child_histogram":{"1":126276,"2":60574,"3":30714,"4":8736,"5":7473,"6":4551,"7":3732,"8":1950,"9":1830,"10":2313,"11":1362,"12":4488,"13":1734,"14":3312,"15":888,"16":3042,"17":1452,"18":1404,"19":846,"20":5148,"21":14988,"22":6672,"36":18,"39":48,"54":13,"57":3,"60":2200},"deficit_per_vertex":1936,"foreign_producer_replays":0,"frames_not_contained_in_original":257,"h":24,"loss":528,"m":72,"maxchild":60,"physical_internal_histogram":{"1":35920,"2":18110,"3":10234,"4":2912,"5":2491,"6":1509,"7":1220,"8":622,"9":606,"10":771,"11":454,"12":1480,"13":530,"14":1104,"15":296,"16":1014,"17":484,"18":444,"19":282,"21":4160,"22":1344},"physical_target_histogram":{"1":4412,"2":28,"3":4,"6":8,"7":24,"8":28,"9":4,"12":16,"13":48,"18":24,"20":836,"21":836},"rank_per_vertex":1486160,"reused_registers":1760,"selected_rank_histogram":{"12":18,"13":48,"18":13,"19":3,"20":2200},"selected_roles":2282,"v":1760,"virtual_R":18908},"candidate_frames_sha256":"387ae1b5e69004426d094f4947b29c532d60fdf31f727f5e8830468278fe0f92","candidate_profile":{"R":17148,"W_per_vertex":20668,"changed_operation_frames":11904,"child_histogram":{"1":126249,"2":60538,"3":30765,"4":8706,"5":7473,"6":4551,"7":3762,"8":1935,"9":1815,"10":2337,"11":1338,"12":4518,"13":1734,"14":3312,"15":873,"16":3042,"17":1452,"18":1404,"19":846,"20":5148,"21":14988,"22":6672,"36":18,"39":48,"54":13,"57":3,"60":2200},"deficit_per_vertex":1936,"foreign_producer_replays":0,"frames_not_contained_in_original":258,"h":24,"loss":528,"m":72,"maxchild":60,"physical_internal_histogram":{"1":35911,"2":18098,"3":10251,"4":2902,"5":2491,"6":1509,"7":1230,"8":617,"9":601,"10":779,"11":446,"12":1490,"13":530,"14":1104,"15":291,"16":1014,"17":484,"18":444,"19":282,"21":4160,"22":1344},"physical_target_histogram":{"1":4412,"2":28,"3":4,"6":8,"7":24,"8":28,"9":4,"12":16,"13":48,"18":24,"20":836,"21":836},"rank_per_vertex":1486160,"reused_registers":1760,"selected_rank_histogram":{"12":18,"13":48,"18":13,"19":3,"20":2200},"selected_roles":2282,"v":1760,"virtual_R":18908},"changed_operation_frames_from_parent":19,"changed_operation_frames_from_word":11904,"child_histogram_delta":{"1":-27,"10":24,"11":-24,"12":30,"15":-15,"2":-36,"3":51,"4":-30,"7":30,"8":-15,"9":-15},"elapsed_seconds_excluded_from_record":true,"group_count":20584,"objective":"float d*log(d) proposal; exact frame and nested-chain predicates admit moves","parent_commit":"df95878d11190518e45ef9717c9ee05011f88ace","proposal_score_gain":6.379919055549248,"random_seed":null,"randomness":"none; deterministic group order and greedy sweeps","schema":"joint-equal-frame-plateau-v1","search_counters":{"dimension_room":40231,"groups":41168,"intersection_room":3799,"nonpositive":3785,"positive":14},"starting_frames_sha256":"687496f3c15818e7cad1beb5fdf3812b8e38825ea302bfd60aea7ac23f3516b4","status":"EXACT_FRAMES_AND_NESTED_CHAINS_PASS_NOT_FULL_WORD_REPLAY","sweeps":[{"changed_operations":19,"score_gain":6.379919055549248,"sweep":0},{"changed_operations":0,"score_gain":0.0,"sweep":1}]}
diff --git a/research/coordinated-crossover-pr200/search/pr230/joint-plateau-search-prdf958.py b/research/coordinated-crossover-pr200/search/pr230/joint-plateau-search-prdf958.py
new file mode 100644
index 0000000000..2268e884cd
--- /dev/null
+++ b/research/coordinated-crossover-pr200/search/pr230/joint-plateau-search-prdf958.py
@@ -0,0 +1,266 @@
+#!/usr/bin/env python3
+"""Reproduce the equal-frame plateau descent on the joint physical word.
+
+The d*log(d) score proposes moves only. The repository's actual joint
+Candidate checks exact operation-frame spans, nondegeneracy, and chain nesting.
+The paid moment and exponent are certified separately by the finite verifier.
+No random numbers are used. This incremental search was prepared by sennemmi
+with substantial OpenAI Codex assistance. It builds on PR210 by eumemic and
+PR216 by Dugongue; all inherited contributors retain their original credit.
+Apache-2.0.
+"""
+from __future__ import annotations
+
+import argparse
+import hashlib
+import importlib.util
+import json
+import math
+import shutil
+import sys
+import tempfile
+import time
+import zipfile
+from collections import Counter, defaultdict
+from pathlib import Path
+
+sys.dont_write_bytecode = True
+SOURCE_P = Path(__file__).resolve().parents[1]
+PARENT = "df95878d11190518e45ef9717c9ee05011f88ace"
+BASE_SHA256 = "687496f3c15818e7cad1beb5fdf3812b8e38825ea302bfd60aea7ac23f3516b4"
+
+
+def sha256(data: bytes) -> str:
+    return hashlib.sha256(data).hexdigest()
+
+
+def canonical(value: object) -> bytes:
+    return (json.dumps(value, sort_keys=True, separators=(",", ":")) + "\n").encode()
+
+
+def source_package(tmp: Path) -> Path:
+    manifest = json.loads((SOURCE_P / "BASELINE.json").read_text())
+    parts = []
+    skip = set()
+    for entry in manifest["parts"]:
+        part = SOURCE_P / entry["file"]
+        data = part.read_bytes()
+        assert len(data) == entry["bytes"] and sha256(data) == entry["sha256"]
+        parts.append(data)
+        skip.add(entry["file"])
+    archive = b"".join(parts)
+    assert sha256(archive) == manifest["archive_sha256"]
+    root = tmp / "source"
+    root.mkdir()
+    with zipfile.ZipFile(__import__("io").BytesIO(archive)) as z:
+        for entry in z.infolist():
+            try:
+                (root / entry.filename).resolve().relative_to(root.resolve())
+            except ValueError as exc:
+                raise AssertionError("unsafe pinned archive entry") from exc
+        z.extractall(root)
+    package = root / "research" / "coordinated-crossover-pr200"
+    package.mkdir()
+    for file in SOURCE_P.rglob("*"):
+        if file.is_file() and file.name not in skip:
+            target = package / file.relative_to(SOURCE_P)
+            target.parent.mkdir(parents=True, exist_ok=True)
+            shutil.copyfile(file, target)
+    # The source tree contains the committed candidate frames. Reproduction must
+    # begin from PR210's pinned parent input, not from that candidate output.
+    parent_frames = package / "search" / "base-frames-prdf958.json"
+    assert sha256(parent_frames.read_bytes()) == BASE_SHA256
+    shutil.copyfile(parent_frames, package / "frames" / "opframe-bases.json")
+    return package
+
+
+def load_candidate(P: Path):
+    sys.path.insert(0, str(P.parents[1] / "research" / "paired-cube-diagonal-bit-168" / "bit"))
+    spec = importlib.util.spec_from_file_location("joint_word", P / "joint" / "joint_word.py")
+    assert spec and spec.loader
+    module = importlib.util.module_from_spec(spec)
+    sys.modules[spec.name] = module
+    spec.loader.exec_module(module)
+    return module.Candidate
+
+
+def run_search(P: Path):
+    started = time.monotonic()
+    base_path = P / "search" / "base-frames-prdf958.json"
+    base_bytes = base_path.read_bytes()
+    assert sha256(base_bytes) == BASE_SHA256, "parent PR210 frame input changed"
+    assert sha256((P / "frames" / "opframe-bases.json").read_bytes()) == BASE_SHA256
+
+    W = load_candidate(P)()
+    C = W.C
+    W.changed_frames = [i for i, (a, b) in enumerate(zip(W.original_opframe, W.opframe)) if a != b]
+    W.endframe = {s: W.opframe[xs[-1]] for s, xs in W.role_ops.items()}
+    W.exact_frames()
+    baseline = W.row()
+    base = W.opframe[:]
+
+    zero = W.register([])
+    full = W.w["full_frame"]
+    before = [[] for _ in W.ops]
+    after = [[] for _ in W.ops]
+    starts = {b: W.w["source_frame"][x] for x, b in W.source.items()}
+    starts.update({b: z["frame"] for b, z in W.gauge.items()})
+    ends = {s: W.w["root_frame"][j] for j, s in enumerate(W.w["rootroles"])}
+    ends.update({d: W.gauge[b]["frame"] for b, d in W.pairs})
+    for s, xs in W.role_ops.items():
+        chain = [-starts.get(s, zero) - 1] + xs + [-ends.get(s, full) - 1]
+        for j, i in enumerate(xs, 1):
+            before[i].append(chain[j - 1])
+            after[i].append(chain[j + 1])
+    assert all(len(x) == 2 for x in before + after)
+
+    def frame(ref):
+        return W.opframe[ref] if ref >= 0 else -ref - 1
+
+    F = [0] + [r * math.log(r) for r in range(1, W.h + 1)]
+
+    def merit(d, prev, nxt):
+        return sum(F[d - C.dimf[p]] for p in prev) + sum(F[C.dimf[n] - d] for n in nxt)
+
+    nondeg = {}
+    intersections = {}
+
+    def good(f):
+        if f not in nondeg:
+            nondeg[f] = C.nondeg(f)
+        return nondeg[f]
+
+    def intersection(a, b):
+        if C.sub(a, b):
+            return a
+        if C.sub(b, a):
+            return b
+        key = tuple(sorted((a, b)))
+        if key not in intersections:
+            rows, _ = W.module.kernel(C.A[a] + C.A[b], W.h)
+            intersections[key] = W.register(rows)
+        return intersections[key]
+
+    parent = list(range(len(W.ops)))
+
+    def find(i):
+        while parent[i] != i:
+            parent[i] = parent[parent[i]]
+            i = parent[i]
+        return i
+
+    def union(i, j):
+        i, j = find(i), find(j)
+        if i != j:
+            parent[j] = i
+
+    for i, successors in enumerate(after):
+        for j in successors:
+            if j >= 0 and C.dimf[W.opframe[i]] == C.dimf[W.opframe[j]] and C.sub(W.opframe[i], W.opframe[j]):
+                union(i, j)
+    groups = defaultdict(list)
+    for i in range(len(W.ops)):
+        groups[find(i)].append(i)
+
+    stats = Counter()
+    moves = []
+    sweeps = []
+    total_gain = 0.0
+    for sweep in range(8):
+        changed = 0
+        sweep_gain = 0.0
+        for xs in groups.values():
+            members = set(xs)
+            old = W.opframe[xs[0]]
+            d = C.dimf[old]
+            if any(not (C.dimf[W.opframe[i]] == d and C.sub(old, W.opframe[i])) for i in xs):
+                continue
+            prev = [frame(j) for i in xs for j in before[i] if j not in members]
+            nxt = [frame(j) for i in xs for j in after[i] if j not in members]
+            stats["groups"] += 1
+            if min(C.dimf[f] for f in nxt) <= d:
+                continue
+            stats["dimension_room"] += 1
+            f = nxt[0]
+            for next_frame in nxt[1:]:
+                f = intersection(f, next_frame)
+                if C.dimf[f] <= d:
+                    break
+            new_dim = C.dimf[f]
+            if new_dim <= d:
+                continue
+            stats["intersection_room"] += 1
+            if not good(f):
+                stats["degenerate"] += 1
+                continue
+            assert C.sub(old, f)
+            gain = merit(new_dim, prev, nxt) - merit(d, prev, nxt)
+            stats["positive" if gain > 1e-10 else "nonpositive"] += 1
+            if gain <= 1e-10:
+                continue
+            for i in xs:
+                W.opframe[i] = f
+            changed += len(xs)
+            sweep_gain += gain
+            moves.append({"ops": xs, "old_dim": d, "new_dim": new_dim, "score_gain": round(gain, 15)})
+        sweeps.append({"sweep": sweep, "changed_operations": changed, "score_gain": round(sweep_gain, 15)})
+        total_gain += sweep_gain
+        if changed == 0:
+            break
+
+    W.changed_frames = [i for i, (a, b) in enumerate(zip(W.original_opframe, W.opframe)) if a != b]
+    W.endframe = {s: W.opframe[xs[-1]] for s, xs in W.role_ops.items()}
+    W.exact_frames()
+    profile = W.row()
+    delta = Counter({int(k): n for k, n in profile["child_histogram"].items()})
+    delta.subtract({int(k): n for k, n in baseline["child_histogram"].items()})
+    all_frames = [[i, C.B[f]] for i, f in enumerate(W.opframe) if f != W.w["op_frame"][i]]
+    frame_bytes = (json.dumps(all_frames, separators=(",", ":")) + "\n").encode()
+    record = {
+        "schema": "joint-equal-frame-plateau-v1",
+        "parent_commit": PARENT,
+        "starting_frames_sha256": BASE_SHA256,
+        "candidate_frames_sha256": sha256(frame_bytes),
+        "random_seed": None,
+        "randomness": "none; deterministic group order and greedy sweeps",
+        "objective": "float d*log(d) proposal; exact frame and nested-chain predicates admit moves",
+        "status": "EXACT_FRAMES_AND_NESTED_CHAINS_PASS_NOT_FULL_WORD_REPLAY",
+        "group_count": len(groups),
+        "sweeps": sweeps,
+        "accepted_moves": moves,
+        "search_counters": dict(sorted(stats.items())),
+        "proposal_score_gain": round(total_gain, 15),
+        "baseline_profile": baseline,
+        "candidate_profile": profile,
+        "child_histogram_delta": {str(k): v for k, v in sorted(delta.items()) if v},
+        "changed_operation_frames_from_parent": len([i for i, (a, b) in enumerate(zip(base, W.opframe)) if a != b]),
+        "changed_operation_frames_from_word": len(W.changed_frames),
+        "elapsed_seconds_excluded_from_record": True,
+    }
+    return frame_bytes, canonical(record), record, time.monotonic() - started
+
+
+def main():
+    ap = argparse.ArgumentParser(description=__doc__)
+    ap.add_argument("--verify-committed", action="store_true")
+    ap.add_argument("--frames-out", type=Path)
+    ap.add_argument("--record-out", type=Path)
+    args = ap.parse_args()
+    with tempfile.TemporaryDirectory(prefix="joint-plateau-") as tmp_name:
+        P = source_package(Path(tmp_name))
+        frames, record_bytes, record, elapsed = run_search(P)
+    if args.verify_committed:
+        assert frames == (SOURCE_P / "frames" / "opframe-bases.json").read_bytes(), "frame witness differs"
+        assert record_bytes == (SOURCE_P / "search" / "joint-plateau-result-prdf958.json").read_bytes(), "search record differs"
+        print(f"PASS deterministic plateau reproduction; {len(record['accepted_moves'])} moves, {record['changed_operation_frames_from_parent']} operation frames; {elapsed:.1f}s")
+        return
+    assert args.frames_out and args.record_out, "supply output paths or use --verify-committed"
+    args.frames_out.parent.mkdir(parents=True, exist_ok=True)
+    args.record_out.parent.mkdir(parents=True, exist_ok=True)
+    args.frames_out.write_bytes(frames)
+    args.record_out.write_bytes(record_bytes)
+    print(f"Wrote candidate frames and record; {len(record['accepted_moves'])} moves in {elapsed:.1f}s")
+
+
+if __name__ == "__main__":
+    main()
diff --git a/research/coordinated-crossover-pr200/search/pr230/opframe-bases.json b/research/coordinated-crossover-pr200/search/pr230/opframe-bases.json
new file mode 100644
index 0000000000..1957343527
--- /dev/null
+++ b/research/coordinated-crossover-pr200/search/pr230/opframe-bases.json
@@ -0,0 +1 @@
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index 0000000000..abaeb087c6
--- /dev/null
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+      "ops": [
+        17806
+      ],
+      "gain": 0.0784184704655293
+    },
+    {
+      "direction": "up",
+      "seed": 21781,
+      "ops": [
+        21781
+      ],
+      "gain": 0.21662404215523923
+    },
+    {
+      "direction": "up",
+      "seed": 19577,
+      "ops": [
+        19577,
+        19578
+      ],
+      "gain": 0.020792795137733577
+    },
+    {
+      "direction": "down",
+      "seed": 23342,
+      "ops": [
+        23342
+      ],
+      "gain": 0.43612160938945976
+    },
+    {
+      "direction": "up",
+      "seed": 24997,
+      "ops": [
+        24997,
+        24998,
+        25019
+      ],
+      "gain": 1.2831647313920813
+    },
+    {
+      "direction": "up",
+      "seed": 21947,
+      "ops": [
+        21947
+      ],
+      "gain": 0.1451346078493434
+    },
+    {
+      "direction": "up",
+      "seed": 27335,
+      "ops": [
+        27335,
+        29619
+      ],
+      "gain": 0.21541639971155457
+    },
+    {
+      "direction": "up",
+      "seed": 26114,
+      "ops": [
+        26114,
+        28200
+      ],
+      "gain": 0.3296450008433516
+    },
+    {
+      "direction": "down",
+      "seed": 29202,
+      "ops": [
+        29202
+      ],
+      "gain": 0.21662404215523923
+    },
+    {
+      "direction": "down",
+      "seed": 19220,
+      "ops": [
+        19220
+      ],
+      "gain": 0.09856395226252301
+    },
+    {
+      "direction": "up",
+      "seed": 30230,
+      "ops": [
+        30230
+      ],
+      "gain": 0.09856395226252324
+    },
+    {
+      "direction": "down",
+      "seed": 29284,
+      "ops": [
+        29284
+      ],
+      "gain": 0.21662404215523923
+    }
+  ],
+  "entropy_gain": 15.931145423120375,
+  "elapsed": 79.81174416700378
+}
diff --git a/research/coordinated-crossover-pr200/search/relaxed_search.py b/research/coordinated-crossover-pr200/search/relaxed_search.py
new file mode 100644
index 0000000000..510c1aaca2
--- /dev/null
+++ b/research/coordinated-crossover-pr200/search/relaxed_search.py
@@ -0,0 +1,183 @@
+"""Seeded cascade perturbation and exact admissibility search.
+
+The frozen output frames require independent paid-moment and scalar-word replay.
+Float entropy is used only to propose candidates. The final output passes exact
+rational value-span, nondegeneracy and complete physical-chain checks.
+
+Prepared with OpenAI Codex assistance; Apache-2.0. Uses PR200's physical word
+and checkers (Chafik Boukhalfa), PR211's cascade frame witness (Rohan Arun with
+Anthropic Claude assistance), and our independently found plateau additions.
+"""
+from pathlib import Path
+from collections import Counter,defaultdict
+import sys,json,time,math,argparse
+sys.dont_write_bytecode=True
+ap=argparse.ArgumentParser(description=__doc__)
+ap.add_argument('--bit-package',required=True,type=Path)
+ap.add_argument('--frames',required=True,type=Path)
+ap.add_argument('--additional-frames',required=True,type=Path)
+ap.add_argument('--out',required=True,type=Path)
+args=ap.parse_args();P=args.bit_package.resolve();D=args.out.resolve();D.mkdir(parents=True,exist_ok=True)
+sys.path.insert(0,str(P/'bit'))
+from word import Candidate
+started=time.monotonic();W=Candidate();C=W.C
+for i,rows in json.loads((args.frames).read_text()):W.opframe[i]=W.register(rows)
+
+for i,rows in json.loads((args.additional_frames).read_text()):W.opframe[i]=W.register(rows)
+base=W.opframe[:]
+W.changed_frames=[i for i,(a,b) in enumerate(zip(W.original_opframe,W.opframe))if a!=b]
+W.endframe={s:W.opframe[xs[-1]]for s,xs in W.role_ops.items()}
+W.exact_frames();baseline=W.row();print('initialized',len(W.ops),len(C.B),'frames',time.monotonic()-started,flush=True)
+# Fixed boundary refs are encoded as -frame-1, while zero frame gets its own id.
+zero=W.register([]);full=W.w['full_frame'];before=[[]for _ in W.ops];after=[[]for _ in W.ops]
+starts={b:W.w['source_frame'][x]for x,b in W.source.items()};starts.update({b:z['frame']for b,z in W.gauge.items()})
+ends={s:W.w['root_frame'][j]for j,s in enumerate(W.w['rootroles'])}
+ends.update({d:W.gauge[b]['frame']for b,d in W.pairs})
+for s,xs in W.role_ops.items():
+ chain=[-starts.get(s,zero)-1]+xs+[-ends.get(s,full)-1]
+ for j,i in enumerate(xs,1):before[i].append(chain[j-1]);after[i].append(chain[j+1])
+assert all(len(x)==2 for x in before+after)
+def frame(ref):return W.opframe[ref]if ref>=0 else-ref-1
+F=[0]+[r*math.log(r)for r in range(1,W.h+1)]
+def merit(d,prev,nxt):return sum(F[d-C.dimf[p]]for p in prev)+sum(F[C.dimf[n]-d]for n in nxt)
+nondeg={};intersections={};changes=[];tot=0
+
+def good(f):
+ if f not in nondeg:nondeg[f]=C.nondeg(f)
+ return nondeg[f]
+def intersection(a,b):
+ if C.sub(a,b):return a
+ if C.sub(b,a):return b
+ k=tuple(sorted((a,b)))
+ if k not in intersections:
+  rows,_=W.module.kernel(C.A[a]+C.A[b],W.h)
+  intersections[k]=W.register(rows)
+ return intersections[k]
+unions={}
+def join(a,b):
+ if C.sub(a,b):return b
+ if C.sub(b,a):return a
+ k=tuple(sorted((a,b)))
+ if k not in unions:unions[k]=W.register(list(C.B[a])+list(C.B[b]))
+ return unions[k]
+# Complete upper envelope determined solely by immutable endpoint boundaries.
+cap=[None]*len(W.ops)
+for i in range(len(W.ops)-1,-1,-1):
+ ns=[cap[j]if j>=0 else frame(j)for j in after[i]]
+ cap[i]=intersection(*ns)
+ assert C.sub(W.opframe[i],cap[i])
+print('global caps built',time.monotonic()-started,flush=True)
+
+def transaction(seed,f):
+ old=W.opframe[seed]
+ if C.sub(f,old):return None
+ if not C.sub(old,f):f=join(old,f)
+ if not C.sub(f,cap[seed]) or not good(f):return None
+ update={seed:f};pending=[seed]
+ while pending:
+  i=pending.pop();f=update[i]
+  for j in after[i]:
+   old=update.get(j,frame(j))
+   if C.sub(f,old):continue
+   if j<0:return None
+   z=join(f,old)
+   if not C.sub(z,cap[j])or not good(z):return None
+   update[j]=z;pending.append(j)
+  if len(update)>1000:return None
+ gain=0.0
+ for i,f in update.items():
+  d=C.dimf[f];od=C.dimf[W.opframe[i]]
+  for j in before[i]:
+   pd=C.dimf[update.get(j,frame(j))];opd=C.dimf[frame(j)]
+   gain+=F[d-pd]-F[od-opd]
+  for j in after[i]:
+   if j not in update:
+    nd=C.dimf[frame(j)];gain+=F[nd-d]-F[nd-od]
+ return gain,update
+
+valueframes={i:f for i,f in enumerate(W.w['source_frame'])}
+def valueframe(n):
+ if n not in valueframes:
+  a,b=W.g['args'][n];valueframes[n]=join(valueframe(a),valueframe(b))
+ return valueframes[n]
+floor=[None]*len(W.ops)
+for i in range(len(W.ops)):
+ ps=[floor[j]if j>=0 else frame(j)for j in before[i]]
+ floor[i]=join(join(*ps),valueframe(W.ops[i][2]))
+ assert C.sub(floor[i],W.opframe[i])
+print('global floors built',time.monotonic()-started,flush=True)
+
+def lower_transaction(seed,f):
+ old=W.opframe[seed]
+ if C.sub(old,f):return None
+ if not C.sub(f,old):f=intersection(old,f)
+ if not C.sub(floor[seed],f) or not good(f):return None
+ update={seed:f};pending=[seed]
+ while pending:
+  i=pending.pop();f=update[i]
+  for j in before[i]:
+   old=update.get(j,frame(j))
+   if C.sub(old,f):continue
+   if j<0:return None
+   z=intersection(f,old)
+   if not C.sub(floor[j],z)or not good(z):return None
+   update[j]=z;pending.append(j)
+  if len(update)>1000:return None
+ gain=0.0
+ for i,f in update.items():
+  d=C.dimf[f];od=C.dimf[W.opframe[i]]
+  for j in before[i]:
+   pd=C.dimf[update.get(j,frame(j))];opd=C.dimf[frame(j)]
+   gain+=F[d-pd]-F[od-opd]
+  for j in after[i]:
+   if j not in update:
+    nd=C.dimf[frame(j)];gain+=F[nd-d]-F[nd-od]
+ return gain,update
+import random
+rng=random.Random(2116205)
+stats=Counter();score=0.;bestscore=0.;bestframes=W.opframe[:];bestchanges=[]
+for sweep,T in enumerate([.8,.4,.15,.05]+[0]*15+[.5,.2,.05,.01]+[0]*15):
+ num=0;gain=0
+ order=list(range(len(W.ops)));rng.shuffle(order)
+ for oi,i in enumerate(order):
+  old=W.opframe[i]
+  direction,bounds,neighbors,fn=rng.choice([('down',floor,before,lower_transaction),('up',cap,after,transaction)])
+  if C.dimf[old]==C.dimf[bounds[i]]:continue
+  candidates={bounds[i]}
+  for chain in range(2):
+   j=neighbors[i][chain]
+   for depth in range(8):
+    candidates.add(frame(j))
+    if j<0:break
+    if depth<3:candidates.update(frame(k)for k in neighbors[j])
+    j=min(neighbors[j],key=lambda k:C.dimf[frame(k)])
+  best=None
+  for f in candidates:
+   if f==old:continue
+   stats['proposed']+=1;r=fn(i,f)
+   if r is None:continue
+   stats['feasible']+=1
+   if best is None or r[0]>best[0]:best=r
+  if best and(best[0]>1e-10 or T and rng.random()<math.exp(min(0,best[0]/T))):
+   improvement,updates=best
+   for j,f in updates.items():W.opframe[j]=f
+   num+=len(updates);gain+=improvement;score+=improvement
+   if improvement>1e-10:stats['improving']+=1
+   else:stats['annealed']+=1
+   changes.append(dict(direction=direction,seed=i,ops=sorted(updates),gain=improvement))
+   if score>bestscore+1e-9:
+    bestscore=score;bestframes=W.opframe[:];bestchanges=changes[:]
+  if oi%10000==0:print('scan',sweep,oi,'T',T,'current_gain',score,'best_gain',bestscore,'seconds',time.monotonic()-started,flush=True)
+ print('sweep',sweep,'T',T,'changes',num,'entropy_gain',gain,'current',score,'best',bestscore,'stats',dict(stats),'seconds',time.monotonic()-started,flush=True)
+ if sweep==18:
+  # Restart second excursion from strongest state after full greedy relaxation.
+  W.opframe=bestframes[:];score=bestscore
+W.opframe=bestframes;tot=bestscore;changes=bestchanges
+W.changed_frames=[i for i,(a,b)in enumerate(zip(W.original_opframe,W.opframe))if a!=b]
+W.endframe={s:W.opframe[xs[-1]]for s,xs in W.role_ops.items()}
+W.exact_frames();row=W.row();delta=Counter({int(k):n for k,n in row['child_histogram'].items()});delta.subtract({int(k):n for k,n in baseline['child_histogram'].items()})
+frames=[[i,C.B[f]]for i,f in enumerate(W.opframe)if f!=base[i]]
+(D/'relaxed-additional-frames.json').write_text(json.dumps(frames,separators=(',',':'))+'\n')
+(D/'relaxed-all-frames.json').write_text(json.dumps([[i,C.B[f]]for i,f in enumerate(W.opframe)if f!=W.w['op_frame'][i]],separators=(',',':'))+'\n')
+out=dict(status='EXACT_FRAMES_AND_NESTED_CHAINS_PASS_NOT_FULL_REPLAY',baseline=baseline,profile=row,child_delta={str(k):v for k,v in sorted(delta.items())if v},changes=changes,entropy_gain=tot,elapsed=time.monotonic()-started)
+(D/'relaxed-result.json').write_text(json.dumps(out,indent=2)+'\n');print('DONE additional',len(frames),'delta',out['child_delta'],'gain',tot,flush=True)
diff --git a/research/coordinated-crossover-pr200/search/source467-selection-full.json b/research/coordinated-crossover-pr200/search/source467-selection-full.json
new file mode 100644
index 0000000000..bfc6812121
--- /dev/null
+++ b/research/coordinated-crossover-pr200/search/source467-selection-full.json
@@ -0,0 +1,572 @@
+{
+  "scope": "Exact best-per-disjoint-target-group matching over every individually positive fixed-frame candidate",
+  "positive_edges": 197,
+  "distinct_target_groups": 24,
+  "selected": 24,
+  "pairwise_disjoint_targets": true,
+  "distinct_source_pairs": true,
+  "distinct_roles": true,
+  "each_group_attains_best_available_gain": true,
+  "local_score": -0.08520954315483209,
+  "selected_first_ranks": {
+    "12": 1,
+    "5": 7,
+    "14": 12,
+    "17": 4
+  },
+  "selected_target_count": 96,
+  "rows": [
+    {
+      "role": 14079,
+      "source": 322,
+      "partner": 321,
+      "mix_frame": 125627,
+      "first_frame": 146011,
+      "rank": 12,
+      "targets": [
+        24,
+        25,
+        26,
+        27
+      ],
+      "local_delta": {
+        "12": -1,
+        "22": -1,
+        "10": 1,
+        "2": 4,
+        "19": 4,
+        "21": -4
+      },
+      "score": -0.0029177649623193247
+    },
+    {
+      "role": 12627,
+      "source": 114,
+      "partner": 113,
+      "mix_frame": 126095,
+      "first_frame": 141019,
+      "rank": 5,
+      "targets": [
+        32,
+        33,
+        34,
+        35
+      ],
+      "local_delta": {
+        "5": -1,
+        "22": -1,
+        "3": 1,
+        "2": 4,
+        "19": 4,
+        "21": -4
+      },
+      "score": -0.0071801182017736664
+    },
+    {
+      "role": 14075,
+      "source": 82,
+      "partner": 81,
+      "mix_frame": 126167,
+      "first_frame": 146059,
+      "rank": 14,
+      "targets": [
+        48,
+        49,
+        50,
+        51
+      ],
+      "local_delta": {
+        "14": -1,
+        "22": -1,
+        "12": 1,
+        "2": 4,
+        "19": 4,
+        "21": -4
+      },
+      "score": -0.0022189077896683784
+    },
+    {
+      "role": 14424,
+      "source": 26,
+      "partner": 25,
+      "mix_frame": 126293,
+      "first_frame": 89451,
+      "rank": 17,
+      "targets": [
+        120,
+        121,
+        122,
+        123
+      ],
+      "local_delta": {
+        "17": -1,
+        "22": -1,
+        "15": 1,
+        "2": 4,
+        "19": 4,
+        "21": -4
+      },
+      "score": -0.0013510143260191398
+    },
+    {
+      "role": 14422,
+      "source": 18,
+      "partner": 17,
+      "mix_frame": 126311,
+      "first_frame": 89451,
+      "rank": 17,
+      "targets": [
+        128,
+        129,
+        130,
+        131
+      ],
+      "local_delta": {
+        "17": -1,
+        "22": -1,
+        "15": 1,
+        "2": 4,
+        "19": 4,
+        "21": -4
+      },
+      "score": -0.0013510143260191398
+    },
+    {
+      "role": 14080,
+      "source": 10,
+      "partner": 9,
+      "mix_frame": 126329,
+      "first_frame": 116002,
+      "rank": 14,
+      "targets": [
+        328,
+        329,
+        330,
+        331
+      ],
+      "local_delta": {
+        "14": -1,
+        "22": -1,
+        "12": 1,
+        "2": 4,
+        "19": 4,
+        "21": -4
+      },
+      "score": -0.0022189077896683784
+    },
+    {
+      "role": 12644,
+      "source": 330,
+      "partner": 329,
+      "mix_frame": 125609,
+      "first_frame": 143766,
+      "rank": 5,
+      "targets": [
+        384,
+        385,
+        386,
+        387
+      ],
+      "local_delta": {
+        "5": -1,
+        "22": -1,
+        "3": 1,
+        "2": 4,
+        "19": 4,
+        "21": -4
+      },
+      "score": -0.0071801182017736664
+    },
+    {
+      "role": 14072,
+      "source": 2,
+      "partner": 1,
+      "mix_frame": 126347,
+      "first_frame": 89739,
+      "rank": 14,
+      "targets": [
+        408,
+        409,
+        410,
+        411
+      ],
+      "local_delta": {
+        "14": -1,
+        "22": -1,
+        "12": 1,
+        "2": 4,
+        "19": 4,
+        "21": -4
+      },
+      "score": -0.0022189077896683784
+    },
+    {
+      "role": 14436,
+      "source": 586,
+      "partner": 585,
+      "mix_frame": 125033,
+      "first_frame": 86539,
+      "rank": 17,
+      "targets": [
+        480,
+        481,
+        482,
+        483
+      ],
+      "local_delta": {
+        "17": -1,
+        "22": -1,
+        "15": 1,
+        "2": 4,
+        "19": 4,
+        "21": -4
+      },
+      "score": -0.0013510143260191398
+    },
+    {
+      "role": 14434,
+      "source": 578,
+      "partner": 577,
+      "mix_frame": 125051,
+      "first_frame": 86539,
+      "rank": 17,
+      "targets": [
+        488,
+        489,
+        490,
+        491
+      ],
+      "local_delta": {
+        "17": -1,
+        "22": -1,
+        "15": 1,
+        "2": 4,
+        "19": 4,
+        "21": -4
+      },
+      "score": -0.0013510143260191398
+    },
+    {
+      "role": 14098,
+      "source": 474,
+      "partner": 473,
+      "mix_frame": 125285,
+      "first_frame": 146110,
+      "rank": 14,
+      "targets": [
+        688,
+        689,
+        690,
+        691
+      ],
+      "local_delta": {
+        "14": -1,
+        "22": -1,
+        "12": 1,
+        "2": 4,
+        "19": 4,
+        "21": -4
+      },
+      "score": -0.0022189077896683784
+    },
+    {
+      "role": 12718,
+      "source": 690,
+      "partner": 689,
+      "mix_frame": 124799,
+      "first_frame": 143808,
+      "rank": 5,
+      "targets": [
+        744,
+        745,
+        746,
+        747
+      ],
+      "local_delta": {
+        "5": -1,
+        "22": -1,
+        "3": 1,
+        "2": 4,
+        "19": 4,
+        "21": -4
+      },
+      "score": -0.0071801182017736664
+    },
+    {
+      "role": 14090,
+      "source": 442,
+      "partner": 441,
+      "mix_frame": 125357,
+      "first_frame": 86832,
+      "rank": 14,
+      "targets": [
+        768,
+        769,
+        770,
+        771
+      ],
+      "local_delta": {
+        "14": -1,
+        "22": -1,
+        "12": 1,
+        "2": 4,
+        "19": 4,
+        "21": -4
+      },
+      "score": -0.0022189077896683784
+    },
+    {
+      "role": 14116,
+      "source": 826,
+      "partner": 825,
+      "mix_frame": 124493,
+      "first_frame": 146150,
+      "rank": 14,
+      "targets": [
+        976,
+        977,
+        978,
+        979
+      ],
+      "local_delta": {
+        "14": -1,
+        "22": -1,
+        "12": 1,
+        "2": 4,
+        "19": 4,
+        "21": -4
+      },
+      "score": -0.0022189077896683784
+    },
+    {
+      "role": 12792,
+      "source": 978,
+      "partner": 977,
+      "mix_frame": 124151,
+      "first_frame": 143859,
+      "rank": 5,
+      "targets": [
+        1032,
+        1033,
+        1034,
+        1035
+      ],
+      "local_delta": {
+        "5": -1,
+        "22": -1,
+        "3": 1,
+        "2": 4,
+        "19": 4,
+        "21": -4
+      },
+      "score": -0.0071801182017736664
+    },
+    {
+      "role": 14108,
+      "source": 802,
+      "partner": 801,
+      "mix_frame": 124547,
+      "first_frame": 83861,
+      "rank": 14,
+      "targets": [
+        1056,
+        1057,
+        1058,
+        1059
+      ],
+      "local_delta": {
+        "14": -1,
+        "22": -1,
+        "12": 1,
+        "2": 4,
+        "19": 4,
+        "21": -4
+      },
+      "score": -0.0022189077896683784
+    },
+    {
+      "role": 14134,
+      "source": 1250,
+      "partner": 1249,
+      "mix_frame": 123539,
+      "first_frame": 146192,
+      "rank": 14,
+      "targets": [
+        1200,
+        1201,
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diff --git a/research/coordinated-crossover-pr200/targetagg/PROVENANCE.json b/research/coordinated-crossover-pr200/targetagg/PROVENANCE.json
new file mode 100644
index 0000000000..15428edc7e
--- /dev/null
+++ b/research/coordinated-crossover-pr200/targetagg/PROVENANCE.json
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+{
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+  "admission": "Literal F2 and both defining-integer signs,17 whole-word controls including integer sign corruption, independent complete source and target paths in both reflections, local integer transfer matrices, prime witnesses, concrete banks and exact paid assembly."
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+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/coordinated-crossover-pr200/extra/setup.py": "a7789d13eda75e78108cee84206692c913f63aafdb9e3e633fddd14640e4678a",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/coordinated-crossover-pr200/extra/selection.json": "791f6ddeb19add7026ba37461256c13efe3d35bba0bccf1cb9787a01fc16f0f2",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/coordinated-crossover-pr200/rank/selection.json": "5c97c6413f1eb56c7b8f9ba9e839329d212fae7ac3a08c29e9efdee42f5048a0",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/coordinated-crossover-pr200/echelon/selection.json": "8e6ce5e257aa7656898ab46bc3421df4a619497b1e9f5956d2c546b3014050a7",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/coordinated-crossover-pr200/newg/profile.json": "6ace263012a6526d5eccdaaa24575ecf76a72a7c97dc5dd1699daa76ebc8d619",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/coordinated-crossover-pr200/targetagg/boundary.py": "20f4c69043754901193bcf55ec89b8d0dec4c518094c7933b77f5715f163787b"
+  },
+  "old_complete_target_histogram": {
+    "1": 4412,
+    "2": 160,
+    "3": 14,
+    "4": 4,
+    "5": 17,
+    "6": 9,
+    "7": 24,
+    "8": 28,
+    "9": 5,
+    "12": 16,
+    "13": 48,
+    "14": 17,
+    "16": 3,
+    "17": 4,
+    "18": 29,
+    "19": 112,
+    "20": 836,
+    "21": 694
+  },
+  "new_complete_target_histogram": {
+    "1": 4435,
+    "2": 118,
+    "3": 13,
+    "4": 1,
+    "5": 17,
+    "6": 4,
+    "7": 24,
+    "8": 9,
+    "9": 1,
+    "12": 12,
+    "13": 24,
+    "14": 17,
+    "16": 3,
+    "17": 4,
+    "18": 17,
+    "19": 80,
+    "20": 870,
+    "21": 732
+  },
+  "reflected_target_histogram": {
+    "1": 4435,
+    "2": 118,
+    "3": 13,
+    "4": 1,
+    "5": 17,
+    "6": 4,
+    "7": 24,
+    "8": 9,
+    "9": 1,
+    "12": 12,
+    "13": 24,
+    "14": 17,
+    "16": 3,
+    "17": 4,
+    "18": 17,
+    "19": 80,
+    "20": 870,
+    "21": 732
+  },
+  "rank_target_delta": {
+    "1": 28,
+    "8": -23,
+    "9": -4,
+    "12": -3,
+    "13": -24,
+    "20": 27
+  },
+  "rank_restores": [
+    {
+      "group": 5,
+      "before_role": 16720,
+      "frame": 148184
+    },
+    {
+      "group": 4,
+      "before_role": 16700,
+      "frame": 148183
+    },
+    {
+      "group": 3,
+      "before_role": 16680,
+      "frame": 148182
+    },
+    {
+      "group": 2,
+      "before_role": 16660,
+      "frame": 148181
+    },
+    {
+      "group": 1,
+      "before_role": 16640,
+      "frame": 148180
+    },
+    {
+      "group": 0,
+      "before_role": 16620,
+      "frame": 148179
+    },
+    {
+      "group": 6,
+      "before_role": 16598,
+      "frame": 148185
+    }
+  ],
+  "rank_removed_reads": 108,
+  "local_transfer_controls_rejected": [
+    "omit_setup",
+    "omit_inverse",
+    "repeat_dependent"
+  ],
+  "all_local_integer_transfers_both_signs": true,
+  "echelon_restores": [
+    {
+      "group": 3,
+      "before_role": 16590,
+      "frame": 148189
+    },
+    {
+      "group": 0,
+      "before_role": 17148,
+      "frame": 148186
+    },
+    {
+      "group": 1,
+      "before_role": 17188,
+      "frame": 148187
+    },
+    {
+      "group": 2,
+      "before_role": 17220,
+      "frame": 148188
+    }
+  ],
+  "echelon_reads": 48,
+  "echelon_local_controls_rejected": [
+    "omit_setup",
+    "omit_inverse",
+    "flip_sign"
+  ],
+  "echelon_transforms": [
+    {
+      "targets": [
+        0,
+        1,
+        2,
+        3
+      ],
+      "matrix": [
+        [
+          1,
+          0,
+          0,
+          0
+        ],
+        [
+          -1,
+          1,
+          0,
+          0
+        ],
+        [
+          -1,
+          0,
+          1,
+          0
+        ],
+        [
+          0,
+          0,
+          -1,
+          1
+        ]
+      ],
+      "inverse": [
+        [
+          1,
+          0,
+          0,
+          0
+        ],
+        [
+          1,
+          1,
+          0,
+          0
+        ],
+        [
+          1,
+          0,
+          1,
+          0
+        ],
+        [
+          1,
+          0,
+          1,
+          1
+        ]
+      ],
+      "determinant": 1
+    },
+    {
+      "targets": [
+        40,
+        41,
+        42,
+        43
+      ],
+      "matrix": [
+        [
+          1,
+          0,
+          0,
+          0
+        ],
+        [
+          -1,
+          1,
+          0,
+          0
+        ],
+        [
+          -1,
+          0,
+          1,
+          0
+        ],
+        [
+          0,
+          0,
+          -1,
+          1
+        ]
+      ],
+      "inverse": [
+        [
+          1,
+          0,
+          0,
+          0
+        ],
+        [
+          1,
+          1,
+          0,
+          0
+        ],
+        [
+          1,
+          0,
+          1,
+          0
+        ],
+        [
+          1,
+          0,
+          1,
+          1
+        ]
+      ],
+      "determinant": 1
+    },
+    {
+      "targets": [
+        72,
+        73,
+        74,
+        75
+      ],
+      "matrix": [
+        [
+          1,
+          0,
+          0,
+          0
+        ],
+        [
+          -1,
+          1,
+          0,
+          0
+        ],
+        [
+          -1,
+          0,
+          1,
+          0
+        ],
+        [
+          0,
+          0,
+          -1,
+          1
+        ]
+      ],
+      "inverse": [
+        [
+          1,
+          0,
+          0,
+          0
+        ],
+        [
+          1,
+          1,
+          0,
+          0
+        ],
+        [
+          1,
+          0,
+          1,
+          0
+        ],
+        [
+          1,
+          0,
+          1,
+          1
+        ]
+      ],
+      "determinant": 1
+    },
+    {
+      "targets": [
+        864,
+        865,
+        866,
+        867,
+        868,
+        869,
+        870,
+        871
+      ],
+      "matrix": [
+        [
+          1,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0
+        ],
+        [
+          -1,
+          1,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0
+        ],
+        [
+          0,
+          0,
+          1,
+          0,
+          0,
+          0,
+          0,
+          0
+        ],
+        [
+          1,
+          -1,
+          -1,
+          1,
+          0,
+          0,
+          0,
+          0
+        ],
+        [
+          -1,
+          1,
+          0,
+          0,
+          1,
+          -1,
+          0,
+          0
+        ],
+        [
+          0,
+          -1,
+          0,
+          0,
+          0,
+          1,
+          0,
+          0
+        ],
+        [
+          0,
+          -1,
+          -1,
+          0,
+          0,
+          1,
+          1,
+          0
+        ],
+        [
+          1,
+          -2,
+          -1,
+          0,
+          0,
+          1,
+          0,
+          1
+        ]
+      ],
+      "inverse": [
+        [
+          1,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0
+        ],
+        [
+          1,
+          1,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0
+        ],
+        [
+          0,
+          0,
+          1,
+          0,
+          0,
+          0,
+          0,
+          0
+        ],
+        [
+          0,
+          1,
+          1,
+          1,
+          0,
+          0,
+          0,
+          0
+        ],
+        [
+          1,
+          0,
+          0,
+          0,
+          1,
+          1,
+          0,
+          0
+        ],
+        [
+          1,
+          1,
+          0,
+          0,
+          0,
+          1,
+          0,
+          0
+        ],
+        [
+          0,
+          0,
+          1,
+          0,
+          0,
+          -1,
+          1,
+          0
+        ],
+        [
+          0,
+          1,
+          1,
+          0,
+          0,
+          -1,
+          0,
+          1
+        ]
+      ],
+      "determinant": 1
+    }
+  ]
+}
diff --git a/research/coordinated-crossover-pr200/targetagg/boundary.py b/research/coordinated-crossover-pr200/targetagg/boundary.py
new file mode 100644
index 0000000000..f378025fa7
--- /dev/null
+++ b/research/coordinated-crossover-pr200/targetagg/boundary.py
@@ -0,0 +1,214 @@
+"""Independent literal event geometry and complete profile for ordered source493.
+OpenAI Codex assistance; Apache-2.0. Does not mutate scalar witnesses.
+"""
+from pathlib import Path
+from collections import Counter,defaultdict
+import json,hashlib,ast,gzip
+D=Path(__file__).resolve().parent;P=D.parent
+sha=lambda p:hashlib.sha256(p.read_bytes()).hexdigest();read=lambda p:json.loads(p.read_text());compact=lambda H:{str(k):v for k,v in sorted(H.items())if v};counter=lambda H:Counter({int(k):v for k,v in H.items()})
+import importlib.util
+sp=importlib.util.spec_from_file_location('independent_complete_setup',D/'setup.py');setup=importlib.util.module_from_spec(sp);sp.loader.exec_module(setup)
+m=setup.m;W=m.W;C=m.C;extra=setup.extra;gs=m.agg;roles=m.role_groups;owner=m.target_group;rf=m.read_frames;extraroles={r['role']for r in extra};scalar=read(D/'replay.json');rankgroups=m.rankgroups;rankowner=m.rankowner;echelon=m.echelon
+inputs=[D/'replay.py',D/'word.py',D/'setup.py',D/'selection.json',D/'replay.json',P/'extra/setup.py',P/'extra/selection.json',P/'rank/selection.json',P/'echelon/selection.json',P/'newg/profile.json',Path(__file__)];pins={str(p):sha(p)for p in inputs}
+assert len(extra)==22 and len(gs)==82 and len(owner)==316 and len(m.borrow)==493
+assert scalar['source_selection']==extra and scalar['status']=='PASS_SOURCE493_82_EQUAL7_RANK4_ECHELON_PR230'
+assert len(scalar['integer'])==2 and {x['direction']for x in scalar['integer']}=={-1,1}
+for x in[scalar['F2']]+scalar['integer']:assert x['formal_columns']==20141 and x['dirty']==16621 and x['all_targets']and x['all_source_and_dirty_restored']and x['all_forward_centers_at_D0']and x['all_reflected_centers_at_D1']
+assert len(scalar['controls'])==17 and all(x['status']=='REJECTED'for x in scalar['controls'].values())
+for path,pin in scalar['input_pins'].items():assert sha(Path(path))==pin
+
+def hist(chain,start=0):
+ h=Counter();d=start;previous=None
+ for f in chain:
+  assert C.nondeg(f)and(previous is None or C.sub(previous,f));n=C.dimf[f];assert n>=d
+  if n>d:h[n-d]+=1
+  d=n;previous=f
+ return h
+
+def reflected_source(chain):
+ h=Counter()
+ for a,b in zip(reversed(chain),list(reversed(chain))[1:]):
+  assert all(W.module.dot(x,y)==0 for x in C.A[a]for y in C.B[b]);d=len(C.A[b])-len(C.A[a]);assert d>=0
+  if d:h[d]+=1
+ return h
+
+def trace(aggregated,include_extra,ranked=False,echeloned=False):
+ paths=[[]for _ in range(W.v)];source_paths={r['role']:[W.w['source_frame'][r['source']],r['mix_frame']]for r in extra} if include_extra else {};source_events={r:[]for r in source_paths}
+ active=set(range(len(gs)))if aggregated else set();seen=[set()for _ in gs];restores=[];pivotreads=0;removedreads=0;events=0
+ rankactive=set(range(len(rankgroups)))if ranked else set();rankseen=[set()for _ in rankgroups];rankrestores=[];rankremoved=0
+ echactive=set(range(len(echelon)))if echeloned else set();echseen=[set()for _ in echelon];echrestores=[];echreads=0
+ def smove(s,f,event):
+  role=W.phys[s]
+  if role not in source_paths:return
+  assert C.nondeg(f)and C.sub(source_paths[role][-1],f),('independent source event',role,event,source_paths[role][-1],f)
+  source_paths[role].append(f);source_events[role].append(dict(virtual=s,frame=f,event=event))
+ def move(t,f):
+  nonlocal events
+  assert C.nondeg(f)and(not paths[t]or C.sub(paths[t][-1],f));assert all(W.module.dot(C.cov[t],b)==0 for b in C.B[f]);paths[t].append(f);events+=1
+ def gauge(s):
+  nonlocal pivotreads,removedreads,rankremoved,echreads
+  if not include_extra and s in extraroles:return
+  echskip=set()
+  for k,g in enumerate(echelon):
+   if k not in echactive:continue
+   if s not in g['roles']:
+    if set(m.adj[s])&set(g['targets']):
+     assert echseen[k]==g['roles']
+     for t in g['targets']:move(t,g['frame'])
+     echactive.remove(k);echrestores.append(dict(group=k,before_role=s,frame=g['frame']))
+    continue
+   echseen[k].add(s);echskip.update(g['targets'])
+   for t,a in g['response'].items():
+    if a.get(s,0):smove(s,W.gauge[s]['frame'],['echelon_read',t]);move(t,W.gauge[s]['frame']);echreads+=1
+  for k in sorted({rankowner[t]for t in m.adj[s]if t in rankowner}):
+   if k not in rankactive:continue
+   g=rankgroups[k]
+   if s in g['roles']:rankseen[k].add(s)
+   else:
+    assert rankseen[k]==g['roles']
+    for t in g['targets']:move(t,g['frame'])
+    rankactive.remove(k);rankrestores.append(dict(group=k,before_role=s,frame=g['frame']))
+  grouped=roles.get(s,[])if aggregated else []
+  for i in sorted({owner[t]for t in m.adj[s]if t in owner}):
+   if i not in active or i in grouped:continue
+   g=gs[i];assert seen[i]==set(g['roles'])
+   for t in g['targets']:move(t,g['frame'])
+   assert len({paths[t][-1]for t in g['targets']})==1
+   active.remove(i);restores.append(dict(group=i,before_role=s,frame=g['frame']))
+  skipped=set()
+  for i in grouped:
+   assert i in active and s not in seen[i];g=gs[i];f=rf[s,i]
+   smove(s,f,['group_read',i]);move(g['pivot'],f);seen[i].add(s);pivotreads+=1;skipped.update(g['targets']);removedreads+=len(g['targets'])-1
+  for t in m.adj[s]:
+   rk=rankowner.get(t)
+   if rk in rankactive and t in rankgroups[rk]['dependent']:rankremoved+=1;continue
+   if t not in skipped and t not in echskip:smove(s,W.gauge[s]['frame'],['gauge_read',t]);move(t,W.gauge[s]['frame'])
+ # All22 new source roles are physically untouched in the retimed center prefix.
+ assert not(extraroles&{W.phys[s]for i in W.phase1 if i not in m.omitted for s in W.ops[i][:2]})
+ for j,op in enumerate(W.rest):
+  for s in m.at[j]:gauge(s)
+  if op in m.bywrite:
+   smove(W.ops[op][1],W.opframe[op],['terminal_write',op]);move(m.bywrite[op]['pivot'],W.opframe[op])
+  elif op not in m.omitted:
+   for s in W.ops[op][:2]:smove(s,W.opframe[op],['operation',op])
+  if op in m.afterwrite:
+   for t in m.afterwrite[op]['targets']:move(t,m.afterwrite[op]['root_frame'])
+ for s in m.at[len(W.rest)]:gauge(s)
+ assert not active and not rankactive and not echactive
+ for j,(root,role)in enumerate(zip(W.g['roots'],W.w['rootroles'])):
+  if root['kind']=='side'and j not in m.deletedroots:
+   smove(role,W.w['root_frame'][j],['side_root',j])
+   for t in root['targets']:move(t,W.w['root_frame'][j])
+  for e in m.deliveries[j]:
+   for t in e['receivers']:move(t,e['deliver_frame'])
+ for role in source_paths:smove(role,W.w['full_frame'],['finish'])
+ H=Counter();R=Counter()
+ for t,path in enumerate(paths):
+  assert path;H.update(hist(path));d=C.dimf[path[-1]];assert d<=W.h-1
+  if d<W.h-1:H[W.h-1-d]+=1
+  previous=[C.cov[t]];d=1
+  for f in reversed(path):
+   assert all(W.module.dot(a,b)==0 for a in previous for b in C.B[f]);step=len(C.A[f])-d;assert step>=0
+   if step:R[step]+=1
+   previous=C.A[f];d=len(previous)
+  if W.h>d:R[W.h-d]+=1
+ assert H==R and sum(k*n for k,n in H.items())==W.v*(W.h-1)
+ return dict(histogram=H,reflected=R,restores=restores,pivotreads=pivotreads,removedreads=removedreads,events=events,source_paths=source_paths,source_events=source_events,rank_restores=rankrestores,rank_removed=rankremoved,echelon_restores=echrestores,echelon_reads=echreads)
+base=trace(False,False);equal=trace(True,True);ranked=trace(True,True,True);final=trace(True,True,True,True);rankdelta=Counter(ranked['histogram']);rankdelta.subtract(equal['histogram']);assert compact(rankdelta)==read(P/'rank/selection.json')['local_delta'];baseline=read(P/'newg/profile.json')['profile'];assert compact(base['histogram'])==baseline['physical_target_histogram']
+for x in[scalar['F2']]+scalar['integer']:
+ assert compact(final['histogram'])==x['actual_target_histogram']==x['actual_reflected_target_histogram']
+ assert final['restores']==x['aggregation_restores']and final['pivotreads']==x['aggregation_pivot_reads']==92
+ assert {str(k):v for k,v in final['source_paths'].items()}==x['actual_extra_source_paths'];assert final['rank_restores']==x['rank_restores']and final['rank_removed']==x['rank_removed_reads']==108
+ assert final['echelon_restores']==x['echelon_restores']and final['echelon_reads']==x['echelon_reads']==48
+# Each removed ordinary physical register includes its complete later alias.
+recipient={d:b for b,d in W.pairs};roots=defaultdict(list)
+for j,r in enumerate(W.w['rootroles']):roots[r].append(W.w['root_frame'][j])
+positions={i:j for j,i in enumerate(W.phase1+W.rest)}
+rows=[];source_delta=Counter();local=counter(baseline['physical_internal_histogram']);sources=counter(baseline['physical_source_histogram'])
+for r in extra:
+ role,s=r['role'],r['source'];ops=W.role_ops[role];aux=[W.opframe[i]for i in ops]+roots[role]
+ if role in recipient:
+  rec=recipient[role];assert positions[ops[-1]]<positions[W.role_ops[rec][0]]
+  aux += [W.gauge[rec]['frame']]+[W.opframe[i]for i in W.role_ops[rec]]+roots[rec]
+ aux.append(W.w['full_frame']);oldaux=hist(aux);oldsource=hist(m.kpair[s]['passive_chain'][1:],1);chain=final['source_paths'][role]
+ newsource=hist(chain[1:],1);assert newsource==reflected_source(chain)
+ d=Counter(newsource);d.subtract(oldaux);d.subtract(oldsource);assert sum(k*v for k,v in d.items())==-24
+ source_delta.update(d);local.subtract(oldaux);sources.subtract(oldsource);sources.update(newsource)
+ rows.append(dict(role=role,source=s,partner=r['partner'],source_path=chain,old_aux_histogram=compact(oldaux),old_source_histogram=compact(oldsource),complete_source_histogram=compact(newsource),source_delta=compact(d),reflected_equal=True))
+assert all(n>=0 for h in(local,sources)for n in h.values());assert sum(k*n for k,n in source_delta.items())==-528
+child=counter(baseline['child_histogram']);target_delta=Counter(final['histogram']);target_delta.subtract(base['histogram']);totaldelta=Counter(source_delta);totaldelta.update(target_delta)
+for k,n in totaldelta.items():child[k]+=3*n
+assert all(n>=0 for n in child.values());mass=sum(k*n for k,n in child.items());stock=baseline['W_per_vertex']-22;R=baseline['R']-22
+assert stock==20141 and R==16621 and mass==1448216 and 72*stock-mass==1936
+assert sum(k*n for k,n in sources.items())==sum(k*n for k,n in final['histogram'].items())==W.v*23
+profile=dict(baseline,R=R,active_virtual_R=R+baseline['reused_registers'],W_per_vertex=stock,rank_per_vertex=mass,child_histogram=compact(child),physical_internal_histogram=compact(local),physical_source_histogram=compact(sources),physical_target_histogram=compact(final['histogram']),physical_original_sources_borrowed=493,source_backed_new_rank2_gauges=45,source_backed_shared_rank4_gauges=5,target_aggregation_groups=82,target_aggregation_targets=316,target_transformed_compensation_reads=140,equal_response_pivot_reads=92,target_basis_scalar_gates=690)
+# Exact arithmetic prime admission for every newly introduced source and inverse frame.
+pcode=(P/'extra/boundary.py').read_text();tree=ast.parse(pcode);defs=[n for n in tree.body if isinstance(n,ast.FunctionDef)and n.name in('determinant','prime_record')];exec(compile(ast.Module(body=defs,type_ignores=[]),'<independent-primes>','exec'),globals())
+frames={g['frame']for g in gs}|{g['frame']for g in rankgroups}|{g['frame']for g in echelon}|{W.gauge[r['role']]['frame']for r in extra}|{r['mix_frame']for r in extra};primes=[dict(frame=f,basis=C.B[f],annihilator=C.A[f],primal=prime_record(C.B[f]),dual=prime_record(C.A[f],True))for f in sorted(frames)]
+# Exact local integer matrices for all equal and multi-pivot groups.
+def vector_add(a,c,b):
+ z=a.copy()
+ for k,v in b.items():
+  z[k]=z.get(k,0)+c*v
+  if not z[k]:del z[k]
+ return z
+def local_transfer(ts,dep,prefix,mutation=None):
+ for sign in(1,-1):
+  y={t:{('Y',t):1}for t in ts};want={t:v.copy()for t,v in y.items()}
+  for t,cs in dep.items():
+   for p,c in cs.items():
+    if mutation!='omit_setup':y[t]=vector_add(y[t],-c,y[p])
+  for role in prefix:
+   z={('Z',role):1}
+   for t in ts:
+    c=m.adj[role].get(t,0);want[t]=vector_add(want[t],-sign*c,z)
+    if t not in dep or mutation=='repeat_dependent':y[t]=vector_add(y[t],-sign*c,z)
+  for t,cs in dep.items():
+   for p,c in cs.items():
+    if mutation!='omit_inverse':y[t]=vector_add(y[t],c,y[p])
+  assert y==want
+for g in gs:local_transfer(g['targets'],{t:{g['pivot']:1}for t in g['targets']if t!=g['pivot']},g['roles'])
+for g in rankgroups:local_transfer(g['targets'],g['dependent'],g['roles'])
+localcontrols=[]
+for mutation in('omit_setup','omit_inverse','repeat_dependent'):
+ g=rankgroups[0]
+ try:local_transfer(g['targets'],g['dependent'],g['roles'],mutation)
+ except AssertionError:localcontrols.append(mutation)
+ else:raise AssertionError('Local matrix corruption accepted')
+# Independent arbitrary-variable signed integer transfers and determinant-one matrices.
+def echelon_transfer(g,mutation=None):
+ for sign in(1,-1):
+  ts=g['targets'];moves=g['moves'];R={int(t):{int(s):c for s,c in a.items()}for t,a in g['transformed_response'].items()}
+  y={t:{('Y',t):1}for t in ts};want={t:a.copy()for t,a in y.items()}
+  if mutation!='omit_setup':
+   for t,p,c in moves:y[t]=vector_add(y[t],c,y[p])
+  for role in g['prefix_roles']:
+   z={('Z',role):1}
+   for t in ts:
+    want[t]=vector_add(want[t],-sign*m.adj[role].get(t,0),z);c=R[t].get(role,0)
+    if mutation=='flip_sign'and c<0:c=-c
+    y[t]=vector_add(y[t],-sign*c,z)
+  if mutation!='omit_inverse':
+   for t,p,c in reversed(moves):y[t]=vector_add(y[t],-c,y[p])
+  assert y==want
+transforms=[]
+for g in setup.echelonselection['groups']:
+ echelon_transfer(g);ts=g['targets'];U={t:{p:int(t==p)for p in ts}for t in ts};V={t:a.copy()for t,a in U.items()}
+ for t,p,c in g['moves']:U[t]=vector_add(U[t],c,U[p])
+ for t,p,c in reversed(g['moves']):V[t]=vector_add(V[t],-c,V[p])
+ A=[[U[t].get(p,0)for p in ts]for t in ts];B=[[V[t].get(p,0)for p in ts]for t in ts];assert determinant(A)==1
+ assert all(sum(A[i][k]*B[k][j]for k in range(len(ts)))==int(i==j)for i in range(len(ts))for j in range(len(ts)))
+ transforms.append(dict(targets=ts,matrix=A,inverse=B,determinant=1))
+echeloncontrols=[]
+for mutation in('omit_setup','omit_inverse','flip_sign'):
+ try:echelon_transfer(setup.echelonselection['groups'][0],mutation)
+ except AssertionError:echeloncontrols.append(mutation)
+ else:raise AssertionError('Echelon local corruption accepted:'+mutation)
+profile.update(target_echelon_groups=4,target_echelon_targets=20,target_echelon_basis_scalar_gates=48,target_echelon_signed_compensation_reads=48,target_echelon_compensation_unit_adds=56,partially_lifted_source_gauges=3)
+profile['source_owned_gauge_entrances']=53;profile['source_owned_gauge_rank_histogram']={'2':45,'4':5,'19':3};profile['all_gauge_entrances']=2332;profile['all_gauge_rank_histogram']={'2':45,'4':5,'12':18,'13':48,'18':13,'19':3,'20':2200}
+profile['target_rank_compression_groups']=7;profile['target_rank_compression_targets']=56;profile['target_rank_compression_dependent_rows']=27;profile['target_aggregation_basis_scalar_gates']=468;profile['target_rank_basis_scaled_add_gates']=174;profile['target_basis_scalar_gates']=690;profile['target_basis_expanded_unit_adds']=702
+setupcalls=sum(len(g['targets'])-1 for g in gs);assert setupcalls==234
+out=dict(status='PASS_SOURCE493_82_PLUS7_PLUS4_INDEPENDENT_ALL_CHANGED_SOURCE_TARGET_LEDGERS_TRANSFER_PROFILE_AND_PRIMES',selected=22,source_heads=493,groups=82,targets=316,source_local_delta=compact(source_delta),target_local_delta=compact(target_delta),complete_local_delta=compact(totaldelta),raw_W=stock,raw_R=R,raw_rank=mass,raw_deficit=1936,profile=profile,source_rows=rows,target_restores=final['restores'],source_events=final['source_events'],source_paths=final['source_paths'],pivot_reads=final['pivotreads'],setup_calls=setupcalls,inverse_calls=setupcalls,removed_compensation_reads=final['removedreads'],prime_witnesses=primes,maximum_prime_residual=max(z[k]['remaining_factor']for z in primes for k in('primal','dual')),all_new_source_paths_reflected=True,all_target_paths_reflected=True,scope='Independent full actual changed source and target event chronology, former auxiliary complete alias paths, both reflected ledgers, full paid profile and dual prime admission. Whole scalar word owned by parent; complete banks and exact assembly remain separate.',input_pins=pins)
+out.update(old_complete_target_histogram=compact(base['histogram']),new_complete_target_histogram=compact(final['histogram']),reflected_target_histogram=compact(final['reflected']),rank_target_delta=compact(rankdelta),rank_restores=final['rank_restores'],rank_removed_reads=final['rank_removed'],local_transfer_controls_rejected=localcontrols,all_local_integer_transfers_both_signs=True,echelon_restores=final['echelon_restores'],echelon_reads=final['echelon_reads'],echelon_local_controls_rejected=echeloncontrols,echelon_transforms=transforms)
+assert all(sha(Path(p))==h for p,h in pins.items())
+(D/'boundary.json').write_text(json.dumps(out,indent=2)+'\n');print('PASS source493 geometry and full profile',dict(W=stock,R=R,rank=mass,local_delta=compact(totaldelta)),flush=True)
diff --git a/research/coordinated-crossover-pr200/targetagg/derive_profile.py b/research/coordinated-crossover-pr200/targetagg/derive_profile.py
new file mode 100644
index 0000000000..ad75992204
--- /dev/null
+++ b/research/coordinated-crossover-pr200/targetagg/derive_profile.py
@@ -0,0 +1,99 @@
+"""Complete source493/82+7+4-group paid profile from literal endpoint ledgers.
+Prepared with substantial OpenAI Codex assistance; Apache-2.0.
+The full source, auxiliary, target, bank and scalar costs are retained.
+"""
+from pathlib import Path
+from collections import Counter
+from copy import deepcopy
+import hashlib,json
+D=Path(__file__).resolve().parent;P=D.parent
+read=lambda p:json.loads(p.read_text());sha=lambda p:hashlib.sha256(p.read_bytes()).hexdigest()
+counter=lambda h:Counter({int(k):n for k,n in h.items()})
+compact=lambda h:{str(k):n for k,n in sorted(h.items())if n}
+base=read(P/'newg/profile.json');r=deepcopy(base['profile'])
+scalar=read(D/'replay.json');source_boundary=read(P/'extra/boundary.json');target_boundary=read(D/'boundary.json')
+selection=read(P/'extra/selection.json');groups=read(D/'selection.json')['groups'];banks=read(P/'joint/joint-banks.json')
+assert len(selection)==22 and len(groups)==82
+assert {x['role']for x in source_boundary['endpoint_rows']}=={x['role']for x in selection}
+assert len(scalar['integer'])==2 and {x['direction']for x in scalar['integer']}=={-1,1}
+assert len(scalar['controls'])==17 and all(x['status']=='REJECTED'for x in scalar['controls'].values())
+assert target_boundary['all_local_integer_transfers_both_signs']and len(target_boundary['local_transfer_controls_rejected'])==3
+assert len(target_boundary['echelon_local_controls_rejected'])==3 and len(target_boundary['echelon_transforms'])==4
+assert {e['role']:(e['old_aux_histogram'],e['old_source_histogram'],e['complete_source_histogram'])for e in source_boundary['endpoint_rows']}=={e['role']:(e['old_aux_histogram'],e['old_source_histogram'],e['complete_source_histogram'])for e in target_boundary['source_rows']}
+target=counter(scalar['F2']['actual_target_histogram'])
+assert target==counter(target_boundary['new_complete_target_histogram'])
+assert counter(target_boundary['old_complete_target_histogram'])==counter(r['physical_target_histogram'])
+for x in [scalar['F2']]+scalar['integer']:
+ assert x['formal_columns']==20141 and x['dirty']==16621 and x['all_targets']and x['all_source_and_dirty_restored']
+ assert x['all_forward_centers_at_D0']and x['all_reflected_centers_at_D1']and x['copied_center_target_reads']==5280
+ assert target==counter(x['actual_target_histogram'])==counter(x['actual_reflected_target_histogram'])
+ assert x['aggregation_groups']==82 and x['aggregation_pivot_reads']==92 and x['echelon_reads']==48 and len(x['echelon_restores'])==4
+ assert x['rank_groups']==7 and x['rank_targets']==56 and x['rank_removed_reads']==108
+ assert len(x['rank_restores'])==7
+ assert counter(x['actual_extra_source_histogram'])==counter(x['actual_extra_reflected_source_histogram'])
+ assert len(x['aggregation_restores'])==82 and {z['group']for z in x['aggregation_restores']}==set(range(82))
+internal=counter(r['physical_internal_histogram']);source=counter(r['physical_source_histogram']);endpoint_delta=Counter()
+for e in source_boundary['endpoint_rows']:
+ old_aux=counter(e['old_aux_histogram']);old_source=counter(e['old_source_histogram']);new_source=counter(e['complete_source_histogram'])
+ internal.subtract(old_aux);source.subtract(old_source);source.update(new_source)
+ d=Counter(new_source);d.subtract(old_source);d.subtract(old_aux);endpoint_delta.update(d)
+assert sum(k*n for k,n in endpoint_delta.items())==-528
+assert all(n>=0 for h in[internal,source,target]for n in h.values())
+target_delta=Counter(target);target_delta.subtract(counter(r['physical_target_histogram']))
+assert sum(k*n for k,n in target_delta.items())==0
+local_delta=Counter(endpoint_delta);local_delta.update(target_delta)
+child=Counter()
+for h in[internal,source,target]:
+ for k,n in h.items():
+  if k and n:child[k]+=3*n
+for rank,count in r['selected_rank_histogram'].items():child[3*int(rank)]+=count
+child[2]+=2*r['v']
+assert all(n>0 for n in child.values())
+stock=r['W_per_vertex']-22;physical=r['R']-22;mass=sum(k*n for k,n in child.items())
+assert (stock,physical,mass)==(20141,16621,1448216)
+assert stock*72-mass==1936
+assert sum(k*n for k,n in source.items())==sum(k*n for k,n in target.items())==1760*23
+delta=Counter(child);delta.subtract(counter(r['child_histogram']))
+assert compact(delta)==compact({k:3*n for k,n in local_delta.items()})
+profile=dict(r,R=physical,active_virtual_R=r['active_virtual_R']-22,W_per_vertex=stock,rank_per_vertex=mass,
+ child_histogram=compact(child),physical_internal_histogram=compact(internal),physical_source_histogram=compact(source),physical_target_histogram=compact(target),
+ physical_original_sources_borrowed=493,source_backed_new_rank2_gauges=45,source_owned_gauge_entrances=53,all_gauge_entrances=2332,
+ target_aggregation_groups=82,target_aggregation_targets=316,target_transformed_compensation_reads=140,equal_response_pivot_reads=92,target_basis_scalar_gates=690)
+profile['source_owned_gauge_rank_histogram']['2']+=19;profile['all_gauge_rank_histogram']['2']+=19
+profile['source_owned_gauge_rank_histogram']['4']+=3;profile['all_gauge_rank_histogram']['4']+=3;profile['source_backed_shared_rank4_gauges']=5
+profile.update(target_echelon_groups=4,target_echelon_targets=20,target_echelon_basis_scalar_gates=48,target_echelon_signed_compensation_reads=48,target_echelon_compensation_unit_adds=56,partially_lifted_source_gauges=3,source_gauge_frame_semantics='Three source-owned rank4 entrances read private targets first at rank2 and shared targets at rank4; complete source paths are priced.')
+profile.update(target_rank_compression_groups=7,target_rank_compression_targets=56,target_rank_compression_dependent_rows=27,target_aggregation_basis_scalar_gates=468,target_rank_basis_scaled_add_gates=174,target_basis_expanded_unit_adds=702)
+assert profile['active_virtual_R']==18381 and profile['bankable_auxiliary_entrances']==profile['selected_roles']==2279
+normalized=Counter(child)
+for k,n in[(60,2200),(54,13),(36,18),(39,48)]:assert normalized.pop(k)==n
+assert normalized.get(57,0)==0
+normalized=Counter({k:24*n for k,n in normalized.items()if n});normalized_mass=sum(k*n for k,n in normalized.items())
+assert normalized_mass==31511856 and banks['W']==438310 and banks['literal_stock']==1314930
+assert 72*banks['W']-normalized_mass==banks['deficit']==46464
+assert banks['assignments']==3590136 and banks['charts']==231
+assert banks['conservative_extra_selector_calls']==177892046640
+assert set(banks['extra_source_roles'])=={x['role']for x in selection}
+# Conservatively retain the old88-group480-unit charge although six pairs
+# were removed, and the old198 rank-unit charge although one octet was removed.
+# Add all48 echelon basis units and all56 signed response units.
+# No removed operation is credited. The22 source mixes retime existing K work.
+uncompressed_reads=sum(len(x['targets'])for x in selection)
+rankgroups=read(P/'rank/selection.json')['groups'];rankgates=2*sum(abs(c)for g in rankgroups for cs in g['dependent'].values()for c in cs.values());assert rankgates==186
+echelongroups=read(P/'echelon/selection.json')['groups'];echelongates=2*sum(abs(c)for g in echelongroups for t,p,c in g['moves']);echreads=sum(abs(c)for g in echelongroups for row in g['transformed_response'].values()for c in row.values())
+equalgates=sum(2*(len(g['targets'])-1)for g in groups);target_gates=equalgates+rankgates+echelongates
+assert(uncompressed_reads,equalgates,echelongates,echreads,target_gates)==(208,468,48,56,702)
+conservative_equalgates=480;conservative_rankgates=198
+added_fixed=2*3*72*(uncompressed_reads+conservative_equalgates+conservative_rankgates+echelongates+echreads)
+assert added_fixed==427680
+total_fixed=banks['conservative_extra_selector_calls']+added_fixed
+assert total_fixed==177892474320<2**40
+out=dict(status='PASS_COMPLETE_SOURCE493_82_PLUS7_PLUS4_PAID_PROFILE',profile=profile,
+ local_source_aux_delta=compact(endpoint_delta),local_target_delta=compact(target_delta),local_delta=compact(local_delta),
+ child_delta=compact(delta),normalized_child_histogram=compact(normalized),raw_W=stock,raw_rank=mass,raw_deficit=1936,
+ normalized_W=banks['W'],normalized_rank=normalized_mass,normalized_deficit=banks['deficit'],literal_stock=banks['literal_stock'],
+ bankable_chart_families_unchanged=True,new_source_uncompressed_compensation_reads=uncompressed_reads,target_basis_scalar_gates=target_gates,ordinary_target_basis_adds=equalgates,conservatively_charged_ordinary_basis_adds=conservative_equalgates,multipivot_expanded_unit_basis_adds=rankgates,conservatively_charged_multipivot_unit_adds=conservative_rankgates,echelon_unit_basis_adds=echelongates,echelon_signed_compensation_reads=48,echelon_compensation_unit_adds=echreads,
+ retimed_existing_K_mix_unmix_gates=44,removed_operations_credited=False,conservative_added_fixed_calls=added_fixed,
+ conservative_total_extra_selector_calls=total_fixed,existing_fixed_call_bound=2**40,
+ input_pins={str(p):sha(p)for p in[P/'echelon/selection.json',P/'rank/selection.json',P/'newg/profile.json',P/'extra/selection.json',P/'extra/boundary.json',D/'selection.json',D/'boundary.json',D/'replay.json',P/'joint/joint-banks.json',Path(__file__)]})
+(D/'profile.json').write_text(json.dumps(out,indent=2)+'\n')
+print({k:v for k,v in out.items()if k not in['profile','input_pins','normalized_child_histogram']},flush=True)
diff --git a/research/coordinated-crossover-pr200/targetagg/profile.json b/research/coordinated-crossover-pr200/targetagg/profile.json
new file mode 100644
index 0000000000..b8677f633e
--- /dev/null
+++ b/research/coordinated-crossover-pr200/targetagg/profile.json
@@ -0,0 +1,306 @@
+{
+  "status": "PASS_COMPLETE_SOURCE493_82_PLUS7_PLUS4_PAID_PROFILE",
+  "profile": {
+    "h": 24,
+    "v": 1760,
+    "R": 16621,
+    "virtual_R": 18908,
+    "reused_registers": 1760,
+    "W_per_vertex": 20141,
+    "m": 72,
+    "loss": 528,
+    "rank_per_vertex": 1448216,
+    "deficit_per_vertex": 1936,
+    "child_histogram": {
+      "1": 126324,
+      "2": 59419,
+      "3": 30681,
+      "4": 8715,
+      "5": 7446,
+      "6": 4530,
+      "7": 3762,
+      "8": 1884,
+      "9": 1803,
+      "10": 2346,
+      "11": 1341,
+      "12": 4536,
+      "13": 1671,
+      "14": 3285,
+      "15": 888,
+      "16": 3024,
+      "17": 1446,
+      "18": 1368,
+      "19": 1062,
+      "20": 5250,
+      "21": 14676,
+      "22": 5193,
+      "36": 18,
+      "39": 48,
+      "54": 13,
+      "60": 2200
+    },
+    "maxchild": 60,
+    "selected_roles": 2279,
+    "selected_rank_histogram": {
+      "12": 18,
+      "13": 48,
+      "18": 13,
+      "20": 2200
+    },
+    "changed_operation_frames": 11904,
+    "frames_not_contained_in_original": 258,
+    "physical_target_histogram": {
+      "1": 4435,
+      "2": 118,
+      "3": 13,
+      "4": 1,
+      "5": 17,
+      "6": 4,
+      "7": 24,
+      "8": 9,
+      "9": 1,
+      "12": 12,
+      "13": 24,
+      "14": 17,
+      "16": 3,
+      "17": 4,
+      "18": 17,
+      "19": 80,
+      "20": 870,
+      "21": 732
+    },
+    "physical_internal_histogram": {
+      "1": 35120,
+      "2": 17575,
+      "3": 10182,
+      "4": 2881,
+      "5": 2461,
+      "6": 1505,
+      "7": 1230,
+      "8": 617,
+      "9": 599,
+      "10": 778,
+      "11": 445,
+      "12": 1480,
+      "13": 530,
+      "14": 1070,
+      "15": 287,
+      "16": 1004,
+      "17": 475,
+      "18": 372,
+      "19": 274,
+      "21": 3787,
+      "22": 1344
+    },
+    "foreign_producer_replays": 0,
+    "terminal_sinks": 34,
+    "active_virtual_R": 18381,
+    "scalar_role_reserve": 18908,
+    "physical_source_histogram": {
+      "1": 2553,
+      "2": 940,
+      "3": 32,
+      "4": 23,
+      "5": 4,
+      "6": 1,
+      "8": 2,
+      "9": 1,
+      "10": 4,
+      "11": 2,
+      "12": 20,
+      "13": 3,
+      "14": 8,
+      "15": 9,
+      "16": 1,
+      "17": 3,
+      "18": 67,
+      "20": 880,
+      "21": 373,
+      "22": 387
+    },
+    "physical_original_sources_borrowed": 493,
+    "source_backed_new_rank2_gauges": 45,
+    "source_backed_shared_rank4_gauges": 5,
+    "center_prefix_operations": 17080,
+    "deferred_center_independent_operations": 168,
+    "source_heads_exposed_by_retiming": 2,
+    "copied_center_frame_chronology": "D0 forward and D1 complementary reverse",
+    "retained_selected_roles": 2282,
+    "retained_selected_rank_histogram": {
+      "20": 2200,
+      "18": 13,
+      "19": 3,
+      "12": 18,
+      "13": 48
+    },
+    "selected_roles_scope": "Actual remaining bankable auxiliary entrances; source-owned data entrances are counted separately.",
+    "bankable_auxiliary_entrances": 2279,
+    "source_owned_gauge_entrances": 53,
+    "source_owned_gauge_rank_histogram": {
+      "2": 45,
+      "4": 5,
+      "19": 3
+    },
+    "all_gauge_entrances": 2332,
+    "all_gauge_rank_histogram": {
+      "2": 45,
+      "4": 5,
+      "12": 18,
+      "13": 48,
+      "18": 13,
+      "19": 3,
+      "20": 2200
+    },
+    "target_aggregation_groups": 82,
+    "target_aggregation_targets": 316,
+    "target_transformed_compensation_reads": 140,
+    "equal_response_pivot_reads": 92,
+    "target_basis_scalar_gates": 690,
+    "target_echelon_groups": 4,
+    "target_echelon_targets": 20,
+    "target_echelon_basis_scalar_gates": 48,
+    "target_echelon_signed_compensation_reads": 48,
+    "target_echelon_compensation_unit_adds": 56,
+    "partially_lifted_source_gauges": 3,
+    "source_gauge_frame_semantics": "Three source-owned rank4 entrances read private targets first at rank2 and shared targets at rank4; complete source paths are priced.",
+    "target_rank_compression_groups": 7,
+    "target_rank_compression_targets": 56,
+    "target_rank_compression_dependent_rows": 27,
+    "target_aggregation_basis_scalar_gates": 468,
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new file mode 100644
index 0000000000..f93d1f58d8
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+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/paired-cube-diagonal-bit-168/selected/bit/graph_p12.json": "31a09a55920e64c8bfbac8b845dc01754c112aa6e207812e2fd1f4939d8f64cb",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/paired-cube-diagonal-bit-168/selected/bit/profile_p12.json": "c44ef8640cbcbcc9bac3a30e13373bd79efddde55d7cea514354ca2cdb60d66d",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/paired-cube-diagonal-bit-168/selected/bit/word_p12.json.gz": "1cb7e8ed45cf542e304fddc4b12247dbb248a0dc84dedbd832f336b2d172793a",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/paired-cube-diagonal-bit-168/selected/bit/frames_p12.json.gz": "ad8e2705652b68476713d7647a8592505e222f2b3ccd1728dedce153aac6d462",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/paired-cube-diagonal-bit-168/selected/bit/kchron_p12.json": "0fab548e84cf5ca7c04e8bd19b5810850463f36e0f92180efe73aeff17105e39",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/coordinated-crossover-pr200/frames/opframe-bases.json": "9a2bba5c72aebc9ee4f21e59b10ce810b2da98a1fa1748fbc44c50a080e584ff",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/coordinated-crossover-pr200/joint/joint_word.py": "de37cbed753fc6f13a7e5dc25c8346279cfee5f1868df8c12a5fcb6ca8f631c0",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/coordinated-crossover-pr200/joint/joint-selection.json": "13f6420d6c341c22235c9d136d5f7c32896974bb165a893fecd7e583b790d133",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/coordinated-crossover-pr200/joint/pairs.json": "da54b863601b5cbc21d27fc5cd623bf805f974a3e47b81ed29a85aa5bad0fa05",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/paired-cube-diagonal-bit-168/references/pr168-v4/research/paired-cube-bit/check_paired_cube_bit.py": "4d7c86cd999a1519e0dfb6dcf33a8f26e442a353c3665a0f06881216a260fbf3",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/coordinated-crossover-pr200/joint/joint-replay.json": "af72d002a13df9e1d9f5b96c3519a7926ffc0abaa5cf19ecccc01676cd88f879",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/coordinated-crossover-pr200/newg/selection.json": "d94ef4e0b8ca98a7385b364cf245816a7cad08f850a60ce8b67aab4c3a89f3f4",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/coordinated-crossover-pr200/gaugeb/selection.json": "b0f9f0e67f94a3c21069f393ea27f9483d315bff2f53f77270163065a6345d1c",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/coordinated-crossover-pr200/newg/replay.py": "0cbec81a11e758156e9303f63dc8c617601da69523073fdf2a77394d76acf56a",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/coordinated-crossover-pr200/newg/schedule.json": "ab39d61fff7a2cfcb13e2c10d65f2c603e76a91013d197260d6c7b645ccb27a1",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/coordinated-crossover-pr200/borrow/selection.json": "6932749407e14b199651f0a7deff02c6de74e2c1be0a1300263f1e164927fb8b",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/coordinated-crossover-pr200/targetagg/replay.py": "e4c5080aa6b59afb045122170770ad18b9465ac7535b5336fca34c7e222bcea8",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/coordinated-crossover-pr200/targetagg/word.py": "293a0babfac7e92ffccfe1cb69971b26a49fe56f67872b1060c3fe49d759a463",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/coordinated-crossover-pr200/targetagg/setup.py": "fd7adf0b12d0769e06438b2ee4322bfcb180d8f73a1104a3eb805ab88c5cc393",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/coordinated-crossover-pr200/targetagg/selection.json": "b7a4d5eb2a6d2f7df867f9be5af7da1d1aa99cfbd5214195967b0ff60642d77a",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/coordinated-crossover-pr200/extra/setup.py": "a7789d13eda75e78108cee84206692c913f63aafdb9e3e633fddd14640e4678a",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/coordinated-crossover-pr200/extra/selection.json": "791f6ddeb19add7026ba37461256c13efe3d35bba0bccf1cb9787a01fc16f0f2",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/coordinated-crossover-pr200/rank/selection.json": "5c97c6413f1eb56c7b8f9ba9e839329d212fae7ac3a08c29e9efdee42f5048a0",
+    "/private/tmp/crossover-pr200-zolzj8sf/source/research/coordinated-crossover-pr200/echelon/selection.json": "8e6ce5e257aa7656898ab46bc3421df4a619497b1e9f5956d2c546b3014050a7"
+  },
+  "seconds": 116.59272270801011
+}
diff --git a/research/coordinated-crossover-pr200/targetagg/replay.py b/research/coordinated-crossover-pr200/targetagg/replay.py
new file mode 100644
index 0000000000..8df9aa15ed
--- /dev/null
+++ b/research/coordinated-crossover-pr200/targetagg/replay.py
@@ -0,0 +1,22 @@
+"""Complete493+82+7+4 scalar word: all20141 formal columns and both integer signs.
+Prepared with substantial OpenAI Codex assistance; Apache-2.0.
+Integer execution checks the defining decoder; its F2 reduction is identity.
+"""
+from pathlib import Path
+import importlib.util,json,hashlib,sys,time,gc
+sys.dont_write_bytecode=True
+D=Path(__file__).resolve().parent;P=D.parent
+sp=importlib.util.spec_from_file_location('complete493_setup',D/'setup.py');setup=importlib.util.module_from_spec(sp);sp.loader.exec_module(setup);m=setup.m
+word=D/'word.py';exec(compile(word.read_text(),str(word),'exec'),m.__dict__)
+start=time.monotonic();out=dict(status='PASS_SOURCE493_82_EQUAL7_RANK4_ECHELON_PR230',source_selection=setup.extra,groups=setup.groups,rank_groups=setup.rankselection['groups'],echelon_groups=setup.echelonselection['groups'])
+out['F2']=m.replay();print('PASS F2 all20141columns/source/dirty restoration and both source/target ledgers',flush=True)
+bound=m.replay('bound');bits=8*((bound.bit_length()+2+7)//8);assert 1<<bits>2*bound;out['majorant']=dict(residual_bound=bound,packing_bits=bits);print('BOUND',bound,bits,flush=True);out['integer']=[]
+for direction in(1,-1):out['integer'].append(m.replay('Z',direction,bits));gc.collect();print('PASS defining integer',direction,'seconds',time.monotonic()-start,flush=True)
+out['controls']={}
+for mutation in['omit_echelon_setup','omit_echelon_inverse','flip_echelon_sign','omit_rank_setup','omit_rank_inverse','repeat_rank_read','omit_aggregation_setup','omit_aggregation_inverse','repeat_aggregation_read','omit_extra_mix','omit_extra_compensation','omit_early_mix','old_adjoint','undo_before_restore','omit_gauge_mix','omit_gauge_compensation','late_phase1_gauge']:
+ try:m.replay('Z'if mutation=='flip_echelon_sign'else'F2',bits=bits,mutation=mutation)
+ except AssertionError as e:out['controls'][mutation]=dict(status='REJECTED',error=str(e))
+ else:raise AssertionError('Corruption accepted: '+mutation)
+assert all(hashlib.sha256(Path(p).read_bytes()).hexdigest()==h for p,h in m.input_pins.items())
+out['input_pins']=dict(m.input_pins);out['input_pins'].update({str(p):hashlib.sha256(p.read_bytes()).hexdigest()for p in(Path(__file__),word,D/'setup.py',D/'selection.json',P/'extra/setup.py',P/'extra/selection.json',P/'rank/selection.json',P/'echelon/selection.json')});out['seconds']=time.monotonic()-start
+(D/'replay.json').write_text(json.dumps(out,indent=2)+'\n');print('PASS complete493+82+7+4 actual scalar word and17controls',flush=True)
diff --git a/research/coordinated-crossover-pr200/targetagg/selection.json b/research/coordinated-crossover-pr200/targetagg/selection.json
new file mode 100644
index 0000000000..2ac7d748ab
--- /dev/null
+++ b/research/coordinated-crossover-pr200/targetagg/selection.json
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+          1,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          -1
+        ],
+        [
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
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+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          -1
+        ],
+        [
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
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+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          -1
+        ],
+        [
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+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
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+          0,
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+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          -1
+        ],
+        [
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
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+          0,
+          1,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          -1
+        ],
+        [
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
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+          0,
+          0,
+          0,
+          0,
+          0,
+          -1
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+        [
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+          0,
+          0,
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+          1,
+          0,
+          0,
+          0,
+          1
+        ],
+        [
+          0,
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+          0,
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+          0,
+          0,
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+          0,
+          0,
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+          0,
+          0,
+          1,
+          0,
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+          -1
+        ],
+        [
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+          0,
+          0,
+          0,
+          1,
+          0,
+          -1
+        ],
+        [
+          0,
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+          0,
+          0,
+          1,
+          -1
+        ]
+      ],
+      "origin": "extra21"
+    }
+  ]
+}
diff --git a/research/coordinated-crossover-pr200/targetagg/setup.py b/research/coordinated-crossover-pr200/targetagg/setup.py
new file mode 100644
index 0000000000..6c48fbd555
--- /dev/null
+++ b/research/coordinated-crossover-pr200/targetagg/setup.py
@@ -0,0 +1,53 @@
+"""Exact setup for493 source heads,82 equal groups and7 integer rank groups.
+Prepared with substantial OpenAI Codex assistance; Apache-2.0.
+"""
+from pathlib import Path
+from collections import defaultdict
+import importlib.util,json,sys
+sys.dont_write_bytecode=True
+D=Path(__file__).resolve().parent;P=D.parent
+sp=importlib.util.spec_from_file_location('ordered493_setup',P/'extra/setup.py');source=importlib.util.module_from_spec(sp);sp.loader.exec_module(source)
+m=source.m;extra=source.extra;W=m.W;C=m.C;groups=json.loads((D/'selection.json').read_text())['groups'];agg=[];role_groups=defaultdict(list);owner={}
+for k,g in enumerate(groups):
+ F=W.register(g['frame_basis']);ts=g['targets'];assert C.nondeg(F)and g['pivot']in ts
+ for t in ts:assert t not in owner;owner[t]=k
+ for s in g['roles']:
+  assert set(ts)<=set(m.adj[s])and len({m.adj[s][t]for t in ts})==1 and C.sub(W.gauge[s]['frame'],F);role_groups[s].append(k)
+ agg.append(dict(targets=ts,pivot=g['pivot'],roles=g['roles'],frame=F))
+assert len(agg)==82 and len(owner)==316
+byrole={r['role']:r for r in extra};rf={}
+for k,g in enumerate(agg):
+ for role in g['roles']:
+  r=byrole.get(role);f=W.gauge[role]['frame']
+  if r and'split_gauge_basis'in r and r['early_role']not in g['roles']:f=r['mix_frame']
+  assert C.sub(f,g['frame']);rf[role,k]=f
+for role,ks in role_groups.items():ks.sort(key=lambda k:C.dimf[rf[role,k]])
+m.agg=agg;m.role_groups=role_groups;m.target_group=owner;m.read_frames=rf;m.extra_byrole=byrole
+rankselection=json.loads((P/'rank/selection.json').read_text());rankgroups=[];rankowner={}
+for k,g in enumerate(rankselection['groups']):
+ F=W.register(g['frame_basis']);dep={int(t):{int(p):c for p,c in cs.items()}for t,cs in g['dependent'].items()};assert not(set(dep)&{p for cs in dep.values()for p in cs})
+ for t in g['targets']:assert t not in owner and t not in rankowner;rankowner[t]=k;assert all(W.module.dot(C.cov[t],b)==0 for b in C.B[F])
+ for s in g['prefix_roles']:
+  assert C.sub(W.gauge[s]['frame'],F)
+  for t,cs in dep.items():assert m.adj[s].get(t,0)==sum(c*m.adj[s].get(p,0)for p,c in cs.items())
+ rankgroups.append(dict(targets=g['targets'],dependent=dep,roles=set(g['prefix_roles']),frame=F))
+assert len(rankgroups)==7 and len(rankowner)==56 and sum(len(g['dependent'])for g in rankgroups)==27
+m.rankgroups=rankgroups;m.rankowner=rankowner
+
+echelonselection=json.loads((P/'echelon/selection.json').read_text());echelon=[];eowner={}
+def addrow(a,c,b):
+ z=a.copy()
+ for s,v in b.items():
+  z[s]=z.get(s,0)+c*v
+  if not z[s]:del z[s]
+ return z
+for k,r in enumerate(echelonselection['groups']):
+ F=W.register(r['frame_basis']);R={int(t):{int(s):c for s,c in a.items()}for t,a in r['transformed_response'].items()};moves=[tuple(v)for v in r['moves']]
+ A={t:{s:m.adj[s].get(t,0)for s in r['prefix_roles']if m.adj[s].get(t,0)}for t in r['targets']}
+ for t,p,c in moves:assert t!=p;A[t]=addrow(A[t],c,A[p])
+ assert A==R and C.nondeg(F)
+ for t in r['targets']:assert t not in owner and t not in rankowner and t not in eowner;eowner[t]=k;assert all(W.module.dot(C.cov[t],b)==0 for b in C.B[F])
+ for s in r['prefix_roles']:assert C.sub(W.gauge[s]['frame'],F)
+ echelon.append(dict(targets=r['targets'],roles=set(r['prefix_roles']),response=R,moves=moves,frame=F))
+assert len(echelon)==4 and len(eowner)==20
+m.echelon=echelon;m.eowner=eowner
diff --git a/research/coordinated-crossover-pr200/targetagg/word.py b/research/coordinated-crossover-pr200/targetagg/word.py
new file mode 100644
index 0000000000..ac223ba3b6
--- /dev/null
+++ b/research/coordinated-crossover-pr200/targetagg/word.py
@@ -0,0 +1,223 @@
+def replay(mode='F2',direction=1,bits=24,mutation=None):
+ major=mode=='bound';binary=mode=='F2';v=W.v
+ unit=(lambda i:1)if major else(lambda i:1<<i)if binary else(lambda i:1<<(bits*i))
+ def add(x,c,y):return x+abs(c)*y if major else x^y if binary and c&1 else x if binary else x+c*y
+ target_frames=[None]*v;target_paths=[[]for _ in range(v)];center_reads=[];target_hist=Counter()
+ aggregation_active=[True]*len(agg);aggregation_read=[set()for _ in agg];aggregation_restores=[];aggregation_pivot_reads=0
+ rank_active=set(range(len(rankgroups)));rank_seen=[set()for _ in rankgroups];rank_restores=[];rank_skipped=0
+ echelon_active=set(range(len(echelon)));echelon_seen=[set()for _ in echelon];echelon_restores=[];echelon_reads=0
+ def target_move(t,f):
+  previous=target_frames[t];assert previous is None or C.sub(previous,f),('actual target chronology',t,previous,f)
+  assert all(W.module.dot(C.cov[t],b)==0 for b in C.B[f]),('actual target cap',t,f)
+  d=C.dimf[f]-(0 if previous is None else C.dimf[previous]);assert d>=0
+  if d:target_hist[d]+=1
+  target_frames[t]=f;target_paths[t].append(f)
+ X0=[unit(i)for i in range(v)];Y0=[unit(v+i)for i in range(v)];Z0={r:unit(idx[r])for r in regs};x=X0[:];y=Y0[:];z=Z0.copy();top=1;done=[]
+ def track(value):
+  nonlocal top
+  if major:top=max(top,value)
+  return value
+ def value(s):
+  s=W.phys[s];return x[borrow[s]]if s in borrow else z[s]
+ def assign(s,val):
+  s=W.phys[s]
+  if s in borrow:x[borrow[s]]=track(val)
+  else:z[s]=track(val)
+ source_paths={role:[W.w['source_frame'][r['source']],r['mix_frame']]for role,r in extra_byrole.items()}
+ def source_move(s,f):
+  role=W.phys[s]
+  if role not in source_paths:return
+  assert C.sub(source_paths[role][-1],f),('orderedsource chronology',s,role,source_paths[role][-1],f)
+  source_paths[role].append(f)
+ responses=W.adjoint() if mutation=='old_adjoint' else adj
+ def aggregate_restore(k,trigger):
+  if not aggregation_active[k]:return
+  group=agg[k];pivot=group['pivot'];assert aggregation_read[k]==set(group['roles']),('early aggregation inverse',k,trigger,aggregation_read[k])
+  for t in group['targets']:target_move(t,group['frame'])
+  if mutation!='omit_aggregation_inverse' or k!=39:
+   for t in group['targets']:
+    if t!=pivot:y[t]=track(add(y[t],1,y[pivot]))
+  aggregation_active[k]=False;aggregation_restores.append(dict(group=k,before_role=trigger,frame=group['frame']))
+ def rank_restore(k,trigger):
+  group=rankgroups[k];assert k in rank_active and rank_seen[k]==group['roles']
+  for t in group['targets']:target_move(t,group['frame'])
+  if mutation!='omit_rank_inverse'or k!=0:
+   for t,cs in group['dependent'].items():
+    for p,c in cs.items():y[t]=track(add(y[t],c,y[p]))
+  rank_active.remove(k);rank_restores.append(dict(group=k,before_role=trigger,frame=group['frame']))
+ def echelon_restore(k,trigger):
+  group=echelon[k];assert echelon_seen[k]==group['roles']
+  for t in group['targets']:target_move(t,group['frame'])
+  if mutation!='omit_echelon_inverse'or k!=0:
+   for t,p,c in reversed(group['moves']):y[t]=track(add(y[t],-c,y[p]))
+  echelon_active.remove(k);echelon_restores.append(dict(group=k,before_role=trigger,frame=group['frame']))
+ def read(s):
+  nonlocal aggregation_pivot_reads,rank_skipped,echelon_reads
+  echelon_skip=set()
+  if s in W.gauge:
+   for k,group in enumerate(echelon):
+    if k not in echelon_active:continue
+    if s not in group['roles']:
+     if set(responses[s])&set(group['targets']):echelon_restore(k,s)
+     continue
+    echelon_seen[k].add(s);echelon_skip.update(group['targets'])
+    for t,a in group['response'].items():
+     c=a.get(s,0)
+     if not c:continue
+     if mutation=='flip_echelon_sign'and k==0 and c<0:c=-c
+     source_move(s,W.gauge[s]['frame']);target_move(t,W.gauge[s]['frame']);y[t]=track(add(y[t],-direction*c,value(s)));echelon_reads+=1
+  if s in W.gauge:
+   for k in sorted({rankowner[t]for t in responses[s]if t in rankowner}):
+    if k not in rank_active:continue
+    if s not in rankgroups[k]['roles']:rank_restore(k,s)
+    else:rank_seen[k].add(s)
+  if mutation=='omit_gauge_compensation'and s==new_gauge_selection[-1]['role']:return
+  if mutation=='omit_extra_compensation'and s==extra_selection[0]['role']:return
+  grouped=role_groups.get(s,[])
+  for k in sorted({target_group[t]for t in responses[s]if t in target_group}) if s in W.gauge else []:
+   if k not in grouped:aggregate_restore(k,s)
+  skipped=set()
+  for k in grouped:
+   group=agg[k];pivot=group['pivot'];assert aggregation_active[k]
+   assert set(group['targets'])<=set(responses[s])and len({responses[s][t]for t in group['targets']})==1
+   c=responses[s][pivot];source_move(s,read_frames[s,k]);target_move(pivot,read_frames[s,k]);y[pivot]=track(add(y[pivot],-direction*c,value(s)))
+   if mutation=='repeat_aggregation_read'and k==39:
+    t=next(t for t in group['targets']if t!=pivot);target_move(t,read_frames[s,k]);y[t]=track(add(y[t],-direction*c,value(s)))
+   skipped.update(group['targets']);aggregation_read[k].add(s);aggregation_pivot_reads+=1
+  for t,c in responses[s].items():
+   if t in skipped or t in echelon_skip:continue
+   rk=rankowner.get(t)
+   if s in W.gauge and rk in rank_active and t in rankgroups[rk]['dependent']:
+    rank_skipped+=1
+    if mutation!='repeat_rank_read'or rk!=0:continue
+   if s in W.gauge:source_move(s,W.gauge[s]['frame']);target_move(t,W.gauge[s]['frame'])
+   else:assert target_frames[t]is None,'initial correction leaves target atD0'
+   y[t]=track(add(y[t],-direction*c,value(s)))
+ def gate(i,sign):
+  a,b,_=W.ops[i]
+  if sign==1:
+   source_move(a,W.opframe[i]);source_move(b,W.opframe[i])
+  assign(a,add(value(a),sign,value(b)))
+ for s in range(W.R):
+  if s not in W.gauge and s not in borrow and s not in removed:read(s)
+ for n,s in W.source.items():assign(s,add(value(s),1,x[n]))
+ for r in gauge_selection:
+  if mutation=='omit_extra_mix'and r is extra_selection[0]:continue
+  if mutation!='omit_gauge_mix' or r is not new_gauge_selection[-1]:x[r['partner']]=track(add(x[r['partner']],1,x[r['source']]))
+ def forward(i):
+  if i not in omitted:gate(i,1);done.append(i)
+ for k in schedule['center_prefix_events']:
+  event=schedule['events'][k];i=event['op']
+  if event['kind']=='early_mix':
+   r=early[i];a,b=r['partner'],r['source']
+   if mutation!='omit_early_mix' or r is not selection[0]:x[a]=track(add(x[a],1,x[b]))
+  else:forward(i)
+ for r,s in zip(W.g['roots'],W.w['rootroles']):
+  if r['kind']=='center':
+   for t in r['targets']:
+    assert all(f is None for f in target_frames),'copied centers scatter before any nonzero target movement'
+    center_reads.append((s,t));y[t]=track(add(y[t],direction,value(s)))
+ for k,group in enumerate(echelon):
+  assert all(target_frames[t]is None for t in group['targets'])
+  if mutation!='omit_echelon_setup'or k!=0:
+   for t,p,c in group['moves']:y[t]=track(add(y[t],c,y[p]))
+ for k,group in enumerate(rankgroups):
+  assert all(target_frames[t]is None for t in group['targets'])
+  if mutation!='omit_rank_setup'or k!=0:
+   for t,cs in group['dependent'].items():
+    for p,c in cs.items():y[t]=track(add(y[t],-c,y[p]))
+ for k,group in enumerate(agg):
+  assert all(target_frames[t]is None for t in group['targets']), 'aggregation setup after all D0 center scatters'
+  if mutation!='omit_aggregation_setup'or k!=39:
+   for t in group['targets']:
+    if t!=group['pivot']:y[t]=track(add(y[t],-1,y[group['pivot']]))
+ for e in terminal:
+  assert all(target_frames[t]is None for t in e['targets']),'terminal setup afterD0center scatter'
+  for t in e['targets']:
+   if t!=e['pivot']:y[t]=track(add(y[t],-1,y[e['pivot']]))
+ for j,i in enumerate(W.rest):
+  for s in at[j]:
+   if mutation!='late_phase1_gauge' or s not in new_phase_roles:read(s)
+  if mutation=='late_phase1_gauge'and j==168:
+   for s in sorted(new_phase_roles):read(s)
+  if i in bywrite:
+   e=bywrite[i];a,b,n=W.ops[i];source_move(b,W.opframe[i]);target_move(e['pivot'],W.opframe[i]);y[e['pivot']]=track(add(y[e['pivot']],direction,value(b)))
+  else:forward(i)
+  if i in afterwrite:
+   e=afterwrite[i]
+   for t in e['targets']:
+    target_move(t,e['root_frame'])
+    if t!=e['pivot']:y[t]=track(add(y[t],1,y[e['pivot']]))
+ for s in at[len(W.rest)]:read(s)
+ assert not any(aggregation_active),'all quotient inverses precede original side/K reads'
+ assert not rank_active,'rank quotient inverses precede side/K reads'
+ assert not echelon_active,'echelon inverses precede side/K reads'
+ for j,(r,s)in enumerate(zip(W.g['roots'],W.w['rootroles'])):
+  if r['kind']=='side' and j not in deletedroots:
+   for k in sorted({target_group[t]for t in r['targets']if t in target_group}):aggregate_restore(k,('root',j))
+   source_move(s,W.w['root_frame'][j])
+   for t in r['targets']:target_move(t,W.w['root_frame'][j]);y[t]=track(add(y[t],direction,value(s)))
+  for e in deliveries[j]:
+   a,b=e['carrier'],e['passive']
+   if b not in by_source:x[a]=track(add(x[a],1,x[b]))
+   for k in sorted({target_group[t]for t in e['receivers']if t in target_group}):aggregate_restore(k,('K',j))
+   for t in e['receivers']:target_move(t,e['deliver_frame']);y[t]=track(add(y[t],direction,x[a]))
+ assert not any(aggregation_active),'all target inverses completed beforecleanup'
+ for e in W.k['entries']:
+  if e['passive']not in by_source:x[e['carrier']]=track(add(x[e['carrier']],-1,x[e['passive']]))
+ if mutation=='undo_before_restore':
+  for e in W.k['entries']:
+   if e['passive']in by_source:x[e['carrier']]=track(add(x[e['carrier']],-1,x[e['passive']]))
+ for role in source_paths:source_move(role,W.w['full_frame'])
+ for i in reversed(done):gate(i,-1)
+ if mutation!='undo_before_restore':
+  for e in W.k['entries']:
+   if e['passive']in by_source:x[e['carrier']]=track(add(x[e['carrier']],-1,x[e['passive']]))
+ for n,s in W.source.items():assign(s,add(value(s),-1,x[n]))
+ if binary:want=[Y0[t]^X0[t]for t in range(v)]
+ else:
+  want=Y0[:];cache={}
+  for root in W.g['roots']:
+   n=root['node']
+   if n not in cache:
+    mask=C.sup[n];val=0
+    while mask:low=mask&-mask;mask-=low;val=add(val,1,X0[low.bit_length()-1])
+    cache[n]=val
+   for t in root['targets']:want[t]=add(want[t],direction,cache[n])
+  for e in W.k['entries']:
+   val=add(X0[e['carrier']],1,X0[e['passive']])
+   for t in e['receivers']:want[t]=add(want[t],direction,val)
+ # Finish every local target at its codimension-one cap, then reverse the exact
+ # annihilator stream. All copied-center scatters lie before these movements,
+ # hence atD0 forward and atD1 after the complementary movements reverse.
+ reflected=Counter()
+ for t,chain in enumerate(target_paths):
+  assert chain
+  d=W.h-1-C.dimf[chain[-1]];assert d>=0
+  if d:target_hist[d]+=1
+  previous=[C.cov[t]];dimension=1
+  for f in reversed(chain):
+   assert all(W.module.dot(a,b)==0 for a in previous for b in C.B[f])
+   basis=C.A[f];step=len(basis)-dimension;assert step>=0
+   if step:reflected[step]+=1
+   previous=basis;dimension=len(basis)
+  if W.h>dimension:reflected[W.h-dimension]+=1
+ assert reflected==target_hist
+ assert len(center_reads)==sum(len(r['targets'])for r in W.g['roots']if r['kind']=='center')
+ assert sum(k*n for k,n in target_hist.items())==W.v*(W.h-1)
+ source_histogram=Counter();source_reflected=Counter()
+ for role,path in source_paths.items():
+  assert C.dimf[path[0]]==1 and C.dimf[path[-1]]==W.h
+  for a,b in zip(path,path[1:]):
+   assert C.sub(a,b)and C.nondeg(a)and C.nondeg(b)
+   step=C.dimf[b]-C.dimf[a]
+   if step:source_histogram[step]+=1
+  for a,b in zip(reversed(path),list(reversed(path))[1:]):
+   assert all(W.module.dot(x,y)==0 for x in C.A[a]for y in C.B[b]);step=len(C.A[b])-len(C.A[a]);assert step>=0
+   if step:source_reflected[step]+=1
+ assert source_histogram==source_reflected and sum(k*n for k,n in source_histogram.items())==len(source_paths)*(W.h-1)
+ if major:return max(top,max(a+b for a,b in zip(y,want)),max(z[r]+Z0[r]for r in regs),max(a+b for a,b in zip(x,X0)))
+ assert y==want,('wrong targets',mode,[t for t in range(v)if y[t]!=want[t]][:10])
+ assert x==X0,('source restoration',mode,[s for s in range(v)if x[s]!=X0[s]][:10])
+ assert z==Z0,('dirty restoration',mode)
+ return dict(mode=mode,direction=direction,formal_columns=2*v+len(regs),sources=v,targets=v,dirty=len(regs),all_targets=True,all_source_and_dirty_restored=True,copied_center_target_reads=len(center_reads),all_forward_centers_at_D0=True,all_reflected_centers_at_D1=True,actual_target_histogram={str(k):n for k,n in sorted(target_hist.items())},actual_reflected_target_histogram={str(k):n for k,n in sorted(reflected.items())},aggregation_groups=len(agg),aggregation_pivot_reads=aggregation_pivot_reads,aggregation_restores=aggregation_restores,actual_extra_source_paths=source_paths,rank_groups=len(rankgroups),rank_targets=len(rankowner),rank_restores=rank_restores,rank_removed_reads=rank_skipped,actual_extra_source_histogram=dict(source_histogram),actual_extra_reflected_source_histogram=dict(source_reflected),echelon_restores=echelon_restores,echelon_reads=echelon_reads)
diff --git a/research/coordinated-crossover-pr200/verify.py b/research/coordinated-crossover-pr200/verify.py
new file mode 100644
index 0000000000..c7c42930d5
--- /dev/null
+++ b/research/coordinated-crossover-pr200/verify.py
@@ -0,0 +1,29 @@
+#!/usr/bin/env python3
+"""Offline finite certificate replay. Python3.11+, numpy2.3.5, scipy1.17.0."""
+from pathlib import Path
+import argparse,hashlib,json,io,zipfile,tempfile,subprocess,shutil,sys
+HERE=Path(__file__).resolve().parent
+sys.dont_write_bytecode=True
+if hasattr(sys,'set_int_max_str_digits'):sys.set_int_max_str_digits(0)
+def sha(p):return hashlib.sha256(p.read_bytes()).hexdigest()
+def pins():
+ m=json.loads((HERE/'FILES.json').read_text())['files'];actual={p.relative_to(HERE).as_posix():sha(p) for p in HERE.rglob('*') if p.is_file() and p.name!='FILES.json'};assert actual==m,'Missing, changed, or unpinned upload file';return actual
+assert not sys.flags.optimize
+ap=argparse.ArgumentParser(description=__doc__);ap.add_argument('--temp-root',type=Path,help='Optional spacious scratch drive for extraction');args=ap.parse_args();before=pins();baseline=json.loads((HERE/'BASELINE.json').read_text());parts=[]
+for entry in baseline['parts']:
+ p=HERE/entry['file'];assert p.stat().st_size==entry['bytes'] and sha(p)==entry['sha256'];parts.append(p.read_bytes())
+blob=b''.join(parts);assert hashlib.sha256(blob).hexdigest()==baseline['archive_sha256']
+with tempfile.TemporaryDirectory(prefix='crossover-pr200-',dir=args.temp_root) as tmp:
+ root=Path(tmp)/'source';root.mkdir()
+ with zipfile.ZipFile(io.BytesIO(blob)) as z:
+  for entry in z.infolist():assert (root/entry.filename).resolve().is_relative_to(root.resolve()),'Unsafe archive entry'
+  z.extractall(root)
+ package=root/'research/coordinated-crossover-pr200';package.mkdir()
+ skip={x['file'] for x in baseline['parts']}
+ for p in HERE.rglob('*'):
+  if p.is_file() and p.name not in skip:
+   target=package/p.relative_to(HERE);target.parent.mkdir(parents=True,exist_ok=True);shutil.copyfile(p,target)
+ result=subprocess.run([sys.executable,'-B',str(package/'verify_inner.py')],text=True)
+ assert result.returncode==0,'Full finite replay failed'
+assert pins()==before,'Upload files changed during replay'
+print('PASS immutable offline package; no network or git required')
\ No newline at end of file
diff --git a/research/coordinated-crossover-pr200/verify_inner.py b/research/coordinated-crossover-pr200/verify_inner.py
new file mode 100644
index 0000000000..ff19b2524c
--- /dev/null
+++ b/research/coordinated-crossover-pr200/verify_inner.py
@@ -0,0 +1,49 @@
+#!/usr/bin/env python3
+"""Full finite replay; all-size compiler/selector and analytic conditions are inherited."""
+from pathlib import Path
+import subprocess,sys,json,importlib.util,hashlib,gzip
+from fractions import Fraction as Q
+sys.set_int_max_str_digits(0);sys.dont_write_bytecode=True
+HERE=Path(__file__).resolve().parent;ROOT=HERE.parents[1];SA=ROOT/'research/source-assisted';CP=ROOT/'research/source-assisted-v4';WORK=CP/'.work'
+def run(*args):
+ r=subprocess.run([sys.executable,'-B',*map(str,args)],cwd=ROOT,capture_output=True,text=True)
+ if r.returncode:raise RuntimeError('Replay rejected: '+str(args)+'\n'+r.stdout[-6000:]+'\n'+r.stderr[-6000:])
+ print(r.stdout[-2000:],end='',flush=True)
+def load(n,p):
+ s=importlib.util.spec_from_file_location(n,p);m=importlib.util.module_from_spec(s);s.loader.exec_module(m);return m
+def read(p):return json.loads(p.read_text())
+assert not sys.flags.optimize
+before={p.relative_to(ROOT).as_posix():hashlib.sha256(p.read_bytes()).hexdigest() for p in [ROOT/'scripts/paired_cube_assembly.py',SA/'decision/exact_complex_flow_lift.py',CP/'contract_v4.py']}
+# Replay the actual82 new gauges, complete reflected ledgers and all new banks.
+run(HERE/'joint/replay_joint.py');run(HERE/'borrow/replay.py');run(HERE/'borrow/boundary.py');run(HERE/'borrow/derive_profile.py');run(HERE/'gaugeb/replay.py');run(HERE/'gaugeb/boundary.py');run(HERE/'gaugeb/derive_profile.py');run(HERE/'newg/replay.py');run(HERE/'newg/schedule_audit.py');run(HERE/'newg/boundary.py');run(HERE/'newg/derive_profile.py');run(HERE/'extra/boundary.py');run(HERE/'joint/pack_joint.py');run(HERE/'targetagg/replay.py');run(HERE/'targetagg/boundary.py');run(HERE/'targetagg/derive_profile.py')
+# The portable witness is checked again; original numerical search need not be repeated.
+run(SA/'decision/exact_complex_flow_lift.py','--witness',WORK/'flow.witness.json','--profile',WORK/'flow.json','--out',WORK/'lift.json')
+lift=read(WORK/'lift.json');expected=read(CP/'certificate.json');assert lift['exact_scalar_program_sha256']==expected['lift']['exact_scalar_program_sha256']
+for key,name in [('certificate_path','lift.certificate.json.gz'),('witness_path','flow.witness.json')]:lift[key]=(WORK/name).relative_to(ROOT).as_posix()
+lift['checker_path']=(SA/'decision/exact_complex_flow_lift.py').relative_to(ROOT).as_posix()
+(WORK/'lift.json').write_text(json.dumps(lift,indent=2)+'\n',newline='\n')
+run(CP/'contract_v4.py','--tree',WORK/'aligned','--cache',WORK/'aligned/cache','--witness',WORK/'flow.witness.json','--flow-profile',WORK/'flow.json','--lift-profile',WORK/'lift.json','--out',HERE/'complex-profile-replayed.json')
+actual=read(HERE/'complex-profile-replayed.json')
+for key in ['m','W_per_vertex','rank_per_vertex','deficit_per_vertex','child_histogram','contract_checks','exact_scalar_program_sha256']:
+ assert actual[key]==expected['complex_profile'][key],('Complex mathematical output changed',key)
+# Paid two-supplier moments, four finite leaf levels, and unchanged 47 constraints.
+run(HERE/'joint/price_joint.py');assert read(HERE/'joint/assembly.json')==read(HERE/'certificate.json'),'Derived final certificate differs'
+cert=read(HERE/'joint/assembly.json');assert all(Q(x)>0 for x in cert['assembly']['strict_constraints'].values());assert len(cert['assembly']['strict_constraints'])==47
+# Independent base-two enclosure, no imports from the public moment implementation.
+im=load('independent_base_two',HERE/'geometry/base_two_moment.py')
+for side in ['bit','complex']:
+ row=cert[side+'_profile'];a=Q(cert[side]['saving']);hist=[(int(t),n) for t,n in row['child_multiplicities'].items()]
+ lo,hi=im.moment(row['m'],row['W'],hist,a);nl,nu=im.moment(row['m'],row['W'],hist,a+Q(1,10**18))
+ if side=='bit':
+  fallback=[(1,32*row['m']**2*sum(n for _,n in hist))];bl,bu=im.moment(row['m'],row['W'],fallback,a);bnl,bnu=im.moment(row['m'],row['W'],fallback,a+Q(1,10**18));hi+=Q(1,10**16)*bu;nl+=Q(1,10**16)*bnl
+ assert hi<1<nl,side+' independent adjacent moment failed'
+ corrupt=dict(row,total_rank=row['total_rank']+1)
+ public=load('paid_moment_control',ROOT/'research/paired-cube-diagonal-bit-168/arithmetic/interval_moment.py')
+ try:public.moment(corrupt,a)
+ except ValueError:pass
+ else:raise AssertionError('Rank-mass negative control accepted')
+chart=read(HERE/'joint/joint-banks.json');assert chart['conservative_extra_selector_calls']==177892046640<2**40
+aggregation=read(HERE/'targetagg/profile.json');assert aggregation['conservative_added_fixed_calls']==427680 and chart['conservative_extra_selector_calls']+aggregation['conservative_added_fixed_calls']==177892474320<2**40
+assert chart['charts']==231 and chart['assignments']==3590136 and chart['literal_stock']==1314930
+assert before=={p:hashlib.sha256((ROOT/p).read_bytes()).hexdigest() for p in before}
+print('PASS full offline bit/frame/bank/chart/complex-lift/contract replay, independent paid moments, 47 inequalities and controls; kappa='+cert['kappa'])
\ No newline at end of file
diff --git a/research/five-stage-source527-banks/BANK-PROOF.md b/research/five-stage-source527-banks/BANK-PROOF.md
new file mode 100644
index 0000000000..ef9db6d9e8
--- /dev/null
+++ b/research/five-stage-source527-banks/BANK-PROOF.md
@@ -0,0 +1,39 @@
+# Five-stage stage-private completed banks
+
+This is a finite bank-allocation extension of the PR197/PR200 completed-bank construction to PR234's five-stage bit geometry. It uses the source527 independent helper census. It does not claim a new all-size compiler theorem or replace the scalar admission.
+
+For one complete local invocation, the source-bound physical telescope has no data/helper cross blocks. On an independent helper with genuine entrance projector sigma its endpoint is the exact partial swap D_(I24-sigma), supported in the active local window. This holds on arbitrary correlated dirty contents. The complemented inverse invocation has the same residual: its reversed endpoints are 0 and I24-sigma. Source-owned gauges are visible-data paths and are excluded from the independent helper census.
+
+In stage i, PR234 routes the helper at local class d to physical class g=d*tau_i. Relative to g, the residual is P_i=tau_i*diag(I24-sigma,0,...,0)*tau_i, supported only in H_i. We give each stage a disjoint collection of banks. Every bank is filled by disjoint coordinate blocks of total dimension120. For the actual gauge chart B whose columns are residual columns followed by gauge columns, set
+
+    N_b=(b+1)*Pi_b*embed_i(B^-1).
+
+For an ungauged role B=I24. Pi_b is the explicit coordinate permutation sending the first residual coordinates in H_i to the assigned bank interval. Thus N_b*P_i*N_b^-1 is exactly that coordinate projector. The scalar leaves conjugation unchanged. The checker reconstructs every actual chart, checks both inverse identities, the G-orthogonality of its two summands, its residual action and its elementary factorization. Therefore this is a projector identity for the actual bases, not a rank substitution.
+
+The bank route is g -> g*N_b^-1, hence d*tau_i*N_b^-1 in the original local invocation. If the bank class is h, its residual is h*N_b*P_i*N_b^-1*h^-1. Every occurrence visits every bank class once because right multiplication is a bijection on the complete GL cover. If two roles in one local invocation are assigned the same bank family, their distinct N_b give distinct bank classes. Distinctness is literal: choose a coordinate j outside H_i. The embedded inverse fixes e_j; N_b*e_j=(b+1)e_(Pi_b(j)). The resulting columns differ for all b, at every eligible prime q>2^80 and every local ring Z/q^w. All scalars are between1 and30 and remain distinct units. The proof uses the full GL cover, not a projective quotient.
+
+Complete the entire cover sweep for each stage and replica in fixed order before its boundary phase. The complete local source endpoint separates every bank from the data and other banks, so each subsequent invocation may receive arbitrary dirty bank contents. For each bank, its block projectors are disjoint and sum to I120. The literal partial swaps therefore compose to D_I120 on every address column. The common inherited ancestor chart is the same for each block and cancels between blocks. The checker verifies all240 address columns, the reverse schedule, and failures from omitting or repeating a block. No new independence assumption on dirt is used.
+
+The five bank collections remain separate, and each is touched only in its own stage. This gives every physical bank its full required endpoint once. The data stage routes, bridge maps, idle children and final exchanges remain PR234's. Consequently there are no independent exterior rank5a children. Their removal follows from the completed bank endpoint; no internal, data, compensation or copied-center child is removed.
+
+For source527 the actual residual families are4:2200,6:13,11:48,12:18,24:14308. Sixty physical replicas fill each stage with these exact120-dimensional patterns:
+
+| Blocks | Banks per stage |
+| --- | ---: |
+| 8 of dimension11 and8 of dimension4 | 360 |
+| 30 of dimension4 | 4304 |
+| 20 of dimension6 | 39 |
+| 10 of dimension12 | 108 |
+| 5 of dimension24 | 171696 |
+
+There are176507 banks per stage,882535 total banks and422400 data families. Literal stock is1304935. Its per-replica normalized value is260987/12. The checker enumerates each of4976100 actual role/replica/stage assignments exactly once, verifies the actual role's chart rank, and hashes the complete allocation.
+
+Every actual chart uses at most200 factors. At most119 coordinate transpositions and120 scalar-coordinate factors give439 normalizer factors. A conservative added out-and-back selector bill, in addition to all retained PR234 route and scalar bills, is
+
+    2*5*60*((1304935-1)+16587*120*439)=525064856400 < 2^40.
+
+The literal stock, not the rational per-replica stock, must be used for physical role routing and finite storage. All60 scalar replicas execute, and all recursive child counts are multiplied by60 in a literal realization. Any smaller integer normalization used only in the moment arithmetic must be distinguished from the literal replica count. Serial execution may share an external copied-center work stream only after erasure. The fixed bank selector bill is explicitly charged before the inherited positive ordinary-leaf gap absorbs it. After completion removal the maximum child is50 of120, so a single level halves; the histogram remains a one-level profile.
+
+This witness leaves the full source527 scalar/integer/sign audits, frame ledgers, exact global lowering, prime guard, complete fallback profile and47 outer inequalities to the integrator. Its input charts/census are the independently admitted source527 bank package. Existing compiler, common weighted chart, restored selectors, complete-stream routing and all-size analytic hypotheses remain conditional as in the cited sources.
+
+Prepared with OpenAI Codex assistance. Apache-2.0; inherited source attribution remains unchanged.
diff --git a/research/five-stage-source527-banks/LICENSE b/research/five-stage-source527-banks/LICENSE
new file mode 100644
index 0000000000..261eeb9e9f
--- /dev/null
+++ b/research/five-stage-source527-banks/LICENSE
@@ -0,0 +1,201 @@
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diff --git a/research/five-stage-source527-banks/MANIFEST.json b/research/five-stage-source527-banks/MANIFEST.json
new file mode 100644
index 0000000000..b3574d4087
--- /dev/null
+++ b/research/five-stage-source527-banks/MANIFEST.json
@@ -0,0 +1,132 @@
+{
+  "files": {
+    "BANK-PROOF.md": "9f3525c8e8b798119d81ca1af67424107a87aa0e98a326244149602fafa805ea",
+    "LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+    "NOTICE.md": "9eb351079c3b65f85a10e411dfb096822caa91028fb3fefe645d61d4bdb1d4b7",
+    "PR-STATEMENT.md": "5e2bdac58dfb1d5128eaf815fcacc28d9c73de662818e49d052dfd75142cc0bd",
+    "PROOF.md": "1142b4671b8d742f84bba16ec831bcdc337ea88dba8291bcc16429321011c140",
+    "README.md": "2894ee0f4fd9738b939ff1d84afdc23a3124bd196b0ea4c0cc865d28709af9d8",
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+    "verify.py": "1513fda0dfdebcaa682c05eb8a879522ea3f36175472e840a7d8211de20f41c2"
+  },
+  "schema": "source527-parity-fused-five-stage-banks-inputs/1"
+}
diff --git a/research/five-stage-source527-banks/NOTICE.md b/research/five-stage-source527-banks/NOTICE.md
new file mode 100644
index 0000000000..d4ad6c3ac7
--- /dev/null
+++ b/research/five-stage-source527-banks/NOTICE.md
@@ -0,0 +1,9 @@
+# Attribution and licence
+
+Apache-2.0 applies. Original headers, licences, notices and AI assistance disclosures are preserved. `UPSTREAM-PR234-NOTICE.md` contains the complete inherited five-stage credits; the individual source roots retain their original notices. Credits do not imply review or endorsement.
+
+The five-stage architecture and finite interface package come from Henry Grant / hcg890's PR234. Rohan Arun's PR237 published the width-120 bank extension before this integration; no exclusive priority for that bank extension is claimed. The stage-private bank assignment and address adapter in this package were developed independently and are checked explicitly.
+
+The source527 construction builds on eumemic's PR210 source reuse and target transforms, sennemmi's PR230 operation-frame changes, Dugongue's PR216 and PR207, Rohan Arun's PR211, Chafik Boukhalfa's PR200, Evan McKinney's completed bank work, and their retained dependencies. The complex input preserves Jacob Sussman's published five-stage certificate and Avi Eisenberg / ikeboy's PR193 helper, with the original Claude assistance disclosures.
+
+The 31 new zero-response source aliases, 48 retained dirty-anchor paths, physical event tagging and source527 integration were prepared by eumemic with substantial OpenAI Codex assistance. All earlier contributor notices and assistance disclosures remain in the supplied source material.
diff --git a/research/five-stage-source527-banks/PR-STATEMENT.md b/research/five-stage-source527-banks/PR-STATEMENT.md
new file mode 100644
index 0000000000..eedc69c3ac
--- /dev/null
+++ b/research/five-stage-source527-banks/PR-STATEMENT.md
@@ -0,0 +1,35 @@
+κ = 7.10069340338651e-4
+
+Conditional **κ ≈ 7.10069340338651 × 10⁻⁴** from a 527-source helper, PR234's five-stage architecture and completed width-120 entrance banks. This refines the preceding source527 result, **κ ≈ 7.10046193349537 × 10⁻⁴**, by approximately 0.00326%.
+
+The latest improvement removes 1,783,196 even-coefficient F₂ payload additions. Three target paths then combine adjacent rank-8 and rank-12 movements into rank-20 movements. Every address, projector and route calculation retains its original odd-prime arithmetic. The emitted word is checked on all formal F₂ columns in both directions and has its own signed prefix bounds; the retained producer separately passes its original signed integer decoder checks.
+
+The underlying helper uses signed changes of target coordinates and exact cancellations between internal registers. Fourteen pairs have equal downstream responses. Seventeen triples have a signed response relation. Injecting an original source into these registers with the corresponding coefficients eliminates 31 further dirty registers without adding target compensation; reversing the word and undoing each injection restores arbitrary dirty contents. Together with the preceding source and target improvements, 527 original sources are reused.
+
+The package regenerates the complete local word and all five stage templates from pinned source recipes. Independent checks cover all 20,107 source, target and dirty columns over F₂, both defining-integer signs for the producer, the emitted literal inverse, 25 corruptions, every physical frame path and both reflected ledgers. Every consumed basis receives a fresh exact determinant check. Target changes of coordinates, all 120 copied-center calls per replica, source compensation, bridges and cleanup remain paid.
+
+For the banks, five separate stage collections use 60 physical replicas. The verifier enumerates all 4,976,100 role/replica/stage assignments, reconstructs all 231 actual charts, checks both inverse identities and normalizer routes, and binds the callable bank mapping to the lowered word. Each full bank is checked on all 240 address columns, including arbitrary correlated dirt and reverse restoration. Source-owned gauges are excluded from these banks.
+
+| Final bit profile | Value |
+| --- | ---: |
+| Dimension | 120 |
+| Literal physical stock | 1,304,935 |
+| Moment normalization | 12 |
+| Normalized stock | 260,987 |
+| Normalized rank mass | 31,265,640 |
+| Normalized deficit | 52,800 |
+| Largest child | 50 |
+| Additional bank selector calls | 525,064,856,400 |
+
+Finite accounting charges the literal 60 replicas and physical stock. The smaller normalization is used only in the exact moment. Two independent rational moment engines, three completed ordinary-leaf levels, the retained complex supplier, all 47 strict assembly constraints and seven margins pass; the adjacent final grid point is rejected. The bit supplier binds.
+
+```sh
+python3 -m pip install -r research/five-stage-source527-banks/requirements.txt
+python3 -B research/five-stage-source527-banks/verify.py --output /tmp/source527-five-stage-verification
+```
+
+The output directory must be new and outside the package. Verification uses no network, recomputes every mandatory stage and checks that all submitted files remain unchanged. Python 3.11+ and SymPy 1.14.0 are required.
+
+The result retains the inherited all-size compiler, weighted/common-ancestor chart, restored-row, selector, routing, prime-supply, precision/recovery, complex symbolic-correctness and analytic-transfer assumptions. Finite replay does not prove those interfaces from scratch; no new Lean build or practical multiplication benchmark is claimed.
+
+Provenance: hcg890's PR234 and Jacob Sussman's published construction supply the five-stage layout and complex baseline. Rohan Arun's PR237 independently published width-120 entrance banking before this submission. This package uses a separately checked stage-private allocation with 60 replicas. Evan McKinney's PR197 supplies the completed-bank method; Chafik Boukhalfa's PR200 supplies the physical helper/checkers; Avi Eisenberg's PR193 supplies the retained complex helper. Dugongue's PR207/216, Rohan Arun's PR211 and sennemmi's nineteen PR230 operation-frame improvements remain in the helper lineage, with their original notices. The new internal response cancellations, target transformations, source reuse and integration were prepared by eumemic with substantial OpenAI Codex assistance. Upstream licenses, contributor names and AI disclosures are retained.
diff --git a/research/five-stage-source527-banks/PROOF.md b/research/five-stage-source527-banks/PROOF.md
new file mode 100644
index 0000000000..a4a8cd6730
--- /dev/null
+++ b/research/five-stage-source527-banks/PROOF.md
@@ -0,0 +1,25 @@
+κ = 7.10069340338651e-4
+
+# Source-bound finite verification
+
+The 527-source construction is a reversible local helper. All inputs and dirty registers are formal variables. Fourteen internal pairs and seventeen internal triples have exact signed adjoint relations. If one register aliases a source S and the retained registers start at B_i+c_i S, the relation A_alias+Σ c_i A_i=0 cancels the injected source response. The full program is reversed, then each retained register is restored by subtracting c_i S before source unmixing. These changes are admitted on their actual operation frames, and every source and retained dirty path is charged.
+
+`prepare.py` reconstructs the current helper from the retained source471 recipe and explicit later selections, target transformations, source gauges, sparse internal relations and PR230 frame witness. `raw_ledger.py` independently rebuilds every source, internal and target path, checks both reflected inclusions, and derives the complete helper histogram. No saved PASS receipt is consumed as a proof.
+
+`portable_bit.py` observes the actual scalar program and emits its physical frame tags, including target setup/inverse operations, internal source injections/restorations, copied centers and cleanup. Its local scalar projection must equal the independent observer. The five-stage lowering preserves PR234's data routes, bridge maps, idle frames and terminal exchange. Execution is phase-major across the complete cover. Exact rational projector checks establish the routed endpoints; dimension counts alone are not used as frame identities.
+
+`scalar_check.py` checks the producer on every F₂ column and both defining-integer signs, including all source/dirty restoration and 25 corruptions. It independently checks the emitted parity-filtered word and its literal inverse on every F₂ column, rejects a missing surviving odd gate, and bounds its signed lift separately. `prime_check.py` recomputes exact cleared-Gram determinants for every consumed basis and checks all finite prime exclusions, including the new source planes and target transformations. The five disjoint address windows introduce no new determinant prime factors.
+
+The executed word is produced by `parity_transform.py`. It deletes only scalar payload ADD instructions with even coefficients, which are the identity over F₂. No rational address, frame, router or normalizer coefficient is reduced modulo two. A pending MOVE is combined with the next MOVE on the same stream only when no surviving ADD, COPY or ERASE uses that stream between them. The exact nested projector increments are orthogonal and add to the combined increment. The independent required-use census reconstructs every path from surviving operations and mathematical endpoints; it checks the original and reflected inclusions and matches the actual instruction histogram. Exactly three rank-8/rank-12 pairs become rank-20 movements. The all-column F₂ word and its literal inverse are replayed, with cancellation-free signed prefix bounds 37631 and 3295796. The old defining-integer decoder remains checked for the producer. The parity-filtered integer lift need not have that same target decoder; this distinction follows the retained F₂ supplier contract detailed in `proof/PARITY-CONTRACT.md`.
+
+The stage-private bank construction maps each independent helper residual into a literal coordinate block through its actual chart. Distinct fixed normalizers prevent two simultaneous local roles from aliasing the same bank address. Five bank collections remain separate. Each complete cover sweep fills the disjoint blocks of a bank, whose projectors sum to the identity; their partial swaps therefore give the full required arbitrary-dirty endpoint. The verifier checks the actual charts, all 4,976,100 assignments, the callable route map and both endpoint directions. The separate bank proof gives the construction and exact extra route bill. Source-owned gauges remain data paths and are excluded.
+
+The bank endpoint discharges precisely the independent rank-5a exterior completions. All local moves, source/target paths, copied centers, data bridges, fallbacks and other children remain charged. Sixty physical replicas give literal stock 1,304,935. Dividing the literal paid moment by five gives normalization 12, dimension 120, stock 260,987, rank mass 31,265,640 and deficit 52,800. The largest child is 50, less than half the dimension.
+
+`finite_check.py` retains every PR234 primitive and router allowance, uses the literal replica and stock counts, and adds all 525,064,856,400 bank selector calls. The positive ordinary-leaf gap absorbs this explicitly charged fixed coefficient. The displayed numerical cutoff uses the counted recipe coefficient; the theorem uses the same cutoff formula with C_full at least that coefficient times every inherited primitive, wrapper and preceding ordinary-level constant. Those constants are fixed under the inherited interfaces, so this yields a finite cutoff without assuming unit machine cost. The half-shrink bound and literal stock less than q give an internal borrowed-row reserve of at most 14401 w ceil(log2 n) plus the inherited ordinary-leaf reserve. Those rows are restored. The separate external complex row coefficient 20161 is freshly checked by the complex guard and bound to the outer assembly. Two rational moment engines independently check contraction, including the complete 10⁻¹⁶ bad-class fallback. Three completed ordinary-leaf levels feed the unchanged balanced outer assembly. All 47 strict inequalities and seven margins are positive, and the adjacent final grid point fails.
+
+The complex label/scalar/geometry/precision pipeline and its primary inputs are retained from PR234 and rerun. Its coarse saving exceeds the effective bit saving, so the bit supplier binds.
+
+The proof boundary remains conditional: the all-size weighted compiler, common ancestor charts, restored rows, completed ordinary leaves, selectors, exact-tape routing, prime supply, precision/recovery, retained complex symbolic correctness and analytic transfer are inherited interfaces. These checks establish the stated finite construction under those interfaces; they do not rebuild Lean or execute a multiplier at all input sizes.
+
+Credit the five-stage architecture to hcg890's PR234 and Jacob Sussman, the completed-bank method to Evan McKinney's PR197, and the earlier width-120 application to Rohan Arun's PR237. The source helper retains all PR210/200/193/207/211/216/230 lineage and disclosures. New source527 response cancellations and integration were prepared by eumemic with substantial OpenAI Codex assistance.
diff --git a/research/five-stage-source527-banks/README.md b/research/five-stage-source527-banks/README.md
new file mode 100644
index 0000000000..247424087a
--- /dev/null
+++ b/research/five-stage-source527-banks/README.md
@@ -0,0 +1,22 @@
+κ = 7.10069340338651e-4
+
+# Parity-fused source527 five-stage completed banks
+
+This version removes 1,783,196 even-coefficient F₂ payload additions and combines three now-adjacent target movements. It retains the complete source527 producer and bank construction.
+
+A conditional finite construction with **κ ≈ 7.10069340338651 × 10⁻⁴**. The package combines a source527 helper with PR234's five-stage cover and a concrete stage-private width-120 bank allocation. See `PR-STATEMENT.md` for the result and attribution, `PROOF.md` for the verification chain and inherited assumptions, and the retained upstream notices for the full lineage.
+
+Run with Python 3.11+ and assertions enabled:
+
+```sh
+python3 -m pip install -r requirements.txt
+python3 -B verify.py --output /tmp/source527-five-stage-verification
+```
+
+The output directory must be new and outside this immutable package. The verifier reconstructs the source recipe, both physical frame ledgers, local and five-stage scalar programs, all consumed determinants, bank charts and assignments, the complex supplier, the complete finite bill, exact moments and outer assembly. It rejects omitted proof stages and changed package files. Verification itself requires no network or Git.
+
+The source helper has 1,760 original sources, 1,760 targets and 16,587 independent dirty registers. Its 31 newest source aliases use fourteen equal-response pairs and seventeen signed-response triples. The banks execute 60 physical replicas with literal stock 1,304,935; the moment uses an integer normalization of 12 with stock 260,987. These two counts are deliberately distinguished in finite accounting.
+
+The result is conditional on the retained all-size compiler, common weighted chart, restored rows, selectors, tape routing, prime supply, precision/recovery, complex symbolic correctness and analytic reduction. The included finite checks do not replace those hypotheses.
+
+Prepared by eumemic with substantial OpenAI Codex assistance. Apache-2.0. Original authors, licenses and AI disclosures are retained in the source directories and upstream notices.
diff --git a/research/five-stage-source527-banks/UPSTREAM-PR234-NOTICE.md b/research/five-stage-source527-banks/UPSTREAM-PR234-NOTICE.md
new file mode 100644
index 0000000000..5f8fbb25c1
--- /dev/null
+++ b/research/five-stage-source527-banks/UPSTREAM-PR234-NOTICE.md
@@ -0,0 +1,13 @@
+# Attribution and licence
+
+This contribution uses Apache-2.0. Original source headers, licences and notices are preserved. These credits do not imply review or endorsement by the named contributors or OpenAI.
+
+- Jacob Sussman: five-stage complex geometry and published gcert/1 export in wht-power-saving-lean at f010392c923279e3dd59ef3aa5fedad23affc5fd. Original Claude assistance disclosure retained in inputs/complex/NOTICE.
+- Avi Eisenberg / ikeboy: retained PR193 source-assisted complex helper and donor selection at 187e1010ac8b259af8e9b5166f68b64bc27b4b47.
+- eumemic: current PR210 frames, gauges, source borrowing and retimed bit helper at df95878d11190518e45ef9717c9ee05011f88ace.
+- Chafik Boukhalfa / chafreaky: PR200 physical bit construction carried by PR202 at 8d8d67bcf69c5ea67d3a29dbc64ba588156d6e8d.
+- Rohan Arun, Dugongue, evmckinney9, icekylinx, James Chang, Douglas Colkitt, OpenAI and earlier contributors retain the attribution and original assistance disclosures in the supplied source notices. The exact outer assembly also credits Zhihao Chen and the RaD / hipotures balanced reduction in its original header.
+
+The paired-cube helper design predates Sussman’s certificate export. The new contribution here is the explicit source-bound five-stage composition, adapters, paid accounting and reproducible finite verification. No exclusive priority is claimed for the inherited designs or theorem interfaces.
+
+Assisted with ChatGPT.
diff --git a/research/five-stage-source527-banks/UPSTREAM-PR234-PROOF.md b/research/five-stage-source527-banks/UPSTREAM-PR234-PROOF.md
new file mode 100644
index 0000000000..dbd8806967
--- /dev/null
+++ b/research/five-stage-source527-banks/UPSTREAM-PR234-PROOF.md
@@ -0,0 +1,29 @@
+# Five-stage finite composition and retained hypotheses
+
+The expected conditional saving is κ = 703701743496697 / 10^18. The fresh verifier establishes the finite construction and exact arithmetic under the inherited source and all-size interfaces below.
+
+## Bit construction
+
+The pinned source471 helper is checked on all 20,163 independent F2 source, target and dirty columns. Its physical realization is the exact input/output frame telescope, with all source loans, donor/recipient identifications, compensated reads, terminal substitutions and cleanup retained. Copied centers are complete temporary streams; their reverse implementation copies the original anew, transforms and scatters that copy, then erases it. It does not reverse an erasure.
+
+The five phase-major invocations use the explicitly routed source owners X1, Y1, X1, Y2 and X2. Each helper's five separated residuals occupy disjoint windows. One split rank-5a exterior correction completes each genuine independent entrance of rank a. All 2,279 corrections remain paid, and source-owned gauges stay within the visible-data paths. Actual rational matrix checks establish bridge equality, nested idle changes and terminal exchanges. Cover expansion has a barrier after each phase, across every class.
+
+The fresh program has m=120, W=23,683, 495,304 children, rank mass 2,837,560 and deficit 4,400. All 120 rank-22 copied-center calls are included. Exact cleared-Gram determinants and factor witnesses are recomputed for every consumed basis. Fivefold direct sums introduce no new excluded primes. The retained weighted q-local compiler, unit-pivot fallback, prime-supply and complete-stream tape proofs remain hypotheses.
+
+The proper child ratio 5/6 requires halving degree four. The one-level weighted recurrence is retained; its histogram is not multiplied by four. The finite accounting rederives route factors, scalar work, internal row stock, strict moment/linear gaps and explicit finite cutoff expressions. Unknown constants of inherited primitive implementations remain quantified constants, not invented measured costs. See proof/FINITE_BIT_ACCOUNTING.md and the included original bit/row/copied-center source texts.
+
+## Complex construction
+
+The published gcert/1 baseline is pinned to Sussman's f010392 revision and the PR193 paired-cube helper. Its live-label walk, scalar programs, copied-center lifetimes, final climbs before cleanup, exact symbolic splice and finite guard are regenerated. Exact complex frame identities retain their phase representatives. Physical work stock 24,127 is separately charged; the live recursion denominator remains 14,692.
+
+The pinned scalar identities hAi, hBi, hx, hy and hid, the Lab.inv dirty-lifting theorem and five-window geometry/NetCert theorems are inherited source proofs. The command does not rebuild their Lean proof chain or claim a fresh all-column complex scalar proof. General Clifford synthesis, charged complete-stream movement, common-grid precision and restored-row proofs remain stated analytic dependencies. Their consumed declarations and sources are hash-pinned under inputs/complex/.
+
+## Arithmetic and outer reduction
+
+Both rational moments are freshly enclosed. Three finite ordinary-leaf levels start from the retained completed ordinary saving 384599/10^10; the unattained coarse limiting saving is not used as an ordinary leaf. Exact outer assembly checks all 47 strict inequalities and seven margins and rejects the next 10^-18 κ grid point. Precision/recovery, prime supply, completed ordinary leaves and the fixed-tape all-size reduction remain hypotheses.
+
+proof/INTEGRATED_PROOF.md preserves the detailed original composition argument. Its development filenames identify the original evidence; V8's executable dependency closure is exclusively MANIFEST.json and the fresh calls in verify.py. Development receipts and their historical source trees are not runtime inputs.
+
+No claim is made of unconditional all-size formalization, global optimality or measured multiplier performance. Original lineage and assistance disclosures are preserved in NOTICE.md and the supplied source notices.
+
+Assisted with ChatGPT.
diff --git a/research/five-stage-source527-banks/bank_check.py b/research/five-stage-source527-banks/bank_check.py
new file mode 100644
index 0000000000..4a91b79d67
--- /dev/null
+++ b/research/five-stage-source527-banks/bank_check.py
@@ -0,0 +1,254 @@
+"""Fresh context-bound stage-private120-coordinate completed bank admission.
+
+No saved chart, receipt, physical census or old source tree is consumed.
+The fraction-free inverse routine derives from the retained PR197/PR200 bank
+checker. Inherited attribution/notices apply. Apache-2.0.
+Prepared with substantial OpenAI Codex assistance.
+"""
+from collections import Counter
+from fractions import Fraction as Q
+from math import gcd
+from pathlib import Path
+import hashlib,json,struct,time
+from bank_template import BankPlan
+
+assert __debug__, 'Do not run bank admission under -O.'
+HERE=Path(__file__).resolve().parent
+
+def inverse_integer(a):
+    """Fraction-free Gauss-Jordan, returning C,d with aC=Ca=dI."""
+    n=len(a);m=[list(row)+[int(i==j)for j in range(n)]for i,row in enumerate(a)]
+    previous=1;factors=[]
+    for k in range(n):
+        pivot=next((i for i in range(k,n)if m[i][k]),None)
+        assert pivot is not None,'singular actual bank chart'
+        m[k],m[pivot]=m[pivot],m[k]
+        if pivot!=k:factors.append(('swap',k,pivot,1,1))
+        p=m[k][k]
+        if p!=previous:
+            ratio=Q(previous,p);factors.append(('scale',k,k,ratio.numerator,ratio.denominator))
+        for i in range(n):
+            if i==k:continue
+            u=m[i][k]
+            if u:
+                ratio=Q(-u,previous);factors.append(('add',i,k,ratio.numerator,ratio.denominator))
+            for j in range(2*n):
+                if j==k:continue
+                value=p*m[i][j]-u*m[k][j];assert value%previous==0;m[i][j]=value//previous
+            m[i][k]=0
+        previous=p
+    d=m[0][0];assert d and all(m[i][j]==d*int(i==j)for i in range(n)for j in range(n))
+    c=[row[n:]for row in m];common=abs(d)
+    for row in c:
+        for x in row:common=gcd(common,x)
+    d//=common;c=[[x//common for x in row]for row in c]
+    if d<0:d=-d;c=[[-x for x in row]for row in c]
+    return c,d,factors
+
+def census(context):
+    W,C,roles=context['W'],context['C'],context['regs']
+    assert len(roles)==len(set(roles))==16587 and roles==sorted(roles)
+    assert set(roles)==set(W.phys.values())-set(context['borrow'])-context['removed']
+    assert not(set(roles)&set(context['borrow']))
+    gauge_frames={role:W.gauge[role]['frame']for role in roles if role in W.gauge}
+    assert Counter(C.dimf[f]for f in gauge_frames.values())=={12:18,13:48,18:13,20:2200}
+    families={}
+    for role in roles:
+        rank=24-C.dimf[gauge_frames[role]]if role in gauge_frames else 24
+        families.setdefault(rank,[]).append(role)
+    assert {r:len(xs)for r,xs in families.items()}=={4:2200,6:13,11:48,12:18,24:14308}
+    return families,gauge_frames
+
+class BankLowerer:
+    """Final full-address lowering of the already checked logical IR.
+
+    Scalar/projection/geometry fields are retained byte-for-byte; only stream
+    operands change to the proven bank namespaces. Completion discharge is
+    checked separately by run(), and no completion opcode is emitted.
+    """
+    def __init__(self,context,lower):
+        families,gauges=census(context)
+        self.plan=BankPlan(families,gauges,context['regs']);self.lower=lower
+    def iter_stage(self,stage,replica):
+        for row in self.lower.iter_stage(stage):yield self.plan.map_stage_row(stage,replica,row)
+    def iter_idle(self,which,replica):
+        for row in self.lower.iter_idle(which):yield self.plan.map_boundary_row(replica,row)
+    def iter_bridges(self,which,replica):
+        for row in self.lower.iter_bridges(which):yield self.plan.map_boundary_row(replica,row)
+    def iter_exchanges(self,replica):
+        for row in self.lower.iter_exchanges():yield self.plan.map_boundary_row(replica,row)
+    def phases(self):
+        """Each iterator is expanded over all cover classes before next phase."""
+        for stage in range(5):
+            for replica in range(60):
+                yield('helper',stage,replica,lambda s=stage,r=replica:self.iter_stage(s,r))
+            if stage in(1,2,4):
+                which={1:0,2:1,4:2}[stage]
+                for replica in range(60):
+                    yield('idle',which,replica,lambda i=which,r=replica:self.iter_idle(i,r))
+                    yield('bridge',which,replica,lambda i=which,r=replica:self.iter_bridges(i,r))
+        for replica in range(60):yield('terminal_exchange',0,replica,lambda r=replica:self.iter_exchanges(r))
+
+def bind(context,lower):return BankLowerer(context,lower)
+
+def run(context,global_result,lower,progress=lambda text:None):
+    begun=time.monotonic();W,C=context['W'],context['C']
+    api_pin=hashlib.sha256((HERE/'bank_template.py').read_bytes()).hexdigest()
+    families,gauge_frames=census(context);bound=bind(context,lower);plan=bound.plan
+    assert lower.W is W and lower.C is C
+    # Derive all actual chart bases and exact inverse/factor programs afresh.
+    charts=[];maxops=maxnum=maxden=0
+    for frame,count in sorted(Counter(gauge_frames.values()).items()):
+        residual=[]
+        for a in C.A[frame]:
+            row=[15*x-sum(a)for x in a];d=gcd(*row);assert d;residual.append([x//d for x in row])
+        source=C.B[frame];rank=len(residual)
+        assert rank+len(source)==24 and len(source)==C.dimf[frame]
+        assert all(9*sum(x*y for x,y in zip(a,b))-sum(a)*sum(b)==0 for a in residual for b in source)
+        B=list(map(list,zip(*(residual+list(source)))));A,d,ops=inverse_integer(B)
+        for L,R in((A,B),(B,A)):
+            assert all(sum(x*y for x,y in zip(row,col))==d*int(i==j)for i,row in enumerate(L)for j,col in enumerate(zip(*R)))
+        P=[[sum(B[i][k]*A[k][j]for k in range(rank))for j in range(24)]for i in range(24)]
+        assert sum(P[i][i]for i in range(24))==d*rank
+        assert all(sum(P[i][k]*B[k][j]for k in range(24))==(d*B[i][j]if j<rank else 0)for i in range(24)for j in range(24))
+        replay=[list(map(Q,row))for row in B]
+        for op,i,j,a,b in ops:
+            if op=='swap':replay[i],replay[j]=replay[j],replay[i]
+            elif op=='scale':replay[i]=[Q(a,b)*x for x in replay[i]]
+            else:
+                assert op=='add';replay[i]=[x+Q(a,b)*y for x,y in zip(replay[i],replay[j])]
+        assert all(x==int(i==j)for i,row in enumerate(replay)for j,x in enumerate(row))
+        maxops=max(maxops,len(ops));maxnum=max(maxnum,max(abs(op[3])for op in ops));maxden=max(maxden,d,max(op[4]for op in ops))
+        charts.append(dict(frame=frame,count=count,residual_rank=rank,basis=B,inverse_numerator=A,inverse_denominator=d,factors=ops))
+    assert len(charts)==231 and max(maxnum,maxden)<2**80 and maxops==200
+    progress('Rebuilt231 exact actual gauge charts')
+    # Match each consumed completion to one actual independent entrance.
+    expected=[]
+    for j,role in enumerate(context['regs']):
+        f=gauge_frames.get(role,lower.ZERO)
+        assert lower.initial[2*W.v+j]==f
+        if role in gauge_frames:expected.append((4*W.v+j,f,C.dimf[f]))
+    assert list(lower.entrances)==expected and len(expected)==2279
+    completions=list(lower.iter_completions())
+    assert completions==[(6,family,-1,0,5*a,f,-1,0)for family,f,a in expected]
+    removed=Counter(row[4]for row in completions)
+    assert removed=={60:18,65:48,90:13,100:2200}
+    H=Counter({int(r):n for r,n in global_result['paid_histogram'].items()})
+    assert sum(H.values())==global_result['paid_calls']
+    for r,n in removed.items():assert H[r]==n;del H[r]
+    assert max(H)==50 and sum(H.values())==492700 and sum(r*n for r,n in H.items())==2605470
+    literal=Counter({r:60*n for r,n in H.items()})
+    assert 120*plan.live_families-sum(r*n for r,n in literal.items())==60*4400
+    # Independently enumerate each physical role/replica/stage slot.
+    incidence=hashlib.sha256();assignments=0;normalizers_checked=set();endpoint_controls=[]
+    for stage in range(5):
+        used=Counter();bank=0;assigned=set()
+        outside=next(j for j in range(120)if not 24*stage<=j<24*(stage+1))
+        for pattern,(widths,count)in enumerate(plan.patterns):
+            offset=0;witnesses=[]
+            for block,rank in enumerate(widths):
+                original=list(range(24*stage,24*stage+rank));target=list(range(offset,offset+rank))
+                src=original+[j for j in range(120)if j not in original];dst=target+[j for j in range(120)if j not in target];pi=dict(zip(src,dst))
+                witnesses.append((pi[outside],block+1));offset+=rank
+            assert len(set(witnesses))==len(widths)
+            if stage==0:
+                offsets=[sum(widths[:b])for b in range(len(widths))]
+                def endpoint(blocks):
+                    state=list(range(240))
+                    for b in blocks:
+                        for j in range(offsets[b],offsets[b]+widths[b]):state[j],state[120+j]=state[120+j],state[j]
+                    return state
+                blocks=list(range(len(widths)));full=list(range(120,240))+list(range(120))
+                assert endpoint(blocks)==full and endpoint(blocks+blocks[::-1])==list(range(240))
+                assert endpoint(blocks[:-1])!=full and endpoint(blocks+[0])!=full
+                endpoint_controls.append(dict(pattern=pattern,forward_columns=240,inverse_columns=240,omitted_rejected=True,repeated_rejected=True))
+            for b in range(bank,bank+count):
+                offset=0
+                for block,rank in enumerate(widths):
+                    q=used[rank];role=families[rank][q//60];replica=q%60;used[rank]+=1
+                    assert(role,replica)not in assigned;assigned.add((role,replica))
+                    a=plan.assignment(stage,replica,role);frame=gauge_frames.get(role)
+                    assert a==dict(family=plan.data_families+stage*plan.banks_per_stage+b,stage_bank=b,pattern=pattern,block=block,offset=offset,rank=rank,frame=frame,scalar=block+1)
+                    key=stage,pattern,block,frame
+                    if key not in normalizers_checked:
+                        N=plan.normalizer(stage,replica,role)
+                        assert N['unit_column_witness']==witnesses[block]and N['inverse_chart_frame']==frame and N['chart_window']==stage
+                        assert N['permutation'][24*stage:24*stage+rank]==tuple(range(offset,offset+rank))
+                        normalizers_checked.add(key)
+                    incidence.update(f'{stage},{pattern},{role},{replica},{b},{block},{offset},{rank}\n'.encode())
+                    assignments+=1;offset+=rank
+                assert offset==120
+            bank+=count
+        assert bank==plan.banks_per_stage and len(assigned)==60*len(context['regs'])and used=={r:60*len(xs)for r,xs in families.items()}
+    assert assignments==4976100
+    progress('Matched all4976100 literal bank assignments')
+    namespace=hashlib.sha256();collisions=0
+    for stage in range(5):
+        for replica in range(60):
+            addresses=[plan.local_address(stage,replica,i)for i in range(plan.local_work+1)]
+            assert len(set(addresses))==len(addresses)and addresses[-1]==(plan.work_family,('external_work',0))
+            assert all(f<plan.live_families for f,route in addresses[:-1])
+            collisions+=len(addresses)-len({f for f,route in addresses})
+            for f,route in addresses:namespace.update(f'{stage},{replica},{f},{route}\n'.encode())
+    assert collisions==14400
+    # Bind every actual logical operand and mathematical field to the callable
+    # address substitution once per stage. The300 injective namespace maps
+    # above then instantiate all sixty replicas without replaying scalar algebra.
+    stream=struct.Struct('<8i');stage_checks=[];actual_helper_uses=set();paid=Counter()
+    for stage in range(5):
+        digest=hashlib.sha256();rows=0;stage_paid=Counter();kinds=Counter()
+        # Full addresses indexed by the original logical family; wrappers use
+        # exactly this tested local inverse and namespace construction.
+        addresses={}
+        for local in range(plan.local_work+1):
+            family=4*W.v+len(context['regs'])if local==plan.local_work else plan.active[stage][0]*W.v+local if local<W.v else plan.active[stage][1]*W.v+local-W.v if local<2*W.v else 4*W.v+local-2*W.v
+            assert plan.local_from_logical_family(stage,family)==local
+            addresses[family]=plan.local_address(stage,0,local)
+        for row in lower.iter_stage(stage):
+            op,a,b,c,f,z,s,rev=row;assert s==stage and rev==int(stage in(1,3))
+            mapped=plan.map_stage_row(stage,0,row)
+            assert mapped==(op,addresses[a],addresses[b]if op in(1,2)else b,c,f,z,s,rev)
+            for family in([a,b]if op in(1,2)else[a]):
+                if 4*W.v<=family<4*W.v+len(context['regs']):actual_helper_uses.add(family-4*W.v)
+            if op==0 and f:stage_paid[f]+=1
+            digest.update(stream.pack(*row));rows+=1;kinds[op]+=1
+        assert stage_paid==Counter({int(r):n for r,n in lower.ns['result']['paid_histogram'].items()})
+        paid.update(stage_paid)
+        stage_checks.append(dict(stage=stage,records=rows,logical_operand_binding_sha256=digest.hexdigest(),opcode_counts=dict(kinds),all_actual_operands_mapped=True))
+    assert actual_helper_uses==set(range(len(context['regs'])))
+    for which in range(3):
+        for row in lower.iter_idle(which):
+            mapped=plan.map_boundary_row(0,row);assert mapped[0]==4;paid[row[4]]+=1
+        for row in lower.iter_bridges(which):assert plan.map_boundary_row(0,row)[0]==5
+    exchanges=sum(1 for row in lower.iter_exchanges()if plan.map_boundary_row(0,row)[0]==7)
+    assert exchanges==2*W.v and paid==H
+    phases=[(kind,stage,replica)for kind,stage,replica,fn in bound.phases()]
+    assert len(phases)==720 and sum(kind=='helper'for kind,s,r in phases)==300
+    assert all(phases.index(('idle',which,r))<phases.index(('bridge',which,r))for which in range(3)for r in range(60))
+    controls=[]
+    try:plan.map_boundary_row(0,completions[0])
+    except AssertionError:controls.append('silent completion discharge')
+    else:raise AssertionError('completion passed address mapper')
+    assert hashlib.sha256((HERE/'bank_template.py').read_bytes()).hexdigest()==api_pin
+    factors=maxops+119+120;K=2*5*60*((plan.live_families-1)+len(context['regs'])*120*factors)
+    assert factors==439 and K==525064856400<2**40
+    result=dict(status='PASS_FRESH_CONTEXT_BOUND_COMPLETE_BANKS_AND_CALLABLE_LOWERING',
+        all_assignments_match=True,all_logical_helpers_bound=True,independent_completions_removed=len(completions),
+        removed_completion_histogram=dict(removed),literal_paid_histogram=dict(sorted(literal.items())),literal_stock=plan.live_families,
+        normalized_stock=str(Q(plan.live_families,60)),physical_replicas=60,bank_families=5*plan.banks_per_stage,data_families=plan.data_families,
+        assignments=assignments,incidence_sha256=incidence.hexdigest(),charts=len(charts),
+        chart_sha256=hashlib.sha256(json.dumps(charts,sort_keys=True,separators=(',',':')).encode()).hexdigest(),
+        max_chart_factors=maxops,max_factor_numerator=maxnum,max_denominator=maxden,normalizer_factor_bound=factors,
+        conservative_extra_selector_calls=K,distinct_actual_normalizers=len(normalizers_checked),
+        full_namespaces_checked=300,namespace_sha256=namespace.hexdigest(),intentional_family_collisions=collisions,
+        local_instruction_bindings=stage_checks,endpoint_checks=endpoint_controls,phase_schedule=phases,
+        completions_discharged_by='Each stage-private bank partitions all120 coordinates; all240 arbitrary-dirty address columns and inverse checked.',
+        inherited_geometry='Helper originalclass d*tau_i maps to bank d*tau_i*N^-1. N conjugates its actual residual in H_i into the assigned coordinate block. Data routes and boundary projectors remain PR234s.',
+        script_sha256=hashlib.sha256(Path(__file__).read_bytes()).hexdigest(),api_sha256=api_pin,
+        rejected_controls=controls,seconds=time.monotonic()-begun)
+    bank_keys=('literal_stock','normalized_stock','physical_replicas','bank_families','data_families','assignments',
+        'incidence_sha256','charts','chart_sha256','max_chart_factors','max_factor_numerator','max_denominator',
+        'normalizer_factor_bound','conservative_extra_selector_calls','distinct_actual_normalizers','endpoint_checks')
+    banks={k:result[k]for k in bank_keys};banks['status']='PASS_FRESH_ACTUAL_CHARTS_AND_COMPLETE_BANKS'
+    banks.update(source527_roles=len(context['regs']),banks_total=5*plan.banks_per_stage)
+    return dict(banks=banks,banked_result=result)
diff --git a/research/five-stage-source527-banks/bank_template.py b/research/five-stage-source527-banks/bank_template.py
new file mode 100644
index 0000000000..90db0a87ee
--- /dev/null
+++ b/research/five-stage-source527-banks/bank_template.py
@@ -0,0 +1,183 @@
+"""Callable stage-private120-coordinate bank address template.
+
+The caller supplies the freshly admitted actual helper role families and gauge
+chart IDs. This module owns no paths, global frame registry or source loading.
+Prepared with OpenAI Codex assistance; Apache-2.0.
+"""
+from collections import Counter
+
+
+class BankPlan:
+    m, h, stages, replicas, v = 120, 24, 5, 60, 1760
+    patterns = ((tuple([11]*8+[4]*8),360),
+                (tuple([4]*30),4304),
+                (tuple([6]*20),39),
+                (tuple([12]*10),108),
+                (tuple([24]*5),171696))
+
+    active = ((0,1),(1,0),(0,1),(3,2),(2,3))
+
+    def __init__(self, families, gauge_frames, helper_roles=None):
+        self.families = {r:tuple(rows) for r,rows in families.items() if rows}
+        assert {r:len(rows) for r,rows in self.families.items()} == {4:2200,6:13,11:48,12:18,24:14308}
+        self.role_index = {}
+        for rank,rows in self.families.items():
+            assert tuple(sorted(rows)) == rows and len(set(rows)) == len(rows)
+            for i,role in enumerate(rows):
+                assert role not in self.role_index
+                self.role_index[role]=(rank,i)
+        self.gauge_frames = dict(gauge_frames)
+        assert set(self.gauge_frames) == {role for r,rows in self.families.items() if r<self.h for role in rows}
+        self.helper_roles=tuple(sorted(self.role_index) if helper_roles is None else helper_roles)
+        assert len(self.helper_roles)==len(self.role_index) and set(self.helper_roles)==set(self.role_index)
+        self.local_work=2*self.v+len(self.helper_roles)
+        self.banks_per_stage=sum(count for widths,count in self.patterns)
+        self.data_families=self.replicas*4*self.v
+        self.live_families=self.data_families+self.stages*self.banks_per_stage
+        self.work_family=self.live_families
+        assert (self.banks_per_stage,self.data_families,self.live_families)==(176507,422400,1304935)
+        self.segments={}
+        used=Counter();bank_start=0
+        for pattern,(widths,count) in enumerate(self.patterns):
+            assert sum(widths)==self.m
+            for rank in set(widths):
+                blocks=tuple(b for b,r in enumerate(widths) if r==rank)
+                n=len(blocks)*count
+                self.segments.setdefault(rank,[]).append((used[rank],used[rank]+n,pattern,bank_start,blocks))
+                used[rank]+=n
+            bank_start+=count
+        assert used=={r:self.replicas*len(rows) for r,rows in self.families.items()}
+
+    def assignment(self, stage, replica, role):
+        """Exact bank family/block for one actual independent helper occurrence."""
+        assert 0<=stage<self.stages and 0<=replica<self.replicas
+        rank,role_index=self.role_index[role]
+        q=role_index*self.replicas+replica
+        for lo,hi,pattern,bank_start,blocks in self.segments[rank]:
+            if lo<=q<hi:
+                bank_offset,within=divmod(q-lo,len(blocks))
+                bank=bank_start+bank_offset;block=blocks[within]
+                widths=self.patterns[pattern][0]
+                return dict(family=self.data_families+stage*self.banks_per_stage+bank,
+                            stage_bank=bank,pattern=pattern,block=block,
+                            offset=sum(widths[:block]),rank=rank,
+                            frame=self.gauge_frames.get(role),scalar=block+1)
+        raise AssertionError('unallocated actual helper occurrence')
+
+    def data_family(self, replica, bank, port):
+        assert 0<=replica<self.replicas and 0<=bank<4 and 0<=port<self.v
+        return (4*replica+bank)*self.v+port
+
+    def normalizer(self, stage, replica, role):
+        """Finite exact data for N=scalar Pi embed_stage(B^-1).
+
+        frame=None denotes identity B=I24. The caller resolves other IDs in
+        its freshly admitted frame registry/charts. inverse_chart remains an
+        exact rational factor program; no floating matrix is consumed.
+        """
+        a=self.assignment(stage,replica,role)
+        original=tuple(range(self.h*stage,self.h*stage+a['rank']))
+        target=tuple(range(a['offset'],a['offset']+a['rank']))
+        src=original+tuple(j for j in range(self.m) if j not in original)
+        dst=target+tuple(j for j in range(self.m) if j not in target)
+        pi=dict(zip(src,dst))
+        assert set(pi)==set(pi.values())==set(range(self.m))
+        permutation=tuple(pi[j] for j in range(self.m))
+        outside=next(j for j in range(self.m) if not self.h*stage<=j<self.h*(stage+1))
+        return dict(scalar=a['scalar'],permutation=permutation,
+                    inverse_chart_frame=a['frame'],chart_window=stage,
+                    outside_column=outside,unit_column_witness=(pi[outside],a['scalar']))
+
+    def address(self, stage, replica, role, cover='d'):
+        """Original class d*tau_i routed to the actual bank by N^-1.
+
+        Multiplication order is literal. A consumer must lower the factor
+        list right-to-left for inverse N, keeping chart/scalar denominators.
+        """
+        a=self.assignment(stage,replica,role)
+        N=('normalizer',stage,a['pattern'],a['block'],a['frame'])
+        return a['family'],('right',cover,('tau',stage),('inverse',N))
+
+    def local_family(self, stage, replica, local):
+        """Live family for the admitted local physical register index.
+
+        Family equality does not imply address equality for banked helpers.
+        Consumers must use local_address for ADD nonalias/namespace checks.
+        """
+        assert 0<=stage<self.stages and 0<=replica<self.replicas
+        assert 0<=local<=self.local_work
+        if local==self.local_work:return self.work_family
+        if local<self.v:return self.data_family(replica,self.active[stage][0],local)
+        if local<2*self.v:return self.data_family(replica,self.active[stage][1],local-self.v)
+        return self.assignment(stage,replica,self.helper_roles[local-2*self.v])['family']
+
+    def local_address(self, stage, replica, local, cover='d'):
+        family=self.local_family(stage,replica,local)
+        if local==self.local_work:return family,('external_work',0)
+        if local<2*self.v:
+            route=('class',cover) if stage==0 else ('right',cover,('rho',stage,local%self.v))
+            return family,route
+        return self.address(stage,replica,self.helper_roles[local-2*self.v],cover)
+
+    def local_from_logical_family(self, stage, family):
+        """Invert PR234's logical five-stage family map for one active stage."""
+        logical_work=4*self.v+len(self.helper_roles)
+        assert 0<=family<=logical_work
+        if family==logical_work:return self.local_work
+        if family>=4*self.v:return 2*self.v+family-4*self.v
+        bank,port=divmod(family,self.v)
+        if bank==self.active[stage][0]:return port
+        assert bank==self.active[stage][1], 'stage references an inactive data bank'
+        return self.v+port
+
+    def map_stage_row(self, stage, replica, row, cover='d'):
+        """Lower an unchanged logical8-field MOVE/ADD/COPY/ERASE record.
+
+        All mathematical fields remain unchanged. The two operand slots now
+        contain exact (family,route) keys when they denote streams; frame IDs
+        in MOVE's old/new slots are not mistaken for stream operands.
+        """
+        op,a,b,c,f,z,s,rev=row
+        assert op in(0,1,2,3) and s==stage and rev==int(stage in(1,3))
+        def address(family):
+            return self.local_address(stage,replica,self.local_from_logical_family(stage,family),cover)
+        physical_a=address(a)
+        physical_b=address(b) if op in(1,2) else b
+        if op in(1,2):assert physical_a!=physical_b
+        return (op,physical_a,physical_b,c,f,z,s,rev)
+
+    def map_boundary_row(self, replica, row, cover='d'):
+        """Lower an IDLE/BRIDGE/EXCHANGE with its existing exact projector.
+
+        Boundary projectors already include the checked data-route conjugacy;
+        their physical class is d. Independent completion rows must be
+        discharged explicitly by the bank endpoint verifier, never skipped
+        implicitly by this address mapper.
+        """
+        op,a,b,c,f,z,s,rev=row
+        assert op in(4,5,7), 'only proved bank completion may discharge opcode6'
+        def address(family):
+            assert 0<=family<4*self.v
+            bank,port=divmod(family,self.v)
+            return self.data_family(replica,bank,port),('class',cover)
+        physical_a=address(a)
+        physical_b=address(b) if op in(5,7) else b
+        if op in(5,7):assert physical_a!=physical_b
+        return (op,physical_a,physical_b,c,f,z,s,rev)
+
+    def phase_schedule(self):
+        """Phase-major schedule; every cover class is completed per replica.
+
+        Each emitted stage phase means: for the indicated replica, iterate
+        every cover class and execute the entire local scalar telescope.
+        Boundary phases occur after all60 replicas of that stage finish.
+        """
+        for stage in range(self.stages):
+            for replica in range(self.replicas):
+                yield ('stage',stage,replica,'all_cover_classes')
+            if stage in (1,2,4):
+                for replica in range(self.replicas):
+                    yield ('idle',stage,replica,'all_cover_classes')
+                    yield ('bridge',stage,replica,'all_cover_classes')
+        for replica in range(self.replicas):
+            yield ('terminal_exchange',replica,'all_cover_classes')
diff --git a/research/five-stage-source527-banks/base_two_moment.py b/research/five-stage-source527-banks/base_two_moment.py
new file mode 100644
index 0000000000..14a90fa0e7
--- /dev/null
+++ b/research/five-stage-source527-banks/base_two_moment.py
@@ -0,0 +1,63 @@
+#!/usr/bin/env python3
+"""Maintainer arithmetic cross-check of the pinned PR39 certificate.
+
+Uses no producer/checker imports. Counts are inputs, not proved by this script.
+80-term rational logarithms and 12-term exponentials independently enclose
+the two moments. The accompanying written review addresses their realization.
+"""
+import argparse
+from fractions import Fraction as Q
+from hashlib import sha256
+import json
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parents[1]
+GRID = 10**30
+
+
+def require(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def log_unit(x):
+    require(1 <= x <= 2, "logarithm range")
+    z = (x-1)/(x+1)
+    lo = 2*sum((z**(2*j+1)/(2*j+1) for j in range(80)), Q(0))
+    return lo, lo+2*z**161/(161*(1-z*z))
+
+
+def log_bounds(x):
+    k = 0
+    while x > 2:
+        x /= 2
+        k += 1
+    lo, hi = log_unit(x)
+    l2, u2 = log_unit(Q(2))
+    lo, hi = lo+k*l2, hi+k*u2
+    # Outward rational rounding keeps later certificates compact.
+    return Q((lo*GRID).__floor__(), GRID), Q((hi*GRID).__ceil__(), GRID)
+
+
+def exp_bounds(x):
+    require(0 <= x < 1, "exponential range")
+    term = total = Q(1)
+    for j in range(1, 13):
+        term *= x/j
+        total += term
+    tail = term*x/13/(1-x/14)
+    return total, total+tail
+
+
+def moment(m, W, entries, saving):
+    lo = hi = Q(0)
+    for t, count in entries:
+        t, count = int(t), int(count)
+        require(0 < t < m and count > 0, "improper child")
+        l, u = log_bounds(Q(m, t))
+        lower = exp_bounds(saving*l)[0]
+        upper = exp_bounds(saving*u)[1]
+        lo += Q(t*count, m*W)*lower
+        hi += Q(t*count, m*W)*upper
+    return Q((lo*GRID).__floor__(), GRID), Q((hi*GRID).__ceil__(), GRID)
+
diff --git a/research/five-stage-source527-banks/code/complex_basis.py b/research/five-stage-source527-banks/code/complex_basis.py
new file mode 100644
index 0000000000..8c400798a5
--- /dev/null
+++ b/research/five-stage-source527-banks/code/complex_basis.py
@@ -0,0 +1,35 @@
+"""Portable pure functions extracted from the reviewed complex checks; no historical receipt readers."""
+from collections import Counter
+from copy import deepcopy
+from fractions import Fraction
+Q=Fraction
+from hashlib import sha256
+from pathlib import Path
+from math import prod
+import gzip,json,sys
+
+def need(b, why):
+    if not b: raise ValueError(why)
+
+def gl_count_test():
+    # Exhaustively test a literal binary Gaussian-elimination implementation
+    # through m=4. Row swaps cost three selected-bit additions.
+    out=[]
+    for m in range(1,5):
+        good=mx=0
+        for word in range(1<<(m*m)):
+            rows=[(word>>(m*i))&((1<<m)-1) for i in range(m)]
+            ops=0
+            for k in range(m):
+                p=next((i for i in range(k,m) if rows[i]>>k&1),None)
+                if p is None: break
+                if p!=k: rows[p],rows[k]=rows[k],rows[p];ops+=3
+                for i in range(m):
+                    if i!=k and rows[i]>>k&1: rows[i]^=rows[k];ops+=1
+            else:
+                need(rows==[1<<i for i in range(m)],'basis reduction endpoint')
+                need(ops<=m*(m-1)+3*m,'row-addition count')
+                good+=1;mx=max(mx,ops)
+        need(good==prod((1<<m)-(1<<i) for i in range(m)),'GL coverage')
+        out.append(dict(m=m,invertible_matrices=good,maximum_selected_bit_additions=mx))
+    return out
diff --git a/research/five-stage-source527-banks/code/complex_labels.py b/research/five-stage-source527-banks/code/complex_labels.py
new file mode 100644
index 0000000000..1f27d6de34
--- /dev/null
+++ b/research/five-stage-source527-banks/code/complex_labels.py
@@ -0,0 +1,176 @@
+"""Portable pure functions extracted from the reviewed complex checks; no historical receipt readers."""
+from collections import Counter
+from copy import deepcopy
+from fractions import Fraction
+Q=Fraction
+from hashlib import sha256
+from pathlib import Path
+from math import prod
+import gzip,json,sys
+
+BASE = dict(h=22, v=1320, R=9412, cst=440, N=262944)
+EXPECTED_H = {1:29178,2:12398,3:9923,4:4209,5:1950,6:1605,7:706,8:1531,
+              9:429,10:930,11:354,12:1197,13:127,14:742,15:1879,16:88,
+              17:81,18:2653,19:167,20:22}
+class Reject(ValueError):
+    pass
+
+def need(ok, msg):
+    if not ok:
+        raise Reject(msg)
+
+def digest(obj):
+    return sha256(json.dumps(obj, separators=(",",":"), sort_keys=True).encode()).hexdigest()
+
+def hist(counter):
+    return {str(k):v for k,v in sorted(counter.items()) if v}
+
+def role(r,v):
+    return "x" if r < v else "y" if r < 2*v else "s"
+
+def reduce_binary(x, frame):
+    for b in frame:
+        if x & (1 << (b.bit_length()-1)):
+            x ^= b
+    return x
+
+def coeff(a,b, where):
+    need(type(a) is int and type(b) is int and b>0, f"{where}: coefficient shape")
+    q=Fraction(a,b)
+    odd=q.denominator
+    while odd%2==0:
+        odd//=2
+    need(odd in (1,3), f"{where}: literal coefficient outside dyadic/divisor-three grid")
+    return q
+
+def replay_label_counts(c):
+    """Full finite LABEL walk only. Never multiplies the 12,052-square scalar map."""
+    need(c.get("format")=="gcert/1", "format")
+    need(all(c.get(k)==v for k,v in BASE.items()), "baseline parameter/count mismatch")
+    h,v,R=c["h"],c["v"],c["R"]; n=2*v+R
+    need(c.get("ext")==[], "baseline has no exterior-gauged helper inputs")
+    frames=c["frames"]; dim=list(map(len,frames))
+    need(len(frames)==18248, "baseline frame-table count")
+    need(frames[0]==[] and frames[1]==[1<<i for i in range(h-1,-1,-1)], "zero/full frames")
+    need(len({tuple(f) for f in frames})==len(frames), "duplicate frames")
+    for f in frames:
+        need(all(type(b) is int and 0<b<1<<h for b in f), "frame bit range")
+        piv=[b.bit_length()-1 for b in f]
+        need(all(piv[i]>piv[i+1] for i in range(len(piv)-1)), "frame pivot order")
+        need(all(not(b>>p&1) for i,b in enumerate(f) for j,p in enumerate(piv) if i!=j),
+             "frame is not reduced echelon")
+    ports=c["ports"]
+    need(len(ports)==v==len(set(ports)), "port count/uniqueness")
+    need(all(type(p) is int and 0<p<(1<<h)-1 and p.bit_count()%2 for p in ports), "unit port/gap")
+    start=c["start"]; final=c["final"]
+    need(len(start)==len(final)==n, "live invocation role count")
+    need(all(type(f) is int and 0<=f<len(frames) for f in start+final), "endpoint frame ID")
+    for t,p in enumerate(ports):
+        need(frames[start[t]]==[p] and final[t]==1, "X endpoint")
+        need(start[v+t]==0 and dim[final[v+t]]==h-1 and
+             all((p&u).bit_count()%2==0 for u in frames[final[v+t]]), "Y endpoint")
+    need(all(f==0 for f in start[2*v:]) and all(f==1 for f in final[2*v:]), "helper endpoints")
+    need(c["ret"] and len(c["ret"])==22, "temporary-center count")
+    need([x[0] for x in c["ret"]]==list(range(22)), "center numbering")
+    need(len({x[1] for x in c["ret"]})==22, "center retained slots repeat")
+    need(c["scat"]=={"inside":[1,3],"outside":[-1,6]}, "baseline scatter rule")
+    need(len(c["A"])==12158 and len(c["B"])==30878, "phase gate count")
+    cur=start.copy(); pairs=set()
+    by_role={r:Counter() for r in "xysc"}
+    by_phase={p:Counter() for p in ("A","copy","B","final")}
+    literal=Counter(); kind=Counter(); ybracket=Counter(); ypiv=set(); ytarget=set()
+    totals={p:dict(gates=0,scalar_additions=0,register_visits=0,spectator_visits=0)
+            for p in ("A","B")}
+    transitions=0
+    def climb(reg,new,phase):
+        nonlocal transitions
+        old=cur[reg]
+        if old==new:
+            return
+        need(dim[old]<dim[new], f"{phase}: frame does not strictly climb")
+        pair=(old,new)
+        if pair not in pairs:
+            need(all(reduce_binary(b,frames[new])==0 for b in frames[old]), f"{phase}: nonnested frames")
+            pairs.add(pair)
+        rank=dim[new]-dim[old]
+        by_role[role(reg,v)][rank]+=1; by_phase[phase][rank]+=1
+        cur[reg]=new;transitions+=1
+    for phase in ("A","B"):
+        for i,g in enumerate(c[phase]):
+            where=f"{phase}[{i}]"
+            need(isinstance(g,list) and len(g)>=4 and g[0] in ("out","in"), where+": gate shape")
+            tag,f,pivot,terms=g[:4]
+            need(len(g)==(5 if tag=="out" else 4), where+": gate arity")
+            need(type(f) is int and 0<=f<len(frames), where+": frame ID")
+            need(all(isinstance(x,list) and len(x)==3 for x in terms), where+": terms")
+            spectators=g[4] if tag=="out" else []
+            regs=[pivot]+[x[0] for x in terms]+spectators
+            need(all(type(r) is int and 0<=r<n for r in regs) and len(set(regs))==len(regs),
+                 where+": duplicate/out-of-range register")
+            need(phase!="A" or all(not(v<=r<2*v) for r in regs), where+": Y role before scatter")
+            totals[phase]["gates"]+=1
+            totals[phase]["register_visits"]+=len(regs)
+            totals[phase]["spectator_visits"]+=len(spectators)
+            for reg,a,b in terms:
+                target,source=(reg,pivot) if tag=="out" else (pivot,reg)
+                sr,tr=role(source,v),role(target,v)
+                need((sr,tr) in {("x","s"),("s","s"),("s","y"),("x","y"),("x","x"),("y","y")},
+                     where+": forbidden role direction")
+                need(phase!="A" or (sr,tr) in {("x","s"),("s","s")}, where+": phase A role direction")
+                q=coeff(a,b,where)
+                need(q!=0,where+": zero scalar addition")
+                literal[str(q)]+=1;kind[f"{phase}:{sr}->{tr}"]+=1
+                totals[phase]["scalar_additions"]+=1
+                if sr==tr=="y":
+                    ybracket[(source,target)]+=q;ypiv.add(source);ytarget.add(target)
+            for r in regs:
+                climb(r,f,phase)
+        if phase=="A":
+            cut=cur.copy()
+            need(all(cur[v+t]==0 for t in range(v)), "scatter: Y role moved too early")
+            for k,r,f in c["ret"]:
+                need(type(r) is int and 2*v<=r<n, "scatter: center source is not a helper role")
+                need(type(f) is int and 0<=f<len(frames) and cur[r]==f and dim[f]==20,
+                     "scatter: center label/rank mismatch")
+                by_role["c"][20]+=1;by_phase["copy"][20]+=1
+    need(not ypiv.intersection(ytarget) and all(q==0 for q in ybracket.values()), "Y bracket cancellation")
+    before_final=cur.copy()
+    for r,f in enumerate(final):
+        climb(r,f,"final")
+    need(cur==final,"final label state")
+    measured={r:hist(cn) for r,cn in by_role.items()}
+    need(measured==c["blocks"], "role histogram/count mismatch")
+    combined=sum(by_role.values(),Counter())
+    need(dict(combined)==EXPECTED_H, "baseline invocation histogram mismatch")
+    mass=sum(r*k for r,k in combined.items())
+    need(mass==c["N"]==R*h+2*v*(h-1)+c["cst"], "rank mass")
+    need(sum(r*k for r,k in by_role["c"].items())==c["cst"],"temporary-center rank mass")
+    scatter=Counter(str(coeff(*(c["scat"]["inside"] if p>>k&1 else c["scat"]["outside"]),"scatter"))
+                    for p in ports for k in range(22))
+    return {
+        "frame_count":len(frames),"tested_distinct_nested_pairs":len(pairs),
+        "live_registers_per_invocation":n,"temporary_centers":22,
+        "ranked_live_transitions":transitions,
+        "phase_stats":totals,"role_histograms":measured,
+        "phase_histograms":{p:hist(x) for p,x in by_phase.items()},
+        "invocation_histogram":hist(combined),"invocation_calls":sum(combined.values()),
+        "invocation_rank_mass":mass,"retained_centers":c["ret"],
+        "phase_state_sha256":{"start":digest(start),"after_A":digest(cut),
+                              "after_B":digest(before_final),"final":digest(final)},
+        "literal_gate_coefficients":dict(sorted(literal.items())),
+        "scalar_role_directions":dict(sorted(kind.items())),
+        "scatter_coefficients":dict(sorted(scatter.items())),
+        "scatter_additions":sum(scatter.values()),"Cc_copy_additions":22,
+        "two_hosts_metadata_count":len(c["two_hosts"]),
+        "two_hosts_note":"Metadata only; not an independent allocator or host-use proof.",
+        "Y_bracket_pairs":len(ybracket),
+        "literal_inverse":"Each nonself additive shear inverts by negating its coefficient; inverse phase reverses order; inverse transpose keeps chronological order and swaps source/target with negation.",
+        "limits":["No scalar all-column replay","No old-response or cleanup matrix formed",
+                  "No generated Lean data comparison or kernel build","No physical stream compiler"]
+    }
+
+def scalar_word(ops):
+    matrix=[[int(i==j) for j in range(4)] for i in range(4)]
+    for target,source,q in ops:
+        matrix[target]=[a+q*b for a,b in zip(matrix[target],matrix[source])]
+    return matrix
diff --git a/research/five-stage-source527-banks/code/complex_scalars.py b/research/five-stage-source527-banks/code/complex_scalars.py
new file mode 100644
index 0000000000..771057b567
--- /dev/null
+++ b/research/five-stage-source527-banks/code/complex_scalars.py
@@ -0,0 +1,216 @@
+"""Portable pure functions extracted from the reviewed complex checks; no historical receipt readers."""
+from collections import Counter
+from copy import deepcopy
+from fractions import Fraction
+Q=Fraction
+from hashlib import sha256
+from pathlib import Path
+from math import prod
+import gzip,json,sys
+
+V,R,H=1320,9412,22
+def need(x,msg):
+    if not x:raise ValueError(msg)
+
+def kind(r):return "X" if r<V else "Y" if r<2*V else "S"
+
+def live(r):return ("L",r)
+
+def tmp(r):return ("T",r-2*V)
+
+def center(k):return ("C",k)
+
+def add(t,s,q):return ("add",t,s,q.numerator,q.denominator)
+
+def clear(t):return ("clear",t)
+
+def copy(t,s):return ("copy",t,s)
+
+def flat(gs):
+    out=[]
+    for g in gs:
+        need(g[0] in ["in","out"],"literal additive gate")
+        for r,a,b in g[3]:
+            t,s=(r,g[2]) if g[0]=="out" else (g[2],r)
+            need(t!=s and kind(s)!="Y" and not(kind(s)=="S" and kind(t)=="X"),
+                 "projection requires Y never read and no helper-to-X gate")
+            out.append((t,s,Q(a,b)))
+    return out
+
+def old_positive(A,B,c):
+    """Circuit with inputs temp S and live Y; X fixed at zero.
+    tempS output is irrelevant. Y output is Y_in + K tempS_in."""
+    def part(gs):
+        for t,s,q in gs:
+            if kind(s)=="X":continue
+            need(kind(s)=="S" and kind(t) in ["S","Y"],"old-response shape")
+            yield add(tmp(t) if kind(t)=="S" else live(t),tmp(s),q)
+    yield from part(A)
+    for t,p in enumerate(c["ports"]):
+        for k,r,_ in c["ret"]:
+            q=Q(*(c["scat"]["inside"] if p>>k&1 else c["scat"]["outside"]))
+            yield add(live(V+t),tmp(r),q)
+    yield from part(B)
+
+def old_word(orientation,K):
+    if orientation=="forward":
+        yield from (copy(("T",q),live(2*V+q)) for q in range(R))
+        for e in K:
+            op,t,s,a,b=e
+            yield add(t,s,Q(-a if t[0]=="L" else a,b))
+    else:
+        yield from (clear(("T",q)) for q in range(R))
+        for op,t,s,a,b in reversed(K):
+            yield add(s,t,Q(a,b))
+        yield from (add(live(2*V+q),("T",q),Q(1)) for q in range(R))
+    yield from (clear(("T",q)) for q in range(R))
+
+def cleanup(orientation,AB):
+    for t,s,q in reversed(AB):
+        if kind(t)=="Y":continue
+        if orientation=="forward":yield add(live(t),live(s),-q)
+        else:yield add(live(s),live(t),q)
+
+def main_scalar(orientation,A,B,c):
+    def phase(gs):
+        for t,s,q in gs:
+            if orientation=="forward":yield add(live(t),live(s),q)
+            else:yield add(live(s),live(t),-q)
+    yield from phase(A)
+    # Every center is a complete temporary stream and one rank20 child,
+    # initialized/read/discarded before the next center uses the work stock.
+    for k,r,_ in c["ret"]:
+        yield clear(center(k))
+        if orientation=="forward":yield add(center(k),live(r),Q(1))
+        for t,p in enumerate(c["ports"]):
+            q=Q(*(c["scat"]["inside"] if p>>k&1 else c["scat"]["outside"]))
+            if orientation=="backward":yield add(center(k),live(V+t),q)
+        yield ("recursive_center",k,20)
+        if orientation=="forward":
+            for t,p in enumerate(c["ports"]):
+                q=Q(*(c["scat"]["inside"] if p>>k&1 else c["scat"]["outside"]))
+                yield add(live(V+t),center(k),q)
+        else:yield add(live(r),center(k),Q(-1))
+        yield clear(center(k))
+    yield from phase(B)
+
+def inv_schedule(orientation,A,B,c):
+    K=list(old_positive(A,B,c))
+    yield from ((p,e) for p,es in [
+       ("old_response",old_word(orientation,K)),
+       ("main",main_scalar(orientation,A,B,c)),
+       ("cleanup",cleanup(orientation,A+B))] for e in es)
+
+def coeff_steps(q):
+    """Non-destructive coefficient expansion using one complete work stream.
+    Copy source, double/halve/divide by3 as needed, add/subtract, clear.
+    Counts Gaussian operations; multiply by2 for real/imag component primitives."""
+    a,b=abs(q.numerator),q.denominator
+    need(a in (1,2) and b in (1,2,3,6),"unsupported new coefficient")
+    return 3+int(a==2)+int(b%2==0)+int(b%3==0)
+
+def bill(schedule):
+    phases={p:Counter() for p in ("old_response","main","cleanup")}
+    coeffs={p:Counter() for p in phases}
+    tape=sha256()
+    for p,e in schedule:
+        tape.update((json.dumps((p,e),separators=(",",":"))+"\n").encode())
+        phases[p]["all_macros"]+=1
+        if e[0]=="recursive_center":
+            phases[p]["recursive_centers"]+=1
+            phases[p]["recursive_rank_mass"]+=e[2]
+            continue
+        phases[p]["scalar_macros"]+=1
+        phases[p][e[0]]+=1
+        if e[0]=="add":
+            q=Q(e[3],e[4]);coeffs[p][str(q)]+=1
+            phases[p]["Gaussian_primitive_steps"]+=coeff_steps(q)
+            phases[p]["scalar_log2_magnitude_charge"]+=1+int(abs(q)==2)
+            phases[p]["halving_exposure"]+=int(q.denominator%2==0)
+            phases[p]["divisor3_exposure"]+=int(q.denominator%3==0)
+        else:phases[p]["Gaussian_primitive_steps"]+=1
+    for p in phases:phases[p]["real_component_primitive_steps"]=2*phases[p]["Gaussian_primitive_steps"]
+    total=sum((Counter({k:v for k,v in x.items() if k not in ["recursive_rank_mass"]}) for x in phases.values()),Counter())
+    return {"phases":{p:dict(x) for p,x in phases.items()},"coefficient_histograms":{p:dict(x) for p,x in coeffs.items()},
+            "totals":dict(total),"schedule_sha256":tape.hexdigest(),
+            "other_main_recursive_calls":"70147 existing live-label transitions, unchanged; not scalar macros"}
+
+def mm(A,B):return [[sum((x*y for x,y in zip(row,col)),Q(0)) for col in zip(*B)] for row in A]
+
+def lifecycle(schedule):
+    """Verify that new scalar-only work stock never survives into a child."""
+    initialized=set();dirty=set();active_centers=set();seen_main=False;children=0
+    for phase,e in schedule:
+        if phase=="main" and not seen_main:
+            need(not dirty,"old-response tableau survives into recursive main phase")
+            seen_main=True
+        need(e[0] in ("add","copy","clear","recursive_center"),"unknown or unpaid recursive operation")
+        if e[0]=="recursive_center":
+            k=e[1]
+            need(type(k) is int and 0<=k<22 and e[2]==20,"new temporary entered uncharged child")
+            need(phase=="main" and not dirty,"old-response temporary used during child")
+            need(active_centers<=set([k]),"other center was not discarded before next child")
+            children+=1
+            continue
+        target=e[1]
+        source=e[2] if e[0] in ("add","copy") else None
+        if source and source[0]=="T":need(source[1] in initialized,"uninitialized temporary read")
+        if target[0]=="T":
+            need(phase=="old_response","scalar-only tableau touched after recursive phase begins")
+            initialized.add(target[1])
+            if e[0]=="clear":dirty.discard(target[1])
+            else:dirty.add(target[1])
+        if target[0]=="C":
+            if e[0]=="clear":active_centers.discard(target[1])
+            else:active_centers.add(target[1])
+        if e[0]=="add":
+            # Coefficient expansion is deliberately local: copy into W,
+            # optionally double/halve/divide3, accumulate, and clear W.
+            expanded=["copy_work"]+["scale"]*(coeff_steps(Q(e[3],e[4]))-3)+["accumulate","clear_work"]
+            need(expanded[-1]=="clear_work","coefficient work survived scalar macro")
+    need(not dirty and not active_centers,"temporary stock not erased at invocation end")
+    need(children==22,"center child count changed")
+    return {"tableau_cleared_before_entire_main_phase":True,"tableau_absent_from_all_children":True,
+            "coefficient_work_cleared_per_add":True,"center_children":children,
+            "new_recursive_children":0,"moment_live_width":14692,
+            "new_work_stock_only":9413,"complete_stream_fraction":"1/(14692*|Orth(110,2)|)"}
+
+def ident(n):return [[Q(int(i==j)) for j in range(n)] for i in range(n)]
+
+def matrix(n,gs):
+    a=ident(n)
+    for t,s,q in gs:a[t]=[x+q*y for x,y in zip(a[t],a[s])]
+    return a
+
+def toy_checks():
+    # 2 X, 3 S, 2 Y. Explicit nontrivial X mixing is undone; helper map
+    # and K are otherwise arbitrary. Exact Fraction checks cover every column
+    # of THIS TINY TEMPLATE, not of the published 12052-register source.
+    n=7;X={0,1};S={2,3,4};Y={5,6}
+    gs=[(2,0,Q(1)),(3,2,Q(2)),(5,3,Q(-1,6)),(0,1,Q(1,2)),
+        (4,0,Q(2)),(2,4,Q(1,2)),(6,2,Q(1,3)),(0,1,Q(-1,2)),
+        (3,1,Q(-1)),(5,4,Q(1)),(6,0,Q(1))]
+    M=matrix(n,gs)
+    need(all(M[i]==ident(n)[i] for i in X),"toy source rows not restored")
+    Ugs=[e for e in gs if e[0] not in Y]
+    U=matrix(n,Ugs)
+    Uinv=matrix(n,[(t,s,-q) for t,s,q in reversed(Ugs)])
+    need(mm(Uinv,U)==ident(n),"restricted inverse identity")
+    Mi=matrix(n,[(t,s,-q) for t,s,q in reversed(gs)])
+    g8=[Mi[i] if i in S else ident(n)[i] for i in range(n)]
+    h8=[M[i] if i in S else ident(n)[i] for i in range(n)]
+    need(g8==Uinv and h8==U,"cleanup projection identities")
+    Ut=matrix(n,[(s,t,q) for t,s,q in reversed(Ugs)])
+    need(Ut==[list(x) for x in zip(*U)],"backward cleanup transpose")
+    Kgs=[e for e in gs if e[1] not in X]
+    Kmat=matrix(n,Kgs)
+    need(all(Kmat[i][j]==M[i][j] for i in Y for j in S),"zero-source old response")
+    Kt=matrix(n,[(s,t,q) for t,s,q in reversed(Kgs)])
+    need(all(Kt[i][j]==M[j][i] for i in S for j in Y),"old response adjoint")
+    bad=matrix(n,[(t,s,q) for t,s,q in reversed(Ugs)])
+    need(bad!=g8,"wrong inverse sign must fail")
+    bad_order=matrix(n,[(s,t,q) for t,s,q in Ugs])
+    need(bad_order!=Ut,"unreversed transpose must fail")
+    return {"tiny_exact_template_columns_checked":7,"checks_passed":7,
+            "wrong_inverse_sign_rejected":True,"unreversed_transpose_rejected":True,
+            "published_full_scalar_replay":False}
diff --git a/research/five-stage-source527-banks/code/complex_splice.py b/research/five-stage-source527-banks/code/complex_splice.py
new file mode 100644
index 0000000000..11373d1d57
--- /dev/null
+++ b/research/five-stage-source527-banks/code/complex_splice.py
@@ -0,0 +1,277 @@
+"""Portable pure functions extracted from the reviewed complex checks; no historical receipt readers."""
+from collections import Counter
+from copy import deepcopy
+from fractions import Fraction
+Q=Fraction
+from hashlib import sha256
+from pathlib import Path
+from math import prod
+import gzip,json,sys
+
+V,R,H,N,W=1320,9412,22,12052,14692
+PHASES=["old_response","ad0","A","centers","B","final","adF","cleanup","done"]
+def need(x,s):
+    if not x:raise ValueError(s)
+
+def blobhash(p):return sha256(p.read_bytes()).hexdigest()
+
+def digest(x):return sha256(json.dumps(x,separators=(',',':'),sort_keys=True).encode()).hexdigest()
+
+def role(r):return 'X' if r<V else 'Y' if r<2*V else 'S'
+
+def add(t,s,q):return ('add',t,s,q.numerator,q.denominator)
+
+def L(r):return ('L',r)
+
+def T(r):return ('T',r-2*V)
+
+def C(k):return ('C',k)
+
+def clear(t):return ('clear',t)
+
+def copy(t,s):return ('copy',t,s)
+
+def flat(gs):
+    for g in gs:
+        for r,a,b in g[3]:
+            t,s=(r,g[2]) if g[0]=='out' else (g[2],r)
+            need(t!=s and role(s)!='Y' and (role(s),role(t))!=('S','X'),'scalar projection shape')
+            yield t,s,Q(a,b)
+
+def scal(phase,e,frame):return ('scalar',phase,e,frame)
+
+def lower_scalar(e):
+    # The coefficient work stream is disjoint from every live and center role.
+    if e[0] in ['copy','clear']:return [e]
+    _,t,s,a,b=e;work=('W',0)
+    out=[copy(work,s)]
+    if abs(a)==2:out.append(('double',work))
+    if b%2==0:out.append(('halve',work))
+    if b%3==0:out.append(('divide3',work))
+    return out+[('accumulate',t,work,1 if a>0 else -1),clear(work)]
+
+def emit(c,ori):
+    """Construct the complete local parametric invocation, before cover routing.
+    One role stream is its full complete address rectangle. Frame labels denote
+    exact canonical representatives, never just rank or binary matrices.
+    """
+    A=list(flat(c['A']));B=list(flat(c['B']));cur=c['start'].copy()
+    K=[]
+    for part in (A,None,B):
+        if part is None:
+            for t,p in enumerate(c['ports']):
+                for k,r,f in c['ret']:
+                    K.append(add(L(V+t),T(r),Q(*(c['scat']['inside'] if p>>k&1 else c['scat']['outside']))))
+        else:
+            for t,s,q in part:
+                if role(s)=='X':continue
+                K.append(add(T(t) if role(t)=='S' else L(t),T(s),q))
+    yield ('phase','old_response')
+    if ori=='forward':
+        for q in range(R):yield scal('old_response',copy(('T',q),L(2*V+q)),'old0')
+        for _,t,s,a,b in K:yield scal('old_response',add(t,s,Q(-a if t[0]=='L' else a,b)),'old0')
+    else:
+        for q in range(R):yield scal('old_response',clear(('T',q)),'old0')
+        for _,t,s,a,b in reversed(K):yield scal('old_response',add(s,t,Q(a,b)),'old0')
+        for q in range(R):yield scal('old_response',add(L(2*V+q),('T',q),Q(1)),'old0')
+    for q in range(R):yield scal('old_response',clear(('T',q)),'old0')
+    yield ('phase','ad0')
+    for r,f in enumerate(c['start']):yield ('adapter','old_to_new',r,f)
+    for part in ['A','centers','B','final']:
+        yield ('phase',part)
+        if part in ['A','B']:
+            for g in c[part]:
+                regs=[g[2]]+[x[0] for x in g[3]]+(g[4] if g[0]=='out' else [])
+                for r in regs:
+                    if cur[r]!=g[1]:
+                        old=cur[r];cur[r]=g[1]
+                        yield ('climb',part,r,old,g[1],len(c['frames'][g[1]])-len(c['frames'][old]))
+                for t,s,q in flat([g]):
+                    e=add(L(t),L(s),q) if ori=='forward' else add(L(s),L(t),-q)
+                    yield scal('main',e,g[1])
+        elif part=='centers':
+            for k,r,f in c['ret']:
+                yield scal('main',clear(C(k)),f if ori=='forward' else 0)
+                if ori=='forward':yield scal('main',add(C(k),L(r),Q(1)),f)
+                else:
+                    for t,p in enumerate(c['ports']):
+                        yield scal('main',add(C(k),L(V+t),Q(*(c['scat']['inside'] if p>>k&1 else c['scat']['outside']))),0)
+                yield ('center',k,f if ori=='forward' else 0,0 if ori=='forward' else f,20)
+                if ori=='forward':
+                    for t,p in enumerate(c['ports']):
+                        yield scal('main',add(L(V+t),C(k),Q(*(c['scat']['inside'] if p>>k&1 else c['scat']['outside']))),0)
+                else:yield scal('main',add(L(r),C(k),Q(-1)),f)
+                yield scal('main',clear(C(k)),0 if ori=='forward' else f)
+        else:
+            for r,f in enumerate(c['final']):
+                if cur[r]!=f:
+                    old=cur[r];cur[r]=f
+                    yield ('climb','final',r,old,f,len(c['frames'][f])-len(c['frames'][old]))
+    yield ('phase','adF')
+    for r,f in enumerate(c['final']):yield ('adapter','new_to_old',r,f)
+    yield ('phase','cleanup')
+    for t,s,q in reversed(A+B):
+        if role(t)=='Y':continue
+        e=add(L(t),L(s),-q) if ori=='forward' else add(L(s),L(t),q)
+        yield scal('cleanup',e,'old1')
+    yield ('phase','done')
+
+def contains(U,V):
+    for u in U:
+        for v in V:
+            if u>>(v.bit_length()-1)&1:u^=v
+        if u:return False
+    return True
+
+def validate(events,c,ori,manifest,bill):
+    phase=None;seen=[];cur=c['start'].copy();cent={};dirty=set();initialized=set()
+    climbed=Counter();hist=Counter();projection=sha256();whole=sha256();counts=Counter();ad0=[];adF=[]
+    pairs=set();states={};all_frame_incidence=0;primitive=sha256();primitive_count=0
+    for ev in events:
+        whole.update((json.dumps(ev,separators=(',',':'))+'\n').encode())
+        tag=ev[0];counts[tag]+=1
+        if tag=='phase':
+            if phase=='A':states['after_A']=digest(cur)
+            if phase=='B':states['after_B']=digest(cur)
+            if phase=='final':states['final']=digest(cur)
+            phase=ev[1];seen.append(phase)
+            need(seen==PHASES[:len(seen)],'strict old/A/copy/B/final/adapter/cleanup chronology')
+            if phase=='ad0':need(not dirty,'old-response tableau survived before label recursion')
+            if phase=='A':need(ad0==list(enumerate(c['start'])),'missing or reordered exact entry adapter')
+            if phase=='adF':need(cur==c['final'],'final climbs omitted before cleanup')
+            if phase=='cleanup':need(adF==list(enumerate(c['final'])),'missing or reordered exact final adapter')
+            continue
+        if tag=='adapter':
+            _,which,r,f=ev
+            if which=='old_to_new':need(phase=='ad0','entry adapter position');ad0.append((r,f))
+            else:need(which=='new_to_old' and phase=='adF','final adapter position');adF.append((r,f))
+        elif tag=='climb':
+            _,p,r,old,new,d=ev
+            need(phase==p and p in ['A','B','final'],'live child outside source label walk')
+            need(not dirty and not cent,'new work stock or center active during live child')
+            need(cur[r]==old and d==len(c['frames'][new])-len(c['frames'][old]) and d>0,'wrong exact child endpoints/rank')
+            pair=(old,new)
+            if pair not in pairs:need(contains(c['frames'][old],c['frames'][new]),'nonnested child');pairs.add(pair)
+            cur[r]=new;climbed[p]+=1;hist[d]+=1
+        elif tag=='center':
+            _,k,old,new,d=ev
+            need(phase=='centers' and not dirty and set(cent)=={k},'center child ownership')
+            need(cent[k]==old and (k,c['ret'][k][1],c['ret'][k][2])==tuple(c['ret'][k]),'center input frame')
+            f=c['ret'][k][2]
+            need((old,new,d)==((f,0,20) if ori=='forward' else (0,f,20)),'center child direction/rank')
+            cent[k]=new;hist[d]+=1
+            projection.update((json.dumps(('main',('recursive_center',k,d)),separators=(',',':'))+'\n').encode())
+        elif tag=='scalar':
+            _,p,e,f=ev
+            need(p==('main' if phase in ['A','centers','B'] else phase),'scalar phase binding')
+            need(e[0] in ['add','copy','clear'],'unpaid scalar operation')
+            target=e[1];source=e[2] if e[0] in ['add','copy'] else None
+            for x in [target]+([source] if source else []):
+                bank,r=x
+                if bank=='T':need(phase=='old_response' and 0<=r<R,'tableau outside old response')
+                elif bank=='C':need(phase=='centers' and 0<=r<H,'center outside copied block')
+                elif bank=='L':
+                    need(0<=r<N,'live register range')
+                    if phase=='old_response':need(role(r) in ['S','Y'] and f=='old0','old response not at common old0')
+                    elif phase=='cleanup':need(role(r) in ['X','S'] and f=='old1' and cur[r]==1,'cleanup not at common old1')
+                    else:need(cur[r]==f,'scalar incidence uses wrong exact label');all_frame_incidence+=1
+                else:raise ValueError('unknown stream namespace')
+            if source and source[0]=='T':need(source[1] in initialized,'read before tableau initialization')
+            if source and source[0]=='C':need(cent.get(source[1])==f,'center source frame')
+            if target[0]=='T':
+                if e[0]=='add':need(target[1] in initialized,'tableau accumulation before initialization')
+                initialized.add(target[1])
+                if e[0]=='clear':dirty.discard(target[1])
+                else:dirty.add(target[1])
+            if target[0]=='C':
+                if e[0]=='clear':
+                    if target[1] in cent:del cent[target[1]]
+                else:
+                    need(not cent or set(cent)=={target[1]},'overlapping center lifetimes')
+                    need(cent.get(target[1],f)==f,'center target frame');cent[target[1]]=f
+            if e[0]=='add':need(e[4] in [1,2,3,6] and abs(e[3]) in [1,2],'unrecorded coefficient')
+            projection.update((json.dumps((p,e),separators=(',',':'))+'\n').encode())
+            lowered=lower_scalar(e)
+            if e[0]=='add':need(lowered[-1]==('clear',('W',0)),'coefficient work survives scalar macro')
+            for op in lowered:
+                primitive.update((json.dumps((p,op),separators=(',',':'))+'\n').encode())
+                primitive_count+=1
+        else:raise ValueError('unknown physical event')
+    need(seen==PHASES and not dirty and not cent,'incomplete invocation or dirty temporary')
+    need(dict(climbed)=={'A':13798,'B':46937,'final':9412},'live transition count/partition')
+    need(counts['center']==22 and counts['climb']==70147,'paid children omitted or doubled')
+    need({str(k):v for k,v in sorted(hist.items())}==manifest['checks']['invocation_histogram'],'paid source histogram changed')
+    need(states=={k:v for k,v in manifest['checks']['phase_state_sha256'].items() if k!='start'},'exact label states differ')
+    need(projection.hexdigest()==bill[ori]['schedule_sha256'],'complete scalar program differs from checked bill')
+    need(2*primitive_count==bill[ori]['totals']['real_component_primitive_steps'],'independent primitive expansion bill')
+    return dict(events=sum(counts.values()),event_counts=dict(counts),program_sha256=whole.hexdigest(),
+        scalar_projection_sha256=projection.hexdigest(),live_transition_partition=dict(climbed),
+        paid_children=counts['climb']+counts['center'],final_climbs_before_cleanup=True,
+        exact_common_label_live_incidences=all_frame_incidence,distinct_nested_pairs=len(pairs),
+        exact_label_state_sha256=states,old_response_erased_before_all_children=True,
+        entry_and_exit_old_new_adapters=2*N,Gaussian_primitives=primitive_count,
+        real_component_primitives=2*primitive_count,scalar_primitive_sha256=primitive.hexdigest())
+
+def global_contract(manifest):
+    return dict(family='baseline5stageComplex',m=110,live=14692,physical=24127,
+        stage_recipe=[dict(stage=j,window=f'G{j}',orientation=o,active_pair=p,
+            bank_owner='d' if j==0 else f'r{j}(t)^(-1)(d)',helper_owner='d' if j==0 else f's{j}(d)',
+            exterior_subset='empty' if j==0 else f'qS bs{j} l',
+            native_frame=f'(G{j} d).X.Phi(frames[f])',old_frame=f'(G{j} d).Xm.Phi(P)')
+            for j,(o,p) in enumerate([('forward',1),('backward',1),('forward',1),('backward',2),('forward',2)])],
+        chronology=manifest['chronology'],exact_frame_bindings=manifest['source_frame_equalities'],
+        scratch_namespace=dict(centers=[14692,14714],tableau=[14714,24126],coefficient=[24126,24127]),
+        recursive_volume_denominator=14692,primitive_policy='charged_fixed_tape_complete_streams',
+        label_policy='identical_exact_representative_and_paid_rank_zero_reconciliation',
+        helper_policy='arbitrary_input; inherited_dirty_lift; Z5=kernel',
+        odd_grid='2^(-P)3^(-G(D+1)); no return rounding',local_bound_exponent=48,
+        global_bound_exponent=6050,maintained_guard_changed=False,
+        primitive_recipe=dict(scalar='copy source to W; optional double/halve/divide3; signed accumulate; clear W',
+            live_child='exact target representative times inverse current representative; two rank-zero factors around one C tensor child',
+            center_forward='copy at retained frame; exact residual to frame zero; scatter; erase',
+            center_backward='zero at frame zero; accumulate transpose scatter; exact residual to retained frame; subtract into original; erase',
+            entry_exit='WinB.old_to_new and WinB.new_to_old, lowered as paid exact rank-zero Clifford operators',
+            rank_zero='binary Gaussian elimination; affine bit flips; Z4 linear phases and even cross terms; exact scalar fourth root',
+            cross_phase='i^(2xy) = i^x i^y i^(-(x xor y)); at most three parity-phase scans',
+            selected_bit='compact-control-movement proposition, complete restored borrowed fields, paid repair, no RAM lookup',
+            phase_scan='evaluate fixed Z4 quadratic descriptors per record, multiply by fourth root, record suffix pays descriptor arithmetic',
+            role_route='permutation on all complete live/work roles, at most W_physical^2 complete-stream exchanges on fixed tapes',
+            row_stock='split only live complete rows; scalar-only work creates no row coordinate or recursive branch'))
+
+def validate_global(g,m):
+    need(g==global_contract(m),'five-window exact source/tape/namespace contract changed')
+    # Injective embedding of each invocation and its disjoint temporary stock.
+    for s in g['stage_recipe']:
+        p=s['active_pair'];offset=2*V*(p-1)
+        live=[offset+r for r in range(2*V)]+[4*V+q for q in range(R)]
+        work=list(range(14692,24127))
+        need(len(set(live+work))==len(live+work),'global workstream aliases a live role')
+        need(all(0<=r<W for r in live),'global live embedding')
+    return True
+
+def controls(events,c,ori,m,b):
+    def rejected(name,fn):
+        x=list(events);fn(x)
+        try:validate(x,c,ori,m,b)
+        except ValueError as e:return dict(name=name,rejected=True,reason=str(e))
+        raise ValueError('mutation accepted: '+name)
+    out=[]
+    out.append(rejected('drop_final_climb',lambda x:x.pop(next(i for i,e in enumerate(x) if e[:2]==('climb','final')))))
+    def move_cleanup(x):
+        a=next(i for i,e in enumerate(x) if e==('phase','final'))
+        b=next(i for i,e in enumerate(x) if e==('phase','cleanup'))
+        x[a:b]=x[b:-1]+x[a:b];x[b:]=[('phase','done')]
+    out.append(rejected('cleanup_before_final_climbs',move_cleanup))
+    out.append(rejected('drop_final_frame_adapter',lambda x:x.pop(next(i for i,e in enumerate(x) if e[:2]==('adapter','new_to_old')))))
+    def wrong_frame(x):
+        i=next(i for i,e in enumerate(x) if e[0]=='scalar' and e[1]=='main' and isinstance(e[3],int))
+        e=x[i];x[i]=e[:3]+(-1,)
+    out.append(rejected('wrong_common_exact_gate_label',wrong_frame))
+    out.append(rejected('drop_center_child',lambda x:x.pop(next(i for i,e in enumerate(x) if e[0]=='center'))))
+    out.append(rejected('tableau_not_erased_before_child',lambda x:x.pop(max(i for i,e in enumerate(x) if e[0]=='scalar' and e[1]=='old_response' and e[2]==('clear',('T',0))))))
+    def wrong_sign(x):
+        i=next(i for i,e in enumerate(x) if e[0]=='scalar' and e[1]=='cleanup')
+        z=x[i];e=z[2];x[i]=z[:2]+(e[:3]+(-e[3],e[4]),z[3])
+    out.append(rejected('wrong_cleanup_sign',wrong_sign))
+    for entry in out:entry['orientation']=ori
+    return out
diff --git a/research/five-stage-source527-banks/code/geometry527.py b/research/five-stage-source527-banks/code/geometry527.py
new file mode 100644
index 0000000000..2cc2739f82
--- /dev/null
+++ b/research/five-stage-source527-banks/code/geometry527.py
@@ -0,0 +1,103 @@
+"""Portable exact h24/m120 callable geometry; inherited PR234 proof and notices."""
+from array import array
+from collections import Counter
+from functools import lru_cache
+from pathlib import Path
+import hashlib,json,sys,time
+import sympy as sp
+sys.dont_write_bytecode=True
+if not __debug__:raise SystemExit('assertions required')
+def sha(p):return hashlib.sha256(p.read_bytes()).hexdigest()
+def run(ctx,L,source_pins=None):
+ begin=time.monotonic()
+ W,C=ctx['W'],ctx['C'];v=W.v
+ Z=W.register([]);F=W.w['full_frame']
+ initial={i:W.w['source_frame'][i]for i in range(v)}
+ initial.update({v+t:Z for t in range(v)})
+ initial.update({2*v+j:W.gauge[r]['frame']if r in W.gauge else Z for j,r in enumerate(ctx['regs'])})
+ lower=L.Lowerer(array('i'),dict(W=W,C=C,initial_state=initial,ZERO=Z,FULL=F))
+ I=sp.eye(120);zero=sp.zeros(120)
+ S=sp.SparseMatrix
+ all_labels=[list(row)for row in W.g['labels']]
+ assert len(all_labels)==1760 and all(len(q)==len(set(q))==3 and all(0<=j<24 for j in q)for q in all_labels)
+ ports=[0,1759];checked=[]
+ for t in ports:
+     routes=[S(lower.rho(i,t))for i in range(5)]
+     assert all(r*r==I for r in routes)
+     P=lower.frame_projector(W.w['source_frame'][t])
+     assert P*P==P and sp.trace(P)==1
+     @lru_cache(None)
+     def data(i,kind):
+         frame=F if kind=='H'else Z if kind=='0'else W.w['source_frame'][t]
+         A=S(lower.global_stage_projector(i,frame,kind=='K'))
+         return routes[i]*A*routes[i]
+     for i in range(1,5):
+         assert data(i-1,'H')==data(i,'P')
+         assert data(i-1,'K')==data(i,'0')
+     p=data(0,'P')
+     pairs=[(p,data(1,'H'),46),(S(zero),data(1,'K'),46),
+            (data(2,'P'),data(3,'P'),23),(data(2,'0'),data(3,'0'),23),
+            (data(2,'H'),S(I),50),(data(2,'K'),S(I)-p,50),
+            (data(4,'H'),S(I),4),(data(4,'K'),S(I)-p,4)]
+     original_data_boundary=lower.data_boundary
+     assert S(original_data_boundary(3,t,'P'))==data(3,'P')
+     # Feed cached exact boundary matrices through the actual idle selector.
+     # This exercises its eight emitted boundary IDs without recomputing all
+     # dense matrices once per selector call.
+     def cached_boundary(i,port,kind):
+         assert port==t
+         return data(i,kind)
+     lower.data_boundary=cached_boundary
+     for boundary,(a,b,rank) in enumerate(pairs):
+         emitted_a,emitted_b=lower.idle_projectors(boundary,t)
+         assert S(emitted_a)==a and S(emitted_b)==b
+         assert a*b==b*a==a
+         gap=b-a
+         assert gap*gap==gap and sp.trace(gap)==rank
+     lower.data_boundary=original_data_boundary
+     # D_I D_P = D_(I-P), since D_I exchanges the address head/tail blocks.
+     # Check its projector algebra before the final stream pair exchange.
+     assert p*(S(I)-p)==S(zero) and p+(S(I)-p)==S(I)
+     checked.append(dict(port=t,labels=all_labels[t],routes=5,boundaries=8,idle_ranks=[r for a,b,r in pairs]))
+ 
+ unique={}
+ for _,frame,rank in lower.entrances:
+     if rank==20:unique.setdefault(tuple(map(tuple,C.B[frame])),frame)
+ frames=list(unique.values())[:2]
+ assert len(frames)==2
+ sigma=[S(lower.frame_projector(f))for f in frames]
+ assert sigma[0]!=sigma[1]
+ helpers=[]
+ for frame,P in zip(frames,sigma):
+     assert P*P==P and sp.trace(P)==20
+     completion=S(lower.completion_projector(frame))
+     residual=S(I)-completion
+     assert completion*completion==completion and sp.trace(completion)==100
+     assert completion*residual==residual*completion==S(zero)
+     actual=[]
+     for i in range(5):
+         tau=S(lower.tau(i));local=sp.zeros(120);local[:24,:24]=sp.eye(24)-P
+         actual.append(tau*S(local)*tau)
+     assert sum(actual,S(zero))==residual
+     assert all(a*b==S(zero)for i,a in enumerate(actual)for j,b in enumerate(actual)if i!=j)
+     assert sum(actual[:4]+[actual[3]],S(zero))!=residual
+     helpers.append(dict(local_frame=frame,local_rank=20,completion_rank=100,distinct_windows=5))
+ assert lower.completion_projector(frames[0])!=lower.completion_projector(frames[1])
+ # Every new sparse-source injection frame is checked as an exact h24 projector.
+ new_frames={r['mix_frame']for r in ctx['pair_selection']};G=sp.eye(24)-sp.ones(24)/9
+ for f in sorted(new_frames):
+  P=lower.frame_projector(f);assert P*P==P and sp.trace(P)==2 and P.T*G==G*P
+ assert len(new_frames)==31
+ 
+ 
+ result=dict(status='PASS_EXACT_H24_CALLABLE_LOWERING_GEOMETRY',m=120,
+     all_1760_ports_coordinate_permutation_equivalent=True,checked_actual_ports=checked,
+     checked_actual_entrances=helpers,checked_new_sparse_source_frames=len(new_frames),
+     distinct_entrances_noncommuting=sigma[0]*sigma[1]!=sigma[1]*sigma[0],
+     wrong_duplicate_window_rejected=True,wrong_equal_rank_correction_rejected=True,
+     terminal_exchange_frame_identity_checked=True,
+     source_pins=dict(source_pins or {}),clean_fingerprint=dict(scalar_word_sha256=hashlib.sha256(ctx['SOURCE_TEXT'].encode()).hexdigest()),
+     script_sha256=sha(Path(__file__)),seconds=time.monotonic()-begin,
+     scope='Exact rational h24/m120 route and idle matrices for actual source ports and actual independent entrance bases. Coordinate-permutation/direct-sum arguments cover the remaining ports and independent entrances. No scalar payload replay.')
+ 
+ return result
diff --git a/research/five-stage-source527-banks/code/global_lowering.py b/research/five-stage-source527-banks/code/global_lowering.py
new file mode 100644
index 0000000000..92f7818d59
--- /dev/null
+++ b/research/five-stage-source527-banks/code/global_lowering.py
@@ -0,0 +1,315 @@
+# Adapted from Henry Grant / hcg890's PR234; upstream code is unchanged except
+# the exact source527 register count and fresh checked count/digest bindings.
+# Prepared with substantial OpenAI Codex assistance; upstream notices retained.
+"""Source-bound callable five-stage global physical lowering, with streaming QA.
+
+The finite cover variable stays symbolic. One iteration means one cover class;
+iter_program() is the exact template to apply to every class in each phase.
+No formal F2 payload columns are replayed. The complete local tagged word is
+materialized once, then all five actual renamed/reversed instruction streams
+are checked and hashed. The local all-column theorem composes through these
+explicit namespace, frame, copy-lifetime and boundary bindings.
+"""
+from array import array
+from collections import Counter
+from contextlib import redirect_stdout
+from functools import lru_cache
+from pathlib import Path
+import hashlib, importlib.util, io, json, struct, sys, time
+
+BASE = Path(__file__).resolve().parent
+ROOT = BASE.parent
+PHYSICAL = BASE/'five_stage_source471_physical_telescope_20261009.py'
+ACTIVE = ((0,1),(1,0),(0,1),(3,2),(2,3))
+BANKS = ('X1','Y1','X2','Y2')
+V, R, H, M = 1760, 16587, 24, 120
+LOCAL, LIVE, WORK = 2*V+R, 4*V+R, 4*V+R
+# Compact instruction kinds: move, add, copy, erase, idle, bridge, completion.
+MOVE, ADD, COPY, ERASE, IDLE, BRIDGE, COMPLETE, EXCHANGE = range(8)
+LOCAL_BEGIN, LOCAL_END = 2, 3
+PACK = struct.Struct('<8i')
+sha = lambda b: hashlib.sha256(b).hexdigest()
+if not __debug__:
+    raise SystemExit('refusing optimized Python: assertion checks are required')
+
+def family(stage, local):
+    assert 0 <= stage < 5 and 0 <= local <= LOCAL
+    if local == LOCAL: return WORK
+    if local < V: return ACTIVE[stage][0]*V+local
+    if local < 2*V: return ACTIVE[stage][1]*V+local-V
+    return 4*V+local-2*V
+
+def address(stage, local, cover='d'):
+    """An actual global family and an exact right-multiplication route word.
+
+    rho(stage,port) is the involution in the checked rational route template;
+    tau(stage) is the whole-block exchange. A workstream is external and is
+    reused only after ERASE. Cover-class enumeration is supplied by the compiler.
+    """
+    f = family(stage,local)
+    if local == LOCAL: return (f, ('external_work', 0))
+    if stage == 0: return (f, ('class',cover))
+    route = ('rho',stage,local%V) if local < 2*V else ('tau',stage)
+    return (f, ('right',cover,route))
+
+def check_namespace(mapper=family):
+    for stage in range(5):
+        images = [mapper(stage,i) for i in range(LOCAL+1)]
+        assert len(set(images)) == LOCAL+1, 'live/work collision'
+        assert images[-1] == WORK and WORK not in images[:-1], 'external work not preserved'
+        for i in range(LOCAL):
+            expected = (ACTIVE[stage][0]*V+i if i<V else
+                        ACTIVE[stage][1]*V+i-V if i<2*V else 4*V+i-2*V)
+            assert images[i] == expected, 'incorrect data/helper owner'
+    return True
+
+class Lowerer:
+    """Callable IR lowering; op records have exactly eight signed int fields.
+
+    Stage MOVE: (op,family,old_frame,new_frame,rank,0,stage,complement).
+    Stage ADD:  (op,dst,src,coefficient,frame,category,stage,complement).
+    COPY:       (op,temp,src,0,frame,0,stage,complement).
+    ERASE:      (op,temp,-1,0,frame,0,stage,complement).
+    IDLE:       (op,family,-1,0,rank,boundary_id,-1,0).
+    BRIDGE:     (op,dst,src,coefficient,0,boundary_id,-1,0).
+    COMPLETE:   (op,helper,-1,0,rank,entrance_frame,-1,0).
+
+    frame_projector() and global_stage_projector() supply exact rational matrix
+    meanings on demand. Data IDLE boundaries and COMPLETE are defined below.
+    """
+    def __init__(self,records,context):
+        self.records,self.ns = records,context
+        self.W,self.C = context['W'],context['C']
+        self.initial = context['initial_state']
+        self.ZERO,self.FULL = context['ZERO'],context['FULL']
+        self.entrances = [(4*V+j,self.initial[2*V+j],self.C.dimf[self.initial[2*V+j]])
+                         for j in range(R) if self.C.dimf[self.initial[2*V+j]]]
+        assert Counter(a for f,s,a in self.entrances) == {12:18,13:48,18:13,20:2200}
+        check_namespace()
+
+    def local_rows(self,reverse=False):
+        n=len(self.records)
+        for k in (range(n-6,-1,-6) if reverse else range(0,n,6)):
+            yield tuple(self.records[k:k+6])
+
+    def iter_stage(self,stage,mapper=family):
+        assert 0 <= stage < 5
+        rev = stage in (1,3)
+        for op,a,b,c,f,z in self.local_rows(rev):
+            if op == MOVE:
+                yield (MOVE,mapper(stage,a),c if rev else b,b if rev else c,f,0,stage,int(rev))
+            elif op == ADD:
+                yield (ADD,mapper(stage,a),mapper(stage,b),-c if rev else c,f,z,stage,int(rev))
+            elif (op == LOCAL_END if rev else op == LOCAL_BEGIN):
+                # Always COPY freshly from the original center. No operation
+                # inverses an erasure. Reflected source is H-U, target is H.
+                yield (COPY,mapper(stage,b),mapper(stage,a),0,c,0,stage,int(rev))
+                yield (MOVE,mapper(stage,b),c,f,z,0,stage,int(rev))
+            else:
+                assert op == (LOCAL_BEGIN if rev else LOCAL_END)
+                yield (ERASE,mapper(stage,b),-1,0,f,0,stage,int(rev))
+
+    def iter_idle(self,which):
+        # IDs select exact projector pairs in idle_projectors(), not ranks alone.
+        rows = {0:((2,46,0),(3,46,1)),
+                1:((2,23,2),(3,23,3)),
+                2:((0,50,4),(1,50,5),(2,4,6),(3,4,7))}[which]
+        for bank,rank,boundary in rows:
+            for t in range(V): yield (IDLE,bank*V+t,-1,0,rank,boundary,-1,0)
+
+    def iter_bridges(self,which):
+        rows = {0:((0,2,1),(3,1,-1)),1:((3,1,1),(0,2,-1)),2:((1,3,-1),(2,0,1))}[which]
+        for dst,src,coefficient in rows:
+            for t in range(V): yield (BRIDGE,dst*V+t,src*V+t,coefficient,0,which,-1,0)
+
+    def iter_completions(self):
+        for helper,sigma,a in self.entrances:
+            yield (COMPLETE,helper,-1,0,5*a,sigma,-1,0)
+
+    def iter_program(self):
+        for stage in range(5):
+            yield from self.iter_stage(stage)
+            if stage in (1,2,4):
+                which={1:0,2:1,4:2}[stage]
+                yield from self.iter_idle(which)
+                yield from self.iter_bridges(which)
+        yield from self.iter_completions()
+        yield from self.iter_exchanges()
+
+    def iter_exchanges(self):
+        # Complete-stream pair permutation, newX=oldY and newY=-oldX.
+        # Two movements per pair are explicitly reserved in the route bill.
+        for x,y in ((0,1),(2,3)):
+            for t in range(V):yield(EXCHANGE,x*V+t,y*V+t,-1,0,0,-1,0)
+
+    @lru_cache(None)
+    def frame_projector(self,frame):
+        import sympy as sp
+        B=sp.Matrix(self.C.B[frame])
+        if not self.C.dimf[frame]: return sp.zeros(H)
+        G=sp.eye(H)-sp.ones(H)/9
+        return B.T*(B*G*B.T).inv()*B*G
+
+    def rho(self,stage,port):
+        import sympy as sp
+        I=sp.eye(M)
+        if stage==0:return I
+        q=list(self.W.g['labels'][port]);rest=[j for j in range(H) if j not in q]
+        permutation=q+rest
+        pi=sp.zeros(H)
+        for j,dst in enumerate(permutation):pi[dst,j]=1
+        qstar=sp.Matrix([int(i<3)for i in range(H)])
+        G=sp.eye(H)-sp.ones(H)/9
+        pstar=qstar*(qstar.T*G)/2; p=pi*pstar*pi.T;kstar=sp.eye(H)-pstar
+        I[:H,:H]=p;I[stage*H:(stage+1)*H,stage*H:(stage+1)*H]=pstar
+        I[:H,stage*H:(stage+1)*H]=pi*kstar
+        I[stage*H:(stage+1)*H,:H]=kstar*pi.T
+        return I
+
+    def tau(self,stage):
+        import sympy as sp
+        I=sp.eye(M)
+        if stage:
+            for j in range(H):I.row_swap(j,stage*H+j)
+        return I
+
+    def global_stage_projector(self,stage,frame,complement=False,chart=None):
+        """Exact d(P direct_sum Q_stage)d^-1, with d=I by default."""
+        import sympy as sp
+        P=self.frame_projector(frame)
+        if complement:P=sp.eye(H)-P
+        A=sp.zeros(M);A[:H,:H]=P
+        qstar=sp.Matrix([int(i<3)for i in range(H)])
+        G=sp.eye(H)-sp.ones(H)/9
+        K=sp.eye(H)-qstar*(qstar.T*G)/2
+        for j in range(1,stage+1):A[j*H:(j+1)*H,j*H:(j+1)*H]=K
+        return A if chart is None else chart*A*chart.inv()
+
+    def data_boundary(self,stage,port,kind):
+        import sympy as sp
+        if kind=='P':frame=self.W.w['source_frame'][port];complement=False
+        elif kind=='K':frame=self.W.w['source_frame'][port];complement=True
+        else:frame=self.FULL if kind=='H' else self.ZERO;complement=False
+        route=self.rho(stage,port)
+        return route*self.global_stage_projector(stage,frame,complement)*route
+
+    def idle_projectors(self,boundary,port):
+        import sympy as sp
+        P=self.data_boundary(0,port,'P');I=sp.eye(M);Z=sp.zeros(M)
+        pairs=((P,self.data_boundary(1,port,'H')),
+               (Z,self.data_boundary(1,port,'K')),
+               (self.data_boundary(2,port,'P'),self.data_boundary(3,port,'P')),
+               (self.data_boundary(2,port,'0'),self.data_boundary(3,port,'0')),
+               (self.data_boundary(2,port,'H'),I),
+               (self.data_boundary(2,port,'K'),I-P),
+               (self.data_boundary(4,port,'H'),I),
+               (self.data_boundary(4,port,'K'),I-P))
+        return pairs[boundary]
+
+    def completion_projector(self,frame,chart=None):
+        import sympy as sp
+        A=sp.diag(*([self.frame_projector(frame)]*5))
+        return A if chart is None else chart*A*chart.inv()
+
+def verify(lower,raw):
+    physical=lower.ns['result']
+    source_tag=physical['tagged_scalar_sha256']
+    stages=[];paid=Counter();program=hashlib.sha256(b'global-physical-v1\0'+source_tag.encode())
+    opcode_counts=Counter();center_category=physical['category_names'].index('center')
+    for stage in range(5):
+        digest=hashlib.sha256(b'stage-v1\0'+bytes([stage])+source_tag.encode())
+        adds=copies=erases=0;copy_open=False;copy_reads=0;coefficient=Counter();stage_paid=Counter()
+        live_image={family(stage,i)for i in range(LOCAL)}
+        for row in lower.iter_stage(stage):
+            op,a,b,c,f,z,s,reverse=row
+            assert s==stage and reverse==int(stage in(1,3))
+            if op==MOVE:
+                assert a in live_image or a==WORK
+                assert f>=0
+                if f:stage_paid[f]+=1
+                if a==WORK:
+                    assert copy_open and c==lower.ZERO and f==22
+            elif op==ADD:
+                assert a in live_image and (b in live_image or b==WORK) and a!=b
+                assert not(a==WORK)
+                assert (b==WORK)==(z==center_category)
+                if b==WORK:assert copy_open and f==lower.ZERO;copy_reads+=1
+                adds+=1;coefficient[abs(c)]+=1
+            elif op==COPY:
+                assert not copy_open and a==WORK and b in live_image
+                assert b>=4*V and lower.C.dimf[f]==22
+                copy_open=True;copy_reads=0;copies+=1
+            elif op==ERASE:
+                assert copy_open and a==WORK and f==lower.ZERO and copy_reads==220
+                copy_open=False;erases+=1
+            else:raise AssertionError('unexpected stage opcode')
+            data=PACK.pack(*row);digest.update(data);program.update(data);opcode_counts[op]+=1
+        assert not copy_open and adds==physical['weighted_scalar_events'] and copies==erases==24
+        assert stage_paid==Counter({int(k):n for k,n in physical['paid_histogram'].items()})
+        assert coefficient=={int(k):v for k,v in physical['coefficient_histogram'].items()}
+        paid.update(stage_paid)
+        stages.append(dict(stage=stage,sha256=digest.hexdigest(),weighted_additions=adds,
+            fresh_copies=copies,erasures=erases,paid_histogram=dict(sorted(stage_paid.items())),
+            source_owner=BANKS[ACTIVE[stage][0]],target_owner=BANKS[ACTIVE[stage][1]],
+            complemented_reverse=stage in(1,3),work_family=WORK))
+        if stage in(1,2,4):
+            which={1:0,2:1,4:2}[stage]
+            for row in lower.iter_idle(which):
+                paid[row[4]]+=1;program.update(PACK.pack(*row));opcode_counts[IDLE]+=1
+            for row in lower.iter_bridges(which):
+                program.update(PACK.pack(*row));opcode_counts[BRIDGE]+=1
+    for row in lower.iter_completions():
+        paid[row[4]]+=1;program.update(PACK.pack(*row));opcode_counts[COMPLETE]+=1
+    for row in lower.iter_exchanges():
+        program.update(PACK.pack(*row));opcode_counts[EXCHANGE]+=1
+    assert opcode_counts[EXCHANGE]==2*V
+    expected=Counter({int(k):5*n for k,n in raw['one_stage_helper_histogram_including_copies'].items()})
+    expected.update({r:2*V for r in(46,23,50,4)})
+    expected.update({5*int(a):n for a,n in raw['auxiliary_entrance_rank_histogram'].items()})
+    assert paid==expected and sum(paid.values())==raw['five_stage_profile']['calls']
+    assert sum(r*n for r,n in paid.items())==raw['five_stage_profile']['rank_mass'] and LIVE*M-sum(r*n for r,n in paid.items())==4400
+    assert opcode_counts[BRIDGE]==6*V and opcode_counts[COMPLETE]==2279
+    # Independent four-bank all-column F2 composition, no source helper replay.
+    def bank_endpoint(skip_bridge=None):
+        banks=[1<<j for j in range(4)];bindex=0
+        for stage,(src,dst)in enumerate(ACTIVE):
+            banks[dst]^=banks[src]
+            if stage in(1,2,4):
+                for row in lower.iter_bridges({1:0,2:1,4:2}[stage]):
+                    if row[1]%V:continue
+                    if bindex!=skip_bridge:banks[row[1]//V]^=banks[row[2]//V]
+                    bindex+=1
+        return banks
+    assert bank_endpoint()==[2,1,8,4]
+    after_exchange=bank_endpoint()
+    after_exchange[0],after_exchange[1]=after_exchange[1],after_exchange[0]
+    after_exchange[2],after_exchange[3]=after_exchange[3],after_exchange[2]
+    assert after_exchange==[1,2,4,8]
+    assert all(bank_endpoint(j)!=[2,1,8,4]for j in range(6))
+    controls=[]
+    try:check_namespace(lambda s,i:LOCAL if i==LOCAL else family(s,i))
+    except AssertionError:controls.append('unremapped local temporary collides with arbitrary live helper')
+    else:raise AssertionError('collision admitted')
+    # An erased workstream cannot be reversed into the original dirty value.
+    original,dirty=0b1011,0b1100
+    fresh_copy=original;fresh_target=dirty^fresh_copy
+    erased_copy=0;literal_reverse_target=dirty^erased_copy
+    assert fresh_target!=literal_reverse_target
+    controls.append('literal inverse of erased copy loses arbitrary center contribution')
+    controls.extend('omitted bridge '+str(j)for j in range(6))
+    # Direct-sum support proof executed on the actual 5-window mask. It depends
+    # on the routed helper window, not just its rank. Every nonzero entrance is
+    # completed separately; no source-owned gauge enters this list.
+    correct_windows=[1<<j for j in range(5)]
+    assert sum(correct_windows)==31
+    wrong_windows=[1<<j for j in(0,1,2,3,3)]
+    assert len(set(wrong_windows))!=5
+    controls.append('stage4 helper route aliases stage3 window')
+    return dict(stages=stages,program_sha256=program.hexdigest(),paid_histogram=dict(sorted(paid.items())),
+                paid_calls=sum(paid.values()),paid_rank_mass=sum(r*n for r,n in paid.items()),
+                deficit=4400,opcode_counts=dict(sorted(opcode_counts.items())),
+                bank_F2_endpoint_before_terminal_relabel=bank_endpoint(),
+                bank_F2_endpoint_after_terminal_relabel=after_exchange,
+                terminal_exchange_pairs=2*V,terminal_exchange_stream_movements=4*V,
+                controls_rejected=controls)
+
diff --git a/research/five-stage-source527-banks/code/physical527.py b/research/five-stage-source527-banks/code/physical527.py
new file mode 100644
index 0000000000..bfd8fd13a6
--- /dev/null
+++ b/research/five-stage-source527-banks/code/physical527.py
@@ -0,0 +1,102 @@
+"""Fresh source527 physical event lowering. Substantial OpenAI Codex assistance.
+Derived from the PR234 physical telescope; see root provenance and notices.
+"""
+from pathlib import Path
+from collections import Counter
+from array import array
+from types import SimpleNamespace
+import ast,hashlib,json,gzip,sys,time
+sys.dont_write_bytecode=True
+if not __debug__:raise SystemExit('assertions required')
+def run(context,source_text,output_dir=None,source_pins=None):
+ D=Path(output_dir)if output_dir is not None else None
+ if D is not None:D.mkdir(parents=True,exist_ok=True)
+ m=SimpleNamespace(**context);W=m.W;C=m.C;v=W.v;regs=m.regs;n=2*v+len(regs)
+ source=source_text;assert len(regs)==16587
+ ZERO=W.register([]);FULL=W.w['full_frame']
+ sha=lambda b:hashlib.sha256(b).hexdigest()
+ def physical_id(s):
+  r=W.phys[s];return m.borrow[r]if r in m.borrow else m.idx[r]
+ def id_expr(node):
+  if isinstance(node,ast.Subscript)and isinstance(node.value,ast.Name)and node.value.id in('x','y'):
+   return node.slice if node.value.id=='x'else ast.BinOp(left=ast.Name(id='v',ctx=ast.Load()),op=ast.Add(),right=node.slice)
+  if isinstance(node,ast.Call)and isinstance(node.func,ast.Name)and node.func.id=='value':return ast.Call(func=ast.Name(id='physical_id',ctx=ast.Load()),args=node.args,keywords=[])
+  raise AssertionError(('unrecognized physical address',ast.unparse(node)))
+ def digest_add(h,row):
+  h.update(json.dumps(row,separators=(',',':')).encode());h.update(b'\n')
+ scalar_digest=hashlib.sha256();scalar_count=0;scalar_coeff=Counter()
+ def observe(dst,c,src):
+  nonlocal scalar_count
+  assert dst!=src;digest_add(scalar_digest,[dst,src,c]);scalar_count+=1;scalar_coeff[abs(c)]+=1
+ class Observe(ast.NodeTransformer):
+  def visit_Call(self,node):
+   self.generic_visit(node)
+   if isinstance(node.func,ast.Name)and node.func.id=='add'and node.lineno<=205:
+    dx,sy=id_expr(node.args[0]),id_expr(node.args[2]);node.func.id='observed_add';node.args += [dx,sy]
+   return node
+ q=Observe().visit(ast.parse(source));fn=q.body[0];i=next(i for i,z in enumerate(fn.body)if isinstance(z,ast.FunctionDef)and z.name=='add');fn.body.insert(i+1,ast.parse('def observed_add(x,c,y,dx,sy):\n observe(dx,c,sy)\n return add(x,c,y)').body[0]);ast.fix_missing_locations(q)
+ m.physical_id=physical_id;m.observe=observe;exec(compile(q,'<portable527 scalar observer>','exec'),m.__dict__);scalar=m.replay();scalar_sha=scalar_digest.hexdigest();print('PASS independent F2 event observer',scalar_count,scalar_sha,dict(scalar_coeff),flush=True)
+ state={i:W.w['source_frame'][i]for i in range(v)};state.update({v+t:ZERO for t in range(v)});state.update({2*v+j:W.gauge[r]['frame']if r in W.gauge else ZERO for j,r in enumerate(regs)});initial_state=state.copy();caps={t:W.register(W.module.kernel([C.cov[t]],W.h)[0])for t in range(v)}
+ records=array('i');hist=Counter();categories=Counter();coefficients=Counter();centers=[];center=None;used_frames=set(state.values());subcache={};count=0;tag_hash=hashlib.sha256();projection=hashlib.sha256()
+ # The line binding is checked against the exact generated source digest below.
+ tags={43:("group['frame']",'equal_inverse'),50:("group['frame']",'rank_inverse'),56:("group['frame']",'echelon_inverse'),62:("W.gauge[s]['frame']",'paired_alias_corruption'),67:("W.gauge[s]['frame']",'fresh_source_read'),81:("W.gauge[s]['frame']",'echelon_read'),96:("read_frames[s,k]",'equal_read'),98:("read_frames[s,k]",'equal_corruption'),108:("W.gauge[s]['frame']if s in W.gauge else ZERO",'dirty_read'),113:("W.opframe[i]if sign==1 else FULL",'gate'),116:("W.w['source_frame'][n]",'inject'),120:("kpair[r['source']]['mix_frame']",'gauge_mix'),127:("kpair[b]['mix_frame']",'partner_mix'),133:('ZERO','center'),137:('ZERO','echelon_setup'),142:('ZERO','rank_setup'),147:('ZERO','equal_setup'),151:('ZERO','terminal_pre'),159:('f','anchor_inject'),166:('W.opframe[i]','terminal_write'),172:("e['root_frame']",'terminal_post'),181:("W.w['root_frame'][j]",'side_root'),184:("kpair[b]['mix_frame']",'partner_mix'),186:("e['deliver_frame']",'partner_delivery'),189:('FULL','partner_cleanup'),192:('FULL','partner_cleanup'),201:('FULL','anchor_restore'),204:('FULL','partner_cleanup'),205:('FULL','uninject')}
+ category_names=sorted(set(k for f,k in tags.values())|{'forward_gate','cleanup_gate'});kind_ids={k:i for i,k in enumerate(category_names)}
+ def move(i,f):
+  used_frames.add(f);old=state[i]
+  if old==f:return
+  if(old,f)not in subcache:subcache[old,f]=C.sub(old,f)
+  assert subcache[old,f],('physical descent',i,old,f,C.dimf[old],C.dimf[f])
+  d=C.dimf[f]-C.dimf[old];assert d>=0
+  records.extend((0,i,old,f,d,0))
+  if d:hist[d]+=1
+  state[i]=f
+ def physical_add(x,c,y,f,kind):
+  nonlocal count
+  assert x!=y;move(x,f);physical_y=y
+  if kind=='center':assert center is not None and center['source']==y and f==ZERO;physical_y=n
+  else:move(y,f)
+  records.extend((1,x,physical_y,c,f,kind_ids[kind]));digest_add(projection,[x,y,c]);digest_add(tag_hash,[x,physical_y,c,f,kind]);count+=1;categories[kind]+=1;coefficients[abs(c)]+=1
+  return x
+ def center_begin(source,frame):
+  nonlocal center
+  assert center is None;move(source,frame);rank=C.dimf[frame];assert rank==22;hist[rank]+=1
+  center=dict(source=source,frame=frame,rank=rank,first_event=count,temporary=n,copy_output_frame=ZERO);centers.append(center);records.extend((2,source,n,frame,ZERO,rank))
+ def center_end():
+  nonlocal center
+  assert center is not None;center['after_event']=count;center['scatter_reads']=count-center['first_event'];assert center['scatter_reads']==220
+  records.extend((3,center['source'],n,center['frame'],ZERO,center['rank']));center=None
+ class Tag(ast.NodeTransformer):
+  def visit_Call(self,node):
+   self.generic_visit(node)
+   if isinstance(node.func,ast.Name)and node.func.id=='add'and node.lineno<=205:
+    frame,kind=tags[node.lineno];node.func.id='physical_add';kindnode=ast.parse("'forward_gate'if sign==1 else'cleanup_gate'",mode='eval').body if kind=='gate'else ast.Constant(kind);node.args += [ast.parse(frame,mode='eval').body,kindnode]
+   return node
+ q=Tag().visit(ast.parse(source));fn=q.body[0];end=next(i for i,z in enumerate(fn.body)if isinstance(z,ast.If)and ast.unparse(z.test)=='binary');fn.body=fn.body[:end]+[ast.parse('return dict(center_reads=len(center_reads))').body[0]]
+ fn.body=[z for z in fn.body if not(isinstance(z,ast.FunctionDef)and z.name=='add')]
+ for node in ast.walk(fn):
+  if isinstance(node,ast.Assign)and any(isinstance(t,ast.Name)and t.id=='unit'for t in node.targets):node.value=ast.parse('lambda i:i',mode='eval').body
+  if isinstance(node,ast.FunctionDef)and node.name=='target_move':node.body.insert(0,ast.parse('move(v+t,f)').body[0])
+  if isinstance(node,ast.FunctionDef)and node.name=='source_move':node.body.insert(0,ast.parse('move(physical_id(s),f)').body[0])
+ # Modify the center loop without changing any scalar expression or event order.
+ centerloop=next(z for z in fn.body if isinstance(z,ast.For)and z.lineno==129)
+ centerloop.target=ast.parse('(j,(r,s))',mode='eval').body;centerloop.target.ctx=ast.Store();centerloop.target.elts[0].ctx=ast.Store();centerloop.target.elts[1].ctx=ast.Store()
+ for z in centerloop.target.elts[1].elts:z.ctx=ast.Store()
+ centerloop.iter=ast.Call(func=ast.Name(id='enumerate',ctx=ast.Load()),args=[centerloop.iter],keywords=[])
+ block=centerloop.body[0];assert isinstance(block,ast.If);block.body.insert(0,ast.parse("center_begin(value(s),W.w['root_frame'][j])").body[0]);block.body.append(ast.parse('center_end()').body[0])
+ ast.fix_missing_locations(q);body=ast.unparse(q);
+ if D is not None:(D/'generated-physical-word.py').write_text(body+'\n')
+ m.__dict__.update(dict(physical_id=physical_id,move=move,physical_add=physical_add,center_begin=center_begin,center_end=center_end,ZERO=ZERO,FULL=FULL))
+ exec(compile(q,'<portable527 physical word>','exec'),m.__dict__);local=m.replay('token');assert center is None
+ for i in range(v):move(i,FULL)
+ for t in range(v):move(v+t,caps[t])
+ for j,r in enumerate(regs):move(2*v+j,FULL)
+ assert count==scalar_count and projection.hexdigest()==scalar_sha and coefficients==scalar_coeff
+ assert len(centers)==24 and local['center_reads']==5280
+ result=dict(status='PASS_PHYSICAL527_EXACT_SCALAR_PROJECTION_AND_MONOTONE_FRAME_TAGS',scalar_source_sha256=sha(source.encode()),scalar_projection_sha256=scalar_sha,tagged_scalar_sha256=tag_hash.hexdigest(),weighted_scalar_events=count,coefficient_histogram=dict(coefficients),categories=dict(categories),category_names=category_names,physical_registers=n,independent_dirty_registers=len(regs),source_heads=len(m.borrow),positive_rank_moves=sum(hist.values())-24,copied_centers=len(centers),copied_center_blocks=centers,center_scatter_reads=local['center_reads'],paid_histogram=dict(sorted(hist.items())),paid_rank_mass=sum(k*v for k,v in hist.items()),initial_independent_entrances=dict(Counter(C.dimf[initial_state[2*v+j]]for j in range(len(regs))if C.dimf[initial_state[2*v+j]])),used_frames=[dict(frame_id=f,dimension=C.dimf[f],basis_sha256=sha(json.dumps(C.B[f],separators=(',',':')).encode()))for f in sorted(used_frames)],input_pins=dict(source_pins or {}),physical_emitter_sha256=sha(Path(__file__).read_bytes()))
+ 
+ if D is not None:
+  (D/'physical.json').write_text(json.dumps(result,indent=2)+'\n')
+  (D/'records.bin.gz').write_bytes(gzip.compress(records.tobytes(),mtime=0))
+ print('PASS PHYSICAL',count,'records',len(records)//6,'rankmass',result['paid_rank_mass'],'tagged',result['tagged_scalar_sha256'],flush=True)
+ 
+ return dict(context=m.__dict__,W=W,C=C,records=records,physical=result,initial_state=initial_state,ZERO=ZERO,FULL=FULL,result=result,scalar_result=scalar)
diff --git a/research/five-stage-source527-banks/finite_check.py b/research/five-stage-source527-banks/finite_check.py
new file mode 100644
index 0000000000..bcc43d1ea4
--- /dev/null
+++ b/research/five-stage-source527-banks/finite_check.py
@@ -0,0 +1,124 @@
+"""Finite bill for the literal60-replica bankedfive-stage527word.
+Prepared with substantial OpenAI Codex assistance; Apache-2.0.
+Retains PR234exact-tape/prime/ordinary-leaf hypotheses, and pays bankroutes.
+"""
+from collections import Counter
+from fractions import Fraction as Q
+from hashlib import sha256
+from pathlib import Path
+import importlib.util,json
+HERE=Path(__file__).resolve().parent
+THEOREM_PINS={
+ 'proof/three-stage-cover-bit.tex':'47123a756f823c83a0d48cca2bf5add6472cd88410b13aed345e70c8834fb511',
+ 'proof/three-stage-cover-rows.tex':'6ad97a5cd7d6aecf978a81226e1c2763d37840caca130c61f3bf5c4ac111c5c8',
+ 'proof/copied-centers-lemma.tex':'d0ce6d3d504996aadb098227885224e48df46101daec8893523a0a8b7364d8fe'}
+def ceilq(x):return -((-x.numerator)//x.denominator)
+
+def cutoff_log2(delta,C):
+    assert 0<delta<=1 and isinstance(C,int) and C>=1
+    return max(1,ceilq(36/delta**2),ceilq(2*(4+(C-1).bit_length())/delta))
+
+def hist(x):return Counter({int(k):v for k,v in x.items()if v})
+
+def route_template():
+    """Recompute the actual 48-dimensional rational route, not its receipt."""
+    h=24
+    I=lambda n:[[Q(i==j)for j in range(n)]for i in range(n)]
+    q=[int(i<3)for i in range(h)]
+    P=[[Q(q[i])*(Q(q[j])-Q(1,3))/2 for j in range(h)]for i in range(h)]
+    K=[[Q(i==j)-P[i][j]for j in range(h)]for i in range(h)]
+    rho=[P[i]+K[i]for i in range(h)]+[K[i]+P[i]for i in range(h)]
+    a=[r[:]for r in rho];factors=[]
+    for j in range(2*h):
+        p=next(i for i in range(j,2*h)if a[i][j])
+        if p!=j:a[j],a[p]=a[p],a[j];factors.append(('swap',j,p,None))
+        if a[j][j]!=1:
+            c=1/a[j][j];a[j]=[c*x for x in a[j]];factors.append(('scale',j,None,c))
+        for i in range(2*h):
+            if i!=j and a[i][j]:
+                c=-a[i][j];a[i]=[x+c*y for x,y in zip(a[i],a[j])];factors.append(('add',i,j,c))
+    assert a==I(2*h)
+    def replay(omit=None):
+        a=I(2*h)
+        for z in reversed(range(len(factors))):
+            if z==omit:continue
+            kind,i,j,c=factors[z]
+            if kind=='swap':a[i],a[j]=a[j],a[i]
+            elif kind=='scale':a[i]=[x/c for x in a[i]]
+            else:a[i]=[x-c*y for x,y in zip(a[i],a[j])]
+        return a
+    assert replay()==rho
+    assert replay(next(i for i,f in enumerate(factors)if f[0]=='add'))!=rho
+    counts=dict(Counter(f[0]for f in factors));denoms=sorted({c.denominator for kind,i,j,c in factors if c is not None})
+    assert counts==dict(scale=27,add=195,swap=44) and denoms==[1,2,3,6]
+    serialization=[[kind,i,j,None if c is None else str(c)]for kind,i,j,c in factors]
+    return dict(factors=len(factors),factor_counts=counts,denominators=denoms,
+        factor_program_sha256=sha256(json.dumps(serialization,separators=(',',':')).encode()).hexdigest(),
+        exact_inverse_reconstruction=True,missing_addition_rejected=True)
+
+def run(raw,physical,scalar,prime_result,math_result,banks,banked_result,global_result):
+ for name,h in THEOREM_PINS.items():assert sha256((HERE/name).read_bytes()).hexdigest()==h,('changed theorem',name)
+ bit=(HERE/'proof/three-stage-cover-bit.tex').read_text();rows=(HERE/'proof/three-stage-cover-rows.tex').read_text();copied=(HERE/'proof/copied-centers-lemma.tex').read_text()
+ for s in ('constants are independent of $w,n$ and ancestor values','absorbed by the complete $q^w$ target fiber','spectator counters','32m^2','only units are inverted'):assert s in bit
+ for s in ('W(w)^{D(n)}','padding factor is less than two','below $q^2$'):assert s in rows
+ assert 'Copying, pointwise reads and erasure are linear'in copied and'fixed work stream'in copied
+ assert raw['physical_R']==16587 and raw['v']==1760 and raw['h']==24 and raw['source_aliases']==527
+ assert scalar['forward']['event_sha256']==physical['scalar_projection_sha256']==global_result['scalar_projection_sha256']
+ assert physical['tagged_scalar_sha256']==global_result['local_tagged_sha256']
+ f,b=scalar['forward'],scalar['inverse'];n=2*raw['v']+raw['physical_R'];assert n==20107
+ for row in(f,b):
+  assert row['all_source_and_dirty_restored']and row['arbitrary_target_contents_preserved']and row['all_formal_columns']==n
+  assert row['weighted_additions']==physical['weighted_scalar_events']==786621
+  assert hist(row['coefficient_counts'])==hist(physical['coefficient_histogram'])
+  assert row['literal_unit_additions']==sum(c*v for c,v in hist(row['coefficient_counts']).items())==789261
+ assert f['max_intermediate_row_l1']==37631 and b['max_intermediate_row_l1']==3295796
+ assert raw['parity_transform']['both_reflected_ledgers'] and raw['parity_transform']['remaining_payload_additions']==786621
+ assert raw['parity_transform']['removed_even_add_count']==1783196
+ assert physical['copied_centers']==24 and len(physical['copied_center_blocks'])==24
+ assert all(z['rank']==22 and z['scatter_reads']==220 for z in physical['copied_center_blocks'])
+ assert hist(physical['paid_histogram'])==hist(raw['one_stage_helper_histogram_including_copies'])
+ assert prime_result['all_remaining_factors_below_2_power_80']and prime_result['physical_inventory_bound']
+ assert prime_result['independent_entrances']==2279 and prime_result['bundled_unique_bases']==231
+ assert hist(prime_result['entrance_rank_counts'])==hist(raw['auxiliary_entrance_rank_histogram'])
+ assert banks['conservative_extra_selector_calls']==525064856400 and banks['normalizer_factor_bound']==439
+ assert banks['physical_replicas']==60 and banks['literal_stock']==1304935 and banks['assignments']==4976100
+ assert banked_result['all_assignments_match']and banked_result['all_logical_helpers_bound']and banked_result['independent_completions_removed']==2279
+ m=120;v=raw['v'];R=raw['physical_R'];T=banks['physical_replicas'];stock=banks['literal_stock'];d=m*m;N=2*m
+ H=hist(raw['five_stage_profile']['histogram']);assert H==hist(global_result['paid_histogram'])
+ for a,cnt in hist(raw['auxiliary_entrance_rank_histogram']).items():assert H.pop(5*a)==cnt
+ literal=Counter({r:T*cnt for r,cnt in H.items()});E=sum(literal.values());mass=sum(r*cnt for r,cnt in literal.items())
+ assert m*stock-mass==264000 and max(literal)==50 and Q(max(literal),m)<Q(1,2)
+ assert hist(banked_result['literal_paid_histogram'])==literal and banked_result['literal_stock']==stock
+ mathematics=math_result['mathematics'];bp=mathematics['bit_profile'];assert bp['literal_stock']==stock and bp['literal_children']==E and bp['literal_rank_mass']==mass
+ weighted=T*(5*f['weighted_additions']+6*v);unit=T*(5*f['literal_unit_additions']+6*v)
+ assert weighted==T*global_result['total_weighted_additions']
+ # Preserve alloriginalroutes/bridges, paidcopies and finiteprimitiveovercharges.
+ # The actual physicalstock and60replicas enter this bill; normalizedmomentW doesnot.
+ J=T*(24*v+10*R);good=8*m*m+8;high=16*(d+1)**2;K=banks['conservative_extra_selector_calls']
+ payload=64*f['max_intermediate_row_l1']**3*b['max_intermediate_row_l1']**2;payloadbits=payload.bit_length();assert payload<2**104
+ fallback=6*N*(N-1)+3*N+6*(N-1);assert fallback==346314<32*m*m==460800
+ coefficient=unit+J*high+J*(stock+24)+E*good+E*(128*N**3)+16*m**3+120*T+E+K+1
+ assert coefficient<2**80 and 2*m**3*10**16<2**80 and 0<K<2**40
+ # Internal borrowed rows are restored; the external complex row reserve is
+ # independently regenerated in portable_complex and bound by verify.py.
+ assert stock+24 < 2**80
+ row_reserve=dict(internal_coefficient=d+1,halving_degree=1,
+     bound='14401*w*ceil(log2(n)) + inherited ordinary-leaf reserve',
+     restored=True,physical_stock_less_than_prime=True,external_complex_coefficient=20161)
+ assert row_reserve['internal_coefficient']==14401
+ route=route_template();spec=importlib.util.spec_from_file_location('bankedfinite_moment',HERE/'moment.py');mod=importlib.util.module_from_spec(spec);spec.loader.exec_module(mod)
+ c=Q(mathematics['bit_coarse']);lo,hi=mod.moment(dict(literal),m,stock,c,True)
+ assert list(map(Q,mathematics['bit_moment_interval']))==[lo,hi]
+ delta_tau=1-hi;delta1=1-Q(mass,m*stock)-Q(1,10**16)*32*m*E/stock;assert delta_tau>0 and delta1>0
+ chain=[Q(384599,10**10)];gaps=[]
+ for j in range(3):
+  old=chain[-1];a=(1-c)*c+c*old;ds=dict(atom=c-a,borrowing=1-a-c,remainder=1-a-c*(1-old),stock=1-c);assert old<a<c<1-a and min(ds.values())>0
+  delta=min(ds.values());L=cutoff_log2(delta,coefficient);assert Q(L)*delta**2>=36 and Q(L)*delta>=2*(4+(coefficient-1).bit_length())
+  gaps.append(dict(stage=j+1,gaps={k:str(x)for k,x in ds.items()},minimum_gap=str(delta),cutoff_log2_at_charged_coefficient=str(L),full_cutoff_rule='Apply cutoff_log2(delta, C_full), where C_full bounds the displayed coefficient times every inherited primitive, wrapper and preceding ordinary-level constant.'));chain.append(a)
+ assert list(map(Q,mathematics['ordinary_bootstrap_chain']))==chain
+ controls=[]
+ for name,test in [('zero cutoff gap',lambda:cutoff_log2(Q(0),coefficient)),('negative constant',lambda:cutoff_log2(Q(1,2),0))]:
+  try:test()
+  except AssertionError:controls.append(name)
+  else:raise AssertionError('bad finitecutoff accepted')
+ return dict(status='PASS_LITERAL_PARITY_FUSED_BANKED_FIVE_STAGE527_FINITE_BILL',theorem_pins=THEOREM_PINS,source_binding=dict(event=physical['scalar_projection_sha256'],tagged=physical['tagged_scalar_sha256'],global_program=global_result['program_sha256']),paid_inventory=dict(m=m,literal_stock=stock,physical_replicas=T,positive_rank_children=E,rank_mass=mass,weighted_additions=weighted,unit_expanded_additions=unit,bridge_additions=T*6*v,route_families=J,terminal_exchange_stream_movements=T*4*v,high_affine_factors=J*high,low_transposition_coefficient=J*(stock+24),generic_wrappers=E*good,matrix_preparation=E*128*N**3,copy_erase_episodes=120*T,fallback_per_child=32*m*m,extra_bank_selector_calls=K,normalizer_factor_bound=439,simultaneous_extra_work_streams=1,payload_prefix_upper=payload,payload_prefix_bits=payloadbits),q_power_bound=dict(coefficient=coefficient,low_matrix_power=14400,strict_upper='q^14401'),rational_route=route,row_reserve=row_reserve,moment=dict(coarse=str(c),tau=str(1-c),delta_tau_lower=str(delta_tau),delta_linear=str(delta1),fallback_added_in_full=True),recurrence=dict(bound='A*(n+n^(1-c)*w^(1-a_j))',A='C*(1+1/delta_linear+1/delta_tau)',halving_degree=1,one_level_histogram=True),bootstrap=dict(chain=list(map(str,chain)),ordinary=str(chain[-1]),gaps=gaps),rejected_controls=controls,scope='Actual60replicas/literalbankstock andallfixedbanknormalizerschargedbefore finiteordinaryleaf absorption. Inheritedweightedcompiler,commonancestorchart,restoredrow,routing,prime,recoveryandanalyticinterfacesremainconditional.')
diff --git a/research/five-stage-source527-banks/inputs/complex/LICENSE b/research/five-stage-source527-banks/inputs/complex/LICENSE
new file mode 100644
index 0000000000..261eeb9e9f
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/LICENSE
@@ -0,0 +1,201 @@
+                                 Apache License
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+                        http://www.apache.org/licenses/
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+   6. Trademarks. This License does not grant permission to use the trade
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diff --git a/research/five-stage-source527-banks/inputs/complex/NOTICE b/research/five-stage-source527-banks/inputs/complex/NOTICE
new file mode 100644
index 0000000000..63aa295646
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/NOTICE
@@ -0,0 +1,290 @@
+Walsh-Hadamard power saving (wht-power-saving-lean)
+Copyright 2026 Jacob Sussman
+
+Repository: https://github.com/jacobalansussman/wht-power-saving-lean
+
+This repository is not a project of OpenAI. OpenAI did not write, review or
+endorse it. The package name "OAI" and the namespace "OAI.PowerSaving" are
+those of openai/math and are kept only because the new files extend that
+development.
+
+This repository is not a project of CrocSwap/integer-mult-bounds either. Its
+authors did not write, review or endorse it. Section 6 says what is taken
+from their repository.
+
+This repository contains files from, and files derived from, the repository
+"openai/math" (https://github.com/openai/math), family 130, "Exact Fourier
+transforms below n log n", at commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb.
+That repository is published under the Apache License, Version 2.0. A copy of
+the licence is in the file LICENSE. The upstream repository has no NOTICE file
+and its licence text names no copyright holder; the files are attributed here
+to the authors of openai/math (OpenAI).
+
+1. Unmodified upstream files.
+   The 89 files under OAI/Computability/ are byte-for-byte copies of the files
+   with the same paths under lean/ in openai/math at the commit above. They
+   have not been changed. Modified copies of some of them are under
+   Work/Fourier/ and are listed in section 7.
+   One more file of openai/math is here unchanged, outside the Lean build:
+   third-party/openai-math/lean/ComparatorChallenges/UniformFourier.lean is a
+   byte-for-byte copy of lean/ComparatorChallenges/UniformFourier.lean at the
+   commit above (the statement of OpenAI's Fourier challenge). It is included
+   so that the modified copy named in section 2 can be compared with it.
+
+2. Files that contain upstream text.
+   - Work/CarrierCheck/ChallengeB2Ke16x.lean,
+     Work/CarrierCheck/ChallengeB2Ke16xR.lean,
+     Work/GCert/Data/ChallengeB2Gp193x.lean and the text template
+     tools/gen/templates/ChallengeB2L16x.lean.txt each contain, as their
+     lines 22-270, lines 9-257 of
+     lean/ComparatorChallenges/UniformFourier.lean of openai/math (the RAM cost
+     model), unchanged. Everything else in these files was written for this
+     repository.
+   - Work/Fourier/UniformFourierChallenge.lean is a modified copy of
+     lean/ComparatorChallenges/UniformFourier.lean of openai/math (the
+     statement of OpenAI's Fourier challenge, 336 lines). Changes: a comment
+     was added at the top; the exponent 1 - 1/(10^(13:ℕ)) was replaced by
+     1 - 7474546/(10^(10:ℕ)); seven names received the suffix "Z"
+     (decimalExponent, decimalTime, TimeBounds, DFTGoal, ConvGoal,
+     transform_main, convolution_main). Everything else in the file is the
+     upstream text, unchanged. See section 7.
+   - Work/Scratch/Engine.lean and Work/Block/Engine.lean contain copies of
+     proofs of openai/math, changed for this repository: sections 1, 2 and 5
+     of Work/Scratch/Engine.lean are upstream proofs with the batch layout
+     renamed, and both files contain a copy of the private upstream theorem
+     "hills_project" (lean/OAI/Computability/FourierTransform/
+     TensorProgram.lean) with its envelope generalised. The comments in the
+     two files mark these places.
+   - lakefile.lean is adapted from lean/lakefile.lean of openai/math: the same
+     package name, toolchain, package option and Mathlib pin; every other
+     dependency was removed and the libraries of this repository were added.
+     lean-toolchain is a byte-for-byte copy of lean/lean-toolchain.
+     lake-manifest.json lists Mathlib and its dependencies at the revisions of
+     lean/lake-manifest.json.
+   - RELATED-WORK.md reproduces the BibTeX entries of two manuscripts of
+     openai/math.
+
+3. New files that build on upstream proofs.
+   The other files under WHTCheck/ and Work/ were written for this
+   repository. They import the upstream files, and several of them generalise
+   or adapt proofs of upstream files (for example Work/Block/Recursion.lean,
+   which says so in its header). They are changes and additions made for this
+   repository and are not part of openai/math.
+
+4. How the new files were made.
+   Jacob Sussman made the new files on 2026-10-08 and 2026-10-09 by directing
+   a team of AI agents (Claude). The agents wrote the new Lean files, the
+   certificate data and the generator scripts. See README.md, section 4.
+
+5. Ideas taken from other public work.
+   The network of the first result uses ideas published by others, among them
+   the helper circuit "NStar3" (Swapnil-jain/integer-mult-kappa, which credits
+   CrocSwap pull request #36 and Paureel for parts of it), its transfer to
+   family 130 (shea256/fourier-transform-below-nlogn), and "carrier links" and
+   "whole-residual batching" (pull requests to CrocSwap/integer-mult-bounds).
+   No file of Swapnil-jain/integer-mult-kappa or of
+   shea256/fourier-transform-below-nlogn is included here. One comment in
+   Work/Scratch/Engine.lean quotes three counts reported in
+   Swapnil-jain/integer-mult-kappa and names that source. The general frame
+   lemma that Work/GFrame proves is stated in
+   notes/general-clifford-frames.tex of CrocSwap/integer-mult-bounds; its Lean
+   proof was written for this repository. The credits, with links, are in
+   RELATED-WORK.md.
+
+6. Data derived from CrocSwap/integer-mult-bounds (second result).
+   The helper circuit of the second result is the paired-cube circuit of the
+   repository "CrocSwap/integer-mult-bounds"
+   (https://github.com/CrocSwap/integer-mult-bounds), in the state of its pull
+   request #193. It is the design of that community: icekylinx (pull requests
+   #144 and #184), eumemic (#168, and #117 behind its modules), ikeboy /
+   Avi Eisenberg (#191 and #193), with an664 (#128) credited there for the
+   workspace-sharing principle. That repository is published under the Apache
+   License, Version 2.0 (the same licence text as the file LICENSE). Its
+   NOTICE file begins "A sharper exponent for integer multiplication /
+   Copyright 2026 Douglas Colkitt" and says that the community release "is
+   not solely Douglas Colkitt's work". A complete copy of that NOTICE file, as
+   it is at the head of pull request #193, is in
+   third-party/CrocSwap-integer-mult-bounds/NOTICE.
+
+   The data were taken from these files of that repository. They were loaded
+   as data by programs written for this repository; no program of that
+   repository was run, and none is included here.
+   - at the head of pull request #168 (eumemic),
+     4a3c769e5c5430e7114c4d3e099ff34664677f17:
+       references/paired-cube/sources/local_L1.json
+       references/paired-cube/sources/tmod_TE_TD_TB3_1_1_4_full_6.0617964e-4.json
+       references/paired-cube/sources/pmod_J0_full_6.0666810e-4.json
+       references/paired-cube/sources/qmod_climb3u_best.json
+       references/paired-cube/selected-module/matching-arcs.json
+       references/paired-cube/physical/frames.json
+   - at the head of pull request #193 (ikeboy),
+     187e1010ac8b259af8e9b5166f68b64bc27b4b47:
+       research/source-assisted-v4/data/physical-pairs.json
+
+   Files of this repository that contain data derived from them:
+   tools/certificate/gcert1-p11-pr193.json.gz and the Lean data modules
+   generated from it, Work/GCert/Data/Gen/P193*.lean and the files they import
+   from Work/GCert/Data/Gen/P193/; its block counts are also in the generated
+   modules Work/GCert/Data/P193Price.lean, RateB2Gp193.lean, RateB2Gp193x.lean
+   and InstB2Gp193.lean.
+   Changes made: the circuit was rebuilt from those data as one explicit,
+   renumbered list of additions with a table of frames, in the certificate
+   format "gcert/1" of this repository; its 1,980 in-place steps were written
+   as pairs of additions with rescaled coefficients; the Lean data modules
+   were generated from that list. No file of that repository is included
+   unchanged, except the copy of its NOTICE file named above.
+
+   Notices of that NOTICE file that pertain to these data, quoted:
+
+     Paired-cube and shared-core extension, 9 October 2026
+     Copyright 2026 icekylinx
+
+     Source-assisted common-frame extension (2026-10-09)
+     Copyright 2026 icekylinx. Apache-2.0.
+     Substantial dependency: eumemic PR168,
+     fd25adb7fbaa12ee761d02c733c54d1d2a7687ee
+
+   The full text of both notices, and the notices of the work they build on,
+   are in the copy named above. The credits, with links and with the words of
+   pull request #193 about its own sources, are in README.md ("Whose circuit
+   this is") and in RELATED-WORK.md, section 1.
+
+7. Modified copies of upstream files (third result).
+   The files listed below are copies of files of openai/math at the commit
+   above, changed for this repository. Each begins with a comment that names
+   the upstream file it is derived from and states what was changed. The
+   upstream originals under lean/OAI/Computability/ are in this repository
+   as well, unchanged (section 1), and the copies import them. The upstream
+   challenge file lean/ComparatorChallenges/UniformFourier.lean is in this
+   repository in both forms: unchanged under third-party/openai-math/
+   (section 1) and modified (section 2).
+
+   - Work/Fourier/ArbitraryLength.lean: modified copy of
+     lean/OAI/Computability/FourierTransform/ArbitraryLength.lean (224
+     lines; 159 of them unchanged in the copy). Changes: identifiers renamed
+     on 64 lines (18 names: 15 with the suffix Z, 3 with Z inserted); 1
+     import line replaced by 1; 15 comment lines added; no other change to
+     the code.
+   - Work/Fourier/Asymptotics.lean: modified copy of
+     lean/OAI/Computability/FourierTransform/Asymptotics.lean (187 lines;
+     112 of them unchanged in the copy). Changes: identifiers renamed on 74
+     lines (32 names: 26 with the suffix Z, 6 with Z inserted); 1 import
+     line replaced by 1; 15 comment lines added; no other change to the
+     code.
+   - Work/Fourier/Axioms.lean: written for this repository (not a copy of an
+     upstream file).
+   - Work/Fourier/ConvolutionProgram.lean: modified copy of
+     lean/OAI/Computability/FourierTransform/ConvolutionProgram.lean (100
+     lines; 77 of them unchanged in the copy). Changes: identifiers renamed
+     on 22 lines (10 names: 9 with the suffix Z, 1 with Z inserted); 1
+     import line replaced by 1; 15 comment lines added; no other change to
+     the code.
+   - Work/Fourier/Goal.lean: written for this repository; contains lines of
+     upstream files: of its 7 code lines, 6 are lines of
+     lean/OAI/Computability/FourierTransform/Goal.lean (original lines 35,
+     75-77, 79, 81): unchanged: its lines 25; with identifiers renamed: its
+     lines 21, 24, 26, 28, 30.
+   - Work/Fourier/Main.lean: modified copy of
+     lean/OAI/Computability/FourierTransform/Main.lean (151 lines; 90 of
+     them unchanged in the copy). Changes: identifiers renamed on 59 lines
+     (40 names: 35 with the suffix Z, 5 with Z inserted); 1 import line
+     replaced by 2; 24 comment lines added; other code lines changed: 1 of
+     the original, 1 of the copy (copy line 59).
+   - Work/Fourier/Seam.lean: written for this repository; contains lines of
+     upstream files: of its 122 code lines, 15 are lines of
+     lean/OAI/Computability/FourierTransform/TensorProgram.lean (original
+     lines 36-37, 95-96): unchanged: its lines 93-94, 113-114, 141-142,
+     163-164, 185-187, 194-196; with identifiers renamed: its lines 184; 3
+     are lines of lean/OAI/Computability/FourierTransform/TensorSaving.lean
+     (original lines 83-84, 114): with identifiers renamed: its lines 45-46,
+     59; 1 is a line of lean/OAI/Computability/FourierTransform/Main.lean
+     (original line 60): with identifiers renamed: its lines 213.
+   - Work/Fourier/SectorAlgorithm.lean: modified copy of
+     lean/OAI/Computability/FourierTransform/SectorAlgorithm.lean (224
+     lines; 162 of them unchanged in the copy). Changes: identifiers renamed
+     on 61 lines (17 names: 13 with the suffix Z, 4 with Z inserted); 1
+     import line replaced by 1; 15 comment lines added; no other change to
+     the code.
+   - Work/Fourier/SynchronizedAlgorithm.lean: modified copy of
+     lean/OAI/Computability/FourierTransform/SynchronizedAlgorithm.lean (126
+     lines; 94 of them unchanged in the copy). Changes: identifiers renamed
+     on 31 lines (12 names: 10 with the suffix Z, 1 with Z inserted, 1
+     other), the other being "canopy" to "Grove.canopy" (5 lines); 1 import
+     line replaced by 1; 16 comment lines added; no other change to the
+     code.
+   - Work/Fourier/TransformProgram.lean: modified copy of
+     lean/OAI/Computability/FourierTransform/TransformProgram.lean (146
+     lines; 94 of them unchanged in the copy). Changes: identifiers renamed
+     on 51 lines (21 names: 19 with the suffix Z, 2 with Z inserted); 1
+     import line replaced by 1; 15 comment lines added; no other change to
+     the code.
+   - Work/Fourier/UniformBounds.lean: modified copy of
+     lean/OAI/Computability/FourierTransform/UniformBounds.lean (183 lines;
+     115 of them unchanged in the copy). Changes: identifiers renamed on 67
+     lines (29 names: 25 with the suffix Z, 4 with Z inserted); 1 import
+     line replaced by 1; 15 comment lines added; no other change to the
+     code.
+   - Work/Fourier/UniformFourierChallenge.lean: modified copy of
+     lean/ComparatorChallenges/UniformFourier.lean (336 lines; 328 of them
+     unchanged in the copy). Changes: identifiers renamed on 7 lines (7
+     names: 7 with the suffix Z); 12 comment lines added; other code lines
+     changed: 1 of the original, 1 of the copy (copy line 281).
+   - Work/Fourier/WorkingCompiler.lean: modified copy of
+     lean/OAI/Computability/FourierTransform/WorkingCompiler.lean (153
+     lines; 104 of them unchanged in the copy). Changes: identifiers renamed
+     on 48 lines (14 names: 14 with the suffix Z); 1 import line replaced by
+     1; 15 comment lines added; no other change to the code.
+   - Work/Fourier/WorkingPreparation.lean: modified copy of
+     lean/OAI/Computability/FourierTransform/WorkingPreparation.lean (212
+     lines; 156 of them unchanged in the copy). Changes: identifiers renamed
+     on 55 lines (30 names: 28 with the suffix Z, 2 with Z inserted); 1
+     import line replaced by 1; 15 comment lines added; no other change to
+     the code.
+   - Work/Fourier/WorkingTransform.lean: modified copy of
+     lean/OAI/Computability/FourierTransform/WorkingTransform.lean (119
+     lines; 91 of them unchanged in the copy). Changes: identifiers renamed
+     on 27 lines (9 names: 8 with the suffix Z, 1 with Z inserted); 1 import
+     line replaced by 1; 15 comment lines added; no other change to the
+     code.
+
+   Changes made to the eleven copied proof files: every declaration received
+   the letter "Z" in its name (as a suffix of its own name, or of the name
+   of the declaration it belongs to), so that it does not collide with the
+   upstream declaration of the same name; the upstream exponent "alpha"
+   (1 - 2/10^11) and the bound "hills" defined from it were replaced by
+   "alphaZ" (1 - 7474547/10^10) and "hillsZ", which are defined in
+   Work/Fourier/Seam.lean, and the single use of the upstream theorem
+   "hills_program" by "hillsZ_program"; the five definitions of the upstream
+   statement that contain its exponent (decimalExponent, decimalTime,
+   TimeBounds, DFTGoal, ConvGoal) were replaced by the versions of
+   Work/Fourier/Goal.lean, whose exponent is 1 - 7474546/(10^(10:ℕ));
+   the import lines point to the copies; a header comment was added.
+      Two further changes: in the copy of Main.lean the same constant stands
+   once as the literal 2/10^11 and was replaced there too, and the two
+   final theorems of that copy received a documentation comment; in the
+   copy of SynchronizedAlgorithm.lean the name "canopy" was written out as
+   "Grove.canopy" in 5 places, because the short name is shadowed in the
+   larger environment of the copy. The script also renamed three names
+   inside upstream comments; the comments are otherwise the upstream ones
+   and describe the upstream network.
+   The proofs are those of openai/math. The copies were made by the script
+   tools/fourier/mkchain.py of this repository.
+
+   Work/Fourier/Goal.lean contains copies of those five definitions of
+   lean/OAI/Computability/FourierTransform/Goal.lean, with the exponent
+   replaced and the suffix "Z" on the five names.
+
+   Work/Fourier/Seam.lean was written for this repository. It repeats the
+   statement of the upstream theorem "hills_program" (TensorProgram.lean) and
+   the definition of "hills" (TensorSaving.lean) with the two names replaced;
+   its header says so.
+
+   The work of others that the third result follows (danadran01, shea256,
+   eumemic) is credited in README.md ("Whose reduction this is") and in
+   RELATED-WORK.md, section 1. No file of their repositories is included
+   here.
+
+Not included in this repository and fetched by the build: Mathlib
+(https://github.com/leanprover-community/mathlib4, Apache License 2.0) and its
+dependencies. The comparator (https://github.com/leanprover/comparator) is a
+separate tool and is not included.
diff --git a/research/five-stage-source527-banks/inputs/complex/gcert1-p11-pr193.json.gz b/research/five-stage-source527-banks/inputs/complex/gcert1-p11-pr193.json.gz
new file mode 100644
index 0000000000..a98026a72f
Binary files /dev/null and b/research/five-stage-source527-banks/inputs/complex/gcert1-p11-pr193.json.gz differ
diff --git a/research/five-stage-source527-banks/inputs/complex/retained/FINITE_BRIDGE.txt b/research/five-stage-source527-banks/inputs/complex/retained/FINITE_BRIDGE.txt
new file mode 100644
index 0000000000..c1a8843611
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/retained/FINITE_BRIDGE.txt
@@ -0,0 +1,141 @@
+FINITE ROUTER, ROW-STOCK, AND SEMANTIC BRIDGE FOR ROUND 13
+
+Prepared for icekylinx by GPT-6 Astra, 9 October 2026. Apache-2.0.
+
+This bridge applies to a completed, physically proved round-13 profile.
+It cannot make a numerically attractive but unconstructed profile valid.
+The local source/target flags, controlled injections, old-response
+subtraction, inverse cleanup, and retained alias boundaries are separate
+construction obligations. assemble_profiles.py records their role as
+proof dependencies and independently checks the exact arithmetic.
+
+1. The inherited uniform interfaces
+
+Retain the three completed-core cover of PR144/PR168 and all of its
+analytic and fixed-tape assumptions. The complex cover is O(m,2), m<=72,
+with order V<2^(m*m)<=2^5184. No new independent cover coordinates are
+introduced by common-frame scalar lifts or source-controlled local
+rewrites. Every stage still uses one completed invocation at each vertex.
+
+For a supplied profile with W0<10^7 roles per cover vertex, the full
+physical stock W=V W0 is less than2^5208. Its total rank s is less
+than mW<2^5215. The profile's actual proper-child maximum is retained;
+the selected complex maximum must be less than m/2. The script checks
+these inequalities rather than assuming that fewer auxiliary rows imply
+a better contraction or a shorter recursion.
+
+The bit word uses the same weighted q-adic frame interface and the same
+internally borrowed selectors as its pinned parent. New source controls
+must use existing nondegenerate source-meeting planes or separately
+certified new frames. All zero-rank adapters remain charged scalar work.
+The group-size factor is not omitted and is not inserted into an external
+polynomial row reserve. The inherited ordinary leaf saving is exactly
+384599/10^10. The bad-row allowance is1/10^16; the charged fallback per
+edge is32m^2. Its inequality2m^3/2^80<1/10^16 is checked explicitly.
+
+2. A deliberately loose finite scalar bound
+
+The pinned words and round-13 modifications are finite straight-line
+programs with fewer than2^40 stored nodes, roles, coefficients, and
+boundary entries. Scalar sum/difference lifts use actual invertible
+matrices and are executed pointwise at common frame representatives.
+For the selected dyadic rewrite, new elementary coefficients are
+dyadic; the inherited center scatters retain their fixed odd divisor3.
+No unrecorded new odd denominator may be included under this bridge.
+
+Fixed integer multipliers can be expanded into scalar doublings and
+additions; dyadic divisions can be expanded into halvings. The explicit
+low words and full-frame inverses are much smaller than2^4096 primitive
+scalar/movement groups per invocation. Even constructing their fixed
+old-response matrices by exact elimination is dominated by this bound:
+matrix order is below2^40, original finite-word coefficient bit lengths
+are below2^80, and the elementary elimination and coefficient expansion
+bounds are below2^1024. No input-size-dependent dense elimination is
+being performed. These are constants of the chosen finite construction.
+
+Consequently all3V invocation words, endpoint adapters, and finite role
+bookkeeping fit within K<2^9300 logical groups. The retained universal
+router is overcharged by a full W-role permutation for every group:
+
+    64 (m+1)^3 (K+1) (W+1)^2 < 2^20000.
+
+Choose the exact dominating scalar/router budget G=2^30000. This also
+dominates the primitive arithmetic required to realize the fixed scalar
+coefficients and source-controlled inverse cleanup. The router still
+uses a fixed number of tapes, with sequential reuse of row stock.
+
+3. Explicit semantic constants
+
+Choose the exact integer E=2^100000 and put B=s+E. Then
+
+    2 G W^2 + 8s + 4W + 4 + 32m < 2^40500 < E,
+    E <= B < 2^100001.
+
+For the actual maximum child r<m, the induction gap obeys
+
+    2B(m-r)-s-E >= s+E >= 1.
+
+Choose C0=2^210000 and C1=1. These dominate both the retained semantic
+requirement32m B^2 and2B+18. The reported strict_literal_gap=1 and
+induction_gap_lower=1 are proved lower bounds on the positive actual
+integer gaps; they are not claims that those gaps equal one.
+
+Use the inherited common dyadic grid with fixed3^(-K') exposure and
+retain its precision/rounding contract. No child receives an independent
+rounding allowance and no source-controlled injection is treated as an
+unpaid numerical operation.
+
+4. External rows and stopping
+
+The complex child maximum below m/2 gives halving degree1, while
+log2 W<5208<10000. Keep the deliberately conservative inherited bit
+coarse reserve9909 and ordinary-leaf reserve252. Thus the external
+row coefficient is bounded by
+
+    A=10000+9909+252=20161.
+
+Choose degree D=1000000 and suffix slope4D. The exact row gap is
+
+    D - (51/25)A > 0.
+
+The q-adic selector fields and source-control labels remain internally
+borrowed and restored. Increasing these finite overcharges changes
+constants and the all-size bridge, not the selected exponent saving.
+
+5. Exact supplier moments and balanced assembly
+
+For each actual histogram, assemble_profiles.py proves
+
+    sum_r n_r r exp(a log(m/r))/(mW0) + bad_allowance < 1
+
+using rational upper bounds, with upward rounding on the2^120 grid.
+The bit atom stop is the first10^(-12) grid point above
+a/(1+a-old), which implies both required strict toll inequalities.
+Its effective saving is(1-beta)a+beta*old. This stopping rule is the
+already reviewed PR158 edge rule, not a new local-word contribution.
+
+The final assembly uses the substantial PR168 balanced-prefix adapter,
+with eta=10^(-8) and phase stop10^(-9), retaining all47 strict
+constraints and all7 margins. The selected final kappa is rounded
+down on the10^(-10) grid below the minimum complete margin. The
+script checks every constraint exactly and reports whether the fixed
+target0.0009218338 is reached.
+
+This is a conditional bound of the retained theorem form
+
+    T(n)=O(n (log n)^(1-kappa)).
+
+The improvement ratio refers to kappa, the logarithmic exponent saving.
+It is not a measurement of practical multiplication throughput.
+
+Credits
+
+The base supplier, scalar/frame verification, and actual profiles are
+pinned to PR168 fd25adb7fbaa12ee761d02c733c54d1d2a7687ee, eumemic
+(Anthropic Claude assistance), building on icekylinx's PR144. The
+balanced prefix is RaD/hipotures; the semantic constraints are retained
+from the reviewed PR23 line; completed-core sharing is credited to
+an664 PR128; the atom-edge stop is credited to geckods PR158. Existing
+source notices retain their more detailed upstream attributions.
+The new source-controlled boundary construction, this conservative
+bridge, and its integration are GPT-6 Astra's round-13 work.
diff --git a/research/five-stage-source527-banks/inputs/complex/retained/compact-control-layout.tex b/research/five-stage-source527-banks/inputs/complex/retained/compact-control-layout.tex
new file mode 100644
index 0000000000..16f4d23347
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/retained/compact-control-layout.tex
@@ -0,0 +1,118 @@
+\paragraph{Rows and complete compact fields.}
+Let $m,W,s$ denote the complex network constants and keep the layer's global
+$p,d,K$ at every node. Put
+\[
+ G=4\lceil\log_2p\rceil+6,\quad H=dG,\quad
+ q_F=\left\lceil\frac{2H}{K}\right\rceil,\quad
+ q_B=\left\lceil\frac{H}{K}\right\rceil,
+\]
+\[
+ q_0=\lceil\log_2W\rceil\lceil\log_m(2d)\rceil,
+ \quad k_0=\lceil\log_m d\rceil,\quad Q_0=q_0+q_F+q_B.
+\]
+If $D\le Q_0$, apply all $D$ selected kernels individually. Otherwise,
+process individually the first $q_0+q_F$ and last $q_B$ selected axes.
+This uses existing coordinates only. The remaining active axes are a
+consecutive middle block of length $D-Q_0$.
+
+Regard the first $q_0$ chunks as the row index. Immediately after it, the
+next $q_F$ chunks have at least $2H$ bits. Reinterpret these consecutive
+bits as two $H$-bit front fields and a spectator remainder; a remainder
+of length zero is omitted. The last $q_B$ chunks similarly supply one
+$H$-bit back field and a spectator remainder. These reinterpretations move
+no records and preserve the total address space. For a call with $f$ axes
+per slot, carve $nG=(f-1)G\le H$ consecutive bits from each of these
+fields; collapse their unused bits into spectators. The resulting field
+list has a fixed number of entries. Both front fields precede all active
+slots and the back field follows them, so the compact-control proposition
+applies to every ordered pair of slots. For $f=1$ no compact fields are used.
+
+The original row range has length $R_{\rm row}=2^{q_0K}\ge W^{k_0}$.
+For each original prefix, pad this row range to the next multiple of
+$W^{k_0}$ by adding zero rows. A row includes the entire complete front
+fields, active middle block, back field, all spectator coordinates and
+polynomial record suffix. Padding adds whole such rows, so increases total
+volume by at most two and does not change any within-row range.
+
+At a node write the row index as $u=Wg+w$, $0\le w<W$, and send the entire
+row to role $w$. The index $u$ consists solely of the first row region:
+\emph{no compact-field bit is included in it}. Each role therefore has
+the same complete within-row address set and exactly $1/W$ of the parent
+volume. At depth $j$ the remaining row count is divisible by $W^{k_0-j}$.
+This invariant survives further splits, padding, and reassembly. Pointwise
+gates on different roles have aligned $g$ and every remaining coordinate.
+All compact operations restore the temporary fields before a scalar gate,
+recursive child, or returned result uses that layout.
+
+The fixed-tape scheduler is unchanged: park only the other role streams at
+the active stack top, execute the child on a complete role stream, and restore
+the parked streams on return. Every child has its own current logical volume;
+copying, parking, rewinds and clearing cost a fixed multiple of that volume.
+The constant number of compact operations per edge use fixed work areas.
+Descriptors include the global $p,d,K,G,H$ and the carved field lengths;
+all have $O(p)$ bits and there are a fixed number of fields. The existing
+record suffix absorbs polynomial setup at every node and per record.
+
+Each completed invocation acts as the desired $C$ tensor on its active axes
+and as identity on the reserved and spectator coordinates. It therefore
+commutes with the earlier individual kernels, including those on bits now
+interpreted as compact fields. Restoring the row permutation returns the
+result to each original row separately. A padded zero row is consequently
+zero at completion and can be deleted exactly as before. Neither temporary
+field contents nor intermediate scratch payloads were assumed to be zero.
+
+The number of individually processed axes is at most $Q_0$, including the
+fallback $D\le Q_0$. For large $p$, $K\ge d^c/2$ and
+\[
+ q_F+q_B\le\frac{3dG}{K}+2\le6d^{1-c}G+2.
+\]
+Thus preprocessing, padding and reservation cost
+\[
+ O\bigl(V(\log(2d)+d^{\max\{1-c,0\}}\log p+1)\bigr).
+\]
+All reserved axes lie within the original $D$ axes; in particular their
+total number never exceeds $d$. This fact also bounds their coefficient
+dependency depth by the same $8d$ allowance used in the guard proof.
+
+\paragraph{Unrolling the compact-control recurrence.}
+Fix rational $0<\beta<1$. Choose $0<\tau<1$ and
+$\max\{0,\log_m(s/W)\}<\sigma<1$. Stop when $e<d^\beta$ and apply
+the remaining kernels individually. With $F(e)$ denoting time per current
+logical volume, the exact row split gives
+\[
+ F(e)\le(s/W)F(e/m)+O(G^\tau e^\tau+1),\qquad
+ F(e)=O(e)\quad(e<d^\beta).
+\]
+Here $G$ and the compact reservations are global parameters, not rebuilt
+at each node. This is why no factor $K^\tau$ appears in the overhead.
+Repair costs are already included in the node overhead.
+
+Write $b=s/W$. At level $j$ the internal contribution is bounded by
+$O(G^\tau e^\tau(b/m^\tau)^j)$. Summing through the stopping depth gives
+\[
+ O(G^\tau\log(2d)d^\chi),\qquad
+ \chi=\tau+(1-\beta)\max\{\sigma-\tau,0\}.
+\]
+Indeed if $\sigma\le\tau$ the root bounds every summand; otherwise the
+last internal level costs at most the root bound times a fixed constant
+and $d^{(1-\beta)(\sigma-\tau)}$. The additive $1$ is absorbed by
+$G^\tau e^\tau$ at internal nodes. The leaf contribution is at most
+$O(d^{\sigma+\beta(1-\sigma)})$, by the original volume-weighted argument.
+
+Partition $D-Q_0$ into consecutive powers of $m$, using its base-$m$
+expansion. There are $O(\log(2d))$ pieces. Include their costs and the
+reservation cost to obtain, per original volume,
+\[
+ O\!\left((\log p)^3
+ [d^\chi+d^{\sigma+\beta(1-\sigma)}+d^{\max\{1-c,0\}}+1]\right).
+\]
+We used $\log d=\Theta(\log p)$ and $G=O(\log p)$ for each fixed
+$d=\Theta(p^\epsilon)$ family. Any fixed rational exponents satisfying
+\[
+ \max\{\tau,\sigma,\chi\}<\lambda<\lambda'<1,\qquad
+ \max\{\sigma+\beta(1-\sigma),1-c,0\}<\lambda'
+\]
+therefore give the complete $O(Vd^{\lambda'})$ layer bound. The strict gaps
+absorb the logarithms. The small-$D$ fallback obeys the same bound.
+For fixed rational $\beta=u/v$, implement stopping by the integer test
+$e^v<d^u$; its polynomial descriptor cost is already included.
diff --git a/research/five-stage-source527-banks/inputs/complex/retained/compact-control-movement.tex b/research/five-stage-source527-banks/inputs/complex/retained/compact-control-movement.tex
new file mode 100644
index 0000000000..06b6a9ef24
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/retained/compact-control-movement.tex
@@ -0,0 +1,167 @@
+\subsection{Compact dirty controls for selected-bit additions}
+\label{sec:compact-control-movement}
+The following replacement for the wide-slot packed operation moves only
+compact control fields. Temporary address fields have arbitrary contents;
+they are complete ranges of existing address bits, not newly allocated
+independent coordinates. Their availability in every recursive call is
+proved separately below.
+
+\begin{lemma}[Controlled rotations with record-paid offset work]
+\label{lem:record-paid-rotation}
+Consider $M$ records of $R$ bits on a complete rectangular address set, of
+volume $V=MR$. Its fixed number of field lengths and controls have at most
+$A=\lceil\log_2(2V)\rceil$ bits each. A binary target field follows all its
+controls. Suppose its rotation offset, independent of that target, is
+computable from the controls and descriptors in $O(A^C)$ tape steps for
+some fixed $C$. Then the rotation can be performed on fixed tapes in
+$O(V+MA^{C'})$ time for a fixed $C'$. In particular, this is $O(V)$ when
+$A=O(p)$ and $R$ exceeds every fixed power of $p$.
+\end{lemma}
+\begin{proof}
+Fix the prefix before the target. If its range is $Q$ and each target block
+has $B$ suffix bits, rotate by $a$ through the two-piece split at $Q-a$ and
+merge in the opposite piece order. This is the controlled-rotation
+construction from the ordered-affine streaming lemma. Each data bit is
+copied a bounded number of times. There are at most $M$ prefix fibers, since
+every fiber contains at least one complete record. At each fiber, compute
+the offset, initialize its two piece lengths, and reset the finite collection
+of counters; recompute the offset at the merge. Ordinary binary arithmetic
+and descriptor copies cost a fixed polynomial in $A$ per fiber. The controls
+can be wider than the target: their lengths are bounded by $A$, not by the
+target width. Local suffix-length copies and ripple counters are charged as
+in the original streaming proof, so no growing descriptor is scanned per
+payload bit. Two piece tapes, output and fixed arithmetic work tapes suffice;
+rewinds and clearing are over the regions just written. This gives the stated
+bound. Finally $MA^{C'}/V=A^{C'}/R=o(1)$ in the specified record regime.
+\end{proof}
+
+Let $x,y$ be disjoint slots of $fK$ bits, with corresponding selected
+positions $j_i=\rho+iK$, $0\le i<f$, $0\le\rho<K$. Put $n=f-1$.
+For $f=1$ use the elementary two-bit XOR. Otherwise fix a positive integer
+$G$, put $B=2^G$, and assume two disjoint complete $nG$-bit fields $u,t$
+precede both slots and a third such field $b$ follows both slots. Arbitrary
+spectator fields may intervene. Each compact field has $n$ radix-$B$ digits.
+The selected-bit operation will preserve these three entire fields.
+
+\paragraph{Earlier source.}
+Suppose $x$ precedes $y$. The unused field $u$ is a spectator. For an integer
+target segment $v$, a bit $z$, and an arbitrary integer temporary $w$, perform
+\[
+ v\gets v+2zw,\qquad w\gets w+(v\bmod2),\qquad
+ v\gets v+z(1-2w),\qquad w\gets w-((v\bmod2)\mathbin\oplus z).
+\]
+Writing the original parity as $a$, the first update leaves it unchanged.
+The final values are $v+z(1-2a)$ and the original $w$. Thus this toggles
+only the parity of $v$ when $z=1$, and restores the dirty temporary.
+
+Apply these four updates simultaneously, with $z_i=x_{j_i}$, to the low
+$n$ selected segments of $y$ and the digits $t_i$ of $t$. A target-slot
+update is one modular rotation of $y$, whose offsets are respectively
+\[
+ \sum_{i<n}2x_{j_i}t_i2^{j_i},\qquad
+ \sum_{i<n}x_{j_i}(1-2t_i)2^{j_i} \pmod{2^{fK}}.
+\]
+Both control fields precede $y$. For a temporary update, swap $t$ with $b$,
+rotate the back field by the packed selected bits of the current $y$, and
+swap back. For unloading, use minus the packed bits
+$y_{j_i}\mathbin\oplus x_{j_i}$. These controls precede the back field.
+There are four width-$nG$ swaps and four rotations. The arbitrary original
+contents of $b$ return after each pair of swaps.
+
+\paragraph{Later source.}
+If $x$ follows $y$, first execute the earlier-source construction with control
+bits $u_i\bmod2$. Load the selected bits of $x$ into $u$ by swapping $u,b$,
+rotating the back field by their packed radix-$B$ representation, and swapping
+back. Repeat the earlier-source construction with the new parities of $u$,
+then unload the selected bits of $x$ from $u$ by the inverse load.
+When the digit loads do not overflow, the two control parities differ by
+exactly $x_{j_i}$. The resulting target update is the required XOR, with
+$u,t,b$ restored. This version uses twelve compact swaps and ten rotations.
+Every offset depends on a fixed number of fields preceding its target.
+
+\paragraph{Guards and deterministic repair.}
+Temporarily omit the highest selected position. The low $n$ segments
+$[j_i,j_i+K)$ are disjoint and fit inside $y$, including when $\rho=K-1$.
+Write each initial segment as $2g_i+a_i$, $a_i\in\{0,1\}$.
+Declare an address bad if a $t$ digit is $B-1$, or a target guard fails
+\[
+ 2B\le g_i<2^{K-1}-2B.
+\]
+For a later source also declare it bad if any $u$ digit is $B-1$.
+On a good address each temporary load stays within its digit, and the target
+displacement is at most $2B$ in either execution of the four-step identity.
+The conservative guard interval therefore prevents every carry or borrow
+between the wide segments. After each completed identity the guard is
+unchanged. Induction through the later-source sequence proves the desired
+map on the good set, including restoration of every temporary and every
+unselected address bit.
+
+Uniform complete ranges give the common bound
+\[
+ \delta=\min\{1,n(2\cdot2^{-G}+8\cdot2^{G-K})\}
+\]
+on the bad fraction; for an earlier source the first factor 2 can be omitted.
+The ideal map $T$ changes only the selected target parities, so it preserves
+the bad set. The actual sequence $S$ is a bijection on the entire rectangle:
+each rotation is independent of its target and all swaps are bijections.
+Since $S=T$ off the bad set, $S$ preserves that set too. Thus $TS^{-1}$ is
+supported there, exactly as in the original exceptional-stream repair.
+
+Extract the bad records, retaining marked holes in the full stream. For each
+extracted address $q$, reverse the swaps and modular rotations to compute
+$S^{-1}q$, then compute $T(S^{-1}q)$. All fields and descriptors have $O(A)$
+bits. A constant number of offset calculations each scan at most $n\le A$
+digits and combine shifted integers of $O(A)$ bits. Integer arithmetic,
+mixed-radix decoding, inverse offsets and the destination key can consequently
+be computed in $O(A^3)$ per record. Sort only the extracted records by their
+$O(A)$-bit destination keys, using stable binary radix passes, then reinsert
+them in their marked holes. Keys are distinct and fill precisely those holes
+because $TS^{-1}$ permutes the bad set. This uses fixed tapes and costs
+\[
+ O(V+MA^3+\delta MA(R+A)).
+\]
+Every inverse offset is evaluated on the current inverse controls; modular
+overflow on bad addresses is handled exactly, rather than treated as an
+integer identity. Finally, supply the omitted highest selected position
+through the elementary two-bit XOR in $O(V)$ time.
+
+\begin{proposition}[Compact-control selected-bit addition]
+\label{prop:compact-selected-addition}
+Under the stated complete-field hypotheses, take
+\[
+ G=4\lceil\log_2 p\rceil+6,\quad f\le p,\quad A=O(p),
+ \quad K/\log p\longrightarrow\infty,
+\]
+and let $R$ exceed every fixed power of $p$. The exact selected-bit row
+addition, with all temporary fields restored, costs
+\[
+ O\bigl(V((f\log p)^\tau+1)\bigr)
+\]
+on a fixed number of tapes, for either order of the two slots. Constants and
+cutoffs are uniform within each fixed parameter family of the layer theorem.
+\end{proposition}
+\begin{proof}
+Use the retained arbitrary-width chunk-swap primitive for the at most twelve
+swaps, each of width $nG$. Its padding, cleanup and all spectator intervals
+are covered by that interface. For every rotation, apply
+Lemma~\ref{lem:record-paid-rotation}; its offset arithmetic is polynomial
+in $A$. No new record volume is introduced by describing the compact fields.
+For $\ell_p=\lceil\log_2 p\rceil$, eventually
+$K\ge G+4\ell_p+10$. Then
+\[
+ \delta\le \frac{5}{128p^3},\qquad
+ \frac{A^3}{R}=o(1),\qquad
+ \delta A(1+A/R)=O(p^{-2}).
+\]
+The repair is therefore $O(V)$ per call, including predicate scans and key
+arithmetic. The elementary highest-bit operation costs $O(V)$. All calls
+reuse their fixed work tapes and finish in the declared physical field order.
+\end{proof}
+
+\paragraph{Application to a fixed binary change of basis.}
+The original phase-frame construction expresses each binary basis change
+as a fixed number of row additions on $m$ slots of $f=e/m$ axes. Use the
+two slots of each addition as $x,y$ and the reserved front/back compact
+fields as controls. The complete basis change and its inverse cost
+$O(V((e\log p)^\tau+1))$. The reservations below guarantee their complete
+ranges, independently of which slots are involved, at every recursive node.
diff --git a/research/five-stage-source527-banks/inputs/complex/retained/copied-centers-assembly.tex b/research/five-stage-source527-banks/inputs/complex/retained/copied-centers-assembly.tex
new file mode 100644
index 0000000000..b0c69166c3
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/retained/copied-centers-assembly.tex
@@ -0,0 +1,118 @@
+\subsection{Selected semantic guard and product rows}
+For each local producer retain
+\[
+ g_h=4(c_h+v_h)+10v_h+4hv_h+4h^2+8h+8.
+\]
+The two-stage literal scalar count includes two additional groups per
+copied center and one endpoint-addition group per data pair:
+\[
+ G=N_c+2v(g_{28}+2\cdot28)=4793351472.
+\]
+These are local scalar charges; the recursive copied transformations
+are already included in $s_c$. Set
+\[
+\begin{aligned}
+ E&=64(W_c+m_c+G+1)^3=9696565176891520488192506033216,\\
+ \mathcal E&=2GW_c^2+8s_c+4W_c+4+32m_c\\
+           &=2771683137698171397039558468<E,\\
+ B_{
m guard}&=s_c+E,\qquad C_0=32m_cB_{
m guard}^2.
+\end{aligned}
+\]
+A completed $u$-axis child has coefficients in $2^{-u}\mathbb Z[i]$
+and absolute row sum at most $2^u$. Its internal excess stays local;
+other streams, including originals of active copies, are parked.
+No return rounding or encoding change is performed. A copied input
+has the same incoming magnitude/grid bound as its original, and its
+completed output obeys the same semantic bound. Consequently, with
+$f=\lfloor e/m_c\rfloor$,
+\[
+ A(e)\le A(756f)+s_cf+E\le2B_{
m guard}e,
+\]
+using $2B_{
m guard}(m_c-756)\ge s_c+E$ and the direct-leaf bound
+$A(e)\le8e$. Prefix kernels, phases and normalization add the retained
+$18e$, which is less than $C_0e-2B_{
m guard}e$. Thus $C_1=1$,
+with the fixed fine-grid precision proof of RaD/PR23 unchanged.
+
+The actual least halving degrees and wire bitlengths are
+\[
+ (t_b,w_b)=(9,28),\qquad(t_c,w_c)=(20,30),
+\]
+where $m_j^{t_j}>2r_j^{t_j}$ for largest children $r_b=529,r_c=756$.
+At width $e$ reserve the product
+$Q=W_c^{D_c(e)}W_b^{D_b(e)}$, with
+$D_j(e)=t_j\lceil\log_2e\rceil$. Its coefficient is
+$9\cdot28+20\cdot30=852$, and
+\[
+ 2000>\frac{51}{25}\,852.
+\]
+The sufficient stock is therefore $p^{2000}$, supplied by the retained
+physically preceding transformed prefix and a single complete-row
+padding. Nested bit stock is restored before another complex child.
+Sequential copies introduce no independent row field. The supplementary
+suffix condition $b_{\rm in}^{1-\epsilon}>8000(\log_2b_{\rm in}+8)$
+is eventual for fixed $\epsilon<1$.
+
+\subsection{Exact conditional assembly}
+Use the analytic proof map in
+\path{notes/structured-bulk-assembly.tex}: arbitrary-source routing
+with active masking and paid repair scans; one rounded phase-cell
+matrix with preserved period cuts and downward-weight error bounds;
+persistent target halos, sequential overlap tapes and factor pages;
+Baker--Harman--Pintz prime packing; and the retained exact-recovery
+interface. Their full costs are unchanged. In particular the bulk
+bound retains the terms
+\[
+ O\!\left(Tp\left[p^\tau\operatorname{polylog}p+
+ d\operatorname{polylog}p+dp^\delta+w^2p^\delta+
+ dw^2p^\delta/L\right]\right)+n^{o(1)},
+\]
+with $L=2^{\lceil8\log_2p\rceil}$,
+$w^2=O(p^{1-r})$, and normalization exponent requiring
+$\epsilon+r<1$. These arguments, not a change of arithmetic model,
+justify the seven margins below.
+
+\begin{theorem}[Conditional fixed-tape multiplication bound]
+Under the retained analytic and fixed-tape interfaces of PR23/PR29
+and the preceding proof map, the selected copied-center networks give
+\[
+ T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad
+ \boxed{\kappa=\frac{384569}{10000000000}=3.84569\times10^{-5}}.
+\]
+\end{theorem}
+\begin{proof}
+Take the strict finite savings $a=a_b=384599/10^{10}$ and
+$b=a_c=717/10^7$, with $\tau=1-a$, $\sigma=1-b$, and put
+\[
+\begin{gathered}
+ \eta=10^{-8},\quad\beta=\tfrac14,\quad q=a(1-2\eta),\quad
+ c=q(1+\eta),\quad\epsilon=\frac{1-\eta}{1+c+q},\\
+ \lambda'=1-q,\quad\lambda=(\tau+\lambda')/2,\quad
+ g=\epsilon q,\quad r=(g+1-\epsilon)/2,\quad\delta=\eta/8.
+\end{gathered}
+\]
+The stopped complex leaf saving $(1-\beta)b=53.775\times10^{-6}$
+is strictly above $a$, so paid internal adapters use exponent $\tau$
+and both layer exponents lie below $\lambda'$. The exact certificate
+checks all 47 strict constraints of the retained semantic/bulk
+composition, using the new literal scalar charge and product-row gap.
+These include compact reservations, geometry, the linear guard,
+$\epsilon+r<1$, phase-cell/band separation, prime packing, short-record
+fallback and artificial-boundary allowances.
+
+The seven final savings are
+\[
+ 1-\epsilon(1+c),\quad a,\quad g,\quad a,\quad
+ \min(1-\epsilon-\delta,r-\delta),\quad
+ 1-\epsilon-\delta,\quad\epsilon.
+\]
+Their minimum is exactly $g$ and the prefix margin equals $g+\eta$.
+Exact rational comparison gives
+\[
+ g-\frac{384569}{10^{10}}>4.0746056580\times10^{-11}>0.
+\]
+These positive gaps absorb fixed logarithms. The all-size conclusion
+retains constructive finite setup, descriptor and catalogue domination,
+prime existence and analytic recovery as eventual thresholds. The
+strict bit and complex moments, copied-stream recurrence contracts and
+these assembly inequalities prove the claimed conditional bound.
+\end{proof}
diff --git a/research/five-stage-source527-banks/inputs/complex/retained/copied-centers-lemma.tex b/research/five-stage-source527-banks/inputs/complex/retained/copied-centers-lemma.tex
new file mode 100644
index 0000000000..8933aaf508
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/retained/copied-centers-lemma.tex
@@ -0,0 +1,103 @@
+\subsection{Scalar, frame and temporary-stream contracts}
+The retained source-copy, reversible mixer and scatter maps satisfy
+$JLV=I$. Over $\mathbb F_2$ the chronological word
+\[
+ L,\ J,\ L^{-1},\ V,\ L,\ J,\ L^{-1},\ V
+\]
+adds the input to the target and restores every arbitrary auxiliary.
+Over Gaussian dyadics use its corresponding old-value subtraction and
+new-value addition. A designated retained output has a distinct terminal
+output-use carrier: a matching may feed that carrier, but cannot merge
+two designated terminals or append a later middle-producer consumer.
+Its remaining uses before cleanup are its scatter reads, all of which
+leave its scalar value unchanged.
+
+Within one tensor invocation suppress the fixed future line and put
+\[
+ D_0=B\otimes F_h,\quad D_1=A\otimes F_h,\quad
+ A=B\mathbin{\perp}P,\quad D_U=D_0\mathbin{\perp}(P\otimes U),
+\]
+where $P$ and the suppressed future factor are one-dimensional and
+$U$ is a nondegenerate retained-center space of dimension $r$.
+Thus a local change of rank $d$ has ambient rank $d$.
+Write $\mathcal F_Q$ for the address frame: it is the rational partial
+swap $D_Q$ in the bit construction, and the binary phase $C_Q$ in the
+complex construction. Changing a copy from frame $Q_1$ to $Q_2$ means
+applying $\mathcal F_{Q_2}\mathcal F_{Q_1}^{-1}$ to its physical stream.
+Orthogonal backgrounds cancel in this quotient.
+
+Copies use Paureel's retained copy/transform/read/discard interface
+\cite{paureel,pr29}. A copy has precisely the original complete role
+stream, including all control and spectator ranges, and volume $V/W$.
+It is processed while inactive original streams are parked on the fixed
+depth-first stack tapes. Copying, pointwise reads and erasure are linear
+in that stream's volume. Sequential processing needs one additional
+fixed work stream, not an independently varying row coordinate.
+
+\begin{lemma}[Copied retained-center schedule]
+In either invocation orientation a designated retained center's
+physical rank profile can be changed from $z^r+z^h$ to
+$z^r+z^{h-r}$, with the same scalar action, original role endpoints,
+and arbitrary-scratch restoration. The rank-$r$ transform on the copy
+is included in the recursive call list.
+\end{lemma}
+\begin{proof}
+In the forward orientation the original center has scalar value $w$
+at frame $D_U$ after the middle mixer. Its old sequence was
+$D_U\to D_0\to D_1$: scatter at $D_0$, then cleanup at $D_1$,
+with ranks $r,h$.
+Keep the original at $D_U$, copy its physical stream, and transform
+only the copy to $D_0$, paying rank $r$. Supply every scatter read
+from that transformed copy while its data targets remain at their
+unchanged common frame $D_0$, then erase the temporary. The reads
+are exactly those of $w$ in the old schedule. Since they do not change
+$w$, move the original directly from $D_U$ to $D_1$ before its first
+cleanup incidence; this costs $h-r$. Every subsequent original gate
+therefore receives the same scalar value at its required frame.
+
+In the reverse complementary orientation the old sequence was
+$D_0\to D_1\to D_{U^\perp}$, with ranks $h,r$ and scatter at $D_1$.
+After the preceding all-$D_0$ operations, advance the original directly
+to $D_{U^\perp}=D_0\mathbin{\perp}(P\otimes U^\perp)$, paying
+$h-r$. Copy it and transform the copy to $D_1$, paying $r$.
+Use that copy for all early scatter reads, then erase it. The original
+already has the exact frame and unchanged scalar value required by the
+following inverse mixer. Terminal output-use separation ensures no
+intervening producer consumer has been bypassed.
+
+For partial swaps either direction of the rank-$r$ change is the same
+involution. For complex phases the forward copy uses $C_U^{-1}$,
+the direct original growth uses $C_{U^\perp}$, and the reverse copy
+uses $C_FC_{U^\perp}^{-1}=C_U$. These are the same charged ranks,
+implemented by the retained forward/inverse residual interface.
+All tensor backgrounds cancel as stated. Every original scalar
+incidence, final source-copy cancellation and auxiliary source/sink
+operator is unchanged. Thus the scalar identity and all endpoints
+remain exact on arbitrary initial auxiliary values.
+\end{proof}
+
+For the selected centers $r=h-1$, and there are $h$ distinct retained
+output uses per invocation. Their old histogram has rank mass
+$hR_h+2\ell_h$, where $\ell_h=h(h-1)$. The new histogram is exactly
+\[
+ H'_{h,1}=H_{h,1}+h,\qquad H'_{h,h}=H_{h,h}-h,
+ \qquad H'_{h,r}=H_{h,r}\quad(r\ne1,h),
+\]
+and hence $\sum_r rH'_{h,r}=hR_h+\ell_h$.
+Every old rank-$(h-1)$ residual remains as a paid transform of a copy;
+its projector and ordinary bit pivot profile are unchanged. Only the
+full-rank original transition becomes the rank-one complement.
+The physical role count $R_h$ is unchanged.
+
+For two factors, put $N=v_av_b$, $B_a=v_bR_a$, $B_b=v_aR_b$,
+$W=2N+B_a+B_b$ and $L=v_b\ell_a+v_a\ell_b$.
+All exterior, data-front and endpoint-correction calls remain in place.
+The resulting complete rank mass is
+\[
+ s=Wm-N+L,\qquad Wm-s=N-L.
+\]
+This follows by subtracting exactly $L$ from the retained two-stage
+physical mass $Wm-N+2L$. The $N$ rank-one endpoint-copy calls are
+separate and remain fully charged. Because center copies retain the
+same complete role-volume fraction $1/W$, the existing complete-row,
+integer-width remainder and fixed-tape recurrence proofs apply.
diff --git a/research/five-stage-source527-banks/inputs/complex/retained/endpoint-gauge-complex.tex b/research/five-stage-source527-banks/inputs/complex/retained/endpoint-gauge-complex.tex
new file mode 100644
index 0000000000..907a0ceda8
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/retained/endpoint-gauge-complex.tex
@@ -0,0 +1,225 @@
+\subsection{A normal form for every nondegenerate binary residual}
+Let $E$ have binary basis matrix $V$ and nonsingular Gram matrix
+$G=V^tV$. In dual Fourier coordinates put
+$q(z)=\operatorname{wt}(VG^{-1}z)\pmod4$.
+Its polarization is $2z^tG^{-1}w$. The same argument applies to mixed
+signs on orthogonal summands, since reversing a sign preserves this
+polarization. Write $S_q=\sum_z i^{q(z)}$. Completing the square gives
+\[
+\begin{aligned}
+K(x,y)&=2^{-r}\sum_z i^{q(z)}(-1)^{z^t(x+y)}\\
+ &=2^{-r}S_q\,i^{-q(Gx)}(-1)^{x^tGy}i^{-q(Gy)}.
+\end{aligned}
+\]
+The scaled Walsh kernel in the middle is a binary permutation of
+$H_0^{\otimes r}$, where
+\[
+ H_0=\tfrac12\begin{pmatrix}1&1\\1&-1\end{pmatrix}
+     =bSCS,\qquad b=(1-i)/2,\quad S=\operatorname{diag}(1,i).
+\]
+The remaining scalar $S_qb^r$ is a fourth root of unity. To verify
+this, decompose the nondegenerate quadratic module into odd
+one-dimensional and alternating two-dimensional blocks. Their Gauss
+sums are $1\pm i$ and $\pm2$, respectively; multiplying by $b$ and
+$b^2$ gives a fourth root in each case. This is a proof device, not
+an orthonormal-basis assumption on $E$.
+
+Consequently one rank-$r$ child suffices, with binary address changes,
+diagonal fourth-root phases, and one scalar fourth-root phase.
+On $f$ parallel columns it is one $C^{\otimes rf}$ child, the scalar
+raised to power $f$, and columnwise quadratic phases modulo four.
+The retained packed address operations evaluate these fixed formulas.
+These wrappers have modulus one and introduce no denominator loss.
+
+\subsection{The selected Gaussian-dyadic producer}
+Use paired triple exclusion with base threshold two, complete unsplit
+disjoint and pair-star sums, and sharing by actual ordinary support.
+For an ordinary support with at least two common points use the span
+of its triple indicators; these form an orthonormal common-pair star.
+Otherwise use the coordinate space of its union. Support inclusion
+preserves these labels: losing the common pair enters a containing
+coordinate space. Source lines use ordinary triple containment.
+
+Retain every full pair-star $P_{uv}=\sum_{w\ne u,v}X_{uvw}$ and form
+$B_u=\sum_{v\ne u}P_{uv}=2\sum_{T\ni u}X_T$.
+For even $h$ label these central additions by the nondegenerate
+hyperplane $E_u=\{x:\sum_{j\ne u}x_j=0\}$; every contributing star
+lies in it. Retain the global total $T$ with full-space label.
+The central correction at target $S$ is
+\[
+ \frac14\sum_{u\in S}B_u-\frac12T
+       =\sum_U\frac{|S\cap U|-1}{2}X_U.
+\]
+Adding half the disjoint sum and subtracting half the intersection-two
+sum gives exactly $X_S$. All coefficients are Gaussian dyadic.
+The transparent retained-output wrapper and oriented carrier compiler
+apply to these nested nondegenerate labels. The $h$ hyperplane centers
+lose $h(h-1)$ and the global center loses $h$, so loss is $h^2$.
+The selected counts, including all point centers, are
+\[
+\begin{array}{c|rrrrrr}
+h&v_h&c_h&q_h&|\mathcal M_h|&R_h&\text{loss}\\\hline
+32&4960&122512&19873&19241&123144&1024\\
+30&4060&99153&16271&15873&99551&900\\
+40&9880&252023&39561&37642&253942&1600
+\end{array}
+\]
+Their internal histograms satisfy $\sum_r rH_{h,r}=hR_h+2h^2$.
+They are distinct from the bit histograms.
+
+\subsection{Endpoint normalization and the complex characteristic}
+For nondegenerate $U$, Fourier phases give
+\[
+ C_U^2=X_{P_U\mathbf1},\qquad
+ C_{U^\perp}C_U^{-1}=C_FX_{P_U\mathbf1}.
+\]
+Put $v=P_U\mathbf1$. On coordinates in $v$, this endpoint has
+$C^{-1}$ instead of $C$. The identity $C^{-1}=-iZCZ$ shows that
+conjugating by $Z_v$ and multiplying by $i^{\operatorname{wt}(v)}$
+normalizes it to $C_F$; for $f$ columns the exponent is
+$f\operatorname{wt}(v)$. These are diagonal wrappers without a child.
+
+Choose stage-two source $C_{D_0}$ and sink $C_{D_0^\perp}$, where
+$D_1=D_0\mathbin{\perp} A$ and $\dim A=b$. The same refined copy
+schedule removes the entrance on every auxiliary role. With
+$R=D_1^\perp$, the exit is $C_RC_{D_0}^{-1}$ on
+$E=R\mathbin{\perp}D_0=A^\perp$. Its rank is $m-b$ and its
+nondegenerate polar form is unchanged by the mixed signs. The normal
+form above executes it as one child. Arbitrary auxiliary restoration
+justifies the endpoint normalization on the entire $B_2$ bank.
+
+Use the complex counts to define $B_i=NR_{h_i}/v_{h_i}$, and now
+\[
+\begin{aligned}
+W&=2N+B_2+B_3=10390178240000,\\
+L&=\sum_i(N/v_{h_i})h_i^2=117399987200,\\
+s&=Wm-2N+2L=398982681296998400.
+\end{aligned}
+\]
+The whole-residual macro list is
+\[
+\begin{array}{c|rrrrr}
+t&38328&38360&38370&37401&899\\\hline
+n_t&B_1&B_3-B_1&B_2&2N&2N.
+\end{array}
+\]
+Add $(N/v_h)H_{h,r}$ rank-$r$ children for each positive internal rank.
+Leave the remaining physical-data rank, 39393978624000, as singleton
+calls. The resulting list has weighted rank exactly $s$.
+For $a_c=3/125000$ and $\sigma=1-a_c$, the rational logarithm bounds
+above and $e^u\le(1-u)^{-1}$, $0\le u<1$, give
+\[
+ \Psi_c(\sigma)\le\sum_t\frac{n_tt}{Wm(1-a_c\ell_t)}
+ <1-\frac{792981575160764}{10^{23}}.
+\]
+
+\subsection{The dependency-path guard}
+Before gauging, nested producer labels telescope along every arithmetic
+path, including wire switches at common-frame scalar gates. Retained
+center drops occur on one return frontier per stage, crossed at most
+once by a path, and each costs at most that axis dimension. Carrier
+reuse adds only increasing continuations. This gives
+$q_0=m+2(a+b+k)$. Gauging removes an early edge and adds $ab-b$ at
+a late stage-two edge. A path meets at most one such edge, since it
+terminates that stage-two role, which is not shared across stages.
+Thus the conservative total rank budget is
+\[
+ q=m+2(a+b+k)+ab=39564.
+\]
+Every child except the four large macros is at most 899; two large
+macros cannot fit within $q$. Set $\rho=19/10$, $\gamma=9997/10000$.
+The integer comparisons $m^{19}>500000000^{10}$ and
+$899^9<460^{10}$ give $m^\rho>500000000$ and
+$899^{\rho-1}<460$.
+If a path has one large child $t$, its moment is at most
+\[
+ (t/m)^\rho+(q-t)899^{\rho-1}/m^\rho.
+\]
+This expression increases over the large-child range, since its
+numerator derivative is $\rho t^{\rho-1}-899^{\rho-1}>0$.
+Use $t=m-b$ and the concavity bound
+$(1-x)^{19/10}\le1-19x/10+9x^2/10$ to obtain
+\[
+ \sum_{\rm path}(r/m)^\rho
+ <1-\frac{19}{12800}+\frac9{10\cdot1280^2}
+       +\frac{1194\cdot460}{500000000}
+ =\frac{51180270301}{51200000000}<\gamma.
+\]
+Without a large macro the bound is $227493/6250000<\gamma$.
+
+For coefficient-depth accounting take $E_0=128(W+m+1)^3$.
+There are at most $s$ positive-rank residual calls and $W$ endpoint
+normalizations. Combining the quadratic phases with the $S$ factors,
+each uses at most two diagonal phase passes and one scalar phase.
+The inherited gate and remainder count is therefore bounded conservatively by
+\[
+ 36W^3+16s+16W+8m+4<E_0.
+\]
+Binary address changes only permute coefficients; descriptor evaluation
+is paid by the retained packed record work and does not increase
+coefficient depth.
+Stop below width $d^\beta$, where $\beta=1/1000$.
+For $e=mf+r$, $0\le r<m$, recurse on $tf$ and treat the remainder
+individually. For sufficiently large $d$, a leaf descending from an
+internal node has width at least $d^\beta/(2m)$. Choose
+$C_{\rm dep}\ge16m$ and
+$(1-\gamma)C_{\rm dep}\ge E_0+16m+1$.
+The retained elementary-kernel bound $A(e)\le8e$ and strong induction
+using the path moment imply
+\[
+ A(e)\le C_{\rm dep}d^{-\beta(\rho-1)}e^\rho.
+\]
+At a leaf the lower width bound proves this inequality, while at an
+internal node the unused $1-\gamma$ fraction covers local work.
+Small roots are handled directly. With $\zeta=1/10000$ take
+\[
+ C_1=\rho-(\rho-1)\beta+\zeta=\frac{1187}{625},\quad
+ C_0=\left\lceil128m(1+1/\zeta)C_{\rm dep}\right\rceil,
+ \quad\Delta=\lceil C_0d^{C_1}\rceil.
+\]
+The retained exact denominator/magnitude-depth induction gives
+$p+\Delta$ fractional bits and $\Delta+3$ integer/sign bits.
+All selected scalar coefficients are Gaussian dyadic and the new
+wrappers have modulus one, so that induction still applies.
+For $\epsilon C_1<1$ their total width is $O(p)$.
+\subsection{Rows, stopping, and transfer to the layer interface}
+Let $t_*<m$ be the largest child and compute
+$D(e)=0$ for $e<m$ and $D(e)=1+D(t_*\lfloor e/m\rfloor)$ otherwise.
+It is $O(\log e)$ and bounds every descendant depth.
+With the retained compact-control parameters, reserve
+\[
+ q_0=\lceil\log_2W\rceil D(d),\qquad Q_0=q_0+q_F+q_B.
+\]
+Process the first $q_0+q_F$ and last $q_B$ selected axes individually;
+if at most $Q_0$ axes exist, process them all individually. Otherwise
+the middle active interval is consecutive. Only the $q_0$ row chunks
+supply row indices, separate from the two compact fields. Pad their
+count once to a multiple of $W^{D(d)}$, by a factor below two.
+Every child retains complete compact and spectator ranges and has
+exactly $1/W$ of its parent's volume. No internal padding is needed.
+The reserved-axis bound remains
+\[
+ Q_0=O(\log(2d)+d^{\max\{1-c,0\}}\log p+1).
+\]
+Remainders commute with the completed tensor map and cost $O(V)$ per
+node. The fixed-tape depth-first scheduler and complete-row argument
+of the bit construction apply with the retained compact descriptors.
+
+Since $\Psi_c(\sigma)<1$, the total leaf $\sigma$-weighted width,
+including the volume factor $W^{-j}$ at depth $j$, is at most
+$d^\sigma$. Thus leaf work is
+$O(d^{\sigma+\beta(1-\sigma)})$. Internal compact operations cost
+$O(G^\tau e^\tau+1)$ per current volume. Summing the strict moments
+gives the retained complete normalized layer bound
+\[
+ O\!\left((\log p)^3\left[
+ d^\chi+d^{\sigma+\beta(1-\sigma)}+d^{\max\{1-c,0\}}+1
+ \right]\right),\qquad
+ \chi=\tau+(1-\beta)\max\{\sigma-\tau,0\}.
+\]
+For $\sigma\le\tau$, use $\Psi_c(\tau)\le\Psi_c(\sigma)$;
+otherwise use $e^\tau\le d^{\beta(\tau-\sigma)}e^\sigma$ at each
+internal node. These arguments do not assume equal leaf depth.
+The usual strict conditions on $\lambda,\lambda'$ therefore give
+the retained $O(Vd^{\lambda'})$ layer interface, now with the new
+moment and the new $C_{\rm dep}$.
diff --git a/research/five-stage-source527-banks/inputs/complex/retained/general-clifford-frames.tex b/research/five-stage-source527-banks/inputs/complex/retained/general-clifford-frames.tex
new file mode 100644
index 0000000000..0367eeb005
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/retained/general-clifford-frames.tex
@@ -0,0 +1,102 @@
+\subsection{Preimage frames for arbitrary binary subspaces}
+This frame construction is due to icekylinx with OpenAI GPT-6 Astra assistance;
+the present exposition and integration use Codex.
+Let $H=\mathbb F_2^h$ with the standard dot product. In the binary
+symplectic space $H\oplus H$ put
+\[
+ L_0=\{(0,z)\},\qquad
+ F=\begin{pmatrix}I&I\\0&I\end{pmatrix},\qquad
+ L_F=FL_0=\{(z,z)\}.
+\]
+We also write $F$ for the exact Gaussian-dyadic operator $C^{\otimes h}$.
+Its exact square is a Pauli translation; only its binary symplectic square
+is the identity. For an exact Clifford operator $T$ the frame label is
+$L(T)=T^{-1}L_0$. The physical transition from $T$ to $S$ is $ST^{-1}$,
+and its Fourier rank is
+$d(L(S),L(T))$, where $d(L,L')=h-\dim(L\cap L')$.
+This follows by transporting the intersection by $S$: the intersection
+with $L_0$ computes the kernel dimension of the $B$ block of $ST^{-1}$.
+
+\begin{lemma}
+For every subspace $U\subseteq H$, including degenerate subspaces,
+\[
+ L_U=\{(u,u+v):u\in U,\ v\in U^\perp\}
+\]
+is Lagrangian, and
+\[
+ d(L_U,L_V)=\dim U+\dim V-2\dim(U\cap V),\qquad
+ FL_U=L_{U^\perp}.
+\]
+\end{lemma}
+\begin{proof}
+The displayed parametrization is injective and has dimension $h$.
+Its symplectic pairing is $u\cdot v'+v\cdot u'=0$.
+In an intersection, $u$ belongs to $U\cap V$ and $v$ to
+$(U+V)^\perp$, giving the asserted dimension and distance.
+Finally $F(u,u+v)=(v,u+v)$ proves the complement identity.
+\end{proof}
+
+\subsection{Exact representatives and nested children}
+Choose a deterministic invertible binary matrix $G$ carrying the first
+$r=\dim U$ coordinate space $E_r$ to $U$, and set
+\[
+ K_G=\begin{pmatrix}G&0\\G+G^{-T}&G^{-T}\end{pmatrix},\qquad
+ T_U=K_G T_{E_r}K_G^{-1}.
+\]
+Here $T_{E_r}$ is the exact coordinate $C$ operator, and the $K_G$ are
+exact Gaussian-dyadic Clifford lifts. The matrix $K_G$ is symplectic
+because $G^TG+I$ is symmetric. It fixes both $L_0,L_F$ and maps $L_V$
+to $L_{GV}$. Thus both $K_G$ and $FK_GF^{-1}$ have Fourier rank zero;
+their lifts are fixed binary address permutations with unit quadratic
+phases. In particular,
+\[
+ L(T_U)=L_U,\quad \operatorname{rank}B(T_U)=r,\quad
+ \operatorname{rank}B(FT_UF^{-1})=r,\quad T_U^2\text{ is Pauli}.
+\]
+Set $V_U=GE_r^\perp$, a direct complement of $U$. The binary action gives
+$T_U^{-1}L_F=L_{V_U}$ and hence
+$\operatorname{rank}B(FT_U^{-1})=h-r$.
+No assertion $V_U=U^\perp$ is needed.
+For the norm-one source lines and their orthogonal target hyperplanes,
+retain the inherited exact representatives; set $T_0=I,T_H=F$.
+At every common scalar gate use one identical exact representative on all
+incident registers. If separately transported representatives have the
+same preimage, their physical ratio has rank zero and is implemented as
+an actual left adapter before the gate.
+
+For $U\subseteq V$, the exact transition $T_VT_U^{-1}$ has rank
+$\dim V-\dim U$. Clifford elimination expresses it as two rank-zero
+adapters surrounding that many coordinate Fourier factors. These factors
+are a single $C^{\otimes r}$ child: the scaled Hadamard satisfies
+\[
+ \frac{1+i}{2}\begin{pmatrix}1&1\\1&-1\end{pmatrix}
+   =SCS,\qquad S=\operatorname{diag}(1,i).
+\]
+Thus no new irrational field is introduced. All adapters are charged
+finite movement and scalar operations. Exact transitions telescope;
+commutativity between distinct $T_U$ is unnecessary.
+
+\subsection{Dirty tails and reverse/complement words}
+A dirty role entering with gauge $T_\sigma$ and whose last active frame
+is $T_U$, with $\sigma\subseteq U$, is assigned terminal gauge
+$FT_\sigma$. Its logical value is restored by the complete transparent
+scalar word, so its physical action is exactly $F$. The tail is
+$FT_\sigma T_U^{-1}$. Its rank is
+\[
+ d(L_U,L_{V_\sigma})=h-\dim U+\dim\sigma.
+\]
+Indeed $H=\sigma\oplus V_\sigma$ and $U\supseteq\sigma$ imply
+$\dim(U\cap V_\sigma)=\dim U-\dim\sigma$.
+An inactive exterior adds $m-h$, giving $m-\dim U+\dim\sigma$.
+
+For the complete time-reversed word use
+\[
+ D_U=T_UF^{-1},\qquad L(D_U)=L_{U^\perp},\qquad
+ D_UD_V^{-1}=T_UT_V^{-1}.
+\]
+The last equality is exact, so reversal preserves every inverse transition
+rank. Reverse the scalar signs and the whole transparent word, including
+old-value corrections and final cleanup. The endpoints $D_0=F^{-1}$
+and $F$ differ by a Pauli, and the inherited norm-one endpoints are
+reconciled by actual rank-zero adapters. Neither equality of binary
+matrices nor equality of frame cosets substitutes for these operators.
diff --git a/research/five-stage-source527-banks/inputs/complex/retained/stopped-product-complex.tex b/research/five-stage-source527-banks/inputs/complex/retained/stopped-product-complex.tex
new file mode 100644
index 0000000000..4a9037137c
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/retained/stopped-product-complex.tex
@@ -0,0 +1,137 @@
+\subsection{All-disjoint retained centers}
+Use two factors $h=24$. For triple inputs write $G=\sum_TX_T$,
+$T_i=\sum_{T\ni i}X_T$, and retain all $A_i=G-T_i$.
+Each input triple avoids exactly 21 points, so
+\[
+ \sum_{i=1}^{24}A_i=21G,\qquad G=\frac1{21}\sum_i A_i.
+\]
+For a target triple $S$ the central correction is
+\[
+ G-\frac12\sum_{i\in S}A_i
+       =\frac{\sum_{i\in S}T_i-G}{2}.
+\]
+An input triple meeting $S$ in $j$ points has central coefficient
+$(j-1)/2$. Adding half the disjoint sum and subtracting half the
+intersection-two sum leaves coefficient one exactly when $j=3$.
+The scalar source-to-scatter transfer is therefore the identity.
+The transparent old-value subtraction and new-value addition use this
+identity only on fresh contributions and restore arbitrary old values.
+
+The base-two paired exclusion and aligned pair-star producer retains
+nested binary labels: common-pair supports have their orthonormal triple
+span, other ordinary sums have their coordinate cover. Each retained
+$A_i$ has the nondegenerate coordinate hyperplane excluding $i$,
+of dimension 23. Hence all copied-center arguments and general
+nondegenerate binary residuals of the preceding proof apply. The
+selected local counts are
+\[
+ v=2024,\quad c_h=45764,\quad q_h=8120,\quad
+ |\mathcal M|=8966,\quad R=44918,\quad\ell=552.
+\]
+The old internal histogram, starting at rank zero, is
+\begin{small}\begin{align*}
+[&8930,49042,30844,15972,11883,8272,6399,4356,6682,2263,\\
+ &5109,1808,5317,1733,5391,1757,4908,3290,6233,4363,\\
+ &4056,264,564,24,24].
+\end{align*}\end{small}
+It has rank mass $24R+2\ell$. Replace 24 rank-24 original transitions
+by 24 rank-one transitions; all rank-23 copied transforms remain.
+The new mass is $24R+\ell$.
+
+\subsection{Physical endpoints and strict complex moment}
+The copied-center two-stage complex contract in
+\path{notes/copied-centers-complex.tex} is unchanged by the scalar
+center decoding. For the tensor triple line $U$, let
+$T=C_U$, $F=C_I$, $E=C_{U^\perp}=FT^{-1}$.
+The two data outputs are $A=-Fy$ and $B=FT^{-2}x+Ey$.
+Add $T^{-1}$ applied to a copy of $A$ into $B$, paying one rank-one
+child per data pair. The inherited input/output $Z$ phases and
+$i^{9f}$ scalar normalize $FT^{-2}$ to $F$; sign and bank corrections
+complete the exact tensor $C$. Auxiliary source/sink operators remain
+$C_{D_0}$ and $FC_{D_0}$ in the second stage. Every original role
+therefore returns the exact selected-axis tensor operation on arbitrary
+inputs. This algebraic statement holds over $\mathbb Q(i)$ as well as
+over Gaussian dyadics.
+
+With $m_c=576$, $N_c=v^2=4096576$, $B_c=vR=90914032$, put
+\[
+ W_c=2N_c+2B_c=190021216,\quad L_c=2v\ell=2234496,
+\]
+\[
+ s_c=W_cm_c-N_c+L_c=109450358336,\qquad N_c-L_c=1862080.
+\]
+The complete list is $2B_c$ width-552 auxiliary exteriors,
+$2N_c$ width-529 data macros, $2vH'_r$ width-$r$ internal children,
+$4N_c$ width-23 data fronts, and $N_c$ width-one endpoint copies.
+These disjoint classes include both kinds of paid copies and sum to
+$s_c$. All binary residuals, including alternating forms, use the retained
+one-child Gauss normal form with paid bit adapters and unit phases.
+At $a_c=1949/25000000$ the exact rational enclosure gives
+\[
+ \sum_r\frac{n_r}{W_c}(r/m_c)^{1-a_c}
+ <1-1.3338373969\times10^{-10}.
+\]
+
+\subsection{One exact odd-denominator grid}
+The only new odd divisor is 21. Let $d$ bound the root axis count,
+$D$ bound its maximum complex recursion depth, and $G_0$ bound the
+number of grouped scalar operations per node. Choose
+\[
+ K=G_0(D+1),\qquad \text{common grid }2^{-P}21^{-K}.
+\]
+Root inputs and arbitrary dirty auxiliaries start Gaussian dyadic.
+Multiply their integer numerators by $21^K$ once at the outer boundary.
+Every local coefficient contains at most one factor 21 in its denominator,
+so a local finite program adds at most $G_0$ to the odd-denominator
+exponent. A completed $u$-axis child has Gaussian-dyadic coefficients
+and therefore preserves its input's existing odd-denominator exponent;
+it adds no new odd factor. Only the active chain of unfinished calls
+can accumulate new factors. Its length is at most $D+1$, proving that
+all intermediates lie on the common grid and every local division by
+21 is an exact integer division of its encoded numerator.
+
+This divisibility claim uses the root dyadic-input invariant, not an
+assumption about arbitrary numerators on the finest grid. Internal values
+may have odd denominators. They are neither rounded nor re-encoded when
+a child returns. At the completed outer return the operator is the exact
+Gaussian-dyadic tensor on root dyadic inputs, so the final numerators
+are divisible by $21^K$; exact unscaling restores dyadic encoding.
+
+Since the largest child is $552<576$, $D=O(\log d)$ and
+$K=O(\log d)$ with fixed constants. Constant-divisor long division
+is linear in numerator length. Initial and final scaling by $K$
+successive constant multiplications/divisions cost $O(V\log d)$ and
+occur only at the outer boundary. The extra numerator length is
+$O(\log d)$, preserving the retained $O(p)$ word bound. This overhead
+is subordinate to the positive-power complex layer bound.
+
+For explicit semantic constants use
+\[
+ g_h=4(c_h+v)+10v+4hv+4h^2+8h+8=408200,
+\]
+\[
+ G_0=N_c+2v(g_h+2h)=1656684480,
+\]
+where endpoint additions and copied-center scalar work are included.
+The complete child contributes at most $u$ binary denominator bits
+and row-norm factor $2^u$, with no additional odd exponent. A local
+21-division contributes fewer than five denominator-height bits,
+covered by the fixed scalar charge. Set
+\[
+\begin{aligned}
+ E_0&=64(W_c+m_c+G_0+1)^3=403063477262941121443761178688,\\
+ \mathcal E&=2G_0W_c^2+8s_c+4W_c+4+32m_c\\
+           &=119639333593035096270287748<E_0,\\
+ B_0&=s_c+E_0,\qquad C_0=32m_cB_0^2.
+\end{aligned}
+\]
+With $f=\lfloor e/m_c\rfloor$, the completed-child induction is
+\[
+ A(e)\le A(552f)+s_cf+E_0\le2B_0e,
+\]
+since $2B_0(m_c-552)\ge s_c+E_0$. It counts total denominator height
+and magnitude, including the local odd factor. The retained prefix and
+normalization allowance $18e$ fits inside $C_0e$. Together with the
+$O(\log d)$ common-grid encoding overhead this gives $C_1=1$ and
+$O(p)$ records for $d=\lfloor b_{\rm in}^{\epsilon}\rfloor$,
+$p=6b_{\rm in}$, fixed $\epsilon<1$.
diff --git a/research/five-stage-source527-banks/inputs/complex/source-pins.json b/research/five-stage-source527-banks/inputs/complex/source-pins.json
new file mode 100644
index 0000000000..5cb04b0c72
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/source-pins.json
@@ -0,0 +1,215 @@
+{
+  "schema": "primary-complex-input-pins/1",
+  "certificate": {
+    "compressed_sha256": "15251deb0c11e0504d7b297c9786818888a93f08b92b03c6c122784b70488633",
+    "uncompressed_sha256": "c64d2a95431c3cf24452de7391cce780bd71521da44ce23923ebb235f6c7962c",
+    "path": "inputs/complex/gcert1-p11-pr193.json.gz"
+  },
+  "files": {
+    "inputs/complex/sources/Work__Bridge__Cert.lean": {
+      "sha256": "5c24746a6330907b9b37d6064095035a58575d64ce793d6d7c9881121fdc3980",
+      "upstream": "Work/Bridge/Cert.lean",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/Work/Bridge/Cert.lean"
+    },
+    "inputs/complex/sources/Work__Bridge__Net.lean": {
+      "sha256": "cf7d7d330b1bffe2e66712c873e647b13dc32db1ccfc0e0f4cd62ee6caf584f0",
+      "upstream": "Work/Bridge/Net.lean",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/Work/Bridge/Net.lean"
+    },
+    "inputs/complex/sources/Work__Fourier__Main.lean": {
+      "sha256": "2e8e88efd4dcbdf0b1a6e39ce7eea8c5fd5ffee2faf360a8c0c9cba84c0e4ef3",
+      "upstream": "Work/Fourier/Main.lean",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/Work/Fourier/Main.lean"
+    },
+    "inputs/complex/sources/Work__GCert__Chain__Geom.lean": {
+      "sha256": "c302f5834287421f11f3b0da7112fcde876cbc702896f2af4145b2d72ee393a2",
+      "upstream": "Work/GCert/Chain/Geom.lean",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/Work/GCert/Chain/Geom.lean"
+    },
+    "inputs/complex/sources/Work__GCert__Chain__InvThm.lean": {
+      "sha256": "ade7e7873da75ca63a4b5ce2f562dde9c12bfbb69d87d7ae831039ff790e5adb",
+      "upstream": "Work/GCert/Chain/InvThm.lean",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/Work/GCert/Chain/InvThm.lean"
+    },
+    "inputs/complex/sources/Work__GCert__Chain__NetCert.lean": {
+      "sha256": "22ae34809466b4ac2a2c5cdde526f51ff246f5189f7c142b38aa4c0dcffb04a4",
+      "upstream": "Work/GCert/Chain/NetCert.lean",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/Work/GCert/Chain/NetCert.lean"
+    },
+    "inputs/complex/sources/Work__GCert__Chain__NetDef.lean": {
+      "sha256": "6e17a68b1e988969f6eb77000028f7ed877e687b8ad3976ec1707fcfe1d2f520",
+      "upstream": "Work/GCert/Chain/NetDef.lean",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/Work/GCert/Chain/NetDef.lean"
+    },
+    "inputs/complex/sources/Work__GCert__Chain__NetStage.lean": {
+      "sha256": "c122bc01392a6fdcc0bee918fed81804d3c9b1d3e071016dab65a4ed804ef0b6",
+      "upstream": "Work/GCert/Chain/NetStage.lean",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/Work/GCert/Chain/NetStage.lean"
+    },
+    "inputs/complex/sources/Work__GCert__Chain__Phased.lean": {
+      "sha256": "0de429c80231334f1ce92575c1e8ac382ca2d5e7d956767b547639a8941d9a8c",
+      "upstream": "Work/GCert/Chain/Phased.lean",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/Work/GCert/Chain/Phased.lean"
+    },
+    "inputs/complex/sources/Work__GCert__Data__B2Gp193Main.lean": {
+      "sha256": "cb61f304da51f1dd0a2ec25bca4b63eb095eeb2863289e1e2bdee7c82beff48b",
+      "upstream": "Work/GCert/Data/B2Gp193Main.lean",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/Work/GCert/Data/B2Gp193Main.lean"
+    },
+    "inputs/complex/sources/Work__GCert__Data__Glue.lean": {
+      "sha256": "d5314f2b93a7f0e1c949fb502331825bfc6d9d8762d35f865443e7c95b6c7ef7",
+      "upstream": "Work/GCert/Data/Glue.lean",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/Work/GCert/Data/Glue.lean"
+    },
+    "inputs/complex/sources/Work__GCert__Data__InstB2Gp193.lean": {
+      "sha256": "9587ddb9961f44938032b609307ab91c0b0e757e3991f1ad95480dd50fd4f72b",
+      "upstream": "Work/GCert/Data/InstB2Gp193.lean",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/Work/GCert/Data/InstB2Gp193.lean"
+    },
+    "inputs/complex/sources/Work__GCert__Data__P193Cert.lean": {
+      "sha256": "d92ee6889be4753fc99f467bd84039743325123a1165128378b9c906a04f2703",
+      "upstream": "Work/GCert/Data/P193Cert.lean",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/Work/GCert/Data/P193Cert.lean"
+    },
+    "inputs/complex/sources/Work__GCert__Data__P193Price.lean": {
+      "sha256": "0299bcd0fe255960eec682eef0f045f7e5cb1e4711af2aab4393526eae2caea3",
+      "upstream": "Work/GCert/Data/P193Price.lean",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/Work/GCert/Data/P193Price.lean"
+    },
+    "inputs/complex/sources/Work__GCert__Data__P193ScalOK.lean": {
+      "sha256": "479434815822e98f015e335361089c3788d00dc3dd39c2944f75410aa0c158fe",
+      "upstream": "Work/GCert/Data/P193ScalOK.lean",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/Work/GCert/Data/P193ScalOK.lean"
+    },
+    "inputs/complex/sources/Work__GCert__Data__RateB2Gp193x.lean": {
+      "sha256": "6b40cac1481c931d47b1a092e382d04716ede0e785c492a2f2983e886a257c3f",
+      "upstream": "Work/GCert/Data/RateB2Gp193x.lean",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/Work/GCert/Data/RateB2Gp193x.lean"
+    },
+    "inputs/complex/sources/Work__GCert__Data__Raw.lean": {
+      "sha256": "aba94a004c94dcae7cf3047c88a6da9561555f4ebe8ab92e299e6ce09ee7aa66",
+      "upstream": "Work/GCert/Data/Raw.lean",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/Work/GCert/Data/Raw.lean"
+    },
+    "inputs/complex/sources/Work__GCert__Data__SolutionB2Gp193x.lean": {
+      "sha256": "45c768c9d71a7cc18bd6db74cb036db880d89580ca60557bb4a492280296b169",
+      "upstream": "Work/GCert/Data/SolutionB2Gp193x.lean",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/Work/GCert/Data/SolutionB2Gp193x.lean"
+    },
+    "inputs/complex/sources/Work__GFrame__Engine__Adapt.lean": {
+      "sha256": "feec0a3828867968a5c7c2c30b86ed70449abc3c660a60ada67203a75bae2ef7",
+      "upstream": "Work/GFrame/Engine/Adapt.lean",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/Work/GFrame/Engine/Adapt.lean"
+    },
+    "inputs/complex/sources/Work__GFrame__Engine__Letters.lean": {
+      "sha256": "34906dbf5ae16b5174db00b4881a5cc1d2e583b8c8806db6523a5e2f86565f09",
+      "upstream": "Work/GFrame/Engine/Letters.lean",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/Work/GFrame/Engine/Letters.lean"
+    },
+    "inputs/complex/sources/Work__GFrame__Engine__Level.lean": {
+      "sha256": "80fb3daa9a45de0ed6da570eb4476b9104734be65d0116f58bcfcbcd67f771d0",
+      "upstream": "Work/GFrame/Engine/Level.lean",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/Work/GFrame/Engine/Level.lean"
+    },
+    "inputs/complex/sources/tools__gx__ORIGIN.md": {
+      "sha256": "6ba897a9495ddadcf8558b93133fdfaf66fc35ce7fa2c53e2d5667335c315e0b",
+      "upstream": "tools/gx/ORIGIN.md",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/tools/gx/ORIGIN.md"
+    },
+    "inputs/complex/sources/tools__gx__gxbind.py": {
+      "sha256": "9c9dcfac47f0535bf3aa8ebf7c662cdef0ebda50a9f0a9282b8f3cf34809f835",
+      "upstream": "tools/gx/gxbind.py",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/tools/gx/gxbind.py"
+    },
+    "inputs/complex/sources/tools__gx__gxconv.py": {
+      "sha256": "dab7fec9f44c2d368ed334055d7126e202017feb46e41a8d86e2d998720f67c2",
+      "upstream": "tools/gx/gxconv.py",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/tools/gx/gxconv.py"
+    },
+    "inputs/complex/sources/tools__gx__gxcore.py": {
+      "sha256": "eb972c06fe8c322db2d9a8d9a7d05596e8f40c4ca8e5f42eb6df3c7361b5d517",
+      "upstream": "tools/gx/gxcore.py",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd",
+      "url": "https://github.com/jacobalansussman/wht-power-saving-lean/blob/f010392c923279e3dd59ef3aa5fedad23affc5fd/tools/gx/gxcore.py"
+    },
+    "inputs/complex/sources/Work__GCert__Chain__WinB.lean": {
+      "sha256": "c33ec3fce09c0971c32ddcbb86d9383ffbe4c6d05cf36f20460005e6547bb845",
+      "upstream": "Work/GCert/Chain/WinB.lean",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd"
+    },
+    "inputs/complex/sources/Work__Carrier__Live.lean": {
+      "sha256": "6d0649eb62589e66057f044e1e2103fc42f3f57a1498f82a26565bd8eb4026d0",
+      "upstream": "Work/Carrier/Live.lean",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd"
+    },
+    "inputs/complex/sources/Work__Carrier__Phases.lean": {
+      "sha256": "71dc3dda23dfe510d5d53cbada3a6c540cd1dc2825503fda618a5bd6cfc3e812",
+      "upstream": "Work/Carrier/Phases.lean",
+      "source_pin": "f010392c923279e3dd59ef3aa5fedad23affc5fd"
+    },
+    "inputs/complex/retained/general-clifford-frames.tex": {
+      "sha256": "c9bb9655b594583c6b4bab8cb485233f1d1e588f83858cf1e4c3530cee8fed54",
+      "upstream": "notes/general-clifford-frames.tex",
+      "source_pin": "187e1010ac8b259af8e9b5166f68b64bc27b4b47"
+    },
+    "inputs/complex/retained/endpoint-gauge-complex.tex": {
+      "sha256": "dfee33b0281e5551bb6927f757ff0240abade60ae1800f715afd88853ae0191e",
+      "upstream": "notes/endpoint-gauge-complex.tex",
+      "source_pin": "187e1010ac8b259af8e9b5166f68b64bc27b4b47"
+    },
+    "inputs/complex/retained/copied-centers-lemma.tex": {
+      "sha256": "d0ce6d3d504996aadb098227885224e48df46101daec8893523a0a8b7364d8fe",
+      "upstream": "notes/copied-centers-lemma.tex",
+      "source_pin": "187e1010ac8b259af8e9b5166f68b64bc27b4b47"
+    },
+    "inputs/complex/retained/copied-centers-assembly.tex": {
+      "sha256": "00117892971ae766188ace0af03030c44164fda105e42b70d84f52c3339a7dcb",
+      "upstream": "notes/copied-centers-assembly.tex",
+      "source_pin": "187e1010ac8b259af8e9b5166f68b64bc27b4b47"
+    },
+    "inputs/complex/retained/stopped-product-complex.tex": {
+      "sha256": "4738089103a4b263b3b0ef458f5ef35519c696f1ea871ae79e508d3720f5870a",
+      "upstream": "notes/stopped-product-complex.tex",
+      "source_pin": "187e1010ac8b259af8e9b5166f68b64bc27b4b47"
+    },
+    "inputs/complex/retained/compact-control-layout.tex": {
+      "sha256": "2bc28e50f3eb092195294cafa81213e520f6d42e7935e9905ad8a6199e2a7c87",
+      "upstream": "notes/compact-control-layout.tex",
+      "source_pin": "187e1010ac8b259af8e9b5166f68b64bc27b4b47"
+    },
+    "inputs/complex/retained/compact-control-movement.tex": {
+      "sha256": "b963c3d2f4e1b7059aa289bb4b4ac89b688920bcf42f5b20d7698cdef0132cc6",
+      "upstream": "notes/compact-control-movement.tex",
+      "source_pin": "187e1010ac8b259af8e9b5166f68b64bc27b4b47"
+    },
+    "inputs/complex/retained/FINITE_BRIDGE.txt": {
+      "sha256": "66d8f4ea8752abd89952d6674a636575a0db16e18b1c16e293f48f64242f0c46",
+      "upstream": "research/source-assisted/global/FINITE_BRIDGE.txt",
+      "source_pin": "187e1010ac8b259af8e9b5166f68b64bc27b4b47"
+    }
+  }
+}
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/Work__Bridge__Cert.lean b/research/five-stage-source527-banks/inputs/complex/sources/Work__Bridge__Cert.lean
new file mode 100644
index 0000000000..f7bdc3d903
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/Work__Bridge__Cert.lean
@@ -0,0 +1,241 @@
+import Work.Bridge.Net
+
+/-!
+# (key: bridge-net) The certificate of the BRIDGED five-stage network B_2
+
+The three layers of free gates between twins, the signed exchange, and
+
+    theorem bridge2_certificate (N : Bridge2 H T Sl C α Γ) :
+        BCert α (Sum.inl : BLive Γ T Sl → BRole Γ T Sl C) (fun φ => card Γ * N.unit φ)
+
+Scalar chain of one class (`x1 y1 x2 y2` on `X1 Y1 X2 Y2`), as in `bridged-word.md`:
+
+    stage 0  F on pair 1     Y1 = y1 + x1
+    stage 1  B on pair 1     X1 = -y1
+    gates A  X1 += X2,  Y2 -= Y1
+    stage 2  F on pair 1     Y1 = x1 + x2
+    gates B  Y2 += Y1,  X1 -= X2          (X1 = -y1, Y2 = y2 - y1 + x2)
+    stage 3  B on pair 2     X2 = y1 - y2
+    stage 4  F on pair 2     Y2 = x2
+    gates C  Y1 -= Y2,  X2 += X1          (X1 = -y1, X2 = -y2, Y1 = x1, Y2 = x2)
+    exchange X_i := Y_i, Y_i := -X_i
+-/
+
+set_option linter.unusedSectionVars false
+set_option linter.unusedVariables false
+
+namespace OAI
+namespace PowerSaving
+namespace BR
+open Binary Matrix Finset RAM SS CB RF
+noncomputable section
+
+section Gates
+variable {Γ T Sl C : Type}
+
+/-- source of the gates of the layers A and B: `X1 ← X2`, `Y2 ← Y1`. -/
+def pAB : BRole Γ T Sl C → BRole Γ T Sl C :=
+  mkB (fun g t => kX2 (g,t)) (fun g t => kY1 (g,t)) (fun g t => kX2 (g,t))
+    (fun g t => kY1 (g,t)) (fun g q => kS (g,q)) (fun g c => kC (g,c))
+
+/-- layer A: `X1 += X2`, `Y2 -= Y1`. -/
+def kA : BRole Γ T Sl C → ℚ :=
+  mkB (fun _ _ => 1) (fun _ _ => 0) (fun _ _ => 0) (fun _ _ => -1) (fun _ _ => 0) (fun _ _ => 0)
+
+/-- layer B: `X1 -= X2`, `Y2 += Y1`. -/
+def kB : BRole Γ T Sl C → ℚ :=
+  mkB (fun _ _ => -1) (fun _ _ => 0) (fun _ _ => 0) (fun _ _ => 1) (fun _ _ => 0) (fun _ _ => 0)
+
+/-- source of the gates of the layer C: `Y1 ← Y2`, `X2 ← X1`. -/
+def pC : BRole Γ T Sl C → BRole Γ T Sl C :=
+  mkB (fun g t => kX1 (g,t)) (fun g t => kY2 (g,t)) (fun g t => kX1 (g,t))
+    (fun g t => kY2 (g,t)) (fun g q => kS (g,q)) (fun g c => kC (g,c))
+
+/-- layer C: `Y1 -= Y2`, `X2 += X1`. -/
+def kCc : BRole Γ T Sl C → ℚ :=
+  mkB (fun _ _ => 0) (fun _ _ => -1) (fun _ _ => 1) (fun _ _ => 0) (fun _ _ => 0) (fun _ _ => 0)
+
+/-- scalar map of a gate layer. -/
+def gsc (p : BRole Γ T Sl C → BRole Γ T Sl C) (k : BRole Γ T Sl C → ℚ)
+    (x : BRole Γ T Sl C → ℂ) : BRole Γ T Sl C → ℂ := fun i => x i + (k i : ℂ) * x (p i)
+
+/-- the signed exchange `X_i := Y_i`, `Y_i := -X_i` on both pairs. -/
+def bpull : BRole Γ T Sl C → BRole Γ T Sl C :=
+  mkB (fun g t => kY1 (g,t)) (fun g t => kX1 (g,t)) (fun g t => kY2 (g,t))
+    (fun g t => kX2 (g,t)) (fun g q => kS (g,q)) (fun g c => kC (g,c))
+
+def bsgn : BRole Γ T Sl C → ℚ :=
+  mkB (fun _ _ => 1) (fun _ _ => -1) (fun _ _ => 1) (fun _ _ => -1) (fun _ _ => 1) (fun _ _ => 1)
+
+variable [Fintype Γ] [DecidableEq Γ] [Fintype T] [DecidableEq T]
+  [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C]
+
+def brecM : Matrix (BRole Γ T Sl C) (BRole Γ T Sl C) ℚ :=
+  fun i j => if j = bpull i then bsgn i else 0
+
+lemma actPoint_brecM (x : BRole Γ T Sl C → ℂ) (i : BRole Γ T Sl C) :
+    actPoint brecM x i = (bsgn i : ℂ) * x (bpull i) := by
+  unfold actPoint
+  rw [sum_eq_single (bpull i)]
+  · simp [brecM]
+  · intro j _ hj; simp [brecM, hj]
+  · simp
+
+/-- the scalar map of the whole network before the exchange, on one class. -/
+def chain (x : BRole Γ T Sl C → ℂ) : BRole Γ T Sl C → ℂ :=
+  gsc pC kCc (lift2 GF (lift2 GB (gsc pAB kB (lift1 GF (gsc pAB kA (lift1 GB (lift1 GF x)))))))
+
+/-- **The scalar identity of the bridged chain**: `X_i = -y_i`, `Y_i = x_i` on both pairs,
+helper slots unchanged. -/
+lemma chain_eq (x : BRole Γ T Sl C → ℂ) (l : BLive Γ T Sl) :
+    chain x (Sum.inl l) = mkB (fun g t => -x (kY1 (g,t))) (fun g t => x (kX1 (g,t)))
+      (fun g t => -x (kY2 (g,t))) (fun g t => x (kX2 (g,t)))
+      (fun g q => x (kS (g,q))) (fun g c => x (kC (g,c))) (Sum.inl l) := by
+  rcases l with (((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)
+  all_goals
+    simp only [chain, gsc, lift1, lift2, GF, GB, mkB, mkS, pAB, pC, kA, kB, kCc, pos1, pos2,
+      kX1, kY1, kX2, kY2, kS, kC, jX, jY, jS, jC, Function.comp_apply]
+    first
+      | (push_cast; ring)
+      | (push_cast; done)
+      | ring
+
+end Gates
+
+section Net
+variable {H T Sl C α Γ : Type} [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T]
+  [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] [Fintype α] [DecidableEq α]
+  [Fintype Γ] [DecidableEq Γ]
+variable (N : Bridge2 H T Sl C α Γ)
+
+/-- gate layer A, after stage 1 and the first idle climb of pair 2: twins have equal frames. -/
+lemma Bridge2.gateA : XRoute N.S3 N.S3 (gsc pAB kA) (fun _ => 0) := by
+  refine gateP pAB kA N.S3 ?_
+  rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩) hk
+  · rfl
+  · exact (hk rfl).elim
+  · exact (hk rfl).elim
+  · rfl
+  · exact (hk rfl).elim
+  · exact (hk rfl).elim
+
+/-- gate layer B, after stage 2 and the second idle climb of pair 2. -/
+lemma Bridge2.gateB : XRoute N.S5 N.S5 (gsc pAB kB) (fun _ => 0) := by
+  refine gateP pAB kB N.S5 ?_
+  rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩) hk
+  · exact N.x23 g t
+  · exact (hk rfl).elim
+  · exact (hk rfl).elim
+  · exact (N.y23 g t).symm
+  · exact (hk rfl).elim
+  · exact (hk rfl).elim
+
+/-- gate layer C, after the final climbs. -/
+lemma Bridge2.gateC : XRoute N.S8 N.S8 (gsc pC kCc) (fun _ => 0) := by
+  refine gateP pC kCc N.S8 ?_
+  rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩) hk
+  · exact (hk rfl).elim
+  · rfl
+  · rfl
+  · exact (hk rfl).elim
+  · exact (hk rfl).elim
+  · exact (hk rfl).elim
+
+/-- the free shifts at the end. -/
+def Bridge2.corr : BRole Γ T Sl C → Space α :=
+  mkB (fun _ _ => 0) (fun g t => N.u g t) (fun _ _ => 0) (fun g t => N.u g t)
+    (fun _ _ => 0) (fun _ _ => 0)
+
+def Bridge2.S9 : BRole Γ T Sl C → CMat α := fun r => shift (N.corr r) * N.S8 r
+
+/-- the whole word before the exchange: five stages, three idle-climb layers, three gate
+layers. -/
+lemma Bridge2.route :
+    XRoute N.S0 N.S8 chain (fun φ => (Fintype.card Γ : ℝ) * N.unit φ) := by
+  have R := ((((((((((N.stage0.trans N.stage1).trans N.climbA).trans N.gateA).trans
+    N.stage2).trans N.climbB).trans N.gateB).trans N.stage3).trans N.stage4).trans
+    N.fin).trans N.gateC)
+  refine (R.castg ?_).cast (fun φ => ?_)
+  · rfl
+  · simp only [Bridge2.unit]
+    ring
+
+/-- **The scratch certificate of the bridged five-stage network with shared helper sets**,
+in whole blocks, with its exact price for every price list. -/
+theorem bridge2_certificate :
+    BCert α (Sum.inl : BLive Γ T Sl → BRole Γ T Sl C)
+      (fun φ => (Fintype.card Γ : ℝ) * N.unit φ) := by
+  have R2 : XRoute N.S8 N.S9 id (fun _ => 0) := XRoute.shifts N.hm N.S8 N.corr
+  have R3 : XRoute N.S9 (fun r => N.S9 (bpull r)) (actPoint brecM) (fun _ => 0) :=
+    XRoute.gate brecM N.S9 _ (fun i j h => by
+      have hj : j = bpull i := by
+        by_contra hne
+        exact h (by simp [brecM, hne])
+      rw [hj])
+  have Rt := ((N.route.trans R2).trans R3).cast (c' := fun φ => (Fintype.card Γ : ℝ) * N.unit φ)
+    (fun φ => by first | ring | simp)
+  have hSinv : ∀ r, ∃ M, N.S0 r * M = 1 := by
+    rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩)
+    · exact N.inv0 g t
+    · refine ⟨1, ?_⟩
+      show (N.M0 g).Φ 0 * 1 = 1
+      rw [N.z0, Matrix.mul_one]
+    · exact N.inv0 g t
+    · refine ⟨1, ?_⟩
+      show (N.M0 g).Φ 0 * 1 = 1
+      rw [N.z0, Matrix.mul_one]
+    · exact ⟨1, Matrix.mul_one _⟩
+    · exact ⟨1, Matrix.mul_one _⟩
+  choose S' hS' using hSinv
+  have hT : ∀ l : BLive Γ T Sl, N.S9 (bpull (Sum.inl l)) = kernel α * N.S0 (Sum.inl l) := by
+    rintro (((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)
+    · show shift (N.u g t) * N.YF g t = kernel α * (N.M0 g).Φ (tt (N.I.tv t))
+      exact (N.term g t).symm
+    · show shift (0 : Space α) * kernel α = kernel α * (N.M0 g).Φ 0
+      rw [shift_zero, Matrix.one_mul, N.z0, Matrix.mul_one]
+    · show shift (N.u g t) * N.YF g t = kernel α * (N.M0 g).Φ (tt (N.I.tv t))
+      exact (N.term g t).symm
+    · show shift (0 : Space α) * kernel α = kernel α * (N.M0 g).Φ 0
+      rw [shift_zero, Matrix.one_mul, N.z0, Matrix.mul_one]
+    · show shift (0 : Space α) * kernel α = kernel α * 1
+      rw [shift_zero, Matrix.one_mul, Matrix.mul_one]
+  have hg : ∀ x : BRole Γ T Sl C → ℂ,
+      (∀ s, (∀ l : BLive Γ T Sl, (Sum.inl l : BRole Γ T Sl C) ≠ s) → x s = 0) →
+      ∀ l : BLive Γ T Sl,
+        (actPoint brecM ∘ (id ∘ chain)) x (Sum.inl l) = x (Sum.inl l) := by
+    intro x _ l
+    change actPoint brecM (chain x) (Sum.inl l) = _
+    rw [actPoint_brecM]
+    rcases l with (((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)
+    · show ((1:ℚ):ℂ) * chain x (Sum.inl (Sum.inl (Sum.inl (Sum.inr (g,t))))) = _
+      rw [chain_eq]
+      show ((1:ℚ):ℂ) * x (kX1 (g,t)) = x (kX1 (g,t))
+      first | (push_cast; ring) | simp
+    · show ((-1:ℚ):ℂ) * chain x (Sum.inl (Sum.inl (Sum.inl (Sum.inl (g,t))))) = _
+      rw [chain_eq]
+      show ((-1:ℚ):ℂ) * (-x (kY1 (g,t))) = x (kY1 (g,t))
+      first | (push_cast; ring) | simp
+    · show ((1:ℚ):ℂ) * chain x (Sum.inl (Sum.inl (Sum.inr (Sum.inr (g,t))))) = _
+      rw [chain_eq]
+      show ((1:ℚ):ℂ) * x (kX2 (g,t)) = x (kX2 (g,t))
+      first | (push_cast; ring) | simp
+    · show ((-1:ℚ):ℂ) * chain x (Sum.inl (Sum.inl (Sum.inr (Sum.inl (g,t))))) = _
+      rw [chain_eq]
+      show ((-1:ℚ):ℂ) * (-x (kY2 (g,t))) = x (kY2 (g,t))
+      first | (push_cast; ring) | simp
+    · show ((1:ℚ):ℂ) * chain x (Sum.inl (Sum.inr (g,q))) = _
+      rw [chain_eq]
+      show ((1:ℚ):ℂ) * x (kS (g,q)) = x (kS (g,q))
+      first | (push_cast; ring) | simp
+  exact XRoute.liveKernel (Sum.inl : BLive Γ T Sl → BRole Γ T Sl C) N.S0 S'
+    (fun r => N.S9 (bpull r)) hS' _ _ Rt hg hT
+
+end Net
+end
+end BR
+end PowerSaving
+end OAI
+
+#print axioms OAI.PowerSaving.BR.chain_eq
+#print axioms OAI.PowerSaving.BR.Bridge2.route
+#print axioms OAI.PowerSaving.BR.bridge2_certificate
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/Work__Bridge__Net.lean b/research/five-stage-source527-banks/inputs/complex/sources/Work__Bridge__Net.lean
new file mode 100644
index 0000000000..204f6d9cf9
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/Work__Bridge__Net.lean
@@ -0,0 +1,345 @@
+import Work.Bridge.Inv
+
+/-!
+# (key: bridge-net) The BRIDGED five-stage network B_2 with shared helper sets, abstract form
+
+Generalises `RF.Fold3S` / `RF.fold3s_certificate` (`Work.Reframe.SCert`) to the bridged word
+of PLAN2.md B1: every class `(g,t)` has TWO bank pairs `(X1,Y1)`, `(X2,Y2)` with the same
+line; five stages of invocations
+
+    stage 0  forward   on pair 1      invocation of the class (g,t):  g
+    stage 1  backward  on pair 1                                      r1 t g
+    stage 2  forward   on pair 1                                      r2 t g
+    stage 3  backward  on pair 2                                      r3 t g
+    stage 4  forward   on pair 2                                      r4 t g
+
+and three layers of FREE gates between twins at equal frames:
+
+    after stage 1 (pair 2 has climbed to the frames of pair 1):   X1 += X2,  Y2 -= Y1
+    after stage 2 (pair 2 has climbed one more window):           Y2 += Y1,  X1 -= X2
+    after the final climbs (all four roles at their end frames):  Y1 -= Y2,  X2 += X1
+
+Result on both pairs: `X_i = -y_i`, `Y_i = x_i`; then the usual signed exchange.
+
+The idle pair does not move during a stage.  Its climbs are ONE block per role:
+`aX2`, `aY2` (pair 2 before the first gate layer, prices `caX`, `caY`; in B_2 ONE block of
+rank `2h-2` each),
+the window of stage 2 (pair 2 before the second gate layer, rank `card H - 1`, from the
+package: `Inv.climbX`, `Inv.climbY`), `fX1`, `fY1` (pair 1 at the end, prices `cX1`, `cY1`;
+in B_2 ONE block of rank `2h+2` each), `fX2`, `fY2` (pair 2 at the end, prices `cX2`, `cY2`;
+in B_2 one block of rank `4` each).  The prices are arbitrary functions of the price list, so
+a geometry that makes an idle climb in several blocks can be plugged in as well.
+
+`Bridge2 H T Sl C α Γ` is everything the wiring needs to know about the geometry;
+`bridge2_certificate`: a `BCert` on the live roles `BLive Γ T Sl`
+(`card Γ * (4 card T + card Sl)` of them, `BR.card_BLive`) whose price, for EVERY price
+list, is `card Γ * N.unit φ`,
+
+    N.unit φ = 5 * N.I.cost φ + card T * ((caX φ + caY φ) + 2 * bcost φ (card H - 1)
+                 + (cX1 φ + cY1 φ) + (cX2 φ + cY2 φ)).
+-/
+
+set_option linter.unusedSectionVars false
+set_option linter.unusedVariables false
+
+namespace OAI
+namespace PowerSaving
+namespace BR
+open Binary Matrix Finset RAM SS CB RF
+noncomputable section
+
+/-- See the file header.  Fields in the style of `RF.Fold3S`; stages are numbered 0 to 4. -/
+structure Bridge2 (H T Sl C α Γ : Type) [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T]
+    [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] [Fintype α] [DecidableEq α] where
+  /-- the invocation (for a block circuit `X`: `X.inv`). -/
+  I : Inv H T Sl C
+  /-- frame maps of the invocations of the five stages. -/
+  M0 : Γ → XMap H α
+  M1 : Γ → XMap H α
+  M2 : Γ → XMap H α
+  M3 : Γ → XMap H α
+  M4 : Γ → XMap H α
+  /-- the class `(g,t)` is served by the invocation `g` in stage 0 and `r_l t g` in stage `l`. -/
+  r1 : T → Γ ≃ Γ
+  r2 : T → Γ ≃ Γ
+  r3 : T → Γ ≃ Γ
+  r4 : T → Γ ≃ Γ
+  /-- the invocation `d` of stage `l` uses the helper set (and the copies) `s_l d`. -/
+  s1 : Γ ≃ Γ
+  s2 : Γ ≃ Γ
+  s3 : Γ ≃ Γ
+  s4 : Γ ≃ Γ
+  /-- inverse of the frame of the base, and frame of the window, of every invocation. -/
+  N1 : Γ → CMat α
+  W1 : Γ → CMat α
+  N2 : Γ → CMat α
+  W2 : Γ → CMat α
+  N3 : Γ → CMat α
+  W3 : Γ → CMat α
+  N4 : Γ → CMat α
+  W4 : Γ → CMat α
+  u : Γ → T → Space α
+  YF : Γ → T → CMat α
+  hm : 2 ≤ Fintype.card α
+  /-- prices of the idle climbs, for every price list.  B_2 with merged climbs:
+  `caX = caY = fun φ => bcost φ (2h-2)`, `cX1 = cY1 = fun φ => bcost φ (2h+2)`,
+  `cX2 = cY2 = fun φ => bcost φ 4`. -/
+  caX : (ℕ → ℝ) → ℝ
+  caY : (ℕ → ℝ) → ℝ
+  cX1 : (ℕ → ℝ) → ℝ
+  cY1 : (ℕ → ℝ) → ℝ
+  cX2 : (ℕ → ℝ) → ℝ
+  cY2 : (ℕ → ℝ) → ℝ
+  inv0 : ∀ g t, ∃ N : CMat α, (M0 g).Φ (tt (I.tv t)) * N = 1
+  z0 : ∀ g, (M0 g).Φ 0 = 1
+  hN1 : ∀ d, (M1 d).Φ 0 * N1 d = 1
+  hW1 : ∀ d, (M1 d).Φ 1 = W1 d * (M1 d).Φ 0
+  hN2 : ∀ d, (M2 d).Φ 0 * N2 d = 1
+  hW2 : ∀ d, (M2 d).Φ 1 = W2 d * (M2 d).Φ 0
+  hN3 : ∀ d, (M3 d).Φ 0 * N3 d = 1
+  hW3 : ∀ d, (M3 d).Φ 1 = W3 d * (M3 d).Φ 0
+  hN4 : ∀ d, (M4 d).Φ 0 * N4 d = 1
+  hW4 : ∀ d, (M4 d).Φ 1 = W4 d * (M4 d).Φ 0
+  /-- the bank frames a stage leaves are the bank frames the next stage starts from. -/
+  x01 : ∀ g t, (M0 g).Φ 1 = (M1 (r1 t g)).Φ (tt (I.tv t))
+  y01 : ∀ g t, (M0 g).Φ (1 + tt (I.tv t)) = (M1 (r1 t g)).Φ 0
+  x12 : ∀ g t, (M1 (r1 t g)).Φ 1 = (M2 (r2 t g)).Φ (tt (I.tv t))
+  y12 : ∀ g t, (M1 (r1 t g)).Φ (1 + tt (I.tv t)) = (M2 (r2 t g)).Φ 0
+  x23 : ∀ g t, (M2 (r2 t g)).Φ 1 = (M3 (r3 t g)).Φ (tt (I.tv t))
+  y23 : ∀ g t, (M2 (r2 t g)).Φ (1 + tt (I.tv t)) = (M3 (r3 t g)).Φ 0
+  x34 : ∀ g t, (M3 (r3 t g)).Φ 1 = (M4 (r4 t g)).Φ (tt (I.tv t))
+  y34 : ∀ g t, (M3 (r3 t g)).Φ (1 + tt (I.tv t)) = (M4 (r4 t g)).Φ 0
+  /-- the five windows of a helper set make up the kernel. -/
+  sfin : ∀ g, W4 (s4.symm g) * (W3 (s3.symm g) * (W2 (s2.symm g) *
+    (W1 (s1.symm g) * (M0 g).Φ 1))) = kernel α
+  /-- pair 2, idle in stages 0 and 1: ONE block from its start frame to the frame of its
+  twin after stage 1. -/
+  aX2 : ∀ g t, XReach ((M0 g).Φ (tt (I.tv t))) ((M1 (r1 t g)).Φ 1) caX
+  aY2 : ∀ g t, XReach ((M0 g).Φ 0) ((M1 (r1 t g)).Φ (1 + tt (I.tv t))) caY
+  /-- pair 1, idle in stages 3 and 4: ONE block from its frame after stage 2 to the end. -/
+  fX1 : ∀ g t, XReach ((M2 (r2 t g)).Φ 1) (kernel α) cX1
+  fY1 : ∀ g t, XReach ((M2 (r2 t g)).Φ (1 + tt (I.tv t))) (YF g t) cY1
+  /-- pair 2 after stage 4: ONE final block. -/
+  fX2 : ∀ g t, XReach ((M4 (r4 t g)).Φ 1) (kernel α) cX2
+  fY2 : ∀ g t, XReach ((M4 (r4 t g)).Φ (1 + tt (I.tv t))) (YF g t) cY2
+  term : ∀ g t, kernel α * (M0 g).Φ (tt (I.tv t)) = shift (u g t) * YF g t
+
+section Net
+variable {H T Sl C α Γ : Type} [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T]
+  [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] [Fintype α] [DecidableEq α]
+  [Fintype Γ] [DecidableEq Γ]
+variable (N : Bridge2 H T Sl C α Γ)
+
+/-- price of the blocks of one unit (one element of `Γ`): five invocations and the idle
+climbs of the four bank roles of every class. -/
+def Bridge2.unit (φ : ℕ → ℝ) : ℝ :=
+  5 * N.I.cost φ + (Fintype.card T : ℝ) * ((N.caX φ + N.caY φ)
+    + 2 * bcost φ (Fintype.card H - 1)
+    + (N.cX1 φ + N.cY1 φ) + (N.cX2 φ + N.cY2 φ))
+
+/-- scratch copies always enter and leave with the matrix `1` (also: the helper slots at the
+start, with `C := Sl`). -/
+def cOne (Γ C α : Type) [Fintype α] [DecidableEq α] : Γ → C → CMat α := fun _ _ => 1
+
+/-- the bank index of stage 0: the class `(g,t)` belongs to the invocation `g`. -/
+abbrev bid (Γ T : Type) : T → Γ ≃ Γ := fun _ => Equiv.refl Γ
+
+/-! bank frames of the class `(g,t)`: `xi l`, `yi l` at the START of stage `l`; `xo l`,
+`yo l` at the END of stage `l` (only where no next stage follows on that pair). -/
+def Bridge2.xi0 (g : Γ) (t : T) : CMat α := (N.M0 g).Φ (tt (N.I.tv t))
+def Bridge2.yi0 (g : Γ) (t : T) : CMat α := (N.M0 g).Φ 0
+def Bridge2.xi1 (g : Γ) (t : T) : CMat α := (N.M1 (N.r1 t g)).Φ (tt (N.I.tv t))
+def Bridge2.yi1 (g : Γ) (t : T) : CMat α := (N.M1 (N.r1 t g)).Φ 0
+def Bridge2.xi2 (g : Γ) (t : T) : CMat α := (N.M2 (N.r2 t g)).Φ (tt (N.I.tv t))
+def Bridge2.yi2 (g : Γ) (t : T) : CMat α := (N.M2 (N.r2 t g)).Φ 0
+def Bridge2.xo2 (g : Γ) (t : T) : CMat α := (N.M2 (N.r2 t g)).Φ 1
+def Bridge2.yo2 (g : Γ) (t : T) : CMat α := (N.M2 (N.r2 t g)).Φ (1 + tt (N.I.tv t))
+def Bridge2.xi3 (g : Γ) (t : T) : CMat α := (N.M3 (N.r3 t g)).Φ (tt (N.I.tv t))
+def Bridge2.yi3 (g : Γ) (t : T) : CMat α := (N.M3 (N.r3 t g)).Φ 0
+def Bridge2.xi4 (g : Γ) (t : T) : CMat α := (N.M4 (N.r4 t g)).Φ (tt (N.I.tv t))
+def Bridge2.yi4 (g : Γ) (t : T) : CMat α := (N.M4 (N.r4 t g)).Φ 0
+def Bridge2.xo4 (g : Γ) (t : T) : CMat α := (N.M4 (N.r4 t g)).Φ 1
+def Bridge2.yo4 (g : Γ) (t : T) : CMat α := (N.M4 (N.r4 t g)).Φ (1 + tt (N.I.tv t))
+
+/-- helper slots after the stages 0, 1, 2, 3, 4 (at the start: `cOne Γ Sl α`). -/
+def Bridge2.Z1 : Γ → Sl → CMat α := fun g _ => (N.M0 g).Φ 1
+def Bridge2.Z2 : Γ → Sl → CMat α := fun g q => N.W1 (N.s1.symm g) * N.Z1 g q
+def Bridge2.Z3 : Γ → Sl → CMat α := fun g q => N.W2 (N.s2.symm g) * N.Z2 g q
+def Bridge2.Z4 : Γ → Sl → CMat α := fun g q => N.W3 (N.s3.symm g) * N.Z3 g q
+def Bridge2.Z5 : Γ → Sl → CMat α := fun g q => N.W4 (N.s4.symm g) * N.Z4 g q
+
+/-- frames: at the start; after stage 0; after stage 1; after the first idle climb of pair 2
+(first gate layer here); after stage 2; after the second idle climb of pair 2 (second gate
+layer here); after stage 3; after stage 4; after the final climbs (third gate layer here). -/
+def Bridge2.S0 : BRole Γ T Sl C → CMat α := mkB N.xi0 N.yi0 N.xi0 N.yi0 (cOne Γ Sl α) (cOne Γ C α)
+def Bridge2.S1 : BRole Γ T Sl C → CMat α := mkB N.xi1 N.yi1 N.xi0 N.yi0 N.Z1 (cOne Γ C α)
+def Bridge2.S2 : BRole Γ T Sl C → CMat α := mkB N.xi2 N.yi2 N.xi0 N.yi0 N.Z2 (cOne Γ C α)
+def Bridge2.S3 : BRole Γ T Sl C → CMat α := mkB N.xi2 N.yi2 N.xi2 N.yi2 N.Z2 (cOne Γ C α)
+def Bridge2.S4 : BRole Γ T Sl C → CMat α := mkB N.xo2 N.yo2 N.xi2 N.yi2 N.Z3 (cOne Γ C α)
+def Bridge2.S5 : BRole Γ T Sl C → CMat α := mkB N.xo2 N.yo2 N.xi3 N.yi3 N.Z3 (cOne Γ C α)
+def Bridge2.S6 : BRole Γ T Sl C → CMat α := mkB N.xo2 N.yo2 N.xi4 N.yi4 N.Z4 (cOne Γ C α)
+def Bridge2.S7 : BRole Γ T Sl C → CMat α := mkB N.xo2 N.yo2 N.xo4 N.yo4 N.Z5 (cOne Γ C α)
+def Bridge2.S8 : BRole Γ T Sl C → CMat α :=
+  mkB (fun _ _ => kernel α) N.YF (fun _ _ => kernel α) N.YF (fun _ _ => kernel α) (cOne Γ C α)
+
+/-- stage 0: forward on pair 1, every helper set starts at the matrix `1`. -/
+lemma Bridge2.stage0 :
+    XRoute N.S0 N.S1 (lift1 GF) (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := by
+  have h := istage_fwd N.I (bid Γ T) (Equiv.refl Γ) N.M0 (fun _ => 1) (fun g => (N.M0 g).Φ 1)
+    (fun d => by rw [N.z0, Matrix.mul_one]) (fun d => by rw [N.z0, Matrix.mul_one])
+    (cOne Γ Sl α) (cOne Γ C α) (cOne Γ C α)
+  have e : bOut N.I.tv (bid Γ T) N.M0
+      (fun g (q : Sl) => (N.M0 ((Equiv.refl Γ).symm g)).Φ 1 * cOne Γ Sl α g q) (cOne Γ C α)
+      = mkS N.xi1 N.yi1 N.Z1 (cOne Γ C α) := by
+    funext r
+    rcases r with (((⟨g,t⟩|⟨g,t⟩)|⟨g,q⟩)|⟨g,c⟩)
+    · exact N.x01 g t
+    · exact N.y01 g t
+    · show (N.M0 g).Φ 1 * 1 = (N.M0 g).Φ 1
+      rw [Matrix.mul_one]
+    · rfl
+  have h' : XRoute (bIn N.I.tv (bid Γ T) N.M0 (cOne Γ Sl α) (cOne Γ C α))
+      (mkS N.xi1 N.yi1 N.Z1 (cOne Γ C α)) GF (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := by
+    rw [← e]
+    exact h
+  exact blift1 N.xi0 N.yi0 h'
+
+/-- stage 1: backward on pair 1; the helper set `g` serves the invocation `s1⁻¹ g`. -/
+lemma Bridge2.stage1 :
+    XRoute N.S1 N.S2 (lift1 GB) (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := by
+  have h := istage_bwd N.I N.r1 N.s1 N.M1 N.N1 N.W1 N.hN1 N.hW1 N.Z1 (cOne Γ C α) (cOne Γ C α)
+  have e : bOut N.I.tv N.r1 N.M1 (fun g q => N.W1 (N.s1.symm g) * N.Z1 g q) (cOne Γ C α)
+      = mkS N.xi2 N.yi2 N.Z2 (cOne Γ C α) := by
+    funext r
+    rcases r with (((⟨g,t⟩|⟨g,t⟩)|⟨g,q⟩)|⟨g,c⟩)
+    · exact N.x12 g t
+    · exact N.y12 g t
+    · rfl
+    · rfl
+  have h' : XRoute (bIn N.I.tv N.r1 N.M1 N.Z1 (cOne Γ C α))
+      (mkS N.xi2 N.yi2 N.Z2 (cOne Γ C α)) GB (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := by
+    rw [← e]
+    exact h
+  exact blift1 N.xi0 N.yi0 h'
+
+/-- stage 2: forward on pair 1. -/
+lemma Bridge2.stage2 :
+    XRoute N.S3 N.S4 (lift1 GF) (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) :=
+  blift1 N.xi2 N.yi2
+    (istage_fwd N.I N.r2 N.s2 N.M2 N.N2 N.W2 N.hN2 N.hW2 N.Z2 (cOne Γ C α) (cOne Γ C α))
+
+/-- stage 3: backward on pair 2. -/
+lemma Bridge2.stage3 :
+    XRoute N.S5 N.S6 (lift2 GB) (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := by
+  have h := istage_bwd N.I N.r3 N.s3 N.M3 N.N3 N.W3 N.hN3 N.hW3 N.Z3 (cOne Γ C α) (cOne Γ C α)
+  have e : bOut N.I.tv N.r3 N.M3 (fun g q => N.W3 (N.s3.symm g) * N.Z3 g q) (cOne Γ C α)
+      = mkS N.xi4 N.yi4 N.Z4 (cOne Γ C α) := by
+    funext r
+    rcases r with (((⟨g,t⟩|⟨g,t⟩)|⟨g,q⟩)|⟨g,c⟩)
+    · exact N.x34 g t
+    · exact N.y34 g t
+    · rfl
+    · rfl
+  have h' : XRoute (bIn N.I.tv N.r3 N.M3 N.Z3 (cOne Γ C α))
+      (mkS N.xi4 N.yi4 N.Z4 (cOne Γ C α)) GB (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := by
+    rw [← e]
+    exact h
+  exact blift2 N.xo2 N.yo2 h'
+
+/-- stage 4: forward on pair 2. -/
+lemma Bridge2.stage4 :
+    XRoute N.S6 N.S7 (lift2 GF) (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) :=
+  blift2 N.xo2 N.yo2
+    (istage_fwd N.I N.r4 N.s4 N.M4 N.N4 N.W4 N.hN4 N.hW4 N.Z4 (cOne Γ C α) (cOne Γ C α))
+
+/-- first idle climb of pair 2 (stages 0 and 1 in ONE block per role), to the frames of
+pair 1 after stage 1. -/
+lemma Bridge2.climbA :
+    XRoute N.S2 N.S3 id (fun φ => (Fintype.card Γ : ℝ) *
+      ((Fintype.card T : ℝ) * (N.caX φ + N.caY φ))) := by
+  have H1 := XRoute.reach_all N.S2 N.S3
+    (mkB (fun _ _ _ => 0) (fun _ _ _ => 0) (fun _ _ => N.caX)
+      (fun _ _ => N.caY) (fun _ _ _ => 0) (fun _ _ _ => 0)) (by
+      rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩)
+      · exact XReach.refl _
+      · exact XReach.refl _
+      · show XReach ((N.M0 g).Φ (tt (N.I.tv t))) ((N.M2 (N.r2 t g)).Φ (tt (N.I.tv t))) N.caX
+        rw [← N.x12 g t]
+        exact N.aX2 g t
+      · show XReach ((N.M0 g).Φ 0) ((N.M2 (N.r2 t g)).Φ 0) N.caY
+        rw [← N.y12 g t]
+        exact N.aY2 g t
+      · exact XReach.refl _
+      · exact XReach.refl _)
+  refine H1.cast (fun φ => ?_)
+  simp_rw [mkB_map (fun c : (ℕ → ℝ) → ℝ => c φ)]
+  rw [sum_mkB]
+  simp only [sum_const, card_univ, nsmul_eq_mul, mul_zero, add_zero, zero_add]
+  ring
+
+/-- second idle climb of pair 2: the window of stage 2, ONE block of rank `card H - 1` per
+role, to the frames of pair 1 after stage 2. -/
+lemma Bridge2.climbB :
+    XRoute N.S4 N.S5 id (fun φ => (Fintype.card Γ : ℝ) *
+      ((Fintype.card T : ℝ) * (2 * bcost φ (Fintype.card H - 1)))) := by
+  have H1 := XRoute.reach_all N.S4 N.S5
+    (mkB (fun _ _ _ => 0) (fun _ _ _ => 0) (fun _ _ φ => bcost φ (Fintype.card H - 1))
+      (fun _ _ φ => bcost φ (Fintype.card H - 1)) (fun _ _ _ => 0) (fun _ _ _ => 0)) (by
+      rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩)
+      · exact XReach.refl _
+      · exact XReach.refl _
+      · show XReach ((N.M2 (N.r2 t g)).Φ (tt (N.I.tv t))) ((N.M3 (N.r3 t g)).Φ (tt (N.I.tv t)))
+          (fun φ => bcost φ (Fintype.card H - 1))
+        rw [← N.x23 g t]
+        exact N.I.climbX (N.M2 (N.r2 t g)) t
+      · show XReach ((N.M2 (N.r2 t g)).Φ 0) ((N.M3 (N.r3 t g)).Φ 0)
+          (fun φ => bcost φ (Fintype.card H - 1))
+        rw [← N.y23 g t]
+        exact N.I.climbY (N.M2 (N.r2 t g)) t
+      · exact XReach.refl _
+      · exact XReach.refl _)
+  refine H1.cast (fun φ => ?_)
+  simp_rw [mkB_map (fun c : (ℕ → ℝ) → ℝ => c φ)]
+  rw [sum_mkB]
+  simp only [sum_const, card_univ, nsmul_eq_mul, mul_zero, add_zero, zero_add]
+  ring
+
+/-- the final climbs: pair 1 from its frames after stage 2 (ONE block per role for the
+stages 3, 4 and the rest), pair 2 from its frames after stage 4; the helper slots are
+already at the kernel. -/
+lemma Bridge2.fin :
+    XRoute N.S7 N.S8 id (fun φ => (Fintype.card Γ : ℝ) *
+      ((Fintype.card T : ℝ) * ((N.cX1 φ + N.cY1 φ) + (N.cX2 φ + N.cY2 φ)))) := by
+  have H1 := XRoute.reach_all N.S7 N.S8
+    (mkB (fun _ _ => N.cX1) (fun _ _ => N.cY1) (fun _ _ => N.cX2) (fun _ _ => N.cY2)
+      (fun _ _ _ => 0) (fun _ _ _ => 0)) (by
+      rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩)
+      · exact N.fX1 g t
+      · exact N.fY1 g t
+      · exact N.fX2 g t
+      · exact N.fY2 g t
+      · show XReach (N.W4 (N.s4.symm g) * (N.W3 (N.s3.symm g) * (N.W2 (N.s2.symm g) *
+          (N.W1 (N.s1.symm g) * (N.M0 g).Φ 1)))) (kernel α) (fun _ => 0)
+        rw [N.sfin g]
+        exact XReach.refl _
+      · exact XReach.refl _)
+  refine H1.cast (fun φ => ?_)
+  simp_rw [mkB_map (fun c : (ℕ → ℝ) → ℝ => c φ)]
+  rw [sum_mkB]
+  simp only [sum_const, card_univ, nsmul_eq_mul, mul_zero, add_zero, zero_add]
+  ring
+
+end Net
+end
+end BR
+end PowerSaving
+end OAI
+
+#print axioms OAI.PowerSaving.BR.Bridge2.stage0
+#print axioms OAI.PowerSaving.BR.Bridge2.stage1
+#print axioms OAI.PowerSaving.BR.Bridge2.stage2
+#print axioms OAI.PowerSaving.BR.Bridge2.stage3
+#print axioms OAI.PowerSaving.BR.Bridge2.stage4
+#print axioms OAI.PowerSaving.BR.Bridge2.climbA
+#print axioms OAI.PowerSaving.BR.Bridge2.climbB
+#print axioms OAI.PowerSaving.BR.Bridge2.fin
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/Work__Carrier__Live.lean b/research/five-stage-source527-banks/inputs/complex/sources/Work__Carrier__Live.lean
new file mode 100644
index 0000000000..b8306527d6
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/Work__Carrier__Live.lean
@@ -0,0 +1,232 @@
+import Work.Carrier.Pack
+import Work.Carrier.Ring
+
+/-!
+# (key: carrier-thm) The live roles of an invocation, and the four dirt gates as matrices
+
+`L3 T Sl = (T ⊕ T) ⊕ Sl`: the bank roles `x` (`lX`), `y` (`lY`) and the helper slots (`lS`), all
+on the same footing (no scratch copies).  `PX`, `PY`: the coordinate projectors.
+
+For the matrix `M` of all gates of the main phase and its inverse `Mi`:
+
+    g1 M  = 1 - PY * M * PS      `y -= K s`            g1i M = 1 + PY * M * PS   (its inverse)
+    g8 Mi = PX + PY + PS * Mi    slot rows of `M⁻¹`    h8 M  = PX + PY + PS * M  (its inverse)
+
+(`PS = 1 - PX - PY`).  This file: unit rows / columns of these gates (hence between which roles
+their entries can be nonzero), the scatter with the copies eliminated (`scatM`), and the scalar
+maps of `1 + PY * M * PX` and of the transpose of `1 - PY * M * PX`.
+
+No `sorry`.
+-/
+
+set_option linter.unusedSectionVars false
+set_option linter.unusedVariables false
+
+namespace OAI
+namespace PowerSaving
+namespace CR
+open Binary Matrix Finset RAM SS CB BR
+noncomputable section
+
+section Diag
+variable {ρ : Type*} [Fintype ρ] [DecidableEq ρ]
+
+lemma col_zero_of {d : ρ → ℚ} {G : Matrix ρ ρ ℚ}
+    (h : G * Matrix.diagonal d = Matrix.diagonal d) {i j : ρ} (hj : d j = 1) (hij : i ≠ j) :
+    G i j = 0 := by
+  have e := congrFun (congrFun h i) j
+  rw [Matrix.mul_diagonal, Matrix.diagonal_apply_ne _ hij, hj, mul_one] at e
+  exact e
+
+lemma row_zero_of {d : ρ → ℚ} {G : Matrix ρ ρ ℚ}
+    (h : Matrix.diagonal d * G = Matrix.diagonal d) {i j : ρ} (hi : d i = 1) (hij : i ≠ j) :
+    G i j = 0 := by
+  have e := congrFun (congrFun h i) j
+  rw [Matrix.diagonal_mul, Matrix.diagonal_apply_ne _ hij, hi, one_mul] at e
+  exact e
+
+lemma diag_mul_of_rows {d : ρ → ℚ} {G : Matrix ρ ρ ℚ}
+    (h : ∀ i j, d i ≠ 0 → G i j = if i = j then 1 else 0) :
+    Matrix.diagonal d * G = Matrix.diagonal d := by
+  ext i j
+  rw [Matrix.diagonal_mul]
+  by_cases hd : d i = 0
+  · rw [hd, zero_mul]
+    by_cases hij : i = j
+    · subst hij; rw [Matrix.diagonal_apply_eq, hd]
+    · rw [Matrix.diagonal_apply_ne _ hij]
+  · rw [h i j hd]
+    by_cases hij : i = j
+    · subst hij; rw [if_pos rfl, mul_one, Matrix.diagonal_apply_eq]
+    · rw [if_neg hij, mul_zero, Matrix.diagonal_apply_ne _ hij]
+
+lemma mul_diag_of_cols {d : ρ → ℚ} {G : Matrix ρ ρ ℚ}
+    (h : ∀ i j, d j ≠ 0 → G i j = if i = j then 1 else 0) :
+    G * Matrix.diagonal d = Matrix.diagonal d := by
+  ext i j
+  rw [Matrix.mul_diagonal]
+  by_cases hd : d j = 0
+  · rw [hd, mul_zero]
+    by_cases hij : i = j
+    · subst hij; rw [Matrix.diagonal_apply_eq, hd]
+    · rw [Matrix.diagonal_apply_ne _ hij]
+  · rw [h i j hd]
+    by_cases hij : i = j
+    · subst hij; rw [if_pos rfl, one_mul, Matrix.diagonal_apply_eq]
+    · rw [if_neg hij, zero_mul, Matrix.diagonal_apply_ne _ hij]
+
+/-- scalar map of a diagonal gate matrix. -/
+lemma act_diag (d : ρ → ℚ) (u : ρ → ℂ) :
+    actPoint (Matrix.diagonal d) u = fun i => ((d i : ℚ) : ℂ) * u i := by
+  funext i
+  show ∑ j, ((Matrix.diagonal d i j : ℚ) : ℂ) * u j = ((d i : ℚ) : ℂ) * u i
+  rw [Finset.sum_eq_single i]
+  · rw [Matrix.diagonal_apply_eq]
+  · intro j _ hj
+    rw [Matrix.diagonal_apply_ne _ (Ne.symm hj), Rat.cast_zero, zero_mul]
+  · intro h
+    exact absurd (Finset.mem_univ i) h
+
+lemma actPoint_mul (A B : Matrix ρ ρ ℚ) (u : ρ → ℂ) :
+    actPoint (A * B) u = actPoint A (actPoint B u) :=
+  ap_mul A B u
+
+end Diag
+
+section Live
+variable {T Sl : Type} [Fintype T] [DecidableEq T] [Fintype Sl] [DecidableEq Sl]
+
+/-- live roles of one invocation: `x`, `y`, helper slots. -/
+abbrev L3 (T Sl : Type) := (T ⊕ T) ⊕ Sl
+abbrev lX (t : T) : L3 T Sl := Sum.inl (Sum.inl t)
+abbrev lY (t : T) : L3 T Sl := Sum.inl (Sum.inr t)
+abbrev lS (q : Sl) : L3 T Sl := Sum.inr q
+
+/-- indicator of the x roles. -/
+def iX : L3 T Sl → ℚ := Sum.elim (Sum.elim (fun _ => 1) (fun _ => 0)) (fun _ => 0)
+/-- indicator of the y roles. -/
+def iY : L3 T Sl → ℚ := Sum.elim (Sum.elim (fun _ => 0) (fun _ => 1)) (fun _ => 0)
+
+def PX : Matrix (L3 T Sl) (L3 T Sl) ℚ := Matrix.diagonal iX
+def PY : Matrix (L3 T Sl) (L3 T Sl) ℚ := Matrix.diagonal iY
+
+lemma iX_cases (i : L3 T Sl) : iX i = 0 ∨ iX i = 1 := by
+  rcases i with ((t|t)|q)
+  · exact Or.inr rfl
+  · exact Or.inl rfl
+  · exact Or.inl rfl
+
+lemma iY_cases (i : L3 T Sl) : iY i = 0 ∨ iY i = 1 := by
+  rcases i with ((t|t)|q)
+  · exact Or.inl rfl
+  · exact Or.inr rfl
+  · exact Or.inl rfl
+
+lemma PX_sq : (PX : Matrix (L3 T Sl) (L3 T Sl) ℚ) * PX = PX := by
+  unfold PX
+  have e : (fun i : L3 T Sl => iX i * iX i) = iX := by
+    funext i
+    rcases i with ((t|t)|q) <;> simp [iX]
+  rw [Matrix.diagonal_mul_diagonal, e]
+
+lemma PY_sq : (PY : Matrix (L3 T Sl) (L3 T Sl) ℚ) * PY = PY := by
+  unfold PY
+  have e : (fun i : L3 T Sl => iY i * iY i) = iY := by
+    funext i
+    rcases i with ((t|t)|q) <;> simp [iY]
+  rw [Matrix.diagonal_mul_diagonal, e]
+
+lemma PX_PY : (PX : Matrix (L3 T Sl) (L3 T Sl) ℚ) * PY = 0 := by
+  unfold PX PY
+  rw [Matrix.diagonal_mul_diagonal]
+  have e : (fun i : L3 T Sl => iX i * iY i) = fun _ => 0 := by
+    funext i
+    rcases i with ((t|t)|q) <;> simp [iX, iY]
+  rw [e]
+  exact Matrix.diagonal_zero
+
+lemma PY_PX : (PY : Matrix (L3 T Sl) (L3 T Sl) ℚ) * PX = 0 := by
+  unfold PX PY
+  rw [Matrix.diagonal_mul_diagonal]
+  have e : (fun i : L3 T Sl => iY i * iX i) = fun _ => 0 := by
+    funext i
+    rcases i with ((t|t)|q) <;> simp [iX, iY]
+  rw [e]
+  exact Matrix.diagonal_zero
+
+lemma PX_T : (PX : Matrix (L3 T Sl) (L3 T Sl) ℚ)ᵀ = PX := Matrix.diagonal_transpose _
+lemma PY_T : (PY : Matrix (L3 T Sl) (L3 T Sl) ℚ)ᵀ = PY := Matrix.diagonal_transpose _
+
+/-- "no gate into an x role": the x rows are unit rows. -/
+lemma PX_mul_of {M : Matrix (L3 T Sl) (L3 T Sl) ℚ}
+    (hx : ∀ t j, M (lX t) j = if lX t = j then 1 else 0) : PX * M = PX := by
+  unfold PX
+  apply diag_mul_of_rows
+  intro i j hi
+  rcases i with ((t|t)|q)
+  · exact hx t j
+  · exact absurd rfl hi
+  · exact absurd rfl hi
+
+/-- "no gate reads a y role": the y columns are unit columns. -/
+lemma mul_PY_of {M : Matrix (L3 T Sl) (L3 T Sl) ℚ}
+    (hy : ∀ i t, M i (lY t) = if i = lY t then 1 else 0) : M * PY = PY := by
+  unfold PY
+  apply mul_diag_of_cols
+  intro i j hj
+  rcases j with ((t|t)|q)
+  · exact absurd rfl hj
+  · exact hy i t
+  · exact absurd rfl hj
+
+/-! ### the four dirt gates -/
+
+def g1 (M : Matrix (L3 T Sl) (L3 T Sl) ℚ) : Matrix (L3 T Sl) (L3 T Sl) ℚ :=
+  1 - PY * M * (1 - PX - PY)
+def g1i (M : Matrix (L3 T Sl) (L3 T Sl) ℚ) : Matrix (L3 T Sl) (L3 T Sl) ℚ :=
+  1 + PY * M * (1 - PX - PY)
+def g8 (Mi : Matrix (L3 T Sl) (L3 T Sl) ℚ) : Matrix (L3 T Sl) (L3 T Sl) ℚ :=
+  PX + PY + (1 - PX - PY) * Mi
+def h8 (M : Matrix (L3 T Sl) (L3 T Sl) ℚ) : Matrix (L3 T Sl) (L3 T Sl) ℚ :=
+  PX + PY + (1 - PX - PY) * M
+
+/-- a gate with unit x rows and unit x columns joins two different roles only if neither is an
+x role. -/
+lemma offdiag_x {G : Matrix (L3 T Sl) (L3 T Sl) ℚ} (h1 : G * PX = PX) (h2 : PX * G = PX)
+    {i j : L3 T Sl} (hij : G i j ≠ 0) (e : i ≠ j) : iX i = 0 ∧ iX j = 0 := by
+  constructor
+  · by_contra h
+    exact hij (row_zero_of h2 ((iX_cases i).resolve_left h) e)
+  · by_contra h
+    exact hij (col_zero_of h1 ((iX_cases j).resolve_left h) e)
+
+/-- a gate with unit x rows, unit y rows and unit y columns: a nonzero entry off the diagonal
+is in a slot row and not in a y column. -/
+lemma offdiag_s {G : Matrix (L3 T Sl) (L3 T Sl) ℚ} (hx : PX * G = PX) (hy : PY * G = PY)
+    (hyc : G * PY = PY) {i j : L3 T Sl} (hij : G i j ≠ 0) (e : i ≠ j) :
+    iX i = 0 ∧ iY i = 0 ∧ iY j = 0 := by
+  refine ⟨?_, ?_, ?_⟩
+  · by_contra h
+    exact hij (row_zero_of hx ((iX_cases i).resolve_left h) e)
+  · by_contra h
+    exact hij (row_zero_of hy ((iY_cases i).resolve_left h) e)
+  · by_contra h
+    exact hij (col_zero_of hyc ((iY_cases j).resolve_left h) e)
+
+lemma g1_x (M : Matrix (L3 T Sl) (L3 T Sl) ℚ) : g1 M * PX = PX ∧ PX * g1 M = PX := by
+  unfold g1
+  constructor
+  · rw [sub_mul, one_mul, side_n_right (PX_sq (T := T) (Sl := Sl)) PY_PX, sub_zero]
+  · rw [mul_sub, mul_one, side_n_left (PX_PY (T := T) (Sl := Sl)), sub_zero]
+
+lemma g1i_x (M : Matrix (L3 T Sl) (L3 T Sl) ℚ) : g1i M * PX = PX ∧ PX * g1i M = PX := by
+  unfold g1i
+  constructor
+  · rw [add_mul, one_mul, side_n_right (PX_sq (T := T) (Sl := Sl)) PY_PX, add_zero]
+  · rw [mul_add, mul_one, side_n_left (PX_PY (T := T) (Sl := Sl)), add_zero]
+
+end Live
+end
+end CR
+end PowerSaving
+end OAI
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/Work__Carrier__Phases.lean b/research/five-stage-source527-banks/inputs/complex/sources/Work__Carrier__Phases.lean
new file mode 100644
index 0000000000..305da15b5d
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/Work__Carrier__Phases.lean
@@ -0,0 +1,192 @@
+import Work.Carrier.Glue
+
+/-!
+# (key: carrier-thm) A carrier circuit given by its phases: hypotheses and scalar identities
+
+`Phased H T Sl C` is the hypothesis structure of the carrier invocation theorem in the general
+scope (any ordered mix of gates `(x, slot)`, `(slot, slot)`, `(x, y)`, `(slot, y)`):
+
+* phase A (before the scatter), phase B (after it) and the final climbs F are exact block routes
+  on the LIVE roles `L3 T Sl` (`x`, `y`, slots: all on the same footing), for every block frame
+  map, with the gate products `MA`, `MB` (and the transposed inverses for the backward word);
+* the scatter goes through the scratch copies: `c += Cc s` at the labels `cen`, the copies come
+  down, `y += Jr c` at 0;
+* `hx`: no gate INTO an x role, `hy`: no gate READS a y role, `hid`: the y roles receive exactly
+  the x roles — three facts about the ONE matrix `Mt = MB * scatM (Jr * Cc) * MA`.
+
+This file: `Mt`, its inverse `Mti`, and the scalar identities of the forward and of the backward
+word (`fwd_scalar`, `bwd_scalar`): with the two dirt gates `g1`, `g8` (resp. the transposes of
+`g1i`, `h8`) the word does `y += x` (resp. `x -= y`) for ARBITRARY slot contents.
+
+No `sorry`.
+-/
+
+set_option linter.unusedSectionVars false
+set_option linter.unusedVariables false
+
+namespace OAI
+namespace PowerSaving
+namespace CR
+open Binary Matrix Finset RAM SS CB BR
+noncomputable section
+
+/-- labels of the live roles at the START of an invocation. -/
+def lab3 {H T Sl : Type} (tv : T → Space H) : L3 T Sl → Lbl H :=
+  Sum.elim (Sum.elim (fun t => lineL (tv t)) (fun _ => zeroL)) (fun _ => zeroL)
+
+/-- **The hypotheses of the carrier invocation theorem, by phases.**  See the file header. -/
+structure Phased (H T Sl C : Type) [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T]
+    [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] where
+  tv : T → Space H
+  kx : ∀ t, Climb (lineL (tv t)) fullL
+  ky : ∀ t, Climb zeroL (perpL (tv t))
+  cen : C → Lbl H
+  kc : ∀ k, Climb zeroL (cen k)
+  Cc : Matrix C Sl ℚ
+  Jr : Matrix T C ℚ
+  labA : L3 T Sl → Lbl H
+  labB : L3 T Sl → Lbl H
+  labF : L3 T Sl → Lbl H
+  hAy : ∀ t, (labA (lY t)).P = 0
+  hAc : ∀ k q, Cc k q ≠ 0 → (cen k).P = (labA (lS q)).P
+  hFx : ∀ t, (labF (lX t)).P = 1
+  hFy : ∀ t, (labF (lY t)).P = 1 + tt (tv t)
+  hFs : ∀ q, (labF (lS q)).P = 1
+  MA : Matrix (L3 T Sl) (L3 T Sl) ℚ
+  MAi : Matrix (L3 T Sl) (L3 T Sl) ℚ
+  MB : Matrix (L3 T Sl) (L3 T Sl) ℚ
+  MBi : Matrix (L3 T Sl) (L3 T Sl) ℚ
+  hAi : MAi * MA = 1
+  hBi : MBi * MB = 1
+  hx : ∀ t j, (MB * scatM (Jr * Cc) * MA) (lX t) j = if lX t = j then 1 else 0
+  hy : ∀ i t, (MB * scatM (Jr * Cc) * MA) i (lY t) = if i = lY t then 1 else 0
+  hid : ∀ S t, (MB * scatM (Jr * Cc) * MA) (lY S) (lX t) = if S = t then 1 else 0
+  rA : List ℕ
+  rB : List ℕ
+  rF : List ℕ
+  pA : ∀ {α : Type} [Fintype α] [DecidableEq α] (Xm : XMap H α),
+    XRoute (fun r => Xm.Φ (lab3 tv r).P) (fun r => Xm.Φ (labA r).P) (actPoint MA)
+        (fun φ => rcost φ rA) ∧
+    XRoute (fun r => Xm.Φ (lab3 tv r).P) (fun r => Xm.Φ (labA r).P) (actPoint MAiᵀ)
+        (fun φ => rcost φ rA)
+  pB : ∀ {α : Type} [Fintype α] [DecidableEq α] (Xm : XMap H α),
+    XRoute (fun r => Xm.Φ (labA r).P) (fun r => Xm.Φ (labB r).P) (actPoint MB)
+        (fun φ => rcost φ rB) ∧
+    XRoute (fun r => Xm.Φ (labA r).P) (fun r => Xm.Φ (labB r).P) (actPoint MBiᵀ)
+        (fun φ => rcost φ rB)
+  pF : ∀ {α : Type} [Fintype α] [DecidableEq α] (Xm : XMap H α),
+    XRoute (fun r => Xm.Φ (labB r).P) (fun r => Xm.Φ (labF r).P) id (fun φ => rcost φ rF)
+
+section Alg
+variable {H T Sl C : Type} [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T]
+  [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C]
+
+namespace Phased
+variable (K : Phased H T Sl C)
+
+/-- the product of ALL gates of the main phase (scatter with the copies eliminated). -/
+def Mt : Matrix (L3 T Sl) (L3 T Sl) ℚ := K.MB * scatM (K.Jr * K.Cc) * K.MA
+/-- its inverse. -/
+def Mti : Matrix (L3 T Sl) (L3 T Sl) ℚ := K.MAi * scatMi (K.Jr * K.Cc) * K.MBi
+
+/-- price of one invocation: the blocks of the three phases and one block per copy. -/
+def cost (φ : ℕ → ℝ) : ℝ :=
+  rcost φ K.rA + (∑ k, bcost φ (K.cen k).d) + rcost φ K.rB + rcost φ K.rF
+
+lemma im : K.Mti * K.Mt = 1 := by
+  unfold Mti Mt
+  have hB : ∀ X : Matrix (L3 T Sl) (L3 T Sl) ℚ, K.MBi * (K.MB * X) = X := fun X => by
+    rw [← Matrix.mul_assoc, K.hBi, Matrix.one_mul]
+  have hS : ∀ X : Matrix (L3 T Sl) (L3 T Sl) ℚ,
+      scatMi (K.Jr * K.Cc) * (scatM (K.Jr * K.Cc) * X) = X := fun X => by
+    rw [← Matrix.mul_assoc, scat_inv, Matrix.one_mul]
+  rw [Matrix.mul_assoc, Matrix.mul_assoc, Matrix.mul_assoc, hB, hS, K.hAi]
+
+lemma mi : K.Mt * K.Mti = 1 := mul_eq_one_comm.mp K.im
+
+lemma hxm : PX * K.Mt = PX := PX_mul_of (M := K.Mt) K.hx
+lemma hmy : K.Mt * PY = PY := mul_PY_of (M := K.Mt) K.hy
+lemma hiy : K.Mti * PY = PY := dirt_iy K.hmy K.im
+
+/-- **the dirt rule, forward**, for the matrices of the circuit. -/
+lemma hF : g8 K.Mti * K.Mt * g1 K.Mt = 1 + PY * K.Mt * PX := by
+  unfold g8 g1
+  exact dirt_fwd PX_sq PY_sq PX_PY PY_PX K.hxm K.hmy K.im K.mi
+
+/-- **the dirt rule, backward.** -/
+lemma hB : g1i K.Mt * K.Mti * h8 K.Mt = 1 - PY * K.Mt * PX := by
+  unfold g1i h8
+  exact dirt_bwd PX_sq PY_sq PX_PY PY_PX K.hxm K.hmy K.im K.mi
+
+/-- live part of the forward word before the scatter: `y -= K s`, then phase A. -/
+def f1 (u : L3 T Sl → ℂ) : L3 T Sl → ℂ := actPoint K.MA (actPoint (g1 K.Mt) u)
+/-- after the scatter: phase B, then the slot rows of the inverse. -/
+def f2 (u : L3 T Sl → ℂ) : L3 T Sl → ℂ := actPoint (g8 K.Mti) (actPoint K.MB u)
+/-- the same for the backward word (transposed inverses). -/
+def b1 (u : L3 T Sl → ℂ) : L3 T Sl → ℂ := actPoint K.MAiᵀ (actPoint (g1i K.Mt)ᵀ u)
+def b2 (u : L3 T Sl → ℂ) : L3 T Sl → ℂ := actPoint (h8 K.Mt)ᵀ (actPoint K.MBiᵀ u)
+
+lemma fwd_live (u : L3 T Sl → ℂ) :
+    K.f2 (actPoint (scatM (K.Jr * K.Cc)) (K.f1 u)) = actPoint (1 + PY * K.Mt * PX) u := by
+  rw [← K.hF]
+  show actPoint (g8 K.Mti) (actPoint K.MB (actPoint (scatM (K.Jr * K.Cc))
+      (actPoint K.MA (actPoint (g1 K.Mt) u))))
+    = actPoint (g8 K.Mti * (K.MB * scatM (K.Jr * K.Cc) * K.MA) * g1 K.Mt) u
+  rw [actPoint_mul, actPoint_mul, actPoint_mul, actPoint_mul]
+
+lemma bwd_live (u : L3 T Sl → ℂ) :
+    K.b2 (actPoint ((scatMi (K.Jr * K.Cc))ᵀ) (K.b1 u))
+      = actPoint ((1 - PY * K.Mt * PX)ᵀ) u := by
+  rw [← K.hB]
+  show actPoint (h8 K.Mt)ᵀ (actPoint K.MBiᵀ (actPoint (scatMi (K.Jr * K.Cc))ᵀ
+      (actPoint K.MAiᵀ (actPoint (g1i K.Mt)ᵀ u))))
+    = actPoint ((g1i K.Mt * (K.MAi * scatMi (K.Jr * K.Cc) * K.MBi) * h8 K.Mt)ᵀ) u
+  rw [Matrix.transpose_mul, Matrix.transpose_mul, Matrix.transpose_mul, Matrix.transpose_mul,
+    actPoint_mul, actPoint_mul, actPoint_mul, actPoint_mul]
+
+lemma liveAct_glue (φ : (L3 T Sl → ℂ) → (L3 T Sl → ℂ)) (u : L3 T Sl → ℂ) (c : C → ℂ) :
+    liveAct φ (glue u c) = glue (φ u) c :=
+  congrArg (fun v => glue (φ v) c) (glue_up u c)
+
+/-- **Scalar identity of the forward word**: `y += x`, slots restored, for ARBITRARY slot
+contents (the copies are erased before and after). -/
+theorem fwd_scalar (z : Box T Sl C → ℂ) :
+    eraseC (liveAct K.f2 (gateS bY bC K.Jr (gateS bC bS K.Cc (liveAct K.f1 (eraseC z)))))
+      = shearF z := by
+  have h1 : liveAct K.f1 (eraseC z) = glue (K.f1 (z ∘ upE)) (fun _ => 0) := by
+    rw [erase_glue z, liveAct_glue]
+  have hs : gateS bY bC K.Jr (gateS bC bS K.Cc (glue (K.f1 (z ∘ upE)) (fun _ => 0))) ∘ upE
+      = actPoint (scatM (K.Jr * K.Cc)) (K.f1 (z ∘ upE)) := by
+    have h := scat_glue K.Jr K.Cc (glue (K.f1 (z ∘ upE)) (fun _ => 0)) rfl
+    rw [glue_up] at h
+    exact h
+  rw [h1]
+  unfold liveAct
+  rw [erase_of_glue, hs, K.fwd_live, act_shear K.Mt K.hid]
+  exact glue_shearF z
+
+/-- **Scalar identity of the backward word**: `x -= y`, slots restored. -/
+theorem bwd_scalar (z : Box T Sl C → ℂ) :
+    eraseC (liveAct K.b2 (gateS bS bC (-K.Ccᵀ) (gateS bC bY K.Jrᵀ (liveAct K.b1 (eraseC z)))))
+      = shearB z := by
+  have h1 : liveAct K.b1 (eraseC z) = glue (K.b1 (z ∘ upE)) (fun _ => 0) := by
+    rw [erase_glue z, liveAct_glue]
+  have hs : gateS bS bC (-K.Ccᵀ) (gateS bC bY K.Jrᵀ (glue (K.b1 (z ∘ upE)) (fun _ => 0))) ∘ upE
+      = actPoint ((scatMi (K.Jr * K.Cc))ᵀ) (K.b1 (z ∘ upE)) := by
+    have h := scat_glueT K.Jr K.Cc (glue (K.b1 (z ∘ upE)) (fun _ => 0)) rfl
+    rw [glue_up] at h
+    exact h
+  rw [h1]
+  unfold liveAct
+  rw [erase_of_glue, hs, K.bwd_live, act_shearT K.Mt K.hid]
+  exact glue_shearB z
+
+end Phased
+end Alg
+end
+end CR
+end PowerSaving
+end OAI
+
+#print axioms OAI.PowerSaving.CR.Phased.fwd_scalar
+#print axioms OAI.PowerSaving.CR.Phased.bwd_scalar
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/Work__Fourier__Main.lean b/research/five-stage-source527-banks/inputs/complex/sources/Work__Fourier__Main.lean
new file mode 100644
index 0000000000..9aa06b5bf2
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/Work__Fourier__Main.lean
@@ -0,0 +1,176 @@
+import Work.Fourier.Asymptotics
+import OAI.Computability.FourierTransform.RegisterBounds
+import Work.Fourier.Goal
+
+/-
+Modified copy of lean/OAI/Computability/FourierTransform/Main.lean of github.com/openai/math
+(commit fd4aeeb, Apache-2.0), generated by `tools/fourier/mkchain.py`. OpenAI's file itself is
+unmodified in this repository and still imported. Changed for this repository, all mechanically:
+  * every declaration of this file (and of the other 10 copied files) has the suffix `Z`
+    on its name, so that it does not collide with OpenAI's original;
+  * OpenAI's exponent `alpha = 1 - 2/10^11` and envelope `hills` are replaced by
+    `alphaZ = 1 - 7474547/10^10` and `hillsZ` (Work/Fourier/Seam.lean), and the single use of
+    OpenAI's `hills_program` by `hillsZ_program`;
+  * `decimalExponent`, `decimalTime`, `TimeBounds`, `DFTGoal`, `ConvGoal` are replaced by their
+    `Z` versions from Work/Fourier/Goal.lean (exponent `1 - 7474546/10^10`);
+  * imports point to the copies.
+  * literal `let d : ℝ := 2/(10:ℝ)^11` -> `let d : ℝ := 7474547/(10:ℝ)^10` (alpha_thetaZ: the local constant d must equal 1 - alphaZ)
+  * `theorem transform_mainZ : DFTGoalZ :=`: docstring added
+  * `theorem convolution_mainZ : ConvGoalZ :=`: docstring added
+Comments and docstrings below are OpenAI's and describe OpenAI's network, not ours.
+-/
+
+namespace OAI
+
+/-!
+Exact uniform Fourier transforms and convolution.
+
+The construction formalizes the exact arithmetic algorithms of
+*An explicit power saving for the exact discrete Fourier transform* (OpenAI).
+The finite phase network originates in the complex construction of
+*Integer multiplication below n log n*.
+
+`transform_mainZ` and `convolution_mainZ` give fixed finite RAM programs, including
+preparation and index work, polynomial storage bounds, and a canonical root whose
+order is computed from the input length. Each includes the paper exponent, decimal
+power saving, and little-o bound. The `_order` variants also specify the root order.
+
+The implementation uses a smaller concrete tensor network (32 binary coordinates
+per factor, 4060 triples, and a `2^50` role envelope) with a stronger saving that
+implies the stated paper constants. The general rational-algebra extraction,
+symbolic certificate search, circuit printer, and bounded-printer uniformization
+from the synthesis appendix are outside this formalization.
+-/
+namespace PowerSaving
+open RAM Bench Selection Filter Asymptotics Real
+noncomputable section
+namespace Skies
+
+lemma multiplier_rangeZ : 1 < paperMultiplier ∧ paperMultiplier < paperM := by
+  norm_num [paperMultiplier,paperM,paperDelta,paperW]
+
+lemma theta_ltZ : theta < 1 := by
+  have h := multiplier_rangeZ
+  have hg : 0<log paperM := Real.log_pos (lt_trans h.1 h.2)
+  rw [theta,div_lt_iff₀ hg,one_mul]
+  exact Real.log_lt_log (by linarith) h.2
+
+lemma alpha_thetaZ : alphaZ < theta := by
+  let d : ℝ := 7474547/(10:ℝ)^10
+  have hp : 0<d := by dsimp [d]; norm_num
+  have hm : 2 ≤ paperM := by norm_num [paperM]
+  have hx : 0<paperMultiplier := lt_trans zero_lt_one multiplier_rangeZ.1
+  have hu : 0<paperM := by linarith
+  have h2 := (Real.log_le_log (by norm_num : 0 < (2:ℝ)) hm)
+  have he := log2_minZ
+  have hh : 0<log paperM := by linarith
+  have hk := Real.log_le_sub_one_of_pos (show 0<paperM/paperMultiplier by positivity)
+  rw [log_div hu.ne' hx.ne'] at hk
+  have H : paperM/paperMultiplier-1 < d*(2/3) := by
+    norm_num [d,paperM,paperMultiplier,paperW,paperDelta]
+  rw [theta,lt_div_iff₀ hh]
+  change (1-d)*log paperM < log paperMultiplier
+  have H' := mul_le_mul_of_nonneg_left (he.trans h2) hp.le
+  have hy := hk.trans_lt (lt_of_lt_of_le H H')
+  rw [sub_mul,one_mul]
+  linarith only [hy]
+
+lemma decimal_rangeZ : alphaZ < decimalExponentZ ∧ decimalExponentZ < 1 := by
+  norm_num [decimalExponentZ,alphaZ]
+
+def speedZ (v : ℝ) (n : ℕ) := (n:ℝ)*(TmZ n)^v
+
+lemma flight_littleZ {v : ℝ} (h : alphaZ < v) :
+    flightZ =o[atTop] speedZ v := by
+  rw [isLittleO_iff]
+  intro c hc
+  filter_upwards [TtopZ.eventually_ge_atTop 1,dwarfZ TtopZ 2
+    (p:=v-alphaZ) (sub_pos.mpr h) hc] with n hn hh
+  have he : 0<TmZ n := by linarith
+  change ‖(n:ℝ)*TmZ n^alphaZ*QmZ n^2‖ ≤ c*‖(n:ℝ)*TmZ n^v‖
+  rw [Real.norm_of_nonneg (by positivity),Real.norm_of_nonneg (by positivity)]
+  calc
+    _ ≤ ((n:ℝ)*TmZ n^alphaZ)*(c*TmZ n^(v-alphaZ)) :=
+      mul_le_mul_of_nonneg_left hh (by positivity)
+    _ = c*((n:ℝ)*(TmZ n^alphaZ*TmZ n^(v-alphaZ))) := by ring
+    _ = _ := by rw [← rpow_add he, show alphaZ + (v-alphaZ)=v by ring] -- a+(v-a)
+
+lemma speed_littleZ {u v : ℝ} (h : u < v) :
+    speedZ u =o[atTop] speedZ v := by
+  rw [isLittleO_iff]
+  intro c hc
+  filter_upwards [TtopZ.eventually_ge_atTop 1,dwarfZ TtopZ 0
+    (p:=v-u) (sub_pos.mpr h) hc] with n hn hh
+  have he : 0<TmZ n := by linarith
+  change ‖(n:ℝ)*TmZ n^u‖ ≤ c*‖(n:ℝ)*TmZ n^v‖
+  rw [Real.norm_of_nonneg (by positivity),Real.norm_of_nonneg (by positivity)]
+  have hh' : 1 ≤ c * TmZ n^(v-u) := by simpa only [pow_zero] using hh
+  calc
+    _ = ((n:ℝ)*TmZ n^u)*1 := by ring
+    _ ≤ ((n:ℝ)*TmZ n^u)*(c*TmZ n^(v-u)) := mul_le_mul_of_nonneg_left hh' (by positivity)
+    _ = c*((n:ℝ)*(TmZ n^u*TmZ n^(v-u))) := by ring
+    _ = _ := by rw [← rpow_add he, show u+(v-u)=v by ring]
+
+theorem decimal_littleZ : decimalTimeZ =o[atTop] nlogn := by
+  have H := speed_littleZ decimal_rangeZ.2
+  have h : speedZ 1 = nlogn := funext fun n => by rw [speedZ,Real.rpow_one]; rfl
+  rw [h] at H; exact H
+
+lemma speed_paperZ : speedZ theta =O[atTop] paperTime := by
+  rw [isBigO_iff]
+  refine ⟨1,?_⟩
+  filter_upwards [TtopZ.eventually_ge_atTop 1,QtopZ.eventually_ge_atTop 1] with n hn hh
+  have he : 0<TmZ n := by linarith
+  have he' : 0<QmZ n := by linarith
+  change ‖(n:ℝ)*TmZ n^theta‖ ≤ 1*‖(n:ℝ)*TmZ n^theta*QmZ n^(4-theta)‖
+  rw [Real.norm_of_nonneg (by positivity),Real.norm_of_nonneg (by positivity),one_mul]
+  have hb := Real.one_le_rpow hh (show 0 ≤ 4-theta by linarith [theta_ltZ])
+  exact le_mul_of_one_le_right (by positivity) hb
+
+lemma time_boundZ {N} (h : ∀ n:ℕ,n ≤ N n ∧ N n ≤ 2*n) (a b : RationZ) :
+    TimeBoundsZ (lunchZ a b N) := by
+  have H := lunch_bigZ a b h
+  have hp := H.trans_isLittleO (flight_littleZ alpha_thetaZ)
+  have hd := H.trans_isLittleO (flight_littleZ decimal_rangeZ.1)
+  have ht : (fun n => (lunchZ a b N n:ℝ)) =O[atTop] decimalTimeZ := hd.isBigO
+  exact ⟨hp.isBigO.trans speed_paperZ,ht,ht.trans_isLittleO decimal_littleZ⟩
+
+end Skies
+
+/-! Main results. Provide formula for the order besides the existential specification. -/
+theorem transform_main_orderZ : ∃ order solve W,
+    DFTProgram order solve W ∧ TimeBoundsZ W ∧
+    ∀ n, 0<n → (run order n).val = receipt n ∧ (run order n).val < 8*n^3 := by
+  obtain ⟨o,s,a,b,hp,ho⟩ := flagshipOneZ
+  exact ⟨o,s,lunchZ a b id,hp,Skies.time_boundZ (N:=id) (fun n => ⟨le_rfl,by dsimp; omega⟩)
+    a b,fun n hn => ⟨ho n hn,(ho n hn).symm ▸ (receipt_bound n hn).2⟩⟩
+
+theorem convolution_main_orderZ : ∃ order solve W,
+    ConvProgram order solve W ∧ TimeBoundsZ W ∧
+    ∀ n, 0<n → (run order n).val = receipt (2*n-1) ∧ (run order n).val < 8*(2*n-1)^3 := by
+  obtain ⟨o,s,a,b,hp,ho⟩ := flagshipBothZ
+  have h (n : ℕ) : n ≤ stretchZ n ∧ stretchZ n ≤ 2*n := by unfold stretchZ; omega
+  exact ⟨o,s,lunchZ a b stretchZ,hp,Skies.time_boundZ h a b,fun n hn =>
+    ⟨ho n hn,(ho n hn).symm ▸ (receipt_bound _ (stretch_posZ hn)).2⟩⟩
+
+/-- **The discrete Fourier transform of every length in `O(n (log n)^(1 - 7474546/10^10))`.**
+OpenAI's `transform_main` (one fixed program, OpenAI's own `DFTProgram`, `paperTime` and `nlogn`)
+with the decimal exponent `1 - 1/10^13` replaced by `decimalExponentZ = 1 - 7474546/10^10`
+(Work/Fourier/Goal.lean). Sibling of the Walsh-Hadamard theorem `wht_main_block_B2Gp193x`: both
+follow from one kernel program, `RAM.hillsZ_program` (Work/Fourier/Seam.lean). -/
+theorem transform_mainZ : DFTGoalZ :=
+  let ⟨o,s,W,h,g,_⟩ := transform_main_orderZ
+  ⟨o,s,W,h,g⟩
+
+/-- **Convolution of every length with the same bounds** (OpenAI's `convolution_main`, same change). -/
+theorem convolution_mainZ : ConvGoalZ :=
+  let ⟨o,s,W,h,g,_⟩ := convolution_main_orderZ
+  ⟨o,s,W,h,g⟩
+
+lemma matrix_definitionZ (n : ℕ) (x : Vec n) : dft n x = Matrix.mulVec (dftMatrix n) x :=
+  rfl
+
+end
+end PowerSaving
+
+end OAI
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Chain__Geom.lean b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Chain__Geom.lean
new file mode 100644
index 0000000000..eb4490af8b
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Chain__Geom.lean
@@ -0,0 +1,210 @@
+import Work.GCert.Chain.NetCert
+import Work.GCert.Chain.WinB
+import Work.BridgeGeom.B2
+
+/-!
+# (key: gx-chain) THE NETWORK LEMMA (spec statement 6): the bridged word B_2 delivers a `GCert`
+
+GENERATED by checks/wht26/gx-chain/py/netgen.py from Work/BridgeGeom/Net.lean (section Net):
+the geometry of `BG.geomBridge2`, word for word, with the windows `gwinB` (`.Xm = BG.fmapB`).
+
+`bridge2_gcert I hH`: for every generalised invocation package, a `GCert` on
+`card Γ * (4 card T + card Sl)` live roles, label dimension `5h`, unit price
+`5 * I.cost φ + card T * BG.bankPrice h φ` (unchanged).  No `sorry`.
+-/
+
+set_option linter.unusedSectionVars false
+set_option linter.unusedVariables false
+
+namespace OAI
+namespace PowerSaving
+namespace GX
+open Binary Matrix Finset RAM SS CB RF BR GF BG
+noncomputable section
+
+section Net
+variable {H T Sl C : Type} [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T]
+  [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C]
+variable (I : GInv H T Sl C) (base : T → OBase H) (l : H)
+  (hbase : ∀ t, I.tv t = (base t).v l) (hH : 1 ≤ Fintype.card H)
+include hbase
+
+lemma ggeom_x01 (g : Orth (L5 H)) (t : T) :
+    PhiB g ∅ 1 = PhiB (RrB (base t) l (0 : Fin 4) g) (qS bs1 l) (tt (I.tv t)) := by
+  rw [hbase t]
+  exact bankT_x0 (base t) l (0 : Fin 4) g
+
+lemma ggeom_y01 (g : Orth (L5 H)) (t : T) :
+    PhiB g ∅ (1 + tt (I.tv t)) = PhiB (RrB (base t) l (0 : Fin 4) g) (qS bs1 l) 0 := by
+  rw [hbase t]
+  exact bankT_y0 (base t) l (0 : Fin 4) g
+
+/-- consecutive stages `c ∈ bs`, `c' ∉ bs`: `X` leaves where it enters. -/
+lemma ggeom_xx (bs : Finset (Fin 4)) (c c' : Fin 4) (hc : c ∈ bs) (hc' : c' ∉ bs)
+    (g : Orth (L5 H)) (t : T) :
+    PhiB (RrB (base t) l c g) (qS bs l) 1
+      = PhiB (RrB (base t) l c' g) (qS (insert c' bs) l) (tt (I.tv t)) := by
+  rw [hbase t]
+  exact bankT_xx (base t) l bs c c' hc hc' g
+
+/-- consecutive stages: `Y` leaves where it enters. -/
+lemma ggeom_yy (bs : Finset (Fin 4)) (c c' : Fin 4) (hc : c ∈ bs) (hc' : c' ∉ bs)
+    (g : Orth (L5 H)) (t : T) :
+    PhiB (RrB (base t) l c g) (qS bs l) (1 + tt (I.tv t))
+      = PhiB (RrB (base t) l c' g) (qS (insert c' bs) l) 0 := by
+  rw [hbase t]
+  exact bankT_yy (base t) l bs c c' hc hc' g
+
+/-- terminal identity of the bank pairs of the class `(g,t)`. -/
+lemma ggeom_term5 (g : Orth (L5 H)) (t : T) :
+    kernel (L5 H) * PhiB g ∅ (tt (I.tv t))
+      = shift ((BBB (base t) g).v (Sum.inl l))
+          * frame (BBB (base t) g) (univ \ {Sum.inl l}) := by
+  rw [hbase t]
+  exact class_term (base t) l g
+
+include hH
+
+/-- pair 2 before the first gate layer, `X`: ONE block of rank `2h-2`. -/
+lemma ggeom_aX2 (g : Orth (L5 H)) (t : T) :
+    XReach (PhiB g ∅ (tt (I.tv t))) (PhiB (RrB (base t) l (0 : Fin 4) g) (qS bs1 l) 1)
+      (fun φ => bcost φ (2 * Fintype.card H - 2)) := by
+  rw [hbase t]
+  exact idle_X0 (base t) l hB4 hH (0 : Fin 4) g _ (by omega)
+
+/-- pair 2 before the first gate layer, `Y`: ONE block of rank `2h-2`. -/
+lemma ggeom_aY2 (g : Orth (L5 H)) (t : T) :
+    XReach (PhiB g ∅ 0) (PhiB (RrB (base t) l (0 : Fin 4) g) (qS bs1 l) (1 + tt (I.tv t)))
+      (fun φ => bcost φ (2 * Fintype.card H - 2)) := by
+  rw [hbase t]
+  exact idle_Y0 (base t) l hB4 hH (0 : Fin 4) g _ (by omega)
+
+/-- pair 1 at the end (idle in stages 3, 4, then the last four moves), `X`: ONE block `2h+2`. -/
+lemma ggeom_fX1 (g : Orth (L5 H)) (t : T) :
+    XReach (PhiB (RrB (base t) l (1 : Fin 4) g) (qS bs2 l) 1) (kernel (L5 H))
+      (fun φ => bcost φ (2 * Fintype.card H + 2)) :=
+  idle_Xend (base t) l hB4 hH bs2 (1 : Fin 4) (Finset.mem_insert_self _ _) g _
+    (by rw [card_bs2, Fintype.card_fin]; omega)
+
+/-- pair 1 at the end, `Y`: ONE block of rank `2h+2`. -/
+lemma ggeom_fY1 (g : Orth (L5 H)) (t : T) :
+    XReach (PhiB (RrB (base t) l (1 : Fin 4) g) (qS bs2 l) (1 + tt (I.tv t)))
+      (frame (BBB (base t) g) (univ \ {Sum.inl l}))
+      (fun φ => bcost φ (2 * Fintype.card H + 2)) := by
+  rw [hbase t]
+  exact idle_Yend (base t) l hB4 hH bs2 (1 : Fin 4) (Finset.mem_insert_self _ _) g _
+    (by rw [card_bs2, Fintype.card_fin]; omega)
+
+/-- pair 2 after stage 4, `X`: ONE block of rank `4`. -/
+lemma ggeom_fX2 (g : Orth (L5 H)) (t : T) :
+    XReach (PhiB (RrB (base t) l (3 : Fin 4) g) (qS bs4 l) 1) (kernel (L5 H))
+      (fun φ => bcost φ 4) :=
+  idle_Xend (base t) l hB4 hH bs4 (3 : Fin 4) (Finset.mem_insert_self _ _) g _
+    (by rw [card_bs4, Fintype.card_fin]; omega)
+
+/-- pair 2 after stage 4, `Y`: ONE block of rank `4`. -/
+lemma ggeom_fY2 (g : Orth (L5 H)) (t : T) :
+    XReach (PhiB (RrB (base t) l (3 : Fin 4) g) (qS bs4 l) (1 + tt (I.tv t)))
+      (frame (BBB (base t) g) (univ \ {Sum.inl l})) (fun φ => bcost φ 4) := by
+  rw [hbase t]
+  exact idle_Yend (base t) l hB4 hH bs4 (3 : Fin 4) (Finset.mem_insert_self _ _) g _
+    (by rw [card_bs4, Fintype.card_fin]; omega)
+
+/-- **The bridged word B_2 as a term of `BR.Bridge2`**: five stages on the label space
+`H ⊕ (Fin 4 × H)`, for any invocation package whose lines are the vectors of index `l` of
+orthonormal bases. -/
+def ggeomBridge2 : GBridge2 H T Sl C (L5 H) (Orth (L5 H)) where
+  I := I
+  G0 := fun g => gwinB hB4 hH g ∅
+  G1 := fun d => gwinB hB4 hH d (qS bs1 l)
+  G2 := fun d => gwinB hB4 hH d (qS bs2 l)
+  G3 := fun d => gwinB hB4 hH d (qS bs3 l)
+  G4 := fun d => gwinB hB4 hH d (qS bs4 l)
+  r1 := fun t => RrB (base t) l (0 : Fin 4)
+  r2 := fun t => RrB (base t) l (1 : Fin 4)
+  r3 := fun t => RrB (base t) l (2 : Fin 4)
+  r4 := fun t => RrB (base t) l (3 : Fin 4)
+  s1 := (Orth.colPerm (tauB (0 : Fin 4))).symm
+  s2 := (Orth.colPerm (tauB (1 : Fin 4))).symm
+  s3 := (Orth.colPerm (tauB (2 : Fin 4))).symm
+  s4 := (Orth.colPerm (tauB (3 : Fin 4))).symm
+  N1 := fun d => unframeM (d.1 * Matrix.fromBlocks 0 0 0 (diagI (qS bs1 l)) * d.1ᵀ)
+  W1 := fun d => frameM (d.1 * diagI (JB univ ∅) * d.1ᵀ)
+  N2 := fun d => unframeM (d.1 * Matrix.fromBlocks 0 0 0 (diagI (qS bs2 l)) * d.1ᵀ)
+  W2 := fun d => frameM (d.1 * diagI (JB univ ∅) * d.1ᵀ)
+  N3 := fun d => unframeM (d.1 * Matrix.fromBlocks 0 0 0 (diagI (qS bs3 l)) * d.1ᵀ)
+  W3 := fun d => frameM (d.1 * diagI (JB univ ∅) * d.1ᵀ)
+  N4 := fun d => unframeM (d.1 * Matrix.fromBlocks 0 0 0 (diagI (qS bs4 l)) * d.1ᵀ)
+  W4 := fun d => frameM (d.1 * diagI (JB univ ∅) * d.1ᵀ)
+  u := fun g t => (BBB (base t) g).v (Sum.inl l)
+  YF := fun g t => frame (BBB (base t) g) (univ \ {Sum.inl l})
+  hm := two_le_card_LB hB4 hH
+  caX := fun φ => bcost φ (2 * Fintype.card H - 2)
+  caY := fun φ => bcost φ (2 * Fintype.card H - 2)
+  cX1 := fun φ => bcost φ (2 * Fintype.card H + 2)
+  cY1 := fun φ => bcost φ (2 * Fintype.card H + 2)
+  cX2 := fun φ => bcost φ 4
+  cY2 := fun φ => bcost φ 4
+  inv0 := fun g t => ⟨unframeM (g.1 * Matrix.fromBlocks (tt (I.tv t)) 0 0 (diagI ∅) * g.1ᵀ),
+    frameM_unframeM _⟩
+  z0 := fun g => PhiB_zero_empty g
+  hN1 := fun d => frameM_unframeM _
+  hW1 := fun d => PhiB_one d (qS bs1 l)
+  hN2 := fun d => frameM_unframeM _
+  hW2 := fun d => PhiB_one d (qS bs2 l)
+  hN3 := fun d => frameM_unframeM _
+  hW3 := fun d => PhiB_one d (qS bs3 l)
+  hN4 := fun d => frameM_unframeM _
+  hW4 := fun d => PhiB_one d (qS bs4 l)
+  x01 := fun g t => ggeom_x01 I base l hbase g t
+  y01 := fun g t => ggeom_y01 I base l hbase g t
+  x12 := fun g t => ggeom_xx I base l hbase bs1 0 1 (Finset.mem_singleton_self _) mem1 g t
+  y12 := fun g t => ggeom_yy I base l hbase bs1 0 1 (Finset.mem_singleton_self _) mem1 g t
+  x23 := fun g t => ggeom_xx I base l hbase bs2 1 2 (Finset.mem_insert_self _ _) mem2 g t
+  y23 := fun g t => ggeom_yy I base l hbase bs2 1 2 (Finset.mem_insert_self _ _) mem2 g t
+  x34 := fun g t => ggeom_xx I base l hbase bs3 2 3 (Finset.mem_insert_self _ _) mem3 g t
+  y34 := fun g t => ggeom_yy I base l hbase bs3 2 3 (Finset.mem_insert_self _ _) mem3 g t
+  sfin := fun g => helper_fin5 g
+  aX2 := fun g t => ggeom_aX2 I base l hbase hH g t
+  aY2 := fun g t => ggeom_aY2 I base l hbase hH g t
+  fX1 := fun g t => ggeom_fX1 I base l hbase hH g t
+  fY1 := fun g t => ggeom_fY1 I base l hbase hH g t
+  fX2 := fun g t => ggeom_fX2 I base l hbase hH g t
+  fY2 := fun g t => ggeom_fY2 I base l hbase hH g t
+  term := fun g t => ggeom_term5 I base l hbase g t
+
+end Net
+
+section End
+variable {H T Sl C : Type} [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T]
+  [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C]
+
+/-- an orthonormal basis through a unit vector with a zero coordinate, at any index. -/
+theorem base_ex (z : Space H) (hz : dot z z = 1) (hg : ∃ p, z p = 0) (l : H) :
+    ∃ A : OBase H, z = A.v l := by
+  obtain ⟨p, hp⟩ := hg
+  obtain ⟨A, k, hA⟩ := exists_base_one z hz p hp
+  refine ⟨reindex A (Equiv.swap k l), ?_⟩
+  rw [← hA]
+  show A.v k = A.v (Equiv.swap k l l)
+  rw [Equiv.swap_apply_right]
+
+/-- **Generalised certificate of the bridged word B_2** for a generalised invocation package. -/
+theorem bridge2_gcert (I : GInv H T Sl C) (hH : 1 ≤ Fintype.card H) :
+    GCert (L5 H) (Sum.inl : BLive (Orth (L5 H)) T Sl → BRole (Orth (L5 H)) T Sl C)
+      (fun φ => (Fintype.card (Orth (L5 H)) : ℝ)
+        * (5 * I.cost φ + (Fintype.card T : ℝ) * bankPrice (Fintype.card H) φ)) := by
+  obtain ⟨l⟩ : Nonempty H := Fintype.card_pos_iff.mp (by omega)
+  have hb : ∀ t, ∃ A : OBase H, I.tv t = A.v l :=
+    fun t => base_ex (I.tv t) (I.hunit t) (I.hgap t) l
+  choose base hbase using hb
+  exact gbridge2_certificate (ggeomBridge2 I base l hbase hH)
+
+end End
+end
+end GX
+end PowerSaving
+end OAI
+
+#print axioms OAI.PowerSaving.GX.ggeomBridge2
+#print axioms OAI.PowerSaving.GX.bridge2_gcert
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Chain__InvThm.lean b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Chain__InvThm.lean
new file mode 100644
index 0000000000..d69fcb4992
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Chain__InvThm.lean
@@ -0,0 +1,164 @@
+import Work.GCert.Chain.Inv
+
+/-!
+# (key: gx-chain) THE CARRIER INVOCATION THEOREM on the generalised engine: `Lab.inv`
+
+`CR.Phased.inv` (`Work/Carrier/PhasedInv.lean`) with subspace labels and `GRoute`.  Forward word:
+
+    erase | y -= K s (old frame 0) | free adapters to the labels of the certificate | phase A |
+    c += Cc s | copies down | y += Jr c | phase B | final climbs | free adapters to the old
+    frames | slots := slot rows of the inverse (old frame 1) | erase
+
+The backward word is the transposed inverse, gate by gate, on the same labels.  The scalar
+identities are `Scal.fwd_scalar` / `Scal.bwd_scalar` (those of `CR.Phased`).
+
+No `sorry`.
+-/
+
+set_option linter.unusedSectionVars false
+set_option linter.unusedVariables false
+
+namespace OAI
+namespace PowerSaving
+namespace GX
+open Binary Matrix Finset RAM SS CB BR CR GF RF
+noncomputable section
+
+section Thm
+variable {H T Sl C : Type} [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T]
+  [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C]
+variable {α : Type} [Fintype α] [DecidableEq α]
+variable {S : Scal T Sl C} (L : Lab H S) (W : GWin H α)
+
+/-- **Forward word on clean frames.** -/
+theorem Lab.fwd_erased (c0 c1 : C → CMat α) :
+    GRoute (iin W.Xm L.tv (fun _ : Sl => W.Xm.Φ 0) c0)
+      (iout W.Xm L.tv (fun _ : Sl => W.Xm.Φ 1) c1) shearF L.cost := by
+  have a1 : GRoute (fun r : L3 T Sl => W.Xm.Φ (lab3 L.tv r).P)
+      (fun r : L3 T Sl => W.Xm.Φ (lab3 L.tv r).P)
+      (actPoint (g1 S.Mt)) (fun _ => 0) :=
+    GRoute.gate (g1 S.Mt) _ _ (fun i j hij => by
+      by_cases e : i = j
+      · subst e; rfl
+      · obtain ⟨h1, h2⟩ := offdiag_x (g1_x S.Mt).1 (g1_x S.Mt).2 hij e
+        exact start_frames W.Xm L.tv i j h1 h2)
+  have b3 : GRoute (fun r : L3 T Sl => W.Xm.Φ (labE L.tv r))
+      (fun r : L3 T Sl => W.Xm.Φ (labE L.tv r))
+      (actPoint (g8 S.Mti)) (fun _ => 0) :=
+    GRoute.gate (g8 S.Mti) _ _ (fun i j hij => by
+      by_cases e : i = j
+      · subst e; rfl
+      · have f := g8_facts S.hiy
+        obtain ⟨h1, h2, h3⟩ := offdiag_s f.1 f.2.1 f.2.2 hij e
+        exact end_old W.Xm L.tv i j h1 h2 h3)
+  have seg1 : GRoute (fun r : L3 T Sl => W.Xm.Φ (lab3 L.tv r).P) (fun r => W.X.Φ (L.labA r))
+      S.f1 (fun φ => rcost φ L.rA) :=
+    ((a1.trans (L.ad0 W)).trans (L.pA W.X W.big).1).cast (fun φ => by simp)
+  have seg2 : GRoute (fun r => W.X.Φ (L.labA r)) (fun r : L3 T Sl => W.Xm.Φ (labE L.tv r))
+      S.f2 (fun φ => rcost φ L.rB + rcost φ L.rF) :=
+    ((((L.pB W.X W.big).1.trans (L.pF W.X W.big)).trans (L.adF W)).trans b3).cast
+      (fun φ => by simp)
+  have e0 : GRoute (iin W.Xm L.tv (fun _ : Sl => W.Xm.Φ 0) c0)
+      (glue (fun r : L3 T Sl => W.Xm.Φ (lab3 L.tv r).P) (fun k => W.X.Φ (L.cen k)))
+      eraseC (fun _ => 0) :=
+    GRoute.ofX (xerase (fun t => rfl) (fun t => rfl) (fun q => rfl))
+  have L1 := glift_live (fun k => W.X.Φ (L.cen k)) seg1
+  have m1 : GRoute (glue (fun r => W.X.Φ (L.labA r)) (fun k => W.X.Φ (L.cen k)))
+      (glue (fun r => W.X.Φ (L.labA r)) (fun k => W.X.Φ (L.cen k)))
+      (fun z => addBlk bC (ap S.Cc (z ∘ bS)) z) (fun _ => 0) :=
+    GRoute.ofX (xgate_shear bC bS bS_inj S.Cc (fun k q h => congrArg W.X.Φ (L.hAc k q h)))
+  have m2 := L.cdown W (fun r => W.X.Φ (L.labA r))
+  have m3 : GRoute (glue (fun r => W.X.Φ (L.labA r)) (fun _ : C => W.X.Φ L.z0))
+      (glue (fun r => W.X.Φ (L.labA r)) (fun _ : C => W.X.Φ L.z0))
+      (fun z => addBlk bY (ap S.Jr (z ∘ bC)) z) (fun _ => 0) :=
+    GRoute.ofX (xgate_shear bY bC bC_inj S.Jr (fun t k _ => congrArg W.X.Φ (L.hAy t)))
+  have L2 := glift_live (fun _ : C => W.X.Φ L.z0) seg2
+  have e1 : GRoute (glue (fun r : L3 T Sl => W.Xm.Φ (labE L.tv r)) (fun _ : C => W.X.Φ L.z0))
+      (iout W.Xm L.tv (fun _ : Sl => W.Xm.Φ 1) c1) eraseC (fun _ => 0) :=
+    GRoute.ofX (xerase (fun t => rfl) (fun t => rfl) (fun q => rfl))
+  have tot := (((((e0.trans L1).trans m1).trans m2).trans m3).trans L2).trans e1
+  refine (tot.castg (g' := shearF) (funext fun z => S.fwd_scalar z)).cast (fun φ => ?_)
+  simp only [Lab.cost, zero_add, add_zero]
+  ring
+
+/-- **Backward word on clean frames**: the transposed inverse gates, same labels; the copies
+start at the zero frame and climb. -/
+theorem Lab.bwd_erased (c0 c1 : C → CMat α) :
+    GRoute (iin W.Xm L.tv (fun _ : Sl => W.Xm.Φ 0) c0)
+      (iout W.Xm L.tv (fun _ : Sl => W.Xm.Φ 1) c1) shearB L.cost := by
+  have a1 : GRoute (fun r : L3 T Sl => W.Xm.Φ (lab3 L.tv r).P)
+      (fun r : L3 T Sl => W.Xm.Φ (lab3 L.tv r).P)
+      (actPoint (g1i S.Mt)ᵀ) (fun _ => 0) :=
+    GRoute.gate (g1i S.Mt)ᵀ _ _ (fun i j hij => by
+      by_cases e : i = j
+      · subst e; rfl
+      · have hij' : g1i S.Mt j i ≠ 0 := hij
+        obtain ⟨h1, h2⟩ := offdiag_x (g1i_x S.Mt).1 (g1i_x S.Mt).2 hij' (Ne.symm e)
+        exact start_frames W.Xm L.tv i j h2 h1)
+  have b3 : GRoute (fun r : L3 T Sl => W.Xm.Φ (labE L.tv r))
+      (fun r : L3 T Sl => W.Xm.Φ (labE L.tv r))
+      (actPoint (h8 S.Mt)ᵀ) (fun _ => 0) :=
+    GRoute.gate (h8 S.Mt)ᵀ _ _ (fun i j hij => by
+      by_cases e : i = j
+      · subst e; rfl
+      · have hij' : h8 S.Mt j i ≠ 0 := hij
+        have f := h8_facts S.hmy
+        obtain ⟨h1, h2, h3⟩ := offdiag_s f.1 f.2.1 f.2.2 hij' (Ne.symm e)
+        exact (end_old W.Xm L.tv j i h1 h2 h3).symm)
+  have seg1 : GRoute (fun r : L3 T Sl => W.Xm.Φ (lab3 L.tv r).P) (fun r => W.X.Φ (L.labA r))
+      S.b1 (fun φ => rcost φ L.rA) :=
+    ((a1.trans (L.ad0 W)).trans (L.pA W.X W.big).2).cast (fun φ => by simp)
+  have seg2 : GRoute (fun r => W.X.Φ (L.labA r)) (fun r : L3 T Sl => W.Xm.Φ (labE L.tv r))
+      S.b2 (fun φ => rcost φ L.rB + rcost φ L.rF) :=
+    ((((L.pB W.X W.big).2.trans (L.pF W.X W.big)).trans (L.adF W)).trans b3).cast
+      (fun φ => by simp)
+  have e0 : GRoute (iin W.Xm L.tv (fun _ : Sl => W.Xm.Φ 0) c0)
+      (glue (fun r : L3 T Sl => W.Xm.Φ (lab3 L.tv r).P) (fun _ : C => W.X.Φ L.z0))
+      eraseC (fun _ => 0) :=
+    GRoute.ofX (xerase (fun t => rfl) (fun t => rfl) (fun q => rfl))
+  have L1 := glift_live (fun _ : C => W.X.Φ L.z0) seg1
+  have m1 : GRoute (glue (fun r => W.X.Φ (L.labA r)) (fun _ : C => W.X.Φ L.z0))
+      (glue (fun r => W.X.Φ (L.labA r)) (fun _ : C => W.X.Φ L.z0))
+      (fun z => addBlk bC (ap S.Jrᵀ (z ∘ bY)) z) (fun _ => 0) :=
+    GRoute.ofX (xgate_shear bC bY bY_inj S.Jrᵀ (fun k t _ => (congrArg W.X.Φ (L.hAy t)).symm))
+  have m2 := L.cup W (fun r => W.X.Φ (L.labA r))
+  have m3 : GRoute (glue (fun r => W.X.Φ (L.labA r)) (fun k => W.X.Φ (L.cen k)))
+      (glue (fun r => W.X.Φ (L.labA r)) (fun k => W.X.Φ (L.cen k)))
+      (fun z => addBlk bS (ap (-S.Ccᵀ) (z ∘ bC)) z) (fun _ => 0) :=
+    GRoute.ofX (xgate_shear bS bC bC_inj (-S.Ccᵀ)
+      (fun q k h => (congrArg W.X.Φ (L.hAc k q (ne_of_negT S.Cc h))).symm))
+  have L2 := glift_live (fun k => W.X.Φ (L.cen k)) seg2
+  have e1 : GRoute (glue (fun r : L3 T Sl => W.Xm.Φ (labE L.tv r)) (fun k => W.X.Φ (L.cen k)))
+      (iout W.Xm L.tv (fun _ : Sl => W.Xm.Φ 1) c1) eraseC (fun _ => 0) :=
+    GRoute.ofX (xerase (fun t => rfl) (fun t => rfl) (fun q => rfl))
+  have tot := (((((e0.trans L1).trans m1).trans m2).trans m3).trans L2).trans e1
+  refine (tot.castg (g' := shearB) (funext fun z => S.bwd_scalar z)).cast (fun φ => ?_)
+  simp only [Lab.cost, zero_add, add_zero]
+  ring
+
+/-- **THE CARRIER INVOCATION THEOREM on the generalised engine.**  The two halves of a
+certificate are an invocation package for the bridged network, with the price `L.cost`. -/
+def Lab.inv : GInv H T Sl C where
+  tv := L.tv
+  hunit := L.hunit
+  hgap := L.hgap
+  cost := L.cost
+  fwd := fun W N0 Wn hN hW Z c0 c1 =>
+    gdirty_of_erased W.Xm L.tv N0 Wn hN hW Z c0 c1 (L.fwd_erased W c0 c1)
+      (shearF_comm (fun _ => 1) (fun q => N0 * Z q) (fun _ => 1))
+  bwd := fun W N0 Wn hN hW Z c0 c1 =>
+    gdirty_of_erased W.Xm L.tv N0 Wn hN hW Z c0 c1 (L.bwd_erased W c0 c1)
+      (shearB_comm (fun _ => 1) (fun q => N0 * Z q) (fun _ => 1))
+
+@[simp] lemma Lab.inv_cost (φ : ℕ → ℝ) : L.inv.cost φ = L.cost φ := rfl
+@[simp] lemma Lab.inv_tv : L.inv.tv = L.tv := rfl
+
+end Thm
+end
+end GX
+end PowerSaving
+end OAI
+
+#print axioms OAI.PowerSaving.GX.Lab.fwd_erased
+#print axioms OAI.PowerSaving.GX.Lab.bwd_erased
+#print axioms OAI.PowerSaving.GX.Lab.inv
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Chain__NetCert.lean b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Chain__NetCert.lean
new file mode 100644
index 0000000000..6336ba0766
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Chain__NetCert.lean
@@ -0,0 +1,157 @@
+import Work.GCert.Chain.NetStage
+import Work.GFrame.Top.Assemble
+
+/-!
+# (key: gx-chain) The bridged word B_2 on the generalised engine: the certificate `GCert`
+
+GENERATED by checks/wht26/gx-chain/py/netgen.py from Work/Bridge/Cert.lean (section Net).
+`gbridge2_certificate`: a `GCert` on the live roles `BLive Γ T Sl` with price `card Γ * N.unit φ`.
+The gate layers, the idle climbs and the exchange are the old routes (`GRoute.ofX`).  No `sorry`.
+-/
+
+set_option linter.unusedSectionVars false
+set_option linter.unusedVariables false
+
+namespace OAI
+namespace PowerSaving
+namespace GX
+open Binary Matrix Finset RAM SS CB RF BR GF
+noncomputable section
+
+section Net
+variable {H T Sl C α Γ : Type} [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T]
+  [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] [Fintype α] [DecidableEq α]
+  [Fintype Γ] [DecidableEq Γ]
+variable (N : GBridge2 H T Sl C α Γ)
+
+/-- gate layer A, after stage 1 and the first idle climb of pair 2: twins have equal frames. -/
+lemma GBridge2.gateA : XRoute N.S3 N.S3 (gsc pAB kA) (fun _ => 0) := by
+  refine gateP pAB kA N.S3 ?_
+  rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩) hk
+  · rfl
+  · exact (hk rfl).elim
+  · exact (hk rfl).elim
+  · rfl
+  · exact (hk rfl).elim
+  · exact (hk rfl).elim
+
+/-- gate layer B, after stage 2 and the second idle climb of pair 2. -/
+lemma GBridge2.gateB : XRoute N.S5 N.S5 (gsc pAB kB) (fun _ => 0) := by
+  refine gateP pAB kB N.S5 ?_
+  rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩) hk
+  · exact N.x23 g t
+  · exact (hk rfl).elim
+  · exact (hk rfl).elim
+  · exact (N.y23 g t).symm
+  · exact (hk rfl).elim
+  · exact (hk rfl).elim
+
+/-- gate layer C, after the final climbs. -/
+lemma GBridge2.gateC : XRoute N.S8 N.S8 (gsc pC kCc) (fun _ => 0) := by
+  refine gateP pC kCc N.S8 ?_
+  rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩) hk
+  · exact (hk rfl).elim
+  · rfl
+  · rfl
+  · exact (hk rfl).elim
+  · exact (hk rfl).elim
+  · exact (hk rfl).elim
+
+/-- the free shifts at the end. -/
+def GBridge2.corr : BRole Γ T Sl C → Space α :=
+  mkB (fun _ _ => 0) (fun g t => N.u g t) (fun _ _ => 0) (fun g t => N.u g t)
+    (fun _ _ => 0) (fun _ _ => 0)
+
+def GBridge2.S9 : BRole Γ T Sl C → CMat α := fun r => shift (N.corr r) * N.S8 r
+
+/-- the whole word before the exchange: five stages, three idle-climb layers, three gate
+layers. -/
+lemma GBridge2.route :
+    GRoute N.S0 N.S8 chain (fun φ => (Fintype.card Γ : ℝ) * N.unit φ) := by
+  have R := ((((((((((N.stage0.trans N.stage1).trans (GRoute.ofX N.climbA)).trans
+    (GRoute.ofX N.gateA)).trans N.stage2).trans (GRoute.ofX N.climbB)).trans
+    (GRoute.ofX N.gateB)).trans N.stage3).trans N.stage4).trans
+    (GRoute.ofX N.fin)).trans (GRoute.ofX N.gateC))
+  refine (R.castg ?_).cast (fun φ => ?_)
+  · rfl
+  · simp only [GBridge2.unit]
+    ring
+
+/-- **The scratch certificate of the bridged five-stage network with shared helper sets**,
+in whole blocks, with its exact price for every price list. -/
+theorem gbridge2_certificate :
+    GCert α (Sum.inl : BLive Γ T Sl → BRole Γ T Sl C)
+      (fun φ => (Fintype.card Γ : ℝ) * N.unit φ) := by
+  have R2 : GRoute N.S8 N.S9 id (fun _ => 0) := GRoute.shifts N.hm N.S8 N.corr
+  have R3 : GRoute N.S9 (fun r => N.S9 (bpull r)) (actPoint brecM) (fun _ => 0) :=
+    GRoute.gate brecM N.S9 _ (fun i j h => by
+      have hj : j = bpull i := by
+        by_contra hne
+        exact h (by simp [brecM, hne])
+      rw [hj])
+  have Rt := ((N.route.trans R2).trans R3).cast (c' := fun φ => (Fintype.card Γ : ℝ) * N.unit φ)
+    (fun φ => by first | ring | simp)
+  have hSinv : ∀ r, ∃ M, N.S0 r * M = 1 := by
+    rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩)
+    · exact N.inv0 g t
+    · refine ⟨1, ?_⟩
+      show (N.G0 g).Xm.Φ 0 * 1 = 1
+      rw [N.z0, Matrix.mul_one]
+    · exact N.inv0 g t
+    · refine ⟨1, ?_⟩
+      show (N.G0 g).Xm.Φ 0 * 1 = 1
+      rw [N.z0, Matrix.mul_one]
+    · exact ⟨1, Matrix.mul_one _⟩
+    · exact ⟨1, Matrix.mul_one _⟩
+  have hT : ∀ l : BLive Γ T Sl, N.S9 (bpull (Sum.inl l)) = kernel α * N.S0 (Sum.inl l) := by
+    rintro (((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)
+    · show shift (N.u g t) * N.YF g t = kernel α * (N.G0 g).Xm.Φ (tt (N.I.tv t))
+      exact (N.term g t).symm
+    · show shift (0 : Space α) * kernel α = kernel α * (N.G0 g).Xm.Φ 0
+      rw [shift_zero, Matrix.one_mul, N.z0, Matrix.mul_one]
+    · show shift (N.u g t) * N.YF g t = kernel α * (N.G0 g).Xm.Φ (tt (N.I.tv t))
+      exact (N.term g t).symm
+    · show shift (0 : Space α) * kernel α = kernel α * (N.G0 g).Xm.Φ 0
+      rw [shift_zero, Matrix.one_mul, N.z0, Matrix.mul_one]
+    · show shift (0 : Space α) * kernel α = kernel α * 1
+      rw [shift_zero, Matrix.one_mul, Matrix.mul_one]
+  have hg : ∀ x : BRole Γ T Sl C → ℂ,
+      (∀ s, (∀ l : BLive Γ T Sl, (Sum.inl l : BRole Γ T Sl C) ≠ s) → x s = 0) →
+      ∀ l : BLive Γ T Sl,
+        (actPoint brecM ∘ (id ∘ chain)) x (Sum.inl l) = x (Sum.inl l) := by
+    intro x _ l
+    change actPoint brecM (chain x) (Sum.inl l) = _
+    rw [actPoint_brecM]
+    rcases l with (((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)
+    · show ((1:ℚ):ℂ) * chain x (Sum.inl (Sum.inl (Sum.inl (Sum.inr (g,t))))) = _
+      rw [chain_eq]
+      show ((1:ℚ):ℂ) * x (kX1 (g,t)) = x (kX1 (g,t))
+      first | (push_cast; ring) | simp
+    · show ((-1:ℚ):ℂ) * chain x (Sum.inl (Sum.inl (Sum.inl (Sum.inl (g,t))))) = _
+      rw [chain_eq]
+      show ((-1:ℚ):ℂ) * (-x (kY1 (g,t))) = x (kY1 (g,t))
+      first | (push_cast; ring) | simp
+    · show ((1:ℚ):ℂ) * chain x (Sum.inl (Sum.inl (Sum.inr (Sum.inr (g,t))))) = _
+      rw [chain_eq]
+      show ((1:ℚ):ℂ) * x (kX2 (g,t)) = x (kX2 (g,t))
+      first | (push_cast; ring) | simp
+    · show ((-1:ℚ):ℂ) * chain x (Sum.inl (Sum.inl (Sum.inr (Sum.inl (g,t))))) = _
+      rw [chain_eq]
+      show ((-1:ℚ):ℂ) * (-x (kY2 (g,t))) = x (kY2 (g,t))
+      first | (push_cast; ring) | simp
+    · show ((1:ℚ):ℂ) * chain x (Sum.inl (Sum.inr (g,q))) = _
+      rw [chain_eq]
+      show ((1:ℚ):ℂ) * x (kS (g,q)) = x (kS (g,q))
+      first | (push_cast; ring) | simp
+  exact gcert_of_groute (Sum.inl : BLive Γ T Sl → BRole Γ T Sl C) N.S0
+    (fun r => N.S9 (bpull r)) _ _ Rt hSinv hg hT
+
+end Net
+
+end
+end GX
+end PowerSaving
+end OAI
+
+#print axioms OAI.PowerSaving.GX.GBridge2.route
+#print axioms OAI.PowerSaving.GX.gbridge2_certificate
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Chain__NetDef.lean b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Chain__NetDef.lean
new file mode 100644
index 0000000000..c13042eb43
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Chain__NetDef.lean
@@ -0,0 +1,157 @@
+import Work.GCert.Chain.Stage
+
+/-!
+# (key: gx-chain) The bridged five-stage word B_2 on the generalised engine: data and frames
+
+GENERATED by checks/wht26/gx-chain/py/netgen.py from Work/Bridge/Net.lean (`BR.Bridge2`): the
+invocation is a `GInv`, the frame maps of the stages are windows `GWin` (`G0 .. G4`; the old
+frame map of a window is `.Xm`).  Everything else word for word.  No `sorry`.
+-/
+
+set_option linter.unusedSectionVars false
+set_option linter.unusedVariables false
+
+namespace OAI
+namespace PowerSaving
+namespace GX
+open Binary Matrix Finset RAM SS CB RF BR GF
+noncomputable section
+
+/-- See the file header.  Fields in the style of `RF.Fold3S`; stages are numbered 0 to 4. -/
+structure GBridge2 (H T Sl C α Γ : Type) [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T]
+    [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] [Fintype α] [DecidableEq α] where
+  /-- the invocation (for a block circuit `X`: `X.inv`). -/
+  I : GInv H T Sl C
+  /-- frame maps of the invocations of the five stages. -/
+  G0 : Γ → GWin H α
+  G1 : Γ → GWin H α
+  G2 : Γ → GWin H α
+  G3 : Γ → GWin H α
+  G4 : Γ → GWin H α
+  /-- the class `(g,t)` is served by the invocation `g` in stage 0 and `r_l t g` in stage `l`. -/
+  r1 : T → Γ ≃ Γ
+  r2 : T → Γ ≃ Γ
+  r3 : T → Γ ≃ Γ
+  r4 : T → Γ ≃ Γ
+  /-- the invocation `d` of stage `l` uses the helper set (and the copies) `s_l d`. -/
+  s1 : Γ ≃ Γ
+  s2 : Γ ≃ Γ
+  s3 : Γ ≃ Γ
+  s4 : Γ ≃ Γ
+  /-- inverse of the frame of the base, and frame of the window, of every invocation. -/
+  N1 : Γ → CMat α
+  W1 : Γ → CMat α
+  N2 : Γ → CMat α
+  W2 : Γ → CMat α
+  N3 : Γ → CMat α
+  W3 : Γ → CMat α
+  N4 : Γ → CMat α
+  W4 : Γ → CMat α
+  u : Γ → T → Space α
+  YF : Γ → T → CMat α
+  hm : 2 ≤ Fintype.card α
+  /-- prices of the idle climbs, for every price list.  B_2 with merged climbs:
+  `caX = caY = fun φ => bcost φ (2h-2)`, `cX1 = cY1 = fun φ => bcost φ (2h+2)`,
+  `cX2 = cY2 = fun φ => bcost φ 4`. -/
+  caX : (ℕ → ℝ) → ℝ
+  caY : (ℕ → ℝ) → ℝ
+  cX1 : (ℕ → ℝ) → ℝ
+  cY1 : (ℕ → ℝ) → ℝ
+  cX2 : (ℕ → ℝ) → ℝ
+  cY2 : (ℕ → ℝ) → ℝ
+  inv0 : ∀ g t, ∃ N : CMat α, (G0 g).Xm.Φ (tt (I.tv t)) * N = 1
+  z0 : ∀ g, (G0 g).Xm.Φ 0 = 1
+  hN1 : ∀ d, (G1 d).Xm.Φ 0 * N1 d = 1
+  hW1 : ∀ d, (G1 d).Xm.Φ 1 = W1 d * (G1 d).Xm.Φ 0
+  hN2 : ∀ d, (G2 d).Xm.Φ 0 * N2 d = 1
+  hW2 : ∀ d, (G2 d).Xm.Φ 1 = W2 d * (G2 d).Xm.Φ 0
+  hN3 : ∀ d, (G3 d).Xm.Φ 0 * N3 d = 1
+  hW3 : ∀ d, (G3 d).Xm.Φ 1 = W3 d * (G3 d).Xm.Φ 0
+  hN4 : ∀ d, (G4 d).Xm.Φ 0 * N4 d = 1
+  hW4 : ∀ d, (G4 d).Xm.Φ 1 = W4 d * (G4 d).Xm.Φ 0
+  /-- the bank frames a stage leaves are the bank frames the next stage starts from. -/
+  x01 : ∀ g t, (G0 g).Xm.Φ 1 = (G1 (r1 t g)).Xm.Φ (tt (I.tv t))
+  y01 : ∀ g t, (G0 g).Xm.Φ (1 + tt (I.tv t)) = (G1 (r1 t g)).Xm.Φ 0
+  x12 : ∀ g t, (G1 (r1 t g)).Xm.Φ 1 = (G2 (r2 t g)).Xm.Φ (tt (I.tv t))
+  y12 : ∀ g t, (G1 (r1 t g)).Xm.Φ (1 + tt (I.tv t)) = (G2 (r2 t g)).Xm.Φ 0
+  x23 : ∀ g t, (G2 (r2 t g)).Xm.Φ 1 = (G3 (r3 t g)).Xm.Φ (tt (I.tv t))
+  y23 : ∀ g t, (G2 (r2 t g)).Xm.Φ (1 + tt (I.tv t)) = (G3 (r3 t g)).Xm.Φ 0
+  x34 : ∀ g t, (G3 (r3 t g)).Xm.Φ 1 = (G4 (r4 t g)).Xm.Φ (tt (I.tv t))
+  y34 : ∀ g t, (G3 (r3 t g)).Xm.Φ (1 + tt (I.tv t)) = (G4 (r4 t g)).Xm.Φ 0
+  /-- the five windows of a helper set make up the kernel. -/
+  sfin : ∀ g, W4 (s4.symm g) * (W3 (s3.symm g) * (W2 (s2.symm g) *
+    (W1 (s1.symm g) * (G0 g).Xm.Φ 1))) = kernel α
+  /-- pair 2, idle in stages 0 and 1: ONE block from its start frame to the frame of its
+  twin after stage 1. -/
+  aX2 : ∀ g t, XReach ((G0 g).Xm.Φ (tt (I.tv t))) ((G1 (r1 t g)).Xm.Φ 1) caX
+  aY2 : ∀ g t, XReach ((G0 g).Xm.Φ 0) ((G1 (r1 t g)).Xm.Φ (1 + tt (I.tv t))) caY
+  /-- pair 1, idle in stages 3 and 4: ONE block from its frame after stage 2 to the end. -/
+  fX1 : ∀ g t, XReach ((G2 (r2 t g)).Xm.Φ 1) (kernel α) cX1
+  fY1 : ∀ g t, XReach ((G2 (r2 t g)).Xm.Φ (1 + tt (I.tv t))) (YF g t) cY1
+  /-- pair 2 after stage 4: ONE final block. -/
+  fX2 : ∀ g t, XReach ((G4 (r4 t g)).Xm.Φ 1) (kernel α) cX2
+  fY2 : ∀ g t, XReach ((G4 (r4 t g)).Xm.Φ (1 + tt (I.tv t))) (YF g t) cY2
+  term : ∀ g t, kernel α * (G0 g).Xm.Φ (tt (I.tv t)) = shift (u g t) * YF g t
+
+section Net
+variable {H T Sl C α Γ : Type} [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T]
+  [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] [Fintype α] [DecidableEq α]
+  [Fintype Γ] [DecidableEq Γ]
+variable (N : GBridge2 H T Sl C α Γ)
+
+/-- price of the blocks of one unit (one element of `Γ`): five invocations and the idle
+climbs of the four bank roles of every class. -/
+def GBridge2.unit (φ : ℕ → ℝ) : ℝ :=
+  5 * N.I.cost φ + (Fintype.card T : ℝ) * ((N.caX φ + N.caY φ)
+    + 2 * bcost φ (Fintype.card H - 1)
+    + (N.cX1 φ + N.cY1 φ) + (N.cX2 φ + N.cY2 φ))
+
+/-- the bank index of stage 0: the class `(g,t)` belongs to the invocation `g`. -/
+abbrev bid (Γ T : Type) : T → Γ ≃ Γ := fun _ => Equiv.refl Γ
+
+/-! bank frames of the class `(g,t)`: `xi l`, `yi l` at the START of stage `l`; `xo l`,
+`yo l` at the END of stage `l` (only where no next stage follows on that pair). -/
+def GBridge2.xi0 (g : Γ) (t : T) : CMat α := (N.G0 g).Xm.Φ (tt (N.I.tv t))
+def GBridge2.yi0 (g : Γ) (t : T) : CMat α := (N.G0 g).Xm.Φ 0
+def GBridge2.xi1 (g : Γ) (t : T) : CMat α := (N.G1 (N.r1 t g)).Xm.Φ (tt (N.I.tv t))
+def GBridge2.yi1 (g : Γ) (t : T) : CMat α := (N.G1 (N.r1 t g)).Xm.Φ 0
+def GBridge2.xi2 (g : Γ) (t : T) : CMat α := (N.G2 (N.r2 t g)).Xm.Φ (tt (N.I.tv t))
+def GBridge2.yi2 (g : Γ) (t : T) : CMat α := (N.G2 (N.r2 t g)).Xm.Φ 0
+def GBridge2.xo2 (g : Γ) (t : T) : CMat α := (N.G2 (N.r2 t g)).Xm.Φ 1
+def GBridge2.yo2 (g : Γ) (t : T) : CMat α := (N.G2 (N.r2 t g)).Xm.Φ (1 + tt (N.I.tv t))
+def GBridge2.xi3 (g : Γ) (t : T) : CMat α := (N.G3 (N.r3 t g)).Xm.Φ (tt (N.I.tv t))
+def GBridge2.yi3 (g : Γ) (t : T) : CMat α := (N.G3 (N.r3 t g)).Xm.Φ 0
+def GBridge2.xi4 (g : Γ) (t : T) : CMat α := (N.G4 (N.r4 t g)).Xm.Φ (tt (N.I.tv t))
+def GBridge2.yi4 (g : Γ) (t : T) : CMat α := (N.G4 (N.r4 t g)).Xm.Φ 0
+def GBridge2.xo4 (g : Γ) (t : T) : CMat α := (N.G4 (N.r4 t g)).Xm.Φ 1
+def GBridge2.yo4 (g : Γ) (t : T) : CMat α := (N.G4 (N.r4 t g)).Xm.Φ (1 + tt (N.I.tv t))
+
+/-- helper slots after the stages 0, 1, 2, 3, 4 (at the start: `cOne Γ Sl α`). -/
+def GBridge2.Z1 : Γ → Sl → CMat α := fun g _ => (N.G0 g).Xm.Φ 1
+def GBridge2.Z2 : Γ → Sl → CMat α := fun g q => N.W1 (N.s1.symm g) * N.Z1 g q
+def GBridge2.Z3 : Γ → Sl → CMat α := fun g q => N.W2 (N.s2.symm g) * N.Z2 g q
+def GBridge2.Z4 : Γ → Sl → CMat α := fun g q => N.W3 (N.s3.symm g) * N.Z3 g q
+def GBridge2.Z5 : Γ → Sl → CMat α := fun g q => N.W4 (N.s4.symm g) * N.Z4 g q
+
+/-- frames: at the start; after stage 0; after stage 1; after the first idle climb of pair 2
+(first gate layer here); after stage 2; after the second idle climb of pair 2 (second gate
+layer here); after stage 3; after stage 4; after the final climbs (third gate layer here). -/
+def GBridge2.S0 : BRole Γ T Sl C → CMat α := mkB N.xi0 N.yi0 N.xi0 N.yi0 (cOne Γ Sl α) (cOne Γ C α)
+def GBridge2.S1 : BRole Γ T Sl C → CMat α := mkB N.xi1 N.yi1 N.xi0 N.yi0 N.Z1 (cOne Γ C α)
+def GBridge2.S2 : BRole Γ T Sl C → CMat α := mkB N.xi2 N.yi2 N.xi0 N.yi0 N.Z2 (cOne Γ C α)
+def GBridge2.S3 : BRole Γ T Sl C → CMat α := mkB N.xi2 N.yi2 N.xi2 N.yi2 N.Z2 (cOne Γ C α)
+def GBridge2.S4 : BRole Γ T Sl C → CMat α := mkB N.xo2 N.yo2 N.xi2 N.yi2 N.Z3 (cOne Γ C α)
+def GBridge2.S5 : BRole Γ T Sl C → CMat α := mkB N.xo2 N.yo2 N.xi3 N.yi3 N.Z3 (cOne Γ C α)
+def GBridge2.S6 : BRole Γ T Sl C → CMat α := mkB N.xo2 N.yo2 N.xi4 N.yi4 N.Z4 (cOne Γ C α)
+def GBridge2.S7 : BRole Γ T Sl C → CMat α := mkB N.xo2 N.yo2 N.xo4 N.yo4 N.Z5 (cOne Γ C α)
+def GBridge2.S8 : BRole Γ T Sl C → CMat α :=
+  mkB (fun _ _ => kernel α) N.YF (fun _ _ => kernel α) N.YF (fun _ _ => kernel α) (cOne Γ C α)
+
+
+end Net
+end
+end GX
+end PowerSaving
+end OAI
+
+#print axioms OAI.PowerSaving.GX.GBridge2.unit
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Chain__NetStage.lean b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Chain__NetStage.lean
new file mode 100644
index 0000000000..670f4c6a84
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Chain__NetStage.lean
@@ -0,0 +1,186 @@
+import Work.GCert.Chain.NetDef
+
+/-!
+# (key: gx-chain) The bridged word B_2 on the generalised engine: the five stages and the idle climbs
+
+GENERATED by checks/wht26/gx-chain/py/netgen.py from Work/Bridge/Net.lean.  The stages are
+`GRoute`s (`gistage_fwd` / `gistage_bwd`, `gblift1` / `gblift2`); the idle climbs of the bank
+roles are the OLD block routes, unchanged.  No `sorry`.
+-/
+
+set_option linter.unusedSectionVars false
+set_option linter.unusedVariables false
+
+namespace OAI
+namespace PowerSaving
+namespace GX
+open Binary Matrix Finset RAM SS CB RF BR GF
+noncomputable section
+
+section Net
+variable {H T Sl C α Γ : Type} [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T]
+  [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] [Fintype α] [DecidableEq α]
+  [Fintype Γ] [DecidableEq Γ]
+variable (N : GBridge2 H T Sl C α Γ)
+
+/-- stage 0: forward on pair 1, every helper set starts at the matrix `1`. -/
+lemma GBridge2.stage0 :
+    GRoute N.S0 N.S1 (lift1 GF) (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := by
+  have h := gistage_fwd N.I (bid Γ T) (Equiv.refl Γ) N.G0 (fun _ => 1) (fun g => (N.G0 g).Xm.Φ 1)
+    (fun d => by rw [N.z0, Matrix.mul_one]) (fun d => by rw [N.z0, Matrix.mul_one])
+    (cOne Γ Sl α) (cOne Γ C α) (cOne Γ C α)
+  have e : bOut N.I.tv (bid Γ T) (fun d => (N.G0 d).Xm)
+      (fun g (q : Sl) => (N.G0 ((Equiv.refl Γ).symm g)).Xm.Φ 1 * cOne Γ Sl α g q) (cOne Γ C α)
+      = mkS N.xi1 N.yi1 N.Z1 (cOne Γ C α) := by
+    funext r
+    rcases r with (((⟨g,t⟩|⟨g,t⟩)|⟨g,q⟩)|⟨g,c⟩)
+    · exact N.x01 g t
+    · exact N.y01 g t
+    · show (N.G0 g).Xm.Φ 1 * 1 = (N.G0 g).Xm.Φ 1
+      rw [Matrix.mul_one]
+    · rfl
+  have h' : GRoute (bIn N.I.tv (bid Γ T) (fun d => (N.G0 d).Xm) (cOne Γ Sl α) (cOne Γ C α))
+      (mkS N.xi1 N.yi1 N.Z1 (cOne Γ C α)) GF (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := by
+    rw [← e]
+    exact h
+  exact gblift1 N.xi0 N.yi0 h'
+
+/-- stage 1: backward on pair 1; the helper set `g` serves the invocation `s1⁻¹ g`. -/
+lemma GBridge2.stage1 :
+    GRoute N.S1 N.S2 (lift1 GB) (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := by
+  have h := gistage_bwd N.I N.r1 N.s1 N.G1 N.N1 N.W1 N.hN1 N.hW1 N.Z1 (cOne Γ C α) (cOne Γ C α)
+  have e : bOut N.I.tv N.r1 (fun d => (N.G1 d).Xm) (fun g q => N.W1 (N.s1.symm g) * N.Z1 g q) (cOne Γ C α)
+      = mkS N.xi2 N.yi2 N.Z2 (cOne Γ C α) := by
+    funext r
+    rcases r with (((⟨g,t⟩|⟨g,t⟩)|⟨g,q⟩)|⟨g,c⟩)
+    · exact N.x12 g t
+    · exact N.y12 g t
+    · rfl
+    · rfl
+  have h' : GRoute (bIn N.I.tv N.r1 (fun d => (N.G1 d).Xm) N.Z1 (cOne Γ C α))
+      (mkS N.xi2 N.yi2 N.Z2 (cOne Γ C α)) GB (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := by
+    rw [← e]
+    exact h
+  exact gblift1 N.xi0 N.yi0 h'
+
+/-- stage 2: forward on pair 1. -/
+lemma GBridge2.stage2 :
+    GRoute N.S3 N.S4 (lift1 GF) (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) :=
+  gblift1 N.xi2 N.yi2
+    (gistage_fwd N.I N.r2 N.s2 N.G2 N.N2 N.W2 N.hN2 N.hW2 N.Z2 (cOne Γ C α) (cOne Γ C α))
+
+/-- stage 3: backward on pair 2. -/
+lemma GBridge2.stage3 :
+    GRoute N.S5 N.S6 (lift2 GB) (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := by
+  have h := gistage_bwd N.I N.r3 N.s3 N.G3 N.N3 N.W3 N.hN3 N.hW3 N.Z3 (cOne Γ C α) (cOne Γ C α)
+  have e : bOut N.I.tv N.r3 (fun d => (N.G3 d).Xm) (fun g q => N.W3 (N.s3.symm g) * N.Z3 g q) (cOne Γ C α)
+      = mkS N.xi4 N.yi4 N.Z4 (cOne Γ C α) := by
+    funext r
+    rcases r with (((⟨g,t⟩|⟨g,t⟩)|⟨g,q⟩)|⟨g,c⟩)
+    · exact N.x34 g t
+    · exact N.y34 g t
+    · rfl
+    · rfl
+  have h' : GRoute (bIn N.I.tv N.r3 (fun d => (N.G3 d).Xm) N.Z3 (cOne Γ C α))
+      (mkS N.xi4 N.yi4 N.Z4 (cOne Γ C α)) GB (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) := by
+    rw [← e]
+    exact h
+  exact gblift2 N.xo2 N.yo2 h'
+
+/-- stage 4: forward on pair 2. -/
+lemma GBridge2.stage4 :
+    GRoute N.S6 N.S7 (lift2 GF) (fun φ => (Fintype.card Γ : ℝ) * N.I.cost φ) :=
+  gblift2 N.xo2 N.yo2
+    (gistage_fwd N.I N.r4 N.s4 N.G4 N.N4 N.W4 N.hN4 N.hW4 N.Z4 (cOne Γ C α) (cOne Γ C α))
+
+/-- first idle climb of pair 2 (stages 0 and 1 in ONE block per role), to the frames of
+pair 1 after stage 1. -/
+lemma GBridge2.climbA :
+    XRoute N.S2 N.S3 id (fun φ => (Fintype.card Γ : ℝ) *
+      ((Fintype.card T : ℝ) * (N.caX φ + N.caY φ))) := by
+  have H1 := XRoute.reach_all N.S2 N.S3
+    (mkB (fun _ _ _ => 0) (fun _ _ _ => 0) (fun _ _ => N.caX)
+      (fun _ _ => N.caY) (fun _ _ _ => 0) (fun _ _ _ => 0)) (by
+      rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩)
+      · exact XReach.refl _
+      · exact XReach.refl _
+      · show XReach ((N.G0 g).Xm.Φ (tt (N.I.tv t))) ((N.G2 (N.r2 t g)).Xm.Φ (tt (N.I.tv t))) N.caX
+        rw [← N.x12 g t]
+        exact N.aX2 g t
+      · show XReach ((N.G0 g).Xm.Φ 0) ((N.G2 (N.r2 t g)).Xm.Φ 0) N.caY
+        rw [← N.y12 g t]
+        exact N.aY2 g t
+      · exact XReach.refl _
+      · exact XReach.refl _)
+  refine H1.cast (fun φ => ?_)
+  simp_rw [mkB_map (fun c : (ℕ → ℝ) → ℝ => c φ)]
+  rw [sum_mkB]
+  simp only [sum_const, card_univ, nsmul_eq_mul, mul_zero, add_zero, zero_add]
+  ring
+
+/-- second idle climb of pair 2: the window of stage 2, ONE block of rank `card H - 1` per
+role, to the frames of pair 1 after stage 2. -/
+lemma GBridge2.climbB :
+    XRoute N.S4 N.S5 id (fun φ => (Fintype.card Γ : ℝ) *
+      ((Fintype.card T : ℝ) * (2 * bcost φ (Fintype.card H - 1)))) := by
+  have H1 := XRoute.reach_all N.S4 N.S5
+    (mkB (fun _ _ _ => 0) (fun _ _ _ => 0) (fun _ _ φ => bcost φ (Fintype.card H - 1))
+      (fun _ _ φ => bcost φ (Fintype.card H - 1)) (fun _ _ _ => 0) (fun _ _ _ => 0)) (by
+      rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩)
+      · exact XReach.refl _
+      · exact XReach.refl _
+      · show XReach ((N.G2 (N.r2 t g)).Xm.Φ (tt (N.I.tv t))) ((N.G3 (N.r3 t g)).Xm.Φ (tt (N.I.tv t)))
+          (fun φ => bcost φ (Fintype.card H - 1))
+        rw [← N.x23 g t]
+        exact N.I.climbX (N.G2 (N.r2 t g)).Xm t
+      · show XReach ((N.G2 (N.r2 t g)).Xm.Φ 0) ((N.G3 (N.r3 t g)).Xm.Φ 0)
+          (fun φ => bcost φ (Fintype.card H - 1))
+        rw [← N.y23 g t]
+        exact N.I.climbY (N.G2 (N.r2 t g)).Xm t
+      · exact XReach.refl _
+      · exact XReach.refl _)
+  refine H1.cast (fun φ => ?_)
+  simp_rw [mkB_map (fun c : (ℕ → ℝ) → ℝ => c φ)]
+  rw [sum_mkB]
+  simp only [sum_const, card_univ, nsmul_eq_mul, mul_zero, add_zero, zero_add]
+  ring
+
+/-- the final climbs: pair 1 from its frames after stage 2 (ONE block per role for the
+stages 3, 4 and the rest), pair 2 from its frames after stage 4; the helper slots are
+already at the kernel. -/
+lemma GBridge2.fin :
+    XRoute N.S7 N.S8 id (fun φ => (Fintype.card Γ : ℝ) *
+      ((Fintype.card T : ℝ) * ((N.cX1 φ + N.cY1 φ) + (N.cX2 φ + N.cY2 φ)))) := by
+  have H1 := XRoute.reach_all N.S7 N.S8
+    (mkB (fun _ _ => N.cX1) (fun _ _ => N.cY1) (fun _ _ => N.cX2) (fun _ _ => N.cY2)
+      (fun _ _ _ => 0) (fun _ _ _ => 0)) (by
+      rintro ((((⟨g,t⟩|⟨g,t⟩)|(⟨g,t⟩|⟨g,t⟩))|⟨g,q⟩)|⟨g,c⟩)
+      · exact N.fX1 g t
+      · exact N.fY1 g t
+      · exact N.fX2 g t
+      · exact N.fY2 g t
+      · show XReach (N.W4 (N.s4.symm g) * (N.W3 (N.s3.symm g) * (N.W2 (N.s2.symm g) *
+          (N.W1 (N.s1.symm g) * (N.G0 g).Xm.Φ 1)))) (kernel α) (fun _ => 0)
+        rw [N.sfin g]
+        exact XReach.refl _
+      · exact XReach.refl _)
+  refine H1.cast (fun φ => ?_)
+  simp_rw [mkB_map (fun c : (ℕ → ℝ) → ℝ => c φ)]
+  rw [sum_mkB]
+  simp only [sum_const, card_univ, nsmul_eq_mul, mul_zero, add_zero, zero_add]
+  ring
+
+end Net
+end
+end GX
+end PowerSaving
+end OAI
+
+#print axioms OAI.PowerSaving.GX.GBridge2.stage0
+#print axioms OAI.PowerSaving.GX.GBridge2.stage1
+#print axioms OAI.PowerSaving.GX.GBridge2.stage2
+#print axioms OAI.PowerSaving.GX.GBridge2.stage3
+#print axioms OAI.PowerSaving.GX.GBridge2.stage4
+#print axioms OAI.PowerSaving.GX.GBridge2.climbA
+#print axioms OAI.PowerSaving.GX.GBridge2.climbB
+#print axioms OAI.PowerSaving.GX.GBridge2.fin
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Chain__Phased.lean b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Chain__Phased.lean
new file mode 100644
index 0000000000..e68c42377f
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Chain__Phased.lean
@@ -0,0 +1,199 @@
+import Work.Carrier.Phases
+import Work.GFrame.Labels.All
+
+/-!
+# (key: gx-chain) The certificate as two halves: `Scal` (scalar half) and `Lab` (label half)
+
+`Scal T Sl C`: the scalar fields of `CR.Phased` (three facts about ONE matrix
+`Mt = MB * scatM (Jr * Cc) * MA`).  The scalar identities of the forward and of the backward
+word (`fwd_scalar`, `bwd_scalar`) are those of `Work/Carrier/Phases.lean`, word for word.
+
+`Lab H S`: subspace labels (`GF.SLbl H`), routes of the three phases on the generalised engine
+(`GF.GRoute`) through every stage-B frame map (`GF.BXMap H α`), and facts about dimensions at
+the two ends.  No projector, no `Climb`.
+
+No `sorry`.
+-/
+
+set_option linter.unusedSectionVars false
+set_option linter.unusedVariables false
+
+namespace OAI
+namespace PowerSaving
+namespace GX
+open Binary Matrix Finset RAM SS CB BR CR GF
+noncomputable section
+
+/-- **The scalar half.** -/
+structure Scal (T Sl C : Type) [Fintype T] [DecidableEq T]
+    [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] where
+  Cc : Matrix C Sl ℚ
+  Jr : Matrix T C ℚ
+  MA : Matrix (L3 T Sl) (L3 T Sl) ℚ
+  MAi : Matrix (L3 T Sl) (L3 T Sl) ℚ
+  MB : Matrix (L3 T Sl) (L3 T Sl) ℚ
+  MBi : Matrix (L3 T Sl) (L3 T Sl) ℚ
+  hAi : MAi * MA = 1
+  hBi : MBi * MB = 1
+  hx : ∀ t j, (MB * scatM (Jr * Cc) * MA) (lX t) j = if lX t = j then 1 else 0
+  hy : ∀ i t, (MB * scatM (Jr * Cc) * MA) i (lY t) = if i = lY t then 1 else 0
+  hid : ∀ S t, (MB * scatM (Jr * Cc) * MA) (lY S) (lX t) = if S = t then 1 else 0
+
+/-- **The label half.**  See `checks/wht26/shared/gcert-interface.md`, section CHAIN. -/
+structure Lab (H : Type) [Fintype H] [DecidableEq H] {T Sl C : Type} [Fintype T] [DecidableEq T]
+    [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] (S : Scal T Sl C) where
+  tv : T → Space H
+  hunit : ∀ t, dot (tv t) (tv t) = 1
+  hgap : ∀ t, ∃ p, tv t p = 0
+  lab0 : L3 T Sl → SLbl H
+  labA : L3 T Sl → SLbl H
+  labB : L3 T Sl → SLbl H
+  labF : L3 T Sl → SLbl H
+  cen : C → SLbl H
+  z0 : SLbl H
+  hz0 : z0.dim = 0
+  h0x : ∀ t, (lab0 (lX t)).dim = 1 ∧ (lab0 (lX t)).Mem (tv t)
+  h0y : ∀ t, (lab0 (lY t)).dim = 0
+  h0s : ∀ q, (lab0 (lS q)).dim = 0
+  hAy : ∀ t, labA (lY t) = z0
+  hAc : ∀ k q, S.Cc k q ≠ 0 → cen k = labA (lS q)
+  hcen : ∀ k, 0 < (cen k).dim
+  hFx : ∀ t, (labF (lX t)).dim = Fintype.card H
+  hFs : ∀ q, (labF (lS q)).dim = Fintype.card H
+  hFy : ∀ t, (labF (lY t)).dim + 1 = Fintype.card H ∧
+    ∀ x, (labF (lY t)).Mem x → dot (tv t) x = 0
+  rA : List ℕ
+  rB : List ℕ
+  rF : List ℕ
+  pA : ∀ {α : Type} [Fintype α] [DecidableEq α] (X : BXMap H α),
+    Fintype.card H < Fintype.card α →
+    GRoute (fun r => X.Φ (lab0 r)) (fun r => X.Φ (labA r)) (actPoint S.MA)
+        (fun φ => rcost φ rA) ∧
+    GRoute (fun r => X.Φ (lab0 r)) (fun r => X.Φ (labA r)) (actPoint S.MAiᵀ)
+        (fun φ => rcost φ rA)
+  pB : ∀ {α : Type} [Fintype α] [DecidableEq α] (X : BXMap H α),
+    Fintype.card H < Fintype.card α →
+    GRoute (fun r => X.Φ (labA r)) (fun r => X.Φ (labB r)) (actPoint S.MB)
+        (fun φ => rcost φ rB) ∧
+    GRoute (fun r => X.Φ (labA r)) (fun r => X.Φ (labB r)) (actPoint S.MBiᵀ)
+        (fun φ => rcost φ rB)
+  pF : ∀ {α : Type} [Fintype α] [DecidableEq α] (X : BXMap H α),
+    Fintype.card H < Fintype.card α →
+    GRoute (fun r => X.Φ (labB r)) (fun r => X.Φ (labF r)) id (fun φ => rcost φ rF)
+
+/-- price of one invocation: the blocks of the three phases and one block per copy. -/
+def Lab.cost {H : Type} [Fintype H] [DecidableEq H] {T Sl C : Type} [Fintype T] [DecidableEq T]
+    [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C] {S : Scal T Sl C} (L : Lab H S)
+    (φ : ℕ → ℝ) : ℝ :=
+  rcost φ L.rA + (∑ k, bcost φ (L.cen k).dim) + rcost φ L.rB + rcost φ L.rF
+
+section Alg
+variable {H T Sl C : Type} [Fintype H] [DecidableEq H] [Fintype T] [DecidableEq T]
+  [Fintype Sl] [DecidableEq Sl] [Fintype C] [DecidableEq C]
+
+namespace Scal
+variable (K : Scal T Sl C)
+
+/-- the product of ALL gates of the main phase (scatter with the copies eliminated). -/
+def Mt : Matrix (L3 T Sl) (L3 T Sl) ℚ := K.MB * scatM (K.Jr * K.Cc) * K.MA
+/-- its inverse. -/
+def Mti : Matrix (L3 T Sl) (L3 T Sl) ℚ := K.MAi * scatMi (K.Jr * K.Cc) * K.MBi
+
+
+lemma im : K.Mti * K.Mt = 1 := by
+  unfold Mti Mt
+  have hB : ∀ X : Matrix (L3 T Sl) (L3 T Sl) ℚ, K.MBi * (K.MB * X) = X := fun X => by
+    rw [← Matrix.mul_assoc, K.hBi, Matrix.one_mul]
+  have hS : ∀ X : Matrix (L3 T Sl) (L3 T Sl) ℚ,
+      scatMi (K.Jr * K.Cc) * (scatM (K.Jr * K.Cc) * X) = X := fun X => by
+    rw [← Matrix.mul_assoc, scat_inv, Matrix.one_mul]
+  rw [Matrix.mul_assoc, Matrix.mul_assoc, Matrix.mul_assoc, hB, hS, K.hAi]
+
+lemma mi : K.Mt * K.Mti = 1 := mul_eq_one_comm.mp K.im
+
+lemma hxm : PX * K.Mt = PX := PX_mul_of (M := K.Mt) K.hx
+lemma hmy : K.Mt * PY = PY := mul_PY_of (M := K.Mt) K.hy
+lemma hiy : K.Mti * PY = PY := dirt_iy K.hmy K.im
+
+/-- **the dirt rule, forward**, for the matrices of the circuit. -/
+lemma hF : g8 K.Mti * K.Mt * g1 K.Mt = 1 + PY * K.Mt * PX := by
+  unfold g8 g1
+  exact dirt_fwd PX_sq PY_sq PX_PY PY_PX K.hxm K.hmy K.im K.mi
+
+/-- **the dirt rule, backward.** -/
+lemma hB : g1i K.Mt * K.Mti * h8 K.Mt = 1 - PY * K.Mt * PX := by
+  unfold g1i h8
+  exact dirt_bwd PX_sq PY_sq PX_PY PY_PX K.hxm K.hmy K.im K.mi
+
+/-- live part of the forward word before the scatter: `y -= K s`, then phase A. -/
+def f1 (u : L3 T Sl → ℂ) : L3 T Sl → ℂ := actPoint K.MA (actPoint (g1 K.Mt) u)
+/-- after the scatter: phase B, then the slot rows of the inverse. -/
+def f2 (u : L3 T Sl → ℂ) : L3 T Sl → ℂ := actPoint (g8 K.Mti) (actPoint K.MB u)
+/-- the same for the backward word (transposed inverses). -/
+def b1 (u : L3 T Sl → ℂ) : L3 T Sl → ℂ := actPoint K.MAiᵀ (actPoint (g1i K.Mt)ᵀ u)
+def b2 (u : L3 T Sl → ℂ) : L3 T Sl → ℂ := actPoint (h8 K.Mt)ᵀ (actPoint K.MBiᵀ u)
+
+lemma fwd_live (u : L3 T Sl → ℂ) :
+    K.f2 (actPoint (scatM (K.Jr * K.Cc)) (K.f1 u)) = actPoint (1 + PY * K.Mt * PX) u := by
+  rw [← K.hF]
+  show actPoint (g8 K.Mti) (actPoint K.MB (actPoint (scatM (K.Jr * K.Cc))
+      (actPoint K.MA (actPoint (g1 K.Mt) u))))
+    = actPoint (g8 K.Mti * (K.MB * scatM (K.Jr * K.Cc) * K.MA) * g1 K.Mt) u
+  rw [actPoint_mul, actPoint_mul, actPoint_mul, actPoint_mul]
+
+lemma bwd_live (u : L3 T Sl → ℂ) :
+    K.b2 (actPoint ((scatMi (K.Jr * K.Cc))ᵀ) (K.b1 u))
+      = actPoint ((1 - PY * K.Mt * PX)ᵀ) u := by
+  rw [← K.hB]
+  show actPoint (h8 K.Mt)ᵀ (actPoint K.MBiᵀ (actPoint (scatMi (K.Jr * K.Cc))ᵀ
+      (actPoint K.MAiᵀ (actPoint (g1i K.Mt)ᵀ u))))
+    = actPoint ((g1i K.Mt * (K.MAi * scatMi (K.Jr * K.Cc) * K.MBi) * h8 K.Mt)ᵀ) u
+  rw [Matrix.transpose_mul, Matrix.transpose_mul, Matrix.transpose_mul, Matrix.transpose_mul,
+    actPoint_mul, actPoint_mul, actPoint_mul, actPoint_mul]
+
+lemma liveAct_glue (φ : (L3 T Sl → ℂ) → (L3 T Sl → ℂ)) (u : L3 T Sl → ℂ) (c : C → ℂ) :
+    liveAct φ (glue u c) = glue (φ u) c :=
+  congrArg (fun v => glue (φ v) c) (glue_up u c)
+
+/-- **Scalar identity of the forward word**: `y += x`, slots restored, for ARBITRARY slot
+contents (the copies are erased before and after). -/
+theorem fwd_scalar (z : Box T Sl C → ℂ) :
+    eraseC (liveAct K.f2 (gateS bY bC K.Jr (gateS bC bS K.Cc (liveAct K.f1 (eraseC z)))))
+      = shearF z := by
+  have h1 : liveAct K.f1 (eraseC z) = glue (K.f1 (z ∘ upE)) (fun _ => 0) := by
+    rw [erase_glue z, liveAct_glue]
+  have hs : gateS bY bC K.Jr (gateS bC bS K.Cc (glue (K.f1 (z ∘ upE)) (fun _ => 0))) ∘ upE
+      = actPoint (scatM (K.Jr * K.Cc)) (K.f1 (z ∘ upE)) := by
+    have h := scat_glue K.Jr K.Cc (glue (K.f1 (z ∘ upE)) (fun _ => 0)) rfl
+    rw [glue_up] at h
+    exact h
+  rw [h1]
+  unfold liveAct
+  rw [erase_of_glue, hs, K.fwd_live, act_shear K.Mt K.hid]
+  exact glue_shearF z
+
+/-- **Scalar identity of the backward word**: `x -= y`, slots restored. -/
+theorem bwd_scalar (z : Box T Sl C → ℂ) :
+    eraseC (liveAct K.b2 (gateS bS bC (-K.Ccᵀ) (gateS bC bY K.Jrᵀ (liveAct K.b1 (eraseC z)))))
+      = shearB z := by
+  have h1 : liveAct K.b1 (eraseC z) = glue (K.b1 (z ∘ upE)) (fun _ => 0) := by
+    rw [erase_glue z, liveAct_glue]
+  have hs : gateS bS bC (-K.Ccᵀ) (gateS bC bY K.Jrᵀ (glue (K.b1 (z ∘ upE)) (fun _ => 0))) ∘ upE
+      = actPoint ((scatMi (K.Jr * K.Cc))ᵀ) (K.b1 (z ∘ upE)) := by
+    have h := scat_glueT K.Jr K.Cc (glue (K.b1 (z ∘ upE)) (fun _ => 0)) rfl
+    rw [glue_up] at h
+    exact h
+  rw [h1]
+  unfold liveAct
+  rw [erase_of_glue, hs, K.bwd_live, act_shearT K.Mt K.hid]
+  exact glue_shearB z
+
+end Scal
+end Alg
+end
+end GX
+end PowerSaving
+end OAI
+
+#print axioms OAI.PowerSaving.GX.Scal.fwd_scalar
+#print axioms OAI.PowerSaving.GX.Scal.bwd_scalar
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Chain__WinB.lean b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Chain__WinB.lean
new file mode 100644
index 0000000000..951edde5cf
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Chain__WinB.lean
@@ -0,0 +1,132 @@
+import Work.GCert.Chain.Win
+import Work.BridgeGeom.Stage
+
+/-!
+# (key: gx-chain) THE WINDOW LIFT (spec statement 3): the windows of the bridged geometry
+
+For the block frame map `BG.fmapB hB hH d q` of a stage of the bridged word (label space
+`LB B H = H ⊕ (B × H)`, old frame of the projector `P`: `PhiB d q P`) the stage-B frame map is
+`BXMap.ofConj d q` (`U ↦ d (U ⊕ span q)`, arbitrary subspaces `U` of `H`).  At a coordinate
+subspace of an orthonormal basis `A` of `H` the two name the SAME subspace of the address space
+(`insub_old_new`), so the old frame and the representative of any label of that subspace
+differ by a free factor (`frame_core_free`, `rep_change`): `old_to_new`, `new_to_old`.
+
+`gwinB hB hH d q : GWin H (LB B H)` with `(gwinB hB hH d q).Xm = fmapB hB hH d q` (`rfl`).
+
+No `sorry`.
+-/
+
+set_option linter.unusedSectionVars false
+set_option linter.unusedVariables false
+
+namespace OAI
+namespace PowerSaving
+namespace GX
+open Binary Matrix Finset RAM SS CB RF BR BG GF
+noncomputable section
+
+section
+variable {B H : Type} [Fintype B] [DecidableEq B] [Fintype H] [DecidableEq H]
+
+lemma blkT_mulVec (A : OBase H) (z : Space (LB B H)) (k : LB B H) :
+    ((blkB B A)ᵀ *ᵥ z) k = Sum.elim (fun i => dot (A.v i) (fun j => z (Sum.inl j)))
+      (fun b => z (Sum.inr b)) k := by
+  cases k with
+  | inl i =>
+    simp [blkB, Matrix.fromBlocks_transpose, Matrix.mulVec, dotProduct, Fintype.sum_sum_type,
+      cmat, dot]
+  | inr b =>
+    simp [blkB, Matrix.fromBlocks_transpose, Matrix.mulVec, dotProduct, Fintype.sum_sum_type,
+      Matrix.one_apply]
+
+/-- coordinates of `y` in the basis `d (A ⊕ std)`. -/
+lemma dot_cols (d : Orth (LB B H)) (A : OBase H) (y : Space (LB B H)) (k : LB B H) :
+    dot ((ofCols (d.1 * blkB B A) (mul_orth d.2 (blkB_orth A))).v k) y
+      = Sum.elim (fun i => dot (A.v i) (fun j => (d.1ᵀ *ᵥ y) (Sum.inl j)))
+          (fun b => (d.1ᵀ *ᵥ y) (Sum.inr b)) k := by
+  rw [← blkT_mulVec A (d.1ᵀ *ᵥ y) k, Matrix.mulVec_mulVec, ← Matrix.transpose_mul]
+  rfl
+
+/-- **the old frame and the lifted label name the same subspace of the address space.** -/
+theorem insub_old_new (d : Orth (LB B H)) (q : Finset (B × H)) (A : OBase H) (s : Finset H)
+    (U : SLbl H) (h : ∀ x, U.Mem x ↔ InSub (coordPerm A) s x) (y : Space (LB B H)) :
+    InSub (coordPerm (ofCols (d.1 * blkB B A) (mul_orth d.2 (blkB_orth A)))) (JB s q) y ↔
+      InSub (conjPerm d.1 d.2 (Orth.orth' d) U.G) (U.s.disjSum q) y := by
+  rw [insub_conj, insub_coordPerm]
+  have hU := h (fun i => (d.1ᵀ *ᵥ y) (Sum.inl i))
+  constructor
+  · intro hk
+    refine ⟨hU.mpr ((insub_coordPerm A s _).mpr (fun i hi => ?_)), fun b hb => ?_⟩
+    · have h1 := hk (Sum.inl i) (fun hm => hi (Finset.inl_mem_disjSum.mp hm))
+      rw [dot_cols] at h1
+      exact h1
+    · have h1 := hk (Sum.inr b) (fun hm => hb (Finset.inr_mem_disjSum.mp hm))
+      rw [dot_cols] at h1
+      exact h1
+  · rintro ⟨h1, h2⟩ k hk
+    rw [dot_cols]
+    cases k with
+    | inl i =>
+      exact (insub_coordPerm A s _).mp (hU.mp h1) i
+        (fun hm => hk (Finset.inl_mem_disjSum.mpr hm))
+    | inr b => exact h2 b (fun hm => hk (Finset.inr_mem_disjSum.mpr hm))
+
+/-- old frame of a coordinate subspace → representative of any label naming it: FREE. -/
+theorem old_to_new (d : Orth (LB B H)) (q : Finset (B × H)) (A : OBase H) (s : Finset H)
+    (U : SLbl H) (h : ∀ x, U.Mem x ↔ InSub (coordPerm A) s x) :
+    GReach (PhiB d q (A.mat s)) ((BXMap.ofConj d.1 d.2 (Orth.orth' d) q).Φ U) (fun _ => 0) := by
+  rw [PhiB_eq]
+  obtain ⟨N, hN, e⟩ := repChange (LB B H) _ _ _ _ (insub_old_new d q A s U h)
+  have a : GReach (frame (ofCols (d.1 * blkB B A) (mul_orth d.2 (blkB_orth A))) (JB s q))
+      (core (coordPerm (ofCols (d.1 * blkB B A) (mul_orth d.2 (blkB_orth A)))) (JB s q))
+      (fun φ => rcost φ []) :=
+    GReach.ofReach (fmv_old_to_core _ _).reach
+  have b : GReach
+      (core (coordPerm (ofCols (d.1 * blkB B A) (mul_orth d.2 (blkB_orth A)))) (JB s q))
+      (core (conjPerm d.1 d.2 (Orth.orth' d) U.G) (U.s.disjSum q)) (fun φ => rcost φ []) :=
+    GReach.ofReach (fmv_of_free hN e).reach
+  exact (a.trans b).cast (fun φ => by simp [rcost])
+
+/-- the same, backwards. -/
+theorem new_to_old (d : Orth (LB B H)) (q : Finset (B × H)) (A : OBase H) (s : Finset H)
+    (U : SLbl H) (h : ∀ x, U.Mem x ↔ InSub (coordPerm A) s x) :
+    GReach ((BXMap.ofConj d.1 d.2 (Orth.orth' d) q).Φ U) (PhiB d q (A.mat s)) (fun _ => 0) := by
+  rw [PhiB_eq]
+  obtain ⟨N, hN, e⟩ := repChange (LB B H) _ _ _ _
+    (fun y => (insub_old_new d q A s U h y).symm)
+  have a : GReach (core (conjPerm d.1 d.2 (Orth.orth' d) U.G) (U.s.disjSum q))
+      (core (coordPerm (ofCols (d.1 * blkB B A) (mul_orth d.2 (blkB_orth A)))) (JB s q))
+      (fun φ => rcost φ []) :=
+    GReach.ofReach (fmv_of_free hN e).reach
+  have b : GReach
+      (core (coordPerm (ofCols (d.1 * blkB B A) (mul_orth d.2 (blkB_orth A)))) (JB s q))
+      (frame (ofCols (d.1 * blkB B A) (mul_orth d.2 (blkB_orth A))) (JB s q))
+      (fun φ => rcost φ []) :=
+    GReach.ofReach (fmv_core_to_old _ _).reach
+  exact (a.trans b).cast (fun φ => by simp [rcost])
+
+/-- **The window of a stage of the bridged geometry.** -/
+def gwinB (hB : 1 ≤ Fintype.card B) (hH : 1 ≤ Fintype.card H) (d : Orth (LB B H))
+    (q : Finset (B × H)) : GWin H (LB B H) where
+  Xm := fmapB hB hH d q
+  X := BXMap.ofConj d.1 d.2 (Orth.orth' d) q
+  big := by
+    rw [card_LB]
+    have := Nat.mul_le_mul_right (Fintype.card H) (show 2 ≤ Fintype.card B + 1 by omega)
+    omega
+  toNew := fun A s U h => old_to_new d q A s U h
+  toOld := fun A s U h => new_to_old d q A s U h
+
+@[simp] lemma gwinB_Xm (hB : 1 ≤ Fintype.card B) (hH : 1 ≤ Fintype.card H) (d : Orth (LB B H))
+    (q : Finset (B × H)) : (gwinB hB hH d q).Xm = fmapB hB hH d q := rfl
+
+end
+end
+end GX
+end PowerSaving
+end OAI
+
+#print axioms OAI.PowerSaving.GX.insub_old_new
+#print axioms OAI.PowerSaving.GX.old_to_new
+#print axioms OAI.PowerSaving.GX.new_to_old
+#print axioms OAI.PowerSaving.GX.gwinB
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__B2Gp193Main.lean b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__B2Gp193Main.lean
new file mode 100644
index 0000000000..d7ecfebcea
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__B2Gp193Main.lean
@@ -0,0 +1,33 @@
+import Work.GCert.Scalar.Main
+import Work.GCert.Data.P193ScalOK
+import Work.GCert.Data.B2Gp193
+
+/-!
+# (key: gx-data) Bridged word B_2 with the gcert/1 circuit `P193`: THE CLOSED THEOREMS
+GENERATED by checks/wht26/gx-data/py/gxrate.py.  No `sorry`.
+`wht_B2Gp193_of_ids` (label half, price, unit list, rate fact, gx-chain's end theorem) applied to the scalar half
+`GXD.P193.scalOK : GS.ScalOK GXD.P193.raw` (kernel checks `scal<i>_ok`, `ychk_ok`; soundness: gx-scalar).
+-/
+
+namespace OAI
+namespace PowerSaving
+namespace WHT
+open BlockWHT
+
+/-- **Walsh-Hadamard transform in `O(n (log n)^(1 - 74745/10^8))`-type time bounds: whole-block recursion on the bridged
+word B_2 with the kernel-checked gcert/1 circuit `P193`.** -/
+theorem wht_main_B2Gp193 :
+    ∃ solve W, WHTProgram solve W ∧ BlockWHT.WHTTimeBoundsAt (1 - 74745/(10:ℝ)^8) W :=
+  wht_B2Gp193_of_ids GXD.P193.scalOK.hv GXD.P193.scalOK.hAi GXD.P193.scalOK.hBi GXD.P193.scalOK.hx GXD.P193.scalOK.hy GXD.P193.scalOK.hid
+
+/-- **Walsh-Hadamard transform in `O(n (log n)^(1 - 7474547/10^10))`-type time bounds: whole-block recursion on the bridged
+word B_2 with the kernel-checked gcert/1 circuit `P193`.** -/
+theorem wht_main_B2Gp193x :
+    ∃ solve W, WHTProgram solve W ∧ BlockWHT.WHTTimeBoundsAt (1 - 7474547/(10:ℝ)^10) W :=
+  wht_B2Gp193x_of_ids GXD.P193.scalOK.hv GXD.P193.scalOK.hAi GXD.P193.scalOK.hBi GXD.P193.scalOK.hx GXD.P193.scalOK.hy GXD.P193.scalOK.hid
+
+end WHT
+end PowerSaving
+end OAI
+#print axioms OAI.PowerSaving.WHT.wht_main_B2Gp193
+#print axioms OAI.PowerSaving.WHT.wht_main_B2Gp193x
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__Glue.lean b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__Glue.lean
new file mode 100644
index 0000000000..c6cbac38e3
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__Glue.lean
@@ -0,0 +1,97 @@
+import Work.GCert.Scalar.Roles
+import Work.GCert.Labels.Main
+
+/-!
+# (key: gx-data) Glue between the scalar half (gx-scalar) and the label half (gx-labels)
+
+For ANY `c : GXD.Raw` (nothing is evaluated here):
+
+* `roles`   gx-scalar's numbering `GS.emb` / `GS.unemb` is gx-labels' `GLab.Roles`;
+* `scalOf`  gx-chain's record `GX.Scal (Fin v) (Fin R) (Fin ret.length)` from the FIVE scalar
+            identities of the matrices `GS.MA`, `GS.MAi`, `GS.MB`, `GS.MBi`, `GS.Jrm`, `GS.Ccm`;
+* `scalOf_MA .. scalOf_MBi`  its gate matrices are the ordered products of the adds of `c`, in
+            the form asked by `GLab.Valid.lab` (`GLab.mic GS.toCoef`);
+* `reg`, `reg_slot`, `reg_mem`  copy `k` reads the retained register `(c.ret.getD k (0,0)).1`.
+
+So a generated instance needs from the scalar side exactly `hAi hBi hx hy hid` (see
+`wht_<Inst>_of_ids` in `Work/GCert/Data/<Inst>.lean`).  No `sorry`.
+-/
+
+namespace GXD
+open SSC OAI.PowerSaving OAI.PowerSaving.CR Matrix
+
+/-- the numbering of the roles: `x_t = t`, `y_t = v + t`, slot `q = 2 v + q` -/
+theorem roles (c : Raw) (hv : 0 < c.v) : GLab.Roles c.v c.R (GS.emb c) (GS.unemb c hv) where
+  hun := GS.unemb_emb hv
+  hin := GS.emb_unemb hv
+  hlt := fun q => by
+    rcases q with (t | S) | q
+    · have := t.isLt; show t.val < 2 * c.v + c.R; omega
+    · have := S.isLt; show c.v + S.val < 2 * c.v + c.R; omega
+    · have := q.isLt; show 2 * c.v + q.val < 2 * c.v + c.R; omega
+  eX := fun _ => rfl
+  eY := fun _ => rfl
+  eS := fun _ => rfl
+
+/-- **the scalar half as gx-chain's record**, from the five scalar identities -/
+noncomputable def scalOf (c : Raw) (hv : 0 < c.v)
+    (hAi : GS.MAi c hv * GS.MA c hv = 1) (hBi : GS.MBi c hv * GS.MB c hv = 1)
+    (hx : ∀ (t : Fin c.v) (j : GS.Rl c), (GS.MB c hv * scatM (GS.Jrm c * GS.Ccm c) * GS.MA c hv)
+      (lX t) j = if lX t = j then 1 else 0)
+    (hy : ∀ (i : GS.Rl c) (t : Fin c.v), (GS.MB c hv * scatM (GS.Jrm c * GS.Ccm c) * GS.MA c hv)
+      i (lY t) = if i = lY t then 1 else 0)
+    (hid : ∀ S t : Fin c.v, (GS.MB c hv * scatM (GS.Jrm c * GS.Ccm c) * GS.MA c hv)
+      (lY S) (lX t) = if S = t then 1 else 0) :
+    GX.Scal (Fin c.v) (Fin c.R) (Fin c.ret.length) where
+  Cc := GS.Ccm c
+  Jr := GS.Jrm c
+  MA := GS.MA c hv
+  MAi := GS.MAi c hv
+  MB := GS.MB c hv
+  MBi := GS.MBi c hv
+  hAi := hAi
+  hBi := hBi
+  hx := hx
+  hy := hy
+  hid := hid
+
+section
+variable (c : Raw) (hv : 0 < c.v) (hAi) (hBi) (hx) (hy) (hid)
+
+theorem scalOf_MA : (scalOf c hv hAi hBi hx hy hid).MA
+    = matP (GS.unemb c hv) (GLab.mic GS.toCoef (c.gatesA.flatMap Gate.adds)) := rfl
+
+theorem scalOf_MAi : (scalOf c hv hAi hBi hx hy hid).MAi
+    = matPinv (GS.unemb c hv) (GLab.mic GS.toCoef (c.gatesA.flatMap Gate.adds)) := rfl
+
+theorem scalOf_MB : (scalOf c hv hAi hBi hx hy hid).MB
+    = matP (GS.unemb c hv) (GLab.mic GS.toCoef (c.gatesB.flatMap Gate.adds)) := rfl
+
+theorem scalOf_MBi : (scalOf c hv hAi hBi hx hy hid).MBi
+    = matPinv (GS.unemb c hv) (GLab.mic GS.toCoef (c.gatesB.flatMap Gate.adds)) := rfl
+
+end
+
+/-- the register read by copy `k` (the slot of the retained total `k`) -/
+def reg (c : Raw) (k : Fin c.ret.length) : Nat := (c.ret.getD k.val (0, 0)).1
+
+theorem reg_slot (c : Raw) (hv : 0 < c.v) (hAi) (hBi) (hx) (hy) (hid) (k : Fin c.ret.length)
+    (q : Fin c.R) (h : (scalOf c hv hAi hBi hx hy hid).Cc k q ≠ 0) :
+    reg c k = 2 * c.v + q.val := by
+  have h' : (if 2 * c.v + q.val = (c.ret.getD k.val (0, 0)).1 then (1:ℚ) else 0) ≠ 0 := h
+  by_cases e : 2 * c.v + q.val = (c.ret.getD k.val (0, 0)).1
+  · exact e.symm
+  · rw [if_neg e] at h'; exact absurd rfl h'
+
+theorem reg_mem (c : Raw) (k : Fin c.ret.length) : reg c k ∈ c.ret.map Prod.fst := by
+  unfold reg
+  rw [List.getD_eq_getElem _ _ k.isLt]
+  exact List.mem_map.mpr ⟨_, List.getElem_mem k.isLt, rfl⟩
+
+end GXD
+
+#print axioms GXD.roles
+#print axioms GXD.scalOf
+#print axioms GXD.scalOf_MA
+#print axioms GXD.reg_slot
+#print axioms GXD.reg_mem
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__InstB2Gp193.lean b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__InstB2Gp193.lean
new file mode 100644
index 0000000000..9b9925ce26
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__InstB2Gp193.lean
@@ -0,0 +1,55 @@
+import Work.BridgeGeom.B2
+import Work.FoldRate.Group
+import Work.GCert.Data.RateB2Gp193
+import Work.GCert.Data.RateB2Gp193x
+
+/-!
+# (key: gx-data) Bridged word B_2 with the gcert/1 circuit `P193`, h = 22: the unit price is the price of the block list
+
+GENERATED by checks/wht26/gx-data/py/gxrate.py.  No `sorry`.  Arithmetic only (no certificate is imported).
+
+One unit: `W = 14692` live roles (`4 v + R`, `v = 1320`, `R = 9412`), label dimension `m = 110`, deficit `D = 3080`.
+`invPrice φ` = the price of the blocks of ONE invocation (the kernel-counted list `GXD.P193.hist`, by rank);
+`unitPrice φ = 5 * invPrice φ + v * BG.bankPrice h φ = listCost φ BlockAccounting.blocksB2Gp193` (`unit_cost`).
+Whole-block `1 - 74745/10^8` (fails at 74746), ten decimals `1 - 7474547/10^10` (fails at 7474550).
+-/
+
+namespace OAI
+namespace PowerSaving
+namespace BridgeRate
+namespace B2Gp193
+open Finset Binary Matrix RAM CB FoldRate
+noncomputable section
+
+/-- price of the blocks of ONE invocation of `P193` (all roles and scratch copies), by rank -/
+def invPrice (φ : ℕ → ℝ) : ℝ := 29178 * φ 1 + 12398 * φ 2 + 9923 * φ 3 + 4209 * φ 4 + 1950 * φ 5 + 1605 * φ 6 + 706 * φ 7 + 1531 * φ 8 + 429 * φ 9 + 930 * φ 10 + 354 * φ 11 + 1197 * φ 12 + 127 * φ 13 + 742 * φ 14 + 1879 * φ 15 + 88 * φ 16 + 81 * φ 17 + 2653 * φ 18 + 167 * φ 19 + 22 * φ 20
+
+/-- price of ONE unit of the bridged word B_2 -/
+def unitPrice (φ : ℕ → ℝ) : ℝ := 5 * invPrice φ + ((1320:ℕ):ℝ) * BG.bankPrice 22 φ
+
+/-- **The unit price is the price of the unit block list.** -/
+theorem unit_cost (φ : ℕ → ℝ) : unitPrice φ = listCost φ BlockAccounting.blocksB2Gp193 := by
+  unfold unitPrice invPrice
+  simp only [BG.bankPrice, listCost, BlockAccounting.blocksB2Gp193, List.map_cons, List.map_nil,
+    List.sum_cons, List.sum_nil]
+  norm_num [bcost]
+  try ring
+
+theorem blocks_x : BlockAccounting.blocksB2Gp193 = BlockAccounting.blocksB2Gp193x := rfl
+
+theorem unit_cost_x (φ : ℕ → ℝ) : unitPrice φ = listCost φ BlockAccounting.blocksB2Gp193x := by
+  rw [unit_cost, blocks_x]
+
+/-- one unit stands for `W * m - D` unit moves -/
+theorem unit_tally : unitPrice (fun r => (r:ℝ)) = ((14692:ℕ):ℝ) * ((110:ℕ):ℝ) - ((3080:ℕ):ℝ) := by
+  rw [unit_cost]
+  exact BlockAccounting.blocksB2Gp193_tally
+
+end
+end B2Gp193
+end BridgeRate
+end PowerSaving
+end OAI
+
+#print axioms OAI.PowerSaving.BridgeRate.B2Gp193.unit_cost
+#print axioms OAI.PowerSaving.BridgeRate.B2Gp193.unit_tally
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__P193Cert.lean b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__P193Cert.lean
new file mode 100644
index 0000000000..3c3250279e
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__P193Cert.lean
@@ -0,0 +1,81 @@
+import Work.GCert.Labels.Main
+import Work.GCert.Data.P193Price
+import Work.GCert.Data.Gen.P193Tab0
+import Work.GCert.Data.Gen.P193Tab1
+import Work.GCert.Data.Gen.P193Tab2
+import Work.GCert.Data.Gen.P193Tab3
+import Work.GCert.Data.Gen.P193Seg0
+import Work.GCert.Data.Gen.P193Seg1
+import Work.GCert.Data.Gen.P193Seg2
+import Work.GCert.Data.Gen.P193Seg3
+import Work.GCert.Data.Gen.P193Seg4
+import Work.GCert.Data.Gen.P193SegF
+import Work.GCert.Data.Gen.P193End
+/-! # (key: gx-data) gcert/1 certificate `P193` (h = 22): THE LABEL HALF and its price
+GENERATED by checks/wht26/gx-data/py/gxgen.py from gcert1-p11-pr193.json.gz.  No `sorry`.
+`v = 1320` ports, `R = 9412` slots, 22 scratch copies of rank 20, copy cost `cst = 440`, `N = 262944` unit moves.
+* `aux : GLab.Aux`, `valid : GLab.Valid raw aux`: frame table (`tab<i>_ok` composed by `tabK_node`), the replays of phase A
+  and phase B (`seg<i>_ok` composed by `Run.append`), the final climbs (`segF_ok`, `fin_ok`), the ends (`end_ok`);
+* `lab_cost`, `lab_price`: for ANY scalar half `S` the label half `valid.lab S ..` costs
+  `rcost rA + card C * bcost 20 + rcost rB + rcost rF` = `invPrice φ` = 29178 * φ 1 + 12398 * φ 2 + 9923 * φ 3 + 4209 * φ 4 + 1950 * φ 5 + 1605 * φ 6 + 706 * φ 7 + 1531 * φ 8 + 429 * φ 9 + 930 * φ 10 + 354 * φ 11 + 1197 * φ 12 + 127 * φ 13 + 742 * φ 14 + 1879 * φ 15 + 88 * φ 16 + 81 * φ 17 + 2653 * φ 18 + 167 * φ 19 + 22 * φ 20
+  (`Gen.P193Hist`, `P193Price`) when there are 22 copies. -/
+set_option maxRecDepth 100000
+namespace GXD.P193
+open GXD OAI.PowerSaving OAI.PowerSaving.CB
+theorem raw_h : raw.h = 22 := rfl
+theorem raw_v : raw.v = 1320 := rfl
+theorem raw_R : raw.R = 9412 := rfl
+/-- what the label checks read next to `raw` -/
+noncomputable def aux : GLab.Aux := ⟨par, st0, st2, st5, st6, fins, rA, rB, rF⟩
+theorem fin_ok : GLab.noAdds fins = true := by decide +kernel
+theorem cen_ok : (raw.ret.map Prod.fst).all (fun r => Nat.beq (GLab.cdim (par.tab.get (st2.get r))) 20) = true := by
+  decide +kernel
+theorem tab_ok : GLab.tabK par.h par.tab = true := (GLab.tabK_node _ _ _ (GLab.tabK_node _ _ _ tab0_ok tab1_ok) (GLab.tabK_node _ _ _ tab2_ok tab3_ok))
+theorem gatesA_eq : raw.gatesA = seg0 ++ (seg1) := by
+  show List.flatten [seg0, seg1] = _
+  simp only [List.flatten_cons, List.flatten_nil, List.append_nil]
+/-- the replay of phase A, composed from its kernel-checked segments -/
+theorem runA : GLab.Run par st0 raw.gatesA st2 rA := by
+  rw [gatesA_eq]
+  exact GLab.Run.append (GLab.Run.of_seg seg0_ok) (GLab.Run.of_seg seg1_ok)
+theorem gatesB_eq : raw.gatesB = seg2 ++ (seg3 ++ (seg4)) := by
+  show List.flatten [seg2, seg3, seg4] = _
+  simp only [List.flatten_cons, List.flatten_nil, List.append_nil]
+/-- the replay of phase B, composed from its kernel-checked segments -/
+theorem runB : GLab.Run par st2 raw.gatesB st5 rB := by
+  rw [gatesB_eq]
+  exact GLab.Run.append (GLab.Run.of_seg seg2_ok) (GLab.Run.append (GLab.Run.of_seg seg3_ok) (GLab.Run.of_seg seg4_ok))
+/-- **the kernel-checked label facts** of the certificate -/
+theorem valid : GLab.Valid raw aux where
+  hn := rfl
+  tab := tab_ok
+  runA := runA
+  runB := runB
+  runF := GLab.Run.of_seg segF_ok
+  fin := fin_ok
+  ends := end_ok
+theorem cen_dim (r : Nat) (hr : r ∈ raw.ret.map Prod.fst) : GLab.cdim (par.tab.get (st2.get r)) = 20 :=
+  Nat.eq_of_beq_eq_true (List.all_eq_true.mp cen_ok r hr)
+/-- price of the label half over ANY scalar half -/
+theorem lab_cost {C : Type} [Fintype C] [DecidableEq C] (S : GX.Scal (Fin raw.v) (Fin raw.R) C)
+    {emb : CR.L3 (Fin raw.v) (Fin raw.R) → Nat} {un : Nat → CR.L3 (Fin raw.v) (Fin raw.R)}
+    (Ro : GLab.Roles raw.v raw.R emb un) (cf : Co → SSC.Coef) (hMA) (hMAi) (hMB) (hMBi)
+    (reg : C → Nat) (hCc) (hreg) (φ : ℕ → ℝ) :
+    (valid.lab S Ro cf hMA hMAi hMB hMBi reg hCc hreg).cost φ
+      = rcost φ rA + (Fintype.card C : ℝ) * bcost φ 20 + rcost φ rB + rcost φ rF := by
+  obtain ⟨h1, h2, h3, h4⟩ := valid.lab_ranks S Ro cf hMA hMAi hMB hMBi reg hCc hreg
+  have h5 : ∀ k, bcost φ (((valid.lab S Ro cf hMA hMAi hMB hMBi reg hCc hreg).cen k).dim) = bcost φ 20 :=
+    fun k => by rw [h4 k]; exact congrArg (bcost φ) (cen_dim _ (hreg k))
+  rw [GX.Lab.cost, h1, h2, h3, Finset.sum_congr rfl (fun k _ => h5 k), Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
+  rfl
+/-- **the label half costs one invocation price** (`invPrice`, by rank, from the kernel counts) -/
+theorem lab_price {C : Type} [Fintype C] [DecidableEq C] (S : GX.Scal (Fin raw.v) (Fin raw.R) C)
+    {emb : CR.L3 (Fin raw.v) (Fin raw.R) → Nat} {un : Nat → CR.L3 (Fin raw.v) (Fin raw.R)}
+    (Ro : GLab.Roles raw.v raw.R emb un) (cf : Co → SSC.Coef) (hMA) (hMAi) (hMB) (hMBi)
+    (reg : C → Nat) (hCc) (hreg) (hC : Fintype.card C = 22) (φ : ℕ → ℝ) :
+    (valid.lab S Ro cf hMA hMAi hMB hMBi reg hCc hreg).cost φ = invPrice φ := by
+  rw [lab_cost S Ro cf hMA hMAi hMB hMBi reg hCc hreg φ, hC, ← inv_cost φ]
+  norm_num [bcost]
+end GXD.P193
+#print axioms GXD.P193.valid
+#print axioms GXD.P193.lab_price
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__P193Price.lean b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__P193Price.lean
new file mode 100644
index 0000000000..eeed011d64
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__P193Price.lean
@@ -0,0 +1,54 @@
+import Work.GCert.Data.Price
+import Work.GCert.Data.Gen.P193Hist
+
+/-!
+# (key: gx-data) gcert/1 certificate `P193` (h = 22): the price of one invocation
+
+GENERATED by checks/wht26/gx-data/py/gxgen.py from gcert1-p11-pr193.json.gz.  No `sorry`.
+`rA_cost`, `rB_cost`, `rF_cost`: the price of the block rank lists of phase A, phase B and the final climbs
+(lists `GXD.P193.rA / rB / rF`, counted by the kernel in `Gen.P193Hist`);
+`inv_cost`: with the 22 scratch copies (22 * φ 20) the blocks of ONE invocation cost
+`invPrice φ` = 29178 * φ 1 + 12398 * φ 2 + 9923 * φ 3 + 4209 * φ 4 + 1950 * φ 5 + 1605 * φ 6 + 706 * φ 7 + 1531 * φ 8 + 429 * φ 9 + 930 * φ 10 + 354 * φ 11 + 1197 * φ 12 + 127 * φ 13 + 742 * φ 14 + 1879 * φ 15 + 88 * φ 16 + 81 * φ 17 + 2653 * φ 18 + 167 * φ 19 + 22 * φ 20;  `inv_moves`: they stand for `R h + 2 v (h - 1) + cst = 262944` unit moves.
+-/
+
+namespace GXD.P193
+open GXD OAI.PowerSaving.CB
+
+/-- price of the blocks of phase A, from the kernel counts -/
+theorem rA_cost (φ : ℕ → ℝ) : rcost φ rA = 9900 * φ 1 + 500 * φ 2 + 2402 * φ 3 + 82 * φ 4 + 242 * φ 5 + 175 * φ 6 + 88 * φ 7 + 68 * φ 8 + 110 * φ 9 + 132 * φ 10 + 10 * φ 11 + 67 * φ 12 + 3 * φ 13 + 19 * φ 15 := by
+  rw [GXD.rcost_wsumK φ rA 19 rankBoundA]
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero, rankA0_ok, rankA1_ok, rankA2_ok, rankA3_ok, rankA4_ok, rankA5_ok, rankA6_ok, rankA7_ok, rankA8_ok, rankA9_ok, rankA10_ok, rankA11_ok, rankA12_ok, rankA13_ok, rankA14_ok, rankA15_ok, rankA16_ok, rankA17_ok, rankA18_ok, rankA19_ok]
+  norm_num [bcost]
+
+/-- price of the blocks of phase B, from the kernel counts -/
+theorem rB_cost (φ : ℕ → ℝ) : rcost φ rB = 16638 * φ 1 + 10380 * φ 2 + 4111 * φ 3 + 3476 * φ 4 + 1531 * φ 5 + 1137 * φ 6 + 531 * φ 7 + 1346 * φ 8 + 293 * φ 9 + 629 * φ 10 + 343 * φ 11 + 1067 * φ 12 + 123 * φ 13 + 712 * φ 14 + 1860 * φ 15 + 81 * φ 16 + 39 * φ 17 + 2640 * φ 18 := by
+  rw [GXD.rcost_wsumK φ rB 19 rankBoundB]
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero, rankB0_ok, rankB1_ok, rankB2_ok, rankB3_ok, rankB4_ok, rankB5_ok, rankB6_ok, rankB7_ok, rankB8_ok, rankB9_ok, rankB10_ok, rankB11_ok, rankB12_ok, rankB13_ok, rankB14_ok, rankB15_ok, rankB16_ok, rankB17_ok, rankB18_ok, rankB19_ok]
+  norm_num [bcost]
+
+/-- price of the blocks of the final climbs, from the kernel counts -/
+theorem rF_cost (φ : ℕ → ℝ) : rcost φ rF = 2640 * φ 1 + 1518 * φ 2 + 3410 * φ 3 + 651 * φ 4 + 177 * φ 5 + 293 * φ 6 + 87 * φ 7 + 117 * φ 8 + 26 * φ 9 + 169 * φ 10 + 1 * φ 11 + 63 * φ 12 + 1 * φ 13 + 30 * φ 14 + 7 * φ 16 + 42 * φ 17 + 13 * φ 18 + 167 * φ 19 := by
+  rw [GXD.rcost_wsumK φ rF 19 rankBoundF]
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero, rankF0_ok, rankF1_ok, rankF2_ok, rankF3_ok, rankF4_ok, rankF5_ok, rankF6_ok, rankF7_ok, rankF8_ok, rankF9_ok, rankF10_ok, rankF11_ok, rankF12_ok, rankF13_ok, rankF14_ok, rankF15_ok, rankF16_ok, rankF17_ok, rankF18_ok, rankF19_ok]
+  norm_num [bcost]
+
+/-- price of the blocks of ONE invocation (x roles, y roles, slots, scratch copies), by rank -/
+def invPrice (φ : ℕ → ℝ) : ℝ := 29178 * φ 1 + 12398 * φ 2 + 9923 * φ 3 + 4209 * φ 4 + 1950 * φ 5 + 1605 * φ 6 + 706 * φ 7 + 1531 * φ 8 + 429 * φ 9 + 930 * φ 10 + 354 * φ 11 + 1197 * φ 12 + 127 * φ 13 + 742 * φ 14 + 1879 * φ 15 + 88 * φ 16 + 81 * φ 17 + 2653 * φ 18 + 167 * φ 19 + 22 * φ 20
+
+/-- **Price of ONE invocation** in the shape of `GX.Lab.cost`: phase A, the scratch copies, phase B, final climbs. -/
+theorem inv_cost (φ : ℕ → ℝ) : rcost φ rA + (22 * φ 20) + rcost φ rB + rcost φ rF = invPrice φ := by
+  rw [rA_cost, rB_cost, rF_cost]
+  unfold invPrice
+  ring
+
+theorem inv_moves : invPrice (fun r => (r:ℝ)) = 9412 * 22 + 2 * 1320 * (22 - 1) + 440 := by
+  unfold invPrice
+  norm_num
+
+theorem inv_count : invPrice (fun _ => (1:ℝ)) = 70169 := by
+  unfold invPrice
+  norm_num
+
+end GXD.P193
+#print axioms GXD.P193.inv_cost
+#print axioms GXD.P193.inv_moves
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__P193ScalOK.lean b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__P193ScalOK.lean
new file mode 100644
index 0000000000..3539192ac1
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__P193ScalOK.lean
@@ -0,0 +1,17 @@
+import Work.GCert.Scalar.Main
+import Work.GCert.Data.Gen.P193Scal0
+import Work.GCert.Data.Gen.P193Scal1
+import Work.GCert.Data.Gen.P193YChk
+/-! # (key: gx-data) gcert/1 certificate `P193` (h = 22): THE SCALAR HALF `scalOK : GS.ScalOK raw`
+GENERATED by checks/wht26/gx-data/py/gxgen.py from gcert1-p11-pr193.json.gz.  No `sorry`.
+Kernel checks `scal<i>_ok` (2 source batches) and `ychk_ok`. -/
+namespace GXD.P193
+open GXD
+theorem scalOK : GS.ScalOK raw where
+  hv := by decide
+  hs := fun t ht =>
+    if h0 : t < 660 then ⟨0, 660, Nat.zero_le t, h0, GS.XRun.of_check scal0_ok⟩ else
+    ⟨660, 660, Nat.le_of_not_lt h0, Nat.lt_of_lt_of_le ht (by decide), GS.XRun.of_check scal1_ok⟩
+  hyc := fun t ht => ⟨0, 1320, Nat.zero_le t, Nat.lt_of_lt_of_le ht (by decide), GS.YRun.of_check ychk_ok⟩
+end GXD.P193
+#print axioms GXD.P193.scalOK
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__RateB2Gp193x.lean b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__RateB2Gp193x.lean
new file mode 100644
index 0000000000..df563cde73
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__RateB2Gp193x.lean
@@ -0,0 +1,327 @@
+import Work.BlockAccounting.Moment
+import Work.BlockAccounting.Sharp
+import Work.FoldRate.RankTools
+
+/-!
+# (key: gx-data) Rate lemmas for one unit of the bridged word B_2 with the gcert/1 circuit `P193` (h = 22)
+
+GENERATED by checks/wht26/gx-data/py/gxrate.py with the text generators of checks/wht9/bridge-rate/py
+(used read-only).  No `sorry`.  Arithmetic only.
+
+One unit: `W = 14692` live roles, label dimension `m = 110`, deficit `D = 3080`.
+
+* `blocksB2Gp193x`            the block list (rank, count) of ONE unit;
+* `fold_B2Gp193x`             whole-block fact at `1 - 7474547/10^10`, fill parameter `s = 40` (slack 2.37e-12);
+* `fold_B2Gp193x_sharp`       `W ≤ moment` at `1 - 7474550/10^10`: fails for every table (excess 1.40e-10);
+* `foldRank_B2Gp193x`         per-rank fact at `1 - 4058344/10^10`, `s = 40` (slack 1.10e-07);
+* `foldRank_B2Gp193x_sharp`   per-rank: fails for every table at `1 - 4058349/10^10`.
+
+That a Lean word has `G` times this list as its block histogram is NOT proved here.
+-/
+
+set_option maxRecDepth 4000
+
+namespace OAI.PowerSaving.BlockAccounting
+open Finset
+
+/-- Block list `(rank, count)` of ONE unit of the bridged five-stage word B_2 with the gcert/1 circuit `P193` (h = 22, m = 110): five invocations (ALL blocks of one invocation: x roles, y roles, slots, scratch copies; the kernel-counted list `GXD.P193.hist`), and for each of the v = 1320 ports the idle climbs of its two twins (2 blocks each of rank h-1, 2h-2, 2h+2, 4); R = 9412 slots shared by the five stages. -/
+def blocksB2Gp193x : List (ℕ × ℕ) := [(1, 145890), (2, 61990), (3, 49615), (4, 23685), (5, 9750), (6, 8025), (7, 3530), (8, 7655), (9, 2145), (10, 4650), (11, 1770), (12, 5985), (13, 635), (14, 3710), (15, 9395), (16, 440), (17, 405), (18, 13265), (19, 835), (20, 110), (21, 2640), (42, 2640), (46, 2640)]
+
+/-- total rank of one unit (= number of unit moves it stands for): `W * m - D` -/
+theorem blocksB2Gp193x_rank : (blocksB2Gp193x.map fun b => b.1 * b.2).sum = 14692 * 110 - 3080 := by
+  norm_num [blocksB2Gp193x]
+
+/-- the same as a price: the unit list at the price list `r ↦ r` (shape of `hPid` in `RAM.engine_program_group_perRank`) -/
+theorem blocksB2Gp193x_tally : (blocksB2Gp193x.map fun b => (b.2:ℝ) * ((b.1:ℕ):ℝ)).sum = ((14692:ℕ):ℝ) * ((110:ℕ):ℝ) - ((3080:ℕ):ℝ) := by
+  norm_num [blocksB2Gp193x]
+
+theorem blocksB2Gp193x_count : (blocksB2Gp193x.map fun b => b.2).sum = 361405 := by
+  norm_num [blocksB2Gp193x]
+
+theorem blocksB2Gp193x_proper : ∀ b ∈ blocksB2Gp193x, 1 ≤ b.1 ∧ b.1 < 110 := by
+  decide
+
+/-- **whole-block fact** of `blocksB2Gp193x` at exponent `1 - 7474547/10^10`, fill parameter `s = 40`: the hypothesis `hnum` of `RAM.engine_program_group_list` (u = 110, W = 14692).  Relative slack of the rational certificate: 2.37e-12. -/
+theorem fold_B2Gp193x :
+    ((2:ℝ)^40 - 1) * moment 110 (1 - 7474547/(10:ℝ)^10) blocksB2Gp193x
+      + ((14692:ℕ):ℝ) * (T 110 (1 - 7474547/(10:ℝ)^10) (110 - 1) 1 + T 110 (1 - 7474547/(10:ℝ)^10) 1 1)
+      < (2:ℝ)^40 * ((14692:ℕ):ℝ) := by
+  have hsum : moment 110 (1 - 7474547/(10:ℝ)^10) blocksB2Gp193x
+      = T 110 (1 - 7474547/(10:ℝ)^10) 1 145890 + (T 110 (1 - 7474547/(10:ℝ)^10) 2 61990 + (T 110 (1 - 7474547/(10:ℝ)^10) 3 49615 + (T 110 (1 - 7474547/(10:ℝ)^10) 4 23685 + (T 110 (1 - 7474547/(10:ℝ)^10) 5 9750 + (T 110 (1 - 7474547/(10:ℝ)^10) 6 8025 + (T 110 (1 - 7474547/(10:ℝ)^10) 7 3530 + (T 110 (1 - 7474547/(10:ℝ)^10) 8 7655 + (T 110 (1 - 7474547/(10:ℝ)^10) 9 2145 + (T 110 (1 - 7474547/(10:ℝ)^10) 10 4650 + (T 110 (1 - 7474547/(10:ℝ)^10) 11 1770 + (T 110 (1 - 7474547/(10:ℝ)^10) 12 5985 + (T 110 (1 - 7474547/(10:ℝ)^10) 13 635 + (T 110 (1 - 7474547/(10:ℝ)^10) 14 3710 + (T 110 (1 - 7474547/(10:ℝ)^10) 15 9395 + (T 110 (1 - 7474547/(10:ℝ)^10) 16 440 + (T 110 (1 - 7474547/(10:ℝ)^10) 17 405 + (T 110 (1 - 7474547/(10:ℝ)^10) 18 13265 + (T 110 (1 - 7474547/(10:ℝ)^10) 19 835 + (T 110 (1 - 7474547/(10:ℝ)^10) 20 110 + (T 110 (1 - 7474547/(10:ℝ)^10) 21 2640 + (T 110 (1 - 7474547/(10:ℝ)^10) 42 2640 + (T 110 (1 - 7474547/(10:ℝ)^10) 46 2640 + 0)))))))))))))))))))))) := rfl
+  have b0 : T 110 (1 - 7474547/(10:ℝ)^10) 1 145890 ≤ ((16636758090767:ℝ)/12500000000) :=
+    term_le 110 1 145890 (7474547/(10:ℝ)^10) ((2350240182897:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 27 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have b1 : T 110 (1 - 7474547/(10:ℝ)^10) 2 61990 ≤ ((22609438945563:ℝ)/20000000000) :=
+    term_le 110 2 61990 (7474547/(10:ℝ)^10) ((2003666592617:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 26 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have b2 : T 110 (1 - 7474547/(10:ℝ)^10) 3 49615 ≤ ((135678423202471:ℝ)/100000000000) :=
+    term_le 110 3 49615 (7474547/(10:ℝ)^10) ((1800934038563:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 24 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have b3 : T 110 (1 - 7474547/(10:ℝ)^10) 4 23685 ≤ ((43170446048157:ℝ)/50000000000) :=
+    term_le 110 4 23685 (7474547/(10:ℝ)^10) ((1657093002337:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 24 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have b4 : T 110 (1 - 7474547/(10:ℝ)^10) 5 9750 ≤ ((44420693582197:ℝ)/100000000000) :=
+    term_le 110 5 9750 (7474547/(10:ℝ)^10) ((38638030667:ℝ)/12500000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 23 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have b5 : T 110 (1 - 7474547/(10:ℝ)^10) 6 8025 ≤ ((43867998737293:ℝ)/100000000000) :=
+    term_le 110 6 8025 (7474547/(10:ℝ)^10) ((1454360448283:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 22 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have b6 : T 110 (1 - 7474547/(10:ℝ)^10) 7 3530 ≤ ((2250993477097:ℝ)/10000000000) :=
+    term_le 110 7 3530 (7474547/(10:ℝ)^10) ((2754570216739:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 22 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have b7 : T 110 (1 - 7474547/(10:ℝ)^10) 8 7655 ≤ ((55781903078171:ℝ)/100000000000) :=
+    term_le 110 8 7655 (7474547/(10:ℝ)^10) ((1310519412057:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 21 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have b8 : T 110 (1 - 7474547/(10:ℝ)^10) 9 2145 ≤ ((17582868029823:ℝ)/100000000000) :=
+    term_le 110 9 2145 (7474547/(10:ℝ)^10) ((1251627894229:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 21 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have b9 : T 110 (1 - 7474547/(10:ℝ)^10) 10 4650 ≤ ((10587140349809:ℝ)/25000000000) :=
+    term_le 110 10 4650 (7474547/(10:ℝ)^10) ((2997369091:ℝ)/1250000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 21 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have b10 : T 110 (1 - 7474547/(10:ℝ)^10) 11 1770 ≤ ((17730489316207:ℝ)/100000000000) :=
+    term_le 110 11 1770 (7474547/(10:ℝ)^10) ((575646273249:ℝ)/250000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 20 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have b11 : T 110 (1 - 7474547/(10:ℝ)^10) 12 5985 ≤ ((65399123107589:ℝ)/100000000000) :=
+    term_le 110 12 5985 (7474547/(10:ℝ)^10) ((1107786858003:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 20 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have b12 : T 110 (1 - 7474547/(10:ℝ)^10) 13 635 ≤ ((3758266936203:ℝ)/50000000000) :=
+    term_le 110 13 635 (7474547/(10:ℝ)^10) ((533882752083:ℝ)/250000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 20 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have b13 : T 110 (1 - 7474547/(10:ℝ)^10) 14 3710 ≤ ((47290992642379:ℝ)/100000000000) :=
+    term_le 110 14 3710 (7474547/(10:ℝ)^10) ((2061423036179:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 19 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have b14 : T 110 (1 - 7474547/(10:ℝ)^10) 15 9395 ≤ ((128304571925681:ℝ)/100000000000) :=
+    term_le 110 15 9395 (7474547/(10:ℝ)^10) ((498107541173:ℝ)/250000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 19 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have b15 : T 110 (1 - 7474547/(10:ℝ)^10) 16 440 ≤ ((6409229123791:ℝ)/100000000000) :=
+    term_le 110 16 440 (7474547/(10:ℝ)^10) ((963945821777:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 19 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have b16 : T 110 (1 - 7474547/(10:ℝ)^10) 17 405 ≤ ((1253566561349:ℝ)/20000000000) :=
+    term_le 110 17 405 (7474547/(10:ℝ)^10) ((933633510869:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 19 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have b17 : T 110 (1 - 7474547/(10:ℝ)^10) 18 13265 ≤ ((217357516651669:ℝ)/100000000000) :=
+    term_le 110 18 13265 (7474547/(10:ℝ)^10) ((905054303949:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 18 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have b18 : T 110 (1 - 7474547/(10:ℝ)^10) 19 835 ≤ ((14441670418763:ℝ)/100000000000) :=
+    term_le 110 19 835 (7474547/(10:ℝ)^10) ((1756041386627:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 18 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have b19 : T 110 (1 - 7474547/(10:ℝ)^10) 20 110 ≤ ((2002550068509:ℝ)/100000000000) :=
+    term_le 110 20 110 (7474547/(10:ℝ)^10) ((21309351153:ℝ)/12500000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 18 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have b20 : T 110 (1 - 7474547/(10:ℝ)^10) 21 2640 ≤ ((25231210703361:ℝ)/50000000000) :=
+    term_le 110 21 2640 (7474547/(10:ℝ)^10) ((165595792807:ℝ)/100000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 18 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have b21 : T 110 (1 - 7474547/(10:ℝ)^10) 42 2640 ≤ ((100872567578631:ℝ)/100000000000) :=
+    term_le 110 42 2640 (7474547/(10:ℝ)^10) ((962810747511:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 15 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have b22 : T 110 (1 - 7474547/(10:ℝ)^10) 46 2640 ≤ ((22094393345401:ℝ)/20000000000) :=
+    term_le 110 46 2640 (7474547/(10:ℝ)^10) ((174367793861:ℝ)/200000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 14 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have b23 : T 110 (1 - 7474547/(10:ℝ)^10) 109 1 ≤ ((198183171:ℝ)/200000000) :=
+    term_le 110 109 1 (7474547/(10:ℝ)^10) ((1826496713:ℝ)/200000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 5 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have b24 : T 110 (1 - 7474547/(10:ℝ)^10) 1 1 ≤ ((456145263:ℝ)/50000000000) :=
+    term_le 110 1 1 (7474547/(10:ℝ)^10) ((2350240182897:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (log_le_of_le_expsum _ _ 27 (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num) (by norm_num)
+  have e1 : ((2:ℝ)^40 - 1) = 1099511627775 := by norm_num
+  have e2 : (2:ℝ)^40 = 1099511627776 := by norm_num
+  have e3 : ((14692:ℕ):ℝ) = 14692 := by norm_num
+  have e4 : T 110 (1 - 7474547/(10:ℝ)^10) (110 - 1) 1 = T 110 (1 - 7474547/(10:ℝ)^10) 109 1 := rfl
+  rw [e1, e2, e3, e4, hsum]
+  linarith
+
+/-- **sharpness**: at exponent `1 - 7474550/10^10` the moment of one unit is at least its number of live roles `14692`, so the moment criterion fails for EVERY table made of copies of the unit and idle roles (`FoldRate.block_fails`).  Relative excess of the rational certificate: 1.40e-10. -/
+theorem fold_B2Gp193x_sharp : ((14692:ℕ):ℝ) ≤ moment 110 (1 - 7474550/(10:ℝ)^10) blocksB2Gp193x := by
+  have hsum : moment 110 (1 - 7474550/(10:ℝ)^10) blocksB2Gp193x
+      = T 110 (1 - 7474550/(10:ℝ)^10) 1 145890 + (T 110 (1 - 7474550/(10:ℝ)^10) 2 61990 + (T 110 (1 - 7474550/(10:ℝ)^10) 3 49615 + (T 110 (1 - 7474550/(10:ℝ)^10) 4 23685 + (T 110 (1 - 7474550/(10:ℝ)^10) 5 9750 + (T 110 (1 - 7474550/(10:ℝ)^10) 6 8025 + (T 110 (1 - 7474550/(10:ℝ)^10) 7 3530 + (T 110 (1 - 7474550/(10:ℝ)^10) 8 7655 + (T 110 (1 - 7474550/(10:ℝ)^10) 9 2145 + (T 110 (1 - 7474550/(10:ℝ)^10) 10 4650 + (T 110 (1 - 7474550/(10:ℝ)^10) 11 1770 + (T 110 (1 - 7474550/(10:ℝ)^10) 12 5985 + (T 110 (1 - 7474550/(10:ℝ)^10) 13 635 + (T 110 (1 - 7474550/(10:ℝ)^10) 14 3710 + (T 110 (1 - 7474550/(10:ℝ)^10) 15 9395 + (T 110 (1 - 7474550/(10:ℝ)^10) 16 440 + (T 110 (1 - 7474550/(10:ℝ)^10) 17 405 + (T 110 (1 - 7474550/(10:ℝ)^10) 18 13265 + (T 110 (1 - 7474550/(10:ℝ)^10) 19 835 + (T 110 (1 - 7474550/(10:ℝ)^10) 20 110 + (T 110 (1 - 7474550/(10:ℝ)^10) 21 2640 + (T 110 (1 - 7474550/(10:ℝ)^10) 42 2640 + (T 110 (1 - 7474550/(10:ℝ)^10) 46 2640 + 0)))))))))))))))))))))) := rfl
+  have c0 : ((665470322971:ℝ)/500000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 1 145890 :=
+    term_ge 110 1 145890 (7474550/(10:ℝ)^10) ((4700480365791:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (le_log_of_expbound _ _ 4 10 (by norm_num) (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num)
+  have c1 : ((565235973477:ℝ)/500000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 2 61990 :=
+    term_ge 110 2 61990 (7474550/(10:ℝ)^10) ((4007333185231:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (le_log_of_expbound _ _ 4 10 (by norm_num) (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num)
+  have c2 : ((1356784232023:ℝ)/1000000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 3 49615 :=
+    term_ge 110 3 49615 (7474550/(10:ℝ)^10) ((3601868077123:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (le_log_of_expbound _ _ 3 11 (by norm_num) (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num)
+  have c3 : ((431704460547:ℝ)/500000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 4 23685 :=
+    term_ge 110 4 23685 (7474550/(10:ℝ)^10) ((3314186004671:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (le_log_of_expbound _ _ 3 11 (by norm_num) (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num)
+  have c4 : ((44420693593:ℝ)/100000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 5 9750 :=
+    term_ge 110 5 9750 (7474550/(10:ℝ)^10) ((3091042453357:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (le_log_of_expbound _ _ 3 11 (by norm_num) (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num)
+  have c5 : ((87735997501:ℝ)/200000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 6 8025 :=
+    term_ge 110 6 8025 (7474550/(10:ℝ)^10) ((2908720896563:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (le_log_of_expbound _ _ 3 11 (by norm_num) (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num)
+  have c6 : ((112549673893:ℝ)/500000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 7 3530 :=
+    term_ge 110 7 3530 (7474550/(10:ℝ)^10) ((86080319273:ℝ)/31250000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (le_log_of_expbound _ _ 3 11 (by norm_num) (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num)
+  have c7 : ((557819030987:ℝ)/1000000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 8 7655 :=
+    term_ge 110 8 7655 (7474550/(10:ℝ)^10) ((2621038824111:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (le_log_of_expbound _ _ 3 10 (by norm_num) (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num)
+  have c8 : ((87914340183:ℝ)/500000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 9 2145 :=
+    term_ge 110 9 2145 (7474550/(10:ℝ)^10) ((500651157691:ℝ)/200000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (le_log_of_expbound _ _ 3 10 (by norm_num) (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num)
+  have c9 : ((423485614161:ℝ)/1000000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 10 4650 :=
+    term_ge 110 10 4650 (7474550/(10:ℝ)^10) ((2397895272797:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (le_log_of_expbound _ _ 3 10 (by norm_num) (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num)
+  have c10 : ((88652446617:ℝ)/500000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 11 1770 :=
+    term_ge 110 11 1770 (7474550/(10:ℝ)^10) ((2302585092993:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (le_log_of_expbound _ _ 3 10 (by norm_num) (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num)
+  have c11 : ((130798246269:ℝ)/200000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 12 5985 :=
+    term_ge 110 12 5985 (7474550/(10:ℝ)^10) ((2215573716003:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (le_log_of_expbound _ _ 3 10 (by norm_num) (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num)
+  have c12 : ((15033067751:ℝ)/200000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 13 635 :=
+    term_ge 110 13 635 (7474550/(10:ℝ)^10) ((2135531008329:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (le_log_of_expbound _ _ 3 10 (by norm_num) (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num)
+  have c13 : ((23645496331:ℝ)/50000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 14 3710 :=
+    term_ge 110 14 3710 (7474550/(10:ℝ)^10) ((128838939761:ℝ)/62500000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (le_log_of_expbound _ _ 3 10 (by norm_num) (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num)
+  have c14 : ((320761429947:ℝ)/250000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 15 9395 :=
+    term_ge 110 15 9395 (7474550/(10:ℝ)^10) ((1992430164689:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (le_log_of_expbound _ _ 2 12 (by norm_num) (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num)
+  have c15 : ((1001442051:ℝ)/15625000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 16 440 :=
+    term_ge 110 16 440 (7474550/(10:ℝ)^10) ((1927891643551:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (le_log_of_expbound _ _ 2 11 (by norm_num) (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num)
+  have c16 : ((62678328093:ℝ)/1000000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 17 405 :=
+    term_ge 110 17 405 (7474550/(10:ℝ)^10) ((373453404347:ℝ)/200000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (le_log_of_expbound _ _ 2 11 (by norm_num) (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num)
+  have c17 : ((1086787583699:ℝ)/500000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 18 13265 :=
+    term_ge 110 18 13265 (7474550/(10:ℝ)^10) ((362021721579:ℝ)/200000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (le_log_of_expbound _ _ 2 11 (by norm_num) (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num)
+  have c18 : ((28883340849:ℝ)/200000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 19 835 :=
+    term_ge 110 19 835 (7474550/(10:ℝ)^10) ((13719073333:ℝ)/7812500000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (le_log_of_expbound _ _ 2 11 (by norm_num) (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num)
+  have c19 : ((20025500693:ℝ)/1000000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 20 110 :=
+    term_ge 110 20 110 (7474550/(10:ℝ)^10) ((1704748092237:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (le_log_of_expbound _ _ 2 11 (by norm_num) (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num)
+  have c20 : ((63078026783:ℝ)/125000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 21 2640 :=
+    term_ge 110 21 2640 (7474550/(10:ℝ)^10) ((1655957928067:ℝ)/1000000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (le_log_of_expbound _ _ 2 11 (by norm_num) (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num)
+  have c21 : ((126090709507:ℝ)/125000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 42 2640 :=
+    term_ge 110 42 2640 (7474550/(10:ℝ)^10) ((240702686877:ℝ)/250000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (le_log_of_expbound _ _ 1 11 (by norm_num) (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num)
+  have c22 : ((552359833771:ℝ)/500000000) ≤ T 110 (1 - 7474550/(10:ℝ)^10) 46 2640 :=
+    term_ge 110 46 2640 (7474550/(10:ℝ)^10) ((435919484651:ℝ)/500000000000) _ (by norm_num) (by norm_num) (by norm_num) (by norm_num)
+      (le_log_of_expbound _ _ 1 11 (by norm_num) (by norm_num) (by norm_num)
+        (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+      (by norm_num)
+  have e3 : ((14692:ℕ):ℝ) = 14692 := by norm_num
+  rw [e3, hsum]
+  linarith
+
+/-- **per-rank fact** at exponent `1 - 4058344/10^10`, fill parameter `s = 40`: the hypothesis `hnum` of `RAM.engine_program_group_perRank` (u = 110, W = 14692, D = 3080).  Relative slack: 1.10e-07. -/
+theorem foldRank_B2Gp193x :
+    (2:ℝ)^40 * ((14692:ℕ):ℝ) * (((110:ℕ):ℝ) - ((110:ℕ):ℝ)^(1 - 4058344/(10:ℝ)^10)) < ((2:ℝ)^40 - 1) * ((3080:ℕ):ℝ) :=
+  FoldRate.rank_fact3 110 14692 3080 40 (4058344/(10:ℝ)^10) ((2350240182897:ℝ)/500000000000) (by norm_num) (by norm_num) (by norm_num)
+    (log_le_of_le_expsum _ _ 27 (by norm_num) (by norm_num)
+      (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+    (by norm_num) (by norm_num)
+
+/-- **per-rank sharpness**: at exponent `1 - 4058349/10^10` the deficit of one unit is at most `W (m - m^z)`, so the per-rank inequality fails for EVERY table made of copies of the unit and idle roles (`FoldRate.rank_fails`). -/
+theorem foldRank_B2Gp193x_sharp :
+    ((3080:ℕ):ℝ) ≤ ((14692:ℕ):ℝ) * (((110:ℕ):ℝ) - ((110:ℕ):ℝ)^(1 - 4058349/(10:ℝ)^10)) :=
+  FoldRate.rank_fact_sharp 110 14692 3080 (4058349/(10:ℝ)^10) ((4700480365791:ℝ)/1000000000000) (by norm_num) (by norm_num) (by norm_num)
+    (le_log_of_expbound _ _ 4 10 (by norm_num) (by norm_num) (by norm_num)
+      (by norm_num [Finset.sum_range_succ, Nat.factorial]))
+    (by norm_num) (by norm_num)
+
+end OAI.PowerSaving.BlockAccounting
+
+#print axioms OAI.PowerSaving.BlockAccounting.fold_B2Gp193x
+#print axioms OAI.PowerSaving.BlockAccounting.fold_B2Gp193x_sharp
+#print axioms OAI.PowerSaving.BlockAccounting.foldRank_B2Gp193x
+#print axioms OAI.PowerSaving.BlockAccounting.foldRank_B2Gp193x_sharp
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__Raw.lean b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__Raw.lean
new file mode 100644
index 0000000000..499fa67e88
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__Raw.lean
@@ -0,0 +1,93 @@
+/-!
+# (key: gx-data) Raw Lean form of a `gcert/1` file (EXTENDED-CERT-SPEC section 1)
+
+Core Lean only (no Mathlib): the generated data modules import nothing else, so they elaborate
+cheaply.  Field order and meaning are those of the JSON; `pairs` and `cuts` are DERIVED by the
+generator (`checks/wht26/gx-data/py/gxgen.py`) and must be tested by the label check.
+
+Registers: `x_t = t`, `y_t = v + t`, slot `k = 2 v + k`.  Vectors are bit masks.
+-/
+
+namespace GXD
+
+/-- the rational `(-1)^neg * num / den` (`den > 0`) -/
+structure Co where
+  neg : Bool
+  num : Nat
+  den : Nat
+  deriving DecidableEq, Repr
+
+/-- a gate: every register named (extras too) first moves to `frame`; then the adds happen in order -/
+inductive Gate where
+  /-- `tgt += co * src` for each `(tgt, co)` -/
+  | out (frame src : Nat) (tgts : List (Nat × Co)) (extra : List Nat)
+  /-- `tgt += co * src` for each `(src, co)` -/
+  | inn (frame tgt : Nat) (srcs : List (Nat × Co))
+  deriving Repr
+
+/-- the scatter `y_t += sum over k of coef(t, k) * total k` -/
+inductive Scat where
+  /-- star rule: `inside` if coordinate `k` is in port `t`, else `outside` -/
+  | star (inside outside : Co)
+  /-- explicit rows: row `t` = `[(total k, coef)]` -/
+  | table (rows : List (List (Nat × Co)))
+  deriving Repr
+
+/-- a `gcert/1` file -/
+structure Raw where
+  h : Nat
+  v : Nat
+  R : Nat
+  N : Nat
+  cst : Nat
+  /-- `v` masks: the unit vectors of the ports -/
+  ports : List Nat
+  /-- CHUNKED frame table (`flatten` = the table); an entry is a reduced echelon basis, leading bits descending -/
+  frames : List (List (List Nat))
+  /-- frame id per register at the start -/
+  start : List Nat
+  /-- frame id per register at the end -/
+  final : List Nat
+  /-- position `k` = retained total `k`: (slot REGISTER, frame id) -/
+  ret : List (Nat × Nat)
+  scat : Scat
+  /-- CHUNKED gates before the scatter -/
+  A : List (List Gate)
+  /-- CHUNKED gates after the scatter -/
+  B : List (List Gate)
+  /-- CHUNKED, derived: the distinct climbs `(a, b)`, `a ≠ b`, in order of first use -/
+  pairs : List (List (Nat × Nat))
+  /-- derived: replay cut points `(gates of A ++ B before the cut, frame id per register there)` -/
+  cuts : List (Nat × List Nat)
+  /-- slots that start off the zero frame `(slot REGISTER, frame id)` (rule 2); EMPTY on the path -/
+  ext : List (Nat × Nat)
+  /-- derived (questions.md Q1): per port `h` masks, the first is the port, pairwise dots = delta
+  (an orthonormal basis of `F_2^h` through the port) -/
+  obase : List (List Nat)
+
+/-- registers of a gate, in the order they are moved -/
+def Gate.regs : Gate → List Nat
+  | .out _ s ts ex => s :: (ts.map Prod.fst ++ ex)
+  | .inn _ t ss => t :: ss.map Prod.fst
+
+/-- frame of a gate -/
+def Gate.frame : Gate → Nat
+  | .out f _ _ _ => f
+  | .inn f _ _ => f
+
+/-- single adds `(tgt, src, co)` of a gate, in order -/
+def Gate.adds : Gate → List (Nat × Nat × Co)
+  | .out _ s ts _ => ts.map fun p => (p.1, s, p.2)
+  | .inn _ t ss => ss.map fun p => (t, p.1, p.2)
+
+/-- the frame table -/
+def Raw.frameTab (c : Raw) : List (List Nat) := c.frames.flatten
+
+/-- all gates of phase A / phase B -/
+def Raw.gatesA (c : Raw) : List Gate := c.A.flatten
+def Raw.gatesB (c : Raw) : List Gate := c.B.flatten
+
+/-- number of registers -/
+def Raw.nreg (c : Raw) : Nat := 2 * c.v + c.R
+
+end GXD
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__SolutionB2Gp193x.lean b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__SolutionB2Gp193x.lean
new file mode 100644
index 0000000000..0392e1f05b
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/Work__GCert__Data__SolutionB2Gp193x.lean
@@ -0,0 +1,38 @@
+import Work.GCert.Data.B2Gp193Main
+
+/-!
+Comparator solution `Work.GCert.Data.SolutionB2Gp193x` (not part of github.com/openai/math).
+
+The whole proof is in `Work.GCert.Data.B2Gp193Main` (and its imports), which ends with
+
+    theorem wht_main_B2Gp193x :
+        ∃ solve W, WHTProgram solve W ∧ BlockWHT.WHTTimeBoundsAt (1 - 7474547/(10:ℝ)^10) W
+
+This module adds nothing mathematical: it defines `WHT.WHTTimeBoundsAt` with the text of the
+challenge (the same definition as `BlockWHT.WHTTimeBoundsAt` of `Work.Block.WHT`) and restates
+the theorem under the name the challenge uses.  Its proof is that single term.
+`WHTProgram`, `wht`, `whtInKind`, `whtOutKind` come from `WHTCheck.Solution`.
+-/
+
+namespace OAI
+namespace PowerSaving
+namespace WHT
+open Filter Asymptotics
+
+/-- `W = O(2^k (k+1)^β)` and `W = o(2^k k)` (`WHTTimeBounds` is the case `β = 1 - 2/10^11`). -/
+def WHTTimeBoundsAt (β : ℝ) (W : ℕ → ℕ) : Prop :=
+  let W' := fun k : ℕ => (W k : ℝ)
+  W' =O[atTop] (fun k : ℕ => (2:ℝ)^k * ((k:ℝ)+1)^β) ∧
+  W' =o[atTop] (fun k : ℕ => (2:ℝ)^k * (k:ℝ))
+
+/-- **Walsh-Hadamard transform with the improved exponent**: the statement `WHTGoal` of
+`WHTCheck/Challenge.lean` with `1 - 2/10^11` replaced by `1 - 7474547/10^10`. -/
+theorem wht_main_block_B2Gp193x :
+    ∃ solve W, WHTProgram solve W ∧ WHTTimeBoundsAt (1 - 7474547/(10:ℝ)^10) W :=
+  wht_main_B2Gp193x
+
+end WHT
+end PowerSaving
+end OAI
+
+#print axioms OAI.PowerSaving.WHT.wht_main_block_B2Gp193x
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/Work__GFrame__Engine__Adapt.lean b/research/five-stage-source527-banks/inputs/complex/sources/Work__GFrame__Engine__Adapt.lean
new file mode 100644
index 0000000000..24e261ad08
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/Work__GFrame__Engine__Adapt.lean
@@ -0,0 +1,183 @@
+import Work.GFrame.Engine.Letters
+
+/-!
+# GFrame engine, part 2: the free adapter letters in the RAM (agent key: eng-ram)
+
+One free adapter on the array of one role is ONE linear pass, no recursive call:
+
+* `Gear.permuted`     the address of `x ↦ G.π x` (columnwise), O(1) per entry with the XOR table
+                      (as `Gear.shifted`, through `Gear.body_additive` of `Work/Block/Address.lean`);
+* `Gear.phase_row`    the `f`-bit row `(z·x_j + c)_j` of an entry, O(1);
+* `GRig`              a rig together with the table of `lum` over `Bits f` (`i^popcount`);
+* `Rig.perm_cost`, `GRig.phase_cost`   the two letters, work = length of the role table.
+-/
+
+set_option linter.unusedSectionVars false
+
+namespace OAI
+namespace PowerSaving.GF
+open Binary Matrix Finset
+noncomputable section
+section
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+/-- An address permutation acts on an array with `f` columns by moving the address. -/
+lemma act_perm (G : APerm α) (f : ℕ) (x : Sky α f → ℂ) (j : Sky α f) :
+    matAct f (permMat G) x j = x (fun k => G.π (j k)) := by
+  change ∑ i, digitProd (fun _ : Fin f => permMat G) j i * x i = _
+  have h (i : Sky α f) : digitProd (fun _ : Fin f => permMat G) j i =
+      if (fun k => G.π (j k)) = i then 1 else 0 :=
+    prod_delta (κ:=fun _ : Fin f => Space α) _ _
+  simp_rw [h]
+  simp [ite_mul]
+
+/-- A diagonal phase acts by the scalar `lum` of the row of phase bits. -/
+lemma act_phase (z : Space α) (c : F) (f : ℕ) (x : Sky α f → ℂ) (j : Sky α f) :
+    matAct f (phaseMat z c) x j = lum (fun k : Fin f => dot z (j k) + c) * x j := by
+  unfold matAct phaseMat
+  rw [digitProd_diagonal, Matrix.mulVec_diagonal]
+  rfl
+
+end
+end
+end PowerSaving.GF
+
+namespace PowerSaving.RAM
+open Binary Ty Finset GF
+noncomputable section
+universe U V
+variable {ι : Type U}
+
+namespace Gear
+variable {m : Bool} {B f : ι → ℕ} {s t : Ty} {x : ι → T s}
+variable (g : Gear m B f s x)
+include g
+variable {α : Type*} [Fintype α] [DecidableEq α] (X : Layout α) (r : ℕ)
+
+/-- Address moved by an additive bijection (across the `f` columns). -/
+lemma permuted (G : APerm α) (v : ∀ i, Grid α r (f i))
+    (h : Knows m B s x (fun i => (book X r (f i)).loc (v i))) :
+    Knows m B s x (fun i => (book X r (f i)).loc ((v i).1,
+      fun j => G.π ((v i).2 j))) := by
+  obtain ⟨hg, hh⟩ := g.grid_fetch X r (v:=v) h
+  let K (i : ι) : Grid α r (f i) := ((v i).1, fun j => G.π ((v i).2 j))
+  apply g.grid_combine X r (v:=K) hg (g.body_set X (fun i => (K i).2) _)
+  intro a
+  exact g.body_additive X (fun y => G.π y a) (fun p q => by rw [G.add]; rfl)
+    (fun i => (v i).2) hh
+
+/-- The row of phase bits `(z·x_j + c)_j` of an address. -/
+lemma phase_row (z : Space α) (c : F) (v : ∀ i, Grid α r (f i))
+    (h : Knows m B s x (fun i => (book X r (f i)).loc (v i))) :
+    Knows m B s x (fun i => (bits (f i)).loc (fun l : Fin (f i) => dot z ((v i).2 l) + c)) := by
+  obtain ⟨_, hh⟩ := g.grid_fetch X r (v:=v) h
+  have hv := g.body_additive X (fun y => dot z y) (fun p q => dot_add_right z p q)
+    (fun i => (v i).2) hh
+  rcases bit_cases c with hc|hc
+  · refine hv.cong (fun _ => rfl) (fun i => ?_)
+    simp [hc]
+  · let q i : Bits (f i) × Bits (f i) := (fun l => dot z ((v i).2 l), fun _ => 1)
+    have hf : Knows m B s x (fun i => (bits (f i)).loc (q i).2) :=
+      (g.power.minus (Can.wc 1) g.bound).cong (fun _ => rfl) (fun i => by
+        change 2^f i-1=(bits _).loc (fun _ => 1)
+        rw [← bits_lit]
+        simp)
+    refine (g.addr_xor q hv hf).cong (fun _ => rfl) (fun i => ?_)
+    apply congr_arg (bits (f i)).loc
+    ext k; simp [q,hc]
+
+end Gear
+
+/-- A rig together with the table of `lum` over `Bits f` (the fourth roots `i^popcount`). -/
+structure GRig {α ρ : Type*} (m : Bool) (B f : ι → ℕ) (s : Ty) (x : ι → T s)
+    (cl : Paint) (X : Layout α) (R : Layout ρ) (r : ℕ) (v : ∀ i, Grain α ρ r (f i)) : Prop where
+  rig : Rig m B f s x cl X R r v
+  lumT : Has m B s (a sc) x fun i => (bits (f i)).tape (fun y : Bits (f i) => lum y)
+
+section
+variable {α ρ : Type*} [Fintype α] [DecidableEq α] [Fintype ρ] [DecidableEq ρ]
+variable {m : Bool} {B f : ι → ℕ} {s t : Ty} {x : ι → T s}
+    {cl : Paint} {X : Layout α} {R : Layout ρ} {r : ℕ} {v : ∀ i, Grain α ρ r (f i)}
+
+omit [Fintype α] [DecidableEq α] [Fintype ρ] [DecidableEq ρ] in
+lemma GRig.gather (h : GRig m B f s x cl X R r v) {κ : Type V} (j : κ → ι) (y : κ → T t)
+    (g : Has m (fun k => B (j k)) t s y (fun k => x (j k))) :
+    GRig m (fun k => B (j k)) (fun k => f (j k)) t y cl X R r (fun k => v (j k)) :=
+  ⟨h.rig.gather j y g, g.then_do (h.lumT.reindex j)⟩
+
+lemma Rig.ferry_perm (h : Rig m B f s x cl X R r v) (l : ρ) (G : APerm α)
+    (q : ∀ i, Beach α ρ r (f i))
+    (g : Knows m B s x fun i => (fleet X R r (f i)).loc (q i)) :
+    Has m B s (c cl) x fun i => ferryM r (f i) l (permMat G) (v i) (q i) := by
+  obtain ⟨hl,hk⟩ := h.addrs q g
+  have hb := h.read q hl hk
+  let t i : Beach α ρ r (f i) := ((q i).1,(q i).2.1,fun j => G.π ((q i).2.2 j))
+  have hh := h.read t hl (h.gear.permuted X r G _ hk)
+  apply Can.cases_code (q:=fun i => (q i).1) R hl
+  intro p
+  by_cases he : p=l
+  · refine (hh.reindex (fun i : {i // (q i).1=p} => i.1)).cong (fun _ => rfl)
+      (fun i => ?_)
+    have hql : (q i.val).1 = l := i.property.trans he
+    change v i.val ((q i.val).1,(q i.val).2.1,fun j => G.π ((q i.val).2.2 j)) = _
+    simp only [ferryM, roleAct, hql, if_true, act_perm]
+  · refine (hb.reindex (fun i : {i // (q i).1=p} => i.1)).cong (fun _ => rfl)
+      (fun i => ?_)
+    have H : (q i.val).1≠l := fun hh => he (i.property.symm.trans hh)
+    simp only [ferryM, roleAct, H, ite_false, Prod.mk.eta]
+
+/-- **Free address permutation**: one pass over the role table. -/
+lemma Rig.perm_cost (h : Rig m B f s x cl X R r v) (l : ρ) (G : APerm α) :
+    Can m B s (a (c cl)) x
+      (fun i => (fleet X R r (f i)).tape (ferryM r (f i) l (permMat G) (v i)))
+      (fun i => (fleet X R r (f i)).n) := by
+  refine h.chart _ ?_
+  let S := Σ i, Beach α ρ r (f i)
+  have hg := h.gather (fun d : S => d.1)
+    (fun d => ((x d.1, (fleet X R r (f d.1)).loc d.2):T (p s w))) (t:=p s w) Can.first
+  exact hg.ferry_perm l G (fun d => d.2) Can.second
+
+lemma GRig.ferry_phase (h : GRig m B f s x cl X R r v) (l : ρ) (z : Space α) (c' : F)
+    (q : ∀ i, Beach α ρ r (f i))
+    (g : Knows m B s x fun i => (fleet X R r (f i)).loc (q i)) :
+    Has m B s (c cl) x fun i => ferryM r (f i) l (phaseMat z c') (v i) (q i) := by
+  obtain ⟨hl,hk⟩ := h.rig.addrs q g
+  have hb := h.rig.read q hl hk
+  have hrow := h.rig.gear.phase_row X r z c' (fun i => (q i).2) hk
+  have hs : Has m B s sc x
+      (fun i => lum (fun k : Fin (f i) => dot z ((q i).2.2 k) + c')) :=
+    Can.fetch (t:=sc) (L:=fun i => bits (f i)) (v:=fun i (y : Bits (f i)) => lum y)
+      (i:=fun i => (fun k : Fin (f i) => dot z ((q i).2.2 k) + c')) h.lumT hrow
+  have hh := hs.mul hb
+  apply Can.cases_code (q:=fun i => (q i).1) R hl
+  intro p
+  by_cases he : p=l
+  · refine (hh.reindex (fun i : {i // (q i).1=p} => i.1)).cong (fun _ => rfl)
+      (fun i => ?_)
+    have hql : (q i.val).1 = l := i.property.trans he
+    change lum (fun k : Fin (f i.val) => dot z ((q i.val).2.2 k) + c') *
+      v i.val ((q i.val).1,(q i.val).2.1,(q i.val).2.2) = _
+    simp only [ferryM, roleAct, hql, if_true, act_phase]
+  · refine (hb.reindex (fun i : {i // (q i).1=p} => i.1)).cong (fun _ => rfl)
+      (fun i => ?_)
+    have H : (q i.val).1≠l := fun hh => he (i.property.symm.trans hh)
+    simp only [ferryM, roleAct, H, ite_false, Prod.mk.eta]
+
+/-- **Free diagonal phase**: one pass over the role table (one table look-up per entry). -/
+lemma GRig.phase_cost (h : GRig m B f s x cl X R r v) (l : ρ) (z : Space α) (c' : F) :
+    Can m B s (a (c cl)) x
+      (fun i => (fleet X R r (f i)).tape (ferryM r (f i) l (phaseMat z c') (v i)))
+      (fun i => (fleet X R r (f i)).n) := by
+  refine h.rig.chart _ ?_
+  let S := Σ i, Beach α ρ r (f i)
+  have hg := h.gather (fun d : S => d.1)
+    (fun d => ((x d.1, (fleet X R r (f d.1)).loc d.2):T (p s w))) (t:=p s w) Can.first
+  exact hg.ferry_phase l z c' (fun d => d.2) Can.second
+
+end
+end
+end PowerSaving.RAM
+end OAI
+
+#print axioms OAI.PowerSaving.RAM.Rig.perm_cost
+#print axioms OAI.PowerSaving.RAM.GRig.phase_cost
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/Work__GFrame__Engine__Letters.lean b/research/five-stage-source527-banks/inputs/complex/sources/Work__GFrame__Engine__Letters.lean
new file mode 100644
index 0000000000..2eca6e713b
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/Work__GFrame__Engine__Letters.lean
@@ -0,0 +1,199 @@
+import Work.Block.Bridge
+
+/-!
+# GFrame engine, part 1: the generalised word language (agent key: eng-ram)
+
+Letters = the letters of `BWord` (`localM`, `shift`, `block`) plus TWO FREE ADAPTER letters on
+one role:
+
+* `perm l G`      address permutation by an additive bijection `G` of the label space
+                  (matrix `permMat G`, `(permMat G) x y = 1` iff `G.π x = y`);
+* `phase l z c`   the diagonal phase `i^[z·x + c]` IN THE STANDARD BASIS
+                  (matrix `phaseMat z c = diagonal (fun x => tint (dot z x + c))`).
+
+Products of `phaseMat` are exactly the diagonal matrices `i^Q(x)`, `Q` a `ℤ/4`-valued quadratic
+form plus a constant; with the old free letter `shift` the free letters generate every affine
+address permutation times such a phase.  A "twisted block" is NOT a primitive letter: it is the
+word `free adapters ++ [block] ++ free adapters` (see `Free.lean`, `gstep_of_factor`).
+
+The semantics is `gwalk` (no unit-move word behind it: the new letters are not Walsh-diagonal).
+Old words embed by `GWord.ofB`, and `gwalk f (ofB w) = walk f w.flat`.
+-/
+
+set_option linter.unusedSectionVars false
+
+namespace OAI
+namespace PowerSaving.GF
+open Binary Matrix Finset
+noncomputable section
+
+/-- An additive bijection of the label space. -/
+structure APerm (α : Type*) where
+  π : Space α ≃ Space α
+  add : ∀ x y, π (x + y) = π x + π y
+
+section Mats
+variable {α : Type*} [Fintype α] [DecidableEq α]
+
+/-- Permutation matrix of an additive bijection: `(permMat G *ᵥ v) x = v (G.π x)`. -/
+def permMat (G : APerm α) : CMat α := fun x y => if G.π x = y then 1 else 0
+
+/-- Elementary diagonal phase in the standard basis: `i^[z·x + c]`. -/
+def phaseMat (z : Space α) (c : F) : CMat α := Matrix.diagonal fun x => tint (dot z x + c)
+
+end Mats
+
+/-- Letters of the generalised word language. -/
+inductive GMove (α ρ : Type*)
+  | old (mv : BMove α ρ)
+  | perm (l : ρ) (G : APerm α)
+  | phase (l : ρ) (z : Space α) (c : F)
+
+abbrev GWord (α ρ : Type*) := List (GMove α ρ)
+
+namespace GMove
+variable {α ρ : Type*}
+
+/-- The letter of `BWord` with the same price (free adapters ↦ a free letter). -/
+def shadow : GMove α ρ → BMove α ρ
+  | .old mv => mv
+  | .perm _ _ => .localM 0
+  | .phase _ _ _ => .localM 0
+
+/-- Rank paid by a letter (`none` = free). -/
+def rk? (mv : GMove α ρ) : Option ℕ := mv.shadow.rk?
+def costR (φ : ℕ → ℝ) (mv : GMove α ρ) : ℝ := mv.shadow.costR φ
+def Proper (u : ℕ) (mv : GMove α ρ) : Prop := mv.shadow.Proper u
+
+@[simp] lemma costR_perm (φ : ℕ → ℝ) (l : ρ) (G : APerm α) :
+    (GMove.perm l G : GMove α ρ).costR φ = 0 := rfl
+@[simp] lemma costR_phase (φ : ℕ → ℝ) (l : ρ) (z : Space α) (c : F) :
+    (GMove.phase l z c : GMove α ρ).costR φ = 0 := rfl
+@[simp] lemma costR_old (φ : ℕ → ℝ) (mv : BMove α ρ) : (GMove.old mv).costR φ = mv.costR φ := rfl
+lemma proper_perm (u : ℕ) (l : ρ) (G : APerm α) : (GMove.perm l G : GMove α ρ).Proper u :=
+  trivial
+lemma proper_phase (u : ℕ) (l : ρ) (z : Space α) (c : F) :
+    (GMove.phase l z c : GMove α ρ).Proper u := trivial
+@[simp] lemma proper_old (u : ℕ) (mv : BMove α ρ) : (GMove.old mv).Proper u ↔ mv.Proper u :=
+  Iff.rfl
+
+end GMove
+
+namespace GWord
+variable {α ρ : Type*}
+
+def shadow (w : GWord α ρ) : BWord α ρ := w.map GMove.shadow
+def costR (φ : ℕ → ℝ) (w : GWord α ρ) : ℝ := (w.map (GMove.costR φ)).sum
+def Proper (u : ℕ) (w : GWord α ρ) : Prop := ∀ mv ∈ w, mv.Proper u
+def hist (w : GWord α ρ) (r : ℕ) : ℕ := w.shadow.hist r
+/-- Natural-number cost when a block of rank `rk` costs `g rk`. -/
+def cost (g : ℕ → ℕ) (w : GWord α ρ) : ℕ := w.shadow.cost g
+/-- An old block word as a generalised word. -/
+def ofB (w : BWord α ρ) : GWord α ρ := w.map GMove.old
+
+@[simp] lemma shadow_nil : shadow ([] : GWord α ρ) = [] := rfl
+@[simp] lemma shadow_cons (mv : GMove α ρ) (w : GWord α ρ) :
+    shadow (mv :: w) = mv.shadow :: shadow w := rfl
+@[simp] lemma shadow_ofB (w : BWord α ρ) : shadow (ofB w) = w := by
+  simp [shadow, ofB, List.map_map, Function.comp_def, GMove.shadow]
+
+lemma costR_shadow (φ : ℕ → ℝ) (w : GWord α ρ) : costR φ w = BWord.costR φ w.shadow := by
+  unfold costR BWord.costR shadow
+  rw [List.map_map]
+  rfl
+@[simp] lemma costR_nil (φ : ℕ → ℝ) : costR φ ([] : GWord α ρ) = 0 := rfl
+lemma costR_cons (φ : ℕ → ℝ) (mv : GMove α ρ) (w : GWord α ρ) :
+    costR φ (mv :: w) = mv.costR φ + costR φ w := by simp [costR]
+lemma costR_append (φ : ℕ → ℝ) (w w' : GWord α ρ) :
+    costR φ (w ++ w') = costR φ w + costR φ w' := by simp [costR]
+lemma costR_ofB (φ : ℕ → ℝ) (w : BWord α ρ) : costR φ (ofB w) = BWord.costR φ w := by
+  rw [costR_shadow, shadow_ofB]
+@[simp] lemma cost_nil (g : ℕ → ℕ) : cost g ([] : GWord α ρ) = 0 := rfl
+lemma cost_cons (g : ℕ → ℕ) (mv : GMove α ρ) (w : GWord α ρ) :
+    cost g (mv :: w) = mv.shadow.cost g + cost g w := BWord.cost_cons g _ _
+
+lemma Proper.shadow {u : ℕ} {w : GWord α ρ} (h : Proper u w) : BWord.Proper u w.shadow := by
+  intro mv hmv
+  obtain ⟨g, hg, rfl⟩ := List.mem_map.mp hmv
+  exact h g hg
+lemma Proper.tail {u : ℕ} {mv : GMove α ρ} {w : GWord α ρ} (h : Proper u (mv :: w)) :
+    Proper u w := fun x hx => h x (List.mem_cons_of_mem _ hx)
+lemma Proper.head {u : ℕ} {mv : GMove α ρ} {w : GWord α ρ} (h : Proper u (mv :: w)) :
+    mv.Proper u := h mv List.mem_cons_self
+lemma Proper.append {u : ℕ} {w w' : GWord α ρ} (h : Proper u w) (h' : Proper u w') :
+    Proper u (w ++ w') := by
+  intro mv hmv
+  rcases List.mem_append.mp hmv with h1|h1
+  · exact h mv h1
+  · exact h' mv h1
+lemma proper_nil (u : ℕ) : Proper u ([] : GWord α ρ) := fun _ h => by simp at h
+lemma proper_ofB {u : ℕ} {w : BWord α ρ} (h : w.Proper u) : Proper u (ofB w) := by
+  intro mv hmv
+  obtain ⟨g, hg, rfl⟩ := List.mem_map.mp hmv
+  exact h g hg
+
+lemma costR_eq_sum (u : ℕ) (φ : ℕ → ℝ) (w : GWord α ρ) (hw : Proper u w) :
+    costR φ w = ∑ r : Fin u, (w.hist r:ℝ) * φ r := by
+  rw [costR_shadow]; exact BWord.costR_eq_sum u φ _ hw.shadow
+lemma cost_eq_sum (u : ℕ) (g : ℕ → ℕ) (w : GWord α ρ) (hw : Proper u w) :
+    cost g w = ∑ r : Fin u, hist w r * g r := BWord.cost_eq_sum u g _ hw.shadow
+lemma hist_zero (u : ℕ) (w : GWord α ρ) (hw : Proper u w) : hist w 0 = 0 :=
+  BWord.hist_zero u _ hw.shadow
+lemma hist_large (u : ℕ) (w : GWord α ρ) (hw : Proper u w) (r : ℕ) (hr : u ≤ r) :
+    hist w r = 0 := BWord.hist_large u _ hw.shadow r hr
+
+end GWord
+
+section Sem
+variable {α ρ σ : Type*} [Fintype α] [DecidableEq α] [Fintype ρ] [DecidableEq ρ]
+
+/-- The action of one letter. -/
+def GMove.act (f : ℕ) : GMove α ρ → Data α ρ f → Data α ρ f
+  | .old mv => mv.act f
+  | .perm l G => roleAct f l (permMat G)
+  | .phase l z c => roleAct f l (phaseMat z c)
+
+/-- **Semantics of a generalised word.** -/
+def gwalk (f : ℕ) (w : GWord α ρ) (x : Data α ρ f) : Data α ρ f :=
+  w.foldl (fun y mv => mv.act f y) x
+
+@[simp] lemma gwalk_nil (f : ℕ) (x : Data α ρ f) : gwalk f ([] : GWord α ρ) x = x := rfl
+lemma gwalk_cons (f : ℕ) (mv : GMove α ρ) (w : GWord α ρ) (x : Data α ρ f) :
+    gwalk f (mv :: w) x = gwalk f w (mv.act f x) := rfl
+lemma gwalk_append (f : ℕ) (p q : GWord α ρ) (x : Data α ρ f) :
+    gwalk f (p ++ q) x = gwalk f q (gwalk f p x) := by simp [gwalk]
+
+/-- Old words keep their meaning. -/
+lemma gwalk_ofB (f : ℕ) (w : BWord α ρ) (x : Data α ρ f) :
+    gwalk f (GWord.ofB w) x = walk f w.flat x := by
+  induction w generalizing x with
+  | nil => rfl
+  | cons mv w ih =>
+    rw [walk_flat_cons, ← ih]
+    rfl
+
+/-- The scratch certificate for the generalised language (cf. `RAM.LiveKernel`). -/
+def GLiveKernel (e : σ → ρ) (w : GWord α ρ) : Prop :=
+  ∀ (f : ℕ) (v : Data α ρ f), (∀ s, (∀ l, e l ≠ s) → v s = 0) →
+    ∀ l, gwalk f w v (e l) = matAct f (kernel α) (v (e l))
+
+/-- Every old certificate is a generalised certificate. -/
+theorem GLiveKernel.ofB {e : σ → ρ} {w : BWord α ρ} (h : RAM.LiveKernel e w.flat) :
+    GLiveKernel e (GWord.ofB w) := by
+  intro f v hv l
+  rw [gwalk_ofB]
+  exact h f v hv l
+
+/-- One role goes from matrix `M` to `N` by a generalised word (cf. `XStep`). -/
+def GStep (l : ρ) (M N : CMat α) (c : (ℕ → ℝ) → ℝ) : Prop :=
+  ∃ (w : GWord α ρ) (U : CMat α), w.Proper (Fintype.card α) ∧ (∀ φ, w.costR φ = c φ) ∧
+    U * M = N ∧ ∀ (f : ℕ) (x : Data α ρ f), gwalk f w x = roleAct f l U x
+
+end Sem
+end
+end PowerSaving.GF
+end OAI
+
+#print axioms OAI.PowerSaving.GF.gwalk_ofB
+#print axioms OAI.PowerSaving.GF.GLiveKernel.ofB
+#print axioms OAI.PowerSaving.GF.GWord.costR_eq_sum
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/Work__GFrame__Engine__Level.lean b/research/five-stage-source527-banks/inputs/complex/sources/Work__GFrame__Engine__Level.lean
new file mode 100644
index 0000000000..6bd7cfa0c3
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/Work__GFrame__Engine__Level.lean
@@ -0,0 +1,163 @@
+import Work.GFrame.Engine.Sweep
+import WHTCheck.Solution
+
+/-!
+# GFrame engine, part 4: one level of the recursion for generalised words (agent key: eng-ram)
+
+Copy of `amplifyB` (`Work/Block/Recursion.lean`) with two changes:
+
+* the word is a `GWord` with certificate `GLiveKernel e w` (semantics `gwalk`);
+* after the XOR table the level also builds the table of `lum` over `Bits f`
+  (`WHT.lum_tables`, doubling construction, `O(2^f)` work), which the free diagonal phases
+  read; it is part of the environment of the sweep (`GRig`).
+
+Cost: the same `2^k` linear term and the same recursive calls as `amplifyB`.
+-/
+
+set_option linter.unusedSectionVars false
+
+namespace OAI
+namespace PowerSaving.RAM
+open Binary Matrix Ty Finset GF
+noncomputable section
+
+section
+universe U
+variable {α ρ σ : Type*} [Fintype α] [DecidableEq α] [Fintype ρ] [DecidableEq ρ]
+  [Fintype σ] [DecidableEq σ]
+variable {ι : Type U} {m : Bool}
+    {k f : ι → ℕ} {cl : Paint} {b : Bank (hatch cl) ι}
+    {v : ∀ i, Sim σ (k i)→ℂ}
+
+/-- **One level of the recursion for a generalised word** (replaces `amplifyB`). -/
+lemma gamplify (m : Bool) (b : Bank (hatch cl) ι) (v : ∀ i, Sim σ (k i) → ℂ)
+    (X : Layout α) (R : Layout ρ) (Q : Layout σ) (e : σ → ρ) (he : Function.Injective e)
+    (r : ℕ) (hr : 3 ≤ X.n) (hq : ∀ i, X.n*f i+r=k i) (hd : r < X.n) (hQ : 0<Q.n)
+    (n τ : ℕ → ι → ℕ)
+    (hc : ∀ rk, 1 ≤ rk → rk < X.n → ∀ i, n rk i*(sim Q (rk * f i)).n=2^k i)
+    (hk : ∀ rk, 1 ≤ rk → rk < X.n → Contract cl b (fun i => rk * f i) (τ rk) Q)
+    (hB : Small b.B (fun i => 2^k i))
+    (w : GWord α ρ) (hw : GLiveKernel e w) (hP : w.Proper X.n) :
+    Able m b (simIn cl) (bundle cl) (fun i => loaded cl Q (k i) (v i))
+      (fun i => (sim Q (k i)).tape (whole (k i) (v i)))
+      (fun i => 2^k i) (fun i => w.cost (fun rk => n rk i * τ rk i)) := by
+  let W i := 2^k i
+  let s := simIn cl
+  let eQ (i : ι) := ebb X r (f i) (k i) (hq i) (v i)
+  let zQ (i : ι) : Grain α σ r (f i) :=
+    fun j => chords r (f i) (fun l => eQ i (j.1,l)) j.2
+  let Z (i : ι) : Grain α ρ r (f i) := crossAct r (f i) (embM e) (zQ i)
+  let O (i : ι) : Grain α σ r (f i) :=
+    crossAct r (f i) (extM e) (gcoast r (f i) w (Z i))
+  let szQ i := (fleet X Q r (f i)).n
+  let szR i := (fleet X R r (f i)).n
+  have hszQ : Dom W szQ := by
+    refine ⟨Q.n+1, fun i => ?_⟩
+    unfold W szQ fleet
+    rw [Layout.pair_len, book_len,hq]
+    simp only [mul_add,add_mul,mul_one,one_mul]
+    gcongr <;> simp
+  have hszR : Dom W szR := by
+    refine ⟨R.n+1, fun i => ?_⟩
+    unfold W szR fleet
+    rw [Layout.pair_len, book_len,hq]
+    simp only [mul_add,add_mul,mul_one,one_mul]
+    gcongr <;> simp
+  have hn (i : ι) : 0 < W i := pow_pos (by decide) _
+  have hf (i) : f i ≤ k i := by
+    have g := hq i
+    have h := Nat.le_mul_of_pos_left (f i) (show 0 < X.n by omega)
+    omega
+  have hy (i) : k i ≤ W i := Nat.lt_two_pow_self.le
+  have sd := hB.mono hy
+  have sq := sd.mono hf
+  let p2 i := 2^f i
+  have hm : Small b.B p2 :=
+    hB.mono (fun i => Nat.pow_le_pow_right (by omega) (hf i))
+  have hcube (i) : p2 i ^ 3 ≤ W i := by
+    unfold p2 W
+    rw [← pow_mul]
+    apply Nat.pow_le_pow_right (by omega)
+    have hh : 3*f i ≤ X.n*f i := Nat.mul_le_mul_right _ hr
+    have hg := hq i
+    rw [mul_comm]
+    omega
+  let env (i : ι) := loaded cl Q (k i) (v i)
+  -- construct fields (f,p2)
+  have hmf : Can m b.B s Ty.w env f W := by
+    have ht : Can m b.B s Ty.w env k W := Can.first
+    refine (ht.dvdW (Can.wc X.n) sd).cong (fun _ => rfl) ?_
+    intro i
+    rw [← hq,mul_comm X.n]
+    exact Layout.div_join hd
+  have hmp : Can m b.B s (Ty.p Ty.w Ty.w) env (fun i => (f i,p2 i)) W :=
+    hmf.pair (hmf.wTwo sq hm (Dom.of_le (fun i => (hf i).trans (hy i))))
+  let T1 := Ty.p s (Ty.p Ty.w Ty.w)
+  let Y (i : ι) : T T1 := (env i,f i,p2 i)
+  have hmx : Can m b.B T1 (Ty.a Ty.w) Y (fun i => xorTab (f i)) W :=
+    (Can.xor_prepare (m:=m) (f:=f) (B:=b.B) (s:=T1) (x:=Y)
+      (Can.fst Can.second) (Can.snd Can.second) sq hm).mono hcube
+  let T2 := Ty.p T1 (Ty.a Ty.w)
+  let Y' (i : ι) : T T2 := (Y i,xorTab (f i))
+  -- NEW: the table of `lum` over `Bits f`
+  have hE2 : Has m b.B T2 s Y' env := Can.fst Can.first
+  have hml : Can m b.B T2 (Ty.a sc) Y'
+      (fun i => (bits (f i)).tape (fun y : Bits (f i) => lum y)) W :=
+    (WHT.lum_tables (k:=f) (Can.fst (Can.snd Can.first)) sq hm hE2.snd.fst).1.mono
+      (fun i => Nat.pow_le_pow_right (by omega) (hf i))
+  let T2' := Ty.p T2 (Ty.a sc)
+  let Y'' (i : ι) : T T2' := (Y' i,(bits (f i)).tape (fun y : Bits (f i) => lum y))
+  -- the input, read as a table on the slots
+  have hG0 : Rig m b.B f T2 Y' cl X Q r eQ := by
+    refine ⟨⟨?_,?_,Can.second,hm⟩,?_,?_⟩
+    · exact Can.snd (Can.snd Can.first)
+    · exact Can.fst (Can.snd Can.first)
+    · exact hE2.snd.fst
+    · exact hE2.snd.snd.cong (fun _ => rfl) (fun i => (ebb_eq ..).symm)
+  have hG : Rig m b.B f T2' Y'' cl X Q r eQ := hG0.gather id Y'' (t:=T2') Can.first
+  -- leftover bits, still on the slots
+  have HH : Can m b.B T2' (bundle cl) Y'' (fun i => (fleet X Q r (f i)).tape (zQ i)) W :=
+    hG.dots_cost.weaken hszQ
+  let T3 := Ty.p T2' (bundle cl)
+  let A (i) : T T3 := (Y'' i,(fleet X Q r (f i)).tape (zQ i))
+  have Hg := hG.gather id A (t:=T3) Can.first
+  have G : Rig m b.B f T3 A cl X Q r zQ := ⟨Hg.gear,Hg.imag,Can.second⟩
+  -- embed the slots into the table of roles; scratch roles are created as zero
+  have HE : Can m b.B T3 (bundle cl) A (fun i => (fleet X R r (f i)).tape (Z i)) W :=
+    (G.cross_point R (embM e)).weaken hszR
+  let T4 := Ty.p T3 (bundle cl)
+  let A4 (i) : T T4 := (A i,(fleet X R r (f i)).tape (Z i))
+  have Hg4 := G.gather id A4 (t:=T4) Can.first
+  have hL4 : Has m b.B T4 (Ty.a sc) A4
+      (fun i => (bits (f i)).tape (fun y : Bits (f i) => lum y)) :=
+    Can.snd (Can.fst Can.first)
+  have G4 : GRig m b.B f T4 A4 cl X R r Z := ⟨⟨Hg4.gear,Hg4.imag,Can.second⟩,hL4⟩
+  -- the GENERALISED word on the table of roles
+  have Hy := G4.gsweep n τ Q hQ hk
+    (fun rk h1 h2 i => by rw [book_len,hq]; exact hc rk h1 h2 i) w hP
+  -- extract the slots; scratch roles are dropped
+  let T5 := Ty.p T4 (bundle cl)
+  let A5 (i) : T T5 := (A4 i,(fleet X R r (f i)).tape (gcoast r (f i) w (Z i)))
+  have Hg5 := G4.rig.gather id A5 (t:=T5) Can.first
+  have G5 : Rig m b.B f T5 A5 cl X R r (fun i => gcoast r (f i) w (Z i)) :=
+    ⟨Hg5.gear,Hg5.imag,Can.second⟩
+  have HX : Can m b.B T5 (bundle cl) A5 (fun i => (fleet X Q r (f i)).tape (O i)) W :=
+    (G5.cross_point Q (extM e)).weaken hszQ
+  have heq (i) : O i = ebb X r (f i) (k i) (hq i) (whole (k i) (v i)) :=
+    gamplifies X r (f i) (k i) (hq i) e he w hw (v i)
+  have ha (i) : (fleet X Q r (f i)).tape (O i) =
+      (sim Q (k i)).tape (whole (k i) (v i)) := by
+    rw [heq]; exact ebb_eq ..
+  have Ho := ((Hy.weaken hszR (fun _ => le_rfl)).save_keep
+    (Able.of_can (b:=b) HX)).cong (fun _ => rfl) ha
+  have Hx := (Able.of_can (b:=b) hmp).save_keep
+    ((Able.of_can (b:=b) hmx).save_keep ((Able.of_can (b:=b) hml).save_keep
+      ((Able.of_can (b:=b) HH).save_keep ((Able.of_can (b:=b) HE).save_keep Ho))))
+  simpa only [Nat.zero_add,Nat.add_zero] using Hx
+
+end
+end
+end PowerSaving.RAM
+end OAI
+
+#print axioms OAI.PowerSaving.RAM.gamplify
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/tools__gx__ORIGIN.md b/research/five-stage-source527-banks/inputs/complex/sources/tools__gx__ORIGIN.md
new file mode 100644
index 0000000000..e28b355d0a
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/tools__gx__ORIGIN.md
@@ -0,0 +1,64 @@
+# tools/gx/ and tools/certificate/gcert1-p11-pr193.json.gz: where they come from
+
+Everything in this directory is Python 3 (standard library only). No Lean is run by these scripts.
+"The outside repository" means github.com/CrocSwap/integer-mult-bounds (Apache-2.0, `LICENSE` and `NOTICE`
+of that repository; `NOTICE`: "A sharper exponent for integer multiplication, Copyright 2026 Douglas Colkitt").
+
+## The scripts (this project)
+
+| file | what it is |
+|---|---|
+| `gxcore.py` | loads a certificate of format gcert/1, brings it to normal form, and is a Python mirror of both Lean checks (labels and scalars); cut points and label data for the generator |
+| `gxunit.py` | split plan of the kernel evaluations, the unit of the word, expected exponents |
+| `gxbind.py` | the texts of the generated Lean modules (check, histogram, price, certificate and final modules) |
+| `gxgen.py` | certificate -> Lean data modules and one module per kernel evaluation |
+| `gxrate.py` | rate lemmas, unit price list, final theorem modules, comparator challenge / solution / configuration |
+| `gxdry.py`, `gxmut.py` | dry run (mirror, sizes, plan, exponents); mutants that the mirror must reject |
+| `gxconv.py` | converts the earlier format gcert/0 to gcert/1 when a step scales its target (the in-place pair step of #193) |
+| `gsmirror.py` | a second, separately written Python mirror of the scalar check `GS.scalCheck` / `GS.yCheck`; `gxbind.py` calls it for the check parameters |
+| `gx.py`, `gxrun.py` | the reference checker of the format (`gx.check1`: labels and exact scalar identity) and its command line |
+| `gxpaths.py`, `regen_check.py`, `refcheck.py` | written for this repository: relative paths; regenerate and compare; `gx.check1` on a gcert/1 file |
+
+The scripts are the working copies with path edits only (absolute paths of the author's machine replaced through
+`gxpaths.py`; templates read from `tools/gen/templates/`; `foldlib.py` / `foldgen.py` taken from `tools/gen/`).
+The sentences "Generated by checks/..." that the generators print into Lean files name the scripts by their
+working paths; they are kept so that the generated modules come out byte for byte.
+No script of the outside repository is included, imported or executed.
+
+## The certificate `tools/certificate/gcert1-p11-pr193.json.gz`
+
+sha256 of the file: `15251deb0c11e0504d7b297c9786818888a93f08b92b03c6c122784b70488633`
+(gzip stores a time stamp; sha256 of the uncompressed JSON: `c64d2a95431c3cf24452de7391cce780bd71521da44ce23923ebb235f6c7962c`).
+
+It is ONE explicit program (numbered registers, every addition, every frame) for the helper circuit at
+h = 22: 1,320 ports, 9,412 slots, 262,944 unit moves. **The design of that circuit is not this project's.** It is the
+"paired-cube" circuit of the outside repository as of pull request #193 (which builds on #191, #184, #168 and #144).
+The outside repository publishes the design as generator scripts, frozen data files and a ledger; it has no file
+of this kind. This file was produced by this project's own code, which
+
+1. loaded the following files of the outside repository **as data** (JSON; nothing was executed):
+   - at pull request #168, commit `4a3c769e5c5430e7114c4d3e099ff34664677f17`:
+     `references/paired-cube/sources/local_L1.json`,
+     `references/paired-cube/sources/tmod_TE_TD_TB3_1_1_4_full_6.0617964e-4.json`,
+     `references/paired-cube/sources/pmod_J0_full_6.0666810e-4.json`,
+     `references/paired-cube/sources/qmod_climb3u_best.json` (the local modules),
+     `references/paired-cube/selected-module/matching-arcs.json` (carrier arcs),
+     `references/paired-cube/physical/frames.json` (moved frames);
+   - at pull request #193, commit `187e1010ac8b259af8e9b5166f68b64bc27b4b47`:
+     `research/source-assisted-v4/data/physical-pairs.json` (the hand-over pairs);
+2. rebuilt the circuit from them step by step, following the algorithms of the outside scripts (read as text),
+   and checked the result exactly (every published count reproduced; the scalar identity as an identity of
+   linear maps with arbitrary contents of the helper registers);
+3. converted it to gcert/1 with `gxconv.py` (tested by `gx.check1`).
+
+Step 1 and step 2 (the rebuild scripts and the outside inputs they read) are NOT part of this repository;
+the inputs can be fetched from the outside repository at the two commits above.
+What this repository proves about the file does not depend on how it was made: the Lean kernel checks the
+generated modules (labels, scalar identity, price), and `regen_check.py` shows that those modules are exactly
+what `gxgen.py` / `gxrate.py` produce from this file.
+
+## Check it yourself
+
+    python3 tools/gx/regen_check.py                       # regenerates the Lean modules and compares
+    python3 tools/gx/gxdry.py tools/certificate/gcert1-p11-pr193.json.gz      # Python mirror of both checks
+    python3 tools/gx/refcheck.py                          # the reference checker gx.check1 on the certificate
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/tools__gx__gxbind.py b/research/five-stage-source527-banks/inputs/complex/sources/tools__gx__gxbind.py
new file mode 100644
index 0000000000..fcb625619c
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/tools__gx__gxbind.py
@@ -0,0 +1,438 @@
+"""gxbind.py: THE BINDING of the generator to the Lean terms of gx-scalar / gx-labels /
+gx-chain.  Everything that names somebody else's definition is in this file (tables SCAL, LAB, CHAIN)."""
+import os, re, subprocess
+import gxpaths
+ROOT = gxpaths.OUT
+GSM = gxpaths.HERE + '/gsmirror.py'
+
+# gx-scalar (gcert-interface.md, SCALAR v1): GS.scalCheck c p lo n, GS.yCheck c p lo n, p : GS.SPar = <d, sw, ux, us, uy>
+SCAL = dict(imports=['Work.GCert.Scalar.Seg'], scal='GS.scalCheck raw ⟨%(d)d, %(sw)d, %(ux)d, %(us)d, %(uy)d⟩ %(lo)d %(n)d',
+            ychk='GS.yCheck raw ⟨%(d)d, %(sw)d, %(ux)d, %(us)d, %(uy)d⟩ 0 %(v)d')
+# gx-labels (gcert-interface.md LABELS v1 + questions.md N1): namespace GLab; frame = code, table and states = SSC.Trie,
+# GLab.Par = <h, n, d, K, tab>.  `end` = GLab.endK of Labels/EndDef.lean (questions.md N2; no obaseK: Q1 closed).
+LAB = dict(imports=['Work.GCert.Labels.Def'], end_imports=['Work.GCert.Labels.EndDef'], par='⟨%(h)d, %(n)d, %(d)d, %(K)d, tab⟩', ns='GLab',
+           tab='GLab.tabK %(h)d %(sub)s', seg='GLab.segK par %(gs)s st%(i)d st%(j)d %(rs)s',
+           end='GLab.endK par %(v)d ports (raw.ret.map Prod.fst) st0 st%(a)d st%(f)d', noadds='GLab.noAdds fins')
+# gx-chain (CHAIN v1): GX.wht_of_halves; the terms below are PLACEHOLDERS until Labels/Main and Scalar/Main compile
+CHAIN = dict(imports=['Work.GCert.Chain.End', 'Work.GCert.Labels.Main'], scal='(GS.ScalOK.scal GXD.%(c)s.scalOK)',
+             lab='(GXD.%(c)s.valid.lab GXD.%(c)s.scalOK)', lab_cost='GXD.%(c)s.lab_cost')
+
+
+def code(h, basis):
+    """gx-labels' code of a reduced echelon basis: dim + 2^8 M + 2^(8+h) P (M = pivot mask, vector of pivot p at digit p)"""
+    M = P = 0
+    for b in basis:
+        p = b.bit_length() - 1
+        M |= 1 << p
+        P |= b << (h * p)
+    return len(basis) + (M << 8) + (P << (8 + h))
+
+
+MOD = '''import %(imports)s
+/-! Kernel check (%(doc)s) of the gcert/1 certificate `GXD.%(name)s.raw`.  Source: %(src)s.
+Generated by checks/wht26/gx-data/py/gxgen.py (do not edit). -/
+set_option maxRecDepth 100000
+namespace GXD.%(name)s
+open GXD
+theorem %(thm)s : %(stmt)s := by decide +kernel
+end GXD.%(name)s
+#print axioms GXD.%(name)s.%(thm)s
+'''
+
+
+def spar(D, path):
+    """scalar parameters: from gx-scalar's mirror when it exists, else from my own replay (sw is then a GUESS)"""
+    import gxcore as gc
+    n, d = 2 * D['v'] + D['R'], 0
+    while (1 << d) < n:
+        d += 1
+    dg = gc.digits(D)
+    S = dict(d=d, sw=16, ux=dg['x'][0], us=dg['s'][0], uy=dg['y'][0], source='gx-data replay; sw = 16 is a GUESS', digits=dg)
+    if os.path.exists(GSM):
+        try:
+            o = subprocess.run(['python3', '-B', GSM, path], capture_output=True, text=True, timeout=1800).stdout
+            m = re.search(r'SPar[^\d]*(\d+)[, ]+(\d+)[, ]+(\d+)[, ]+(\d+)[, ]+(\d+)', o)
+            if m:
+                S.update(zip(('d', 'sw', 'ux', 'us', 'uy'), map(int, m.groups())), source='gsmirror.py')
+        except Exception as e:
+            S['source'] += ' (gsmirror.py failed: %r)' % (e,)
+    return S
+
+
+def write_checks(D, M, name, src, out, L, S, plan):
+    base = out[len(ROOT):].replace('/', '.')
+    raw = '%s.Gen.%s' % (base, name)
+    h, v, n = D['h'], D['v'], 2 * D['v'] + D['R']
+    res = []
+
+    def mod(suffix, imports, thm, stmt, doc):
+        m = '%s.Gen.%s%s' % (base, name, suffix)
+        open('%s/Gen/%s%s.lean' % (out, name, suffix), 'w').write(MOD % dict(
+            imports='\nimport '.join(imports), doc=doc, name=name, src=src, thm=thm, stmt=stmt))
+        res.append((m, thm, stmt))
+    for i, (lo, hi) in enumerate(M['scal']['batches']):
+        mod('Scal%d' % i, SCAL['imports'] + [raw], 'scal%d_ok' % i, SCAL['scal'] % dict(S, lo=lo, n=hi - lo) + ' = true',
+            'scalar identity E3 E5 for the sources %d .. %d' % (lo, hi - 1))
+    mod('YChk', SCAL['imports'] + [raw], 'ychk_ok', SCAL['ychk'] % dict(S, v=v) + ' = true', 'the y -> y adds, E4')
+    par = '%s.Gen.%sPar' % (base, name)
+    open('%s/Gen/%sPar.lean' % (out, name), 'w').write(
+        'import %s\nimport %s\n/-! Parameters `GLab.Par` of the label check of `GXD.%s.raw` (h, registers, trie depth, K, frame table).\n'
+        'Generated by checks/wht26/gx-data/py/gxgen.py (do not edit). -/\nnamespace GXD.%s\nopen GXD\n'
+        'noncomputable def par : GLab.Par := %s\nend GXD.%s\n'
+        % (LAB['imports'][0], raw, name, name, LAB['par'] % dict(h=h, n=n, d=L['d'], K=L['K']), name))
+    paths = [format(i, '0%db' % L['ts']) for i in range(1 << L['ts'])] if L['ts'] else ['']
+    for i, pth in enumerate(paths):
+        mod('Tab%d' % i, LAB['imports'] + [raw], 'tab%d_ok' % i,
+            LAB['tab'] % dict(h=h, sub='tab_n' + pth if pth else 'tab') + ' = true',
+            'E0: subtrie %d of %d of the frame table (codes of reduced echelon bases)' % (i, len(paths)))
+    for i in range(L['nseg']):
+        mod('Seg%d' % i, [par], 'seg%d_ok' % i,
+            LAB['seg'] % dict(gs='seg%d' % i, i=i, j=i + 1, rs='rs%d' % i) + ' = true',
+            'E1: replay of the gates %d .. %d (phase %s) and their block ranks' % (
+                M['cuts'][i], M['cuts'][i + 1] - 1, 'A' if i < L['nsegA'] else 'B'))
+    mod('SegF', [par], 'segF_ok', LAB['seg'] % dict(gs='fins', i=L['nseg'], j=L['nseg'] + 1, rs='rF') + ' = true',
+        'E1: the final climbs and their block ranks')
+    if LAB['end']:
+        mod('End', LAB['end_imports'] + [par], 'end_ok', LAB['end'] % dict(v=v, a=L['nsegA'], f=L['nseg'] + 1) + ' = true',
+            'E1 E2: frames at the start, at the scatter, at the end')
+    return res
+
+
+def nest(xs):
+    return ' ++ ('.join(xs) + ')' * (len(xs) - 1) if xs else '[]'
+
+
+def write_hist(D, M, name, src, out, L):
+    """kernel counts of the block ranks on the EXPLICIT rank lists rA, rB, rF (no replay)"""
+    base = out[len(ROOT):].replace('/', '.')
+    h, rs = D['h'], M['lab']['rs']
+    ph = {'A': sum(rs[:L['nsegA']], []), 'B': sum(rs[L['nsegA']:], []), 'F': M['lab']['rF']}
+    top = max(max(x) if x else 0 for x in ph.values())
+    cop = [M['dim'][f] for _, f in D['ret']]
+    tot = {r: sum(x.count(r) for x in ph.values()) + cop.count(r) for r in range(1, max(top, max(cop)) + 1)}
+    assert {r: c for r, c in tot.items() if c} == M['hist'], 'histogram of the rank lists differs from the mirror'
+    lines = ['/-- block ranks of phase A / phase B, in order (the lists of the replay segments) -/',
+             'noncomputable def rA : List Nat := ' + nest(['rs%d' % i for i in range(L['nsegA'])]),
+             'noncomputable def rB : List Nat := ' + nest(['rs%d' % i for i in range(L['nsegA'], L['nseg'])])]
+    names = []
+    for X, lst in sorted(ph.items()):
+        l = 'r' + X
+        lines.append('theorem rankBound%s : wsumK (SSC.wGt %d) %s 0 = 0 := by decide +kernel' % (X, top, l))
+        lines += ['theorem rank%s%d_ok : wsumK (SSC.wEq %d) %s 0 = %d := by decide +kernel' % (X, r, r, l, lst.count(r))
+                  for r in range(top + 1)]
+        names += ['rankBound' + X] + ['rank%s%d_ok' % (X, r) for r in range(top + 1)]
+    txt = ('import Work.GCert.Data.Count\nimport %s.Gen.%s\n' % (base, name) +
+           '/-! Kernel count of the BLOCK RANKS of the gcert/1 certificate `GXD.%s.raw` (h = %d) on the explicit rank lists\n'
+           '`rA`, `rB`, `rF` (phase A, phase B, final climbs): `wsumK (SSC.wEq r) l 0` blocks of rank `r`, none above %d\n'
+           '(`GXD.sum_of_wsumK` turns a count into `(l.map wt).sum` = `SSC.wsum wt l`).\n'
+           'The %d scratch copies (ranks %s) are not in these lists.  Whole invocation, by rank: %s.\n'
+           'Source: %s.  Generated by checks/wht26/gx-data/py/gxgen.py (do not edit). -/\n'
+           'set_option maxRecDepth 100000\nnamespace GXD.%s\nopen GXD\n' % (
+               name, h, top, len(cop), sorted(set(cop)), M['hist'], src, name) + '\n'.join(lines) +
+           '\nend GXD.%s\n' % name + '\n'.join('#print axioms GXD.%s.%s' % (name, x) for x in names[:2]) + '\n')
+    open('%s/Gen/%sHist.lean' % (out, name), 'w').write(txt)
+    return '%s.Gen.%sHist' % (base, name)
+
+
+def write_price(D, M, name, src, out, L):
+    """Mathlib module: the price of the rank lists rA, rB, rF from the kernel counts, and of one whole invocation"""
+    base = out[len(ROOT):].replace('/', '.')
+    rs = M['lab']['rs']
+    ph = {'A': sum(rs[:L['nsegA']], []), 'B': sum(rs[L['nsegA']:], []), 'F': M['lab']['rF']}
+    top = max(max(x) if x else 0 for x in ph.values())
+    cop = [M['dim'][f] for _, f in D['ret']]
+    tm = lambda cnt: ' + '.join('%d * φ %d' % (n, r) for r, n in sorted(cnt.items()) if n) or '0'
+    thms = ''
+    for X in 'ABF':
+        cnt = {r: ph[X].count(r) for r in range(1, top + 1)}
+        thms += '''
+/-- price of the blocks of %s, from the kernel counts -/
+theorem r%s_cost (φ : ℕ → ℝ) : rcost φ r%s = %s := by
+  rw [GXD.rcost_wsumK φ r%s %d rankBound%s]
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero, %s]
+  norm_num [bcost]
+''' % ({'A': 'phase A', 'B': 'phase B', 'F': 'the final climbs'}[X], X, X, tm(cnt), X, top, X,
+       ', '.join('rank%s%d_ok' % (X, r) for r in range(top + 1)))
+    copies = tm({r: cop.count(r) for r in set(cop)})
+    txt = '''import Work.GCert.Data.Price
+import %(base)s.Gen.%(name)sHist
+
+/-!
+# (key: gx-data) gcert/1 certificate `%(name)s` (h = %(h)d): the price of one invocation
+
+GENERATED by checks/wht26/gx-data/py/gxgen.py from %(src)s.  No `sorry`.
+`rA_cost`, `rB_cost`, `rF_cost`: the price of the block rank lists of phase A, phase B and the final climbs
+(lists `GXD.%(name)s.rA / rB / rF`, counted by the kernel in `Gen.%(name)sHist`);
+`inv_cost`: with the %(nc)d scratch copies (%(copies)s) the blocks of ONE invocation cost
+`invPrice φ` = %(terms)s;  `inv_moves`: they stand for `R h + 2 v (h - 1) + cst = %(N)d` unit moves.
+-/
+
+namespace GXD.%(name)s
+open GXD OAI.PowerSaving.CB
+%(thms)s
+/-- price of the blocks of ONE invocation (x roles, y roles, slots, scratch copies), by rank -/
+def invPrice (φ : ℕ → ℝ) : ℝ := %(terms)s
+
+/-- **Price of ONE invocation** in the shape of `GX.Lab.cost`: phase A, the scratch copies, phase B, final climbs. -/
+theorem inv_cost (φ : ℕ → ℝ) : rcost φ rA + (%(copies)s) + rcost φ rB + rcost φ rF = invPrice φ := by
+  rw [rA_cost, rB_cost, rF_cost]
+  unfold invPrice
+  ring
+
+theorem inv_moves : invPrice (fun r => (r:ℝ)) = %(R)d * %(h)d + 2 * %(v)d * (%(h)d - 1) + %(cst)d := by
+  unfold invPrice
+  norm_num
+
+theorem inv_count : invPrice (fun _ => (1:ℝ)) = %(count)d := by
+  unfold invPrice
+  norm_num
+
+end GXD.%(name)s
+#print axioms GXD.%(name)s.inv_cost
+#print axioms GXD.%(name)s.inv_moves
+''' % dict(base=base, name=name, h=D['h'], src=src, nc=len(cop), copies=copies, terms=tm(M['hist']), N=D['N'], thms=thms,
+           R=D['R'], v=D['v'], cst=D['cst'], count=sum(M['hist'].values()))
+    open('%s/%sPrice.lean' % (out, name), 'w').write(txt)
+    return '%s.%sPrice' % (base, name)
+
+
+def write_cert(D, M, name, src, out, chk, L):
+    """<Name>Cert: THE LABEL HALF (aux, valid : GLab.Valid raw aux, lab_cost, lab_price) -- compiles without the scalar half;
+    <Name>ScalOK: the scalar half `scalOK : GS.ScalOK raw` (needs Work/GCert/Scalar/Main)."""
+    base = out[len(ROOT):].replace('/', '.')
+    bs = M['scal']['batches']
+    hs = ''
+    for i, (lo, hi) in enumerate(bs):
+        lo_pf = 'Nat.zero_le t' if i == 0 else 'Nat.le_of_not_lt h%d' % (i - 1)
+        if i + 1 < len(bs):
+            hs += '    if h%d : t < %d then ⟨%d, %d, %s, h%d, GS.XRun.of_check scal%d_ok⟩ else\n' % (i, hi, lo, hi - lo, lo_pf, i, i)
+        else:
+            hs += '    ⟨%d, %d, %s, Nat.lt_of_lt_of_le ht (by decide), GS.XRun.of_check scal%d_ok⟩' % (lo, hi - lo, lo_pf, i)
+    terms = ' + '.join('%d * φ %d' % (n, r) for r, n in M['hist'].items())
+    info = dict(base=base, name=name, h=D['h'], src=src, v=D['v'], R=D['R'], nc=len(D['ret']), cst=D['cst'], N=D['N'],
+                nb=len(bs), terms=terms, hs=hs)
+    txt = '''import Work.GCert.Scalar.Main
+import %(imps)s
+/-! # (key: gx-data) gcert/1 certificate `%(name)s` (h = %(h)d): THE SCALAR HALF `scalOK : GS.ScalOK raw`
+GENERATED by checks/wht26/gx-data/py/gxgen.py from %(src)s.  No `sorry`.
+Kernel checks `scal<i>_ok` (%(nb)d source batches) and `ychk_ok`. -/
+namespace GXD.%(name)s
+open GXD
+theorem scalOK : GS.ScalOK raw where
+  hv := by decide
+  hs := fun t ht =>
+%(hs)s
+  hyc := fun t ht => ⟨0, %(v)d, Nat.zero_le t, Nat.lt_of_lt_of_le ht (by decide), GS.YRun.of_check ychk_ok⟩
+end GXD.%(name)s
+#print axioms GXD.%(name)s.scalOK
+''' % dict(info, imps='\nimport '.join(c[0] for c in chk if 'Scal' in c[0] or 'YChk' in c[0]))
+    open('%s/%sScalOK.lean' % (out, name), 'w').write(txt)
+    # ---- the label half
+    nA, ns, ts = L['nsegA'], L['nseg'], L['ts']
+    cop = sorted(set(M['dim'][f] for _, f in D['ret']))
+    assert len(cop) == 1, 'scratch copies of several ranks %s: lab_cost of gxbind.py assumes one' % cop
+
+    def tabpf(path):
+        if len(path) == ts:
+            return 'tab%d_ok' % (int(path, 2) if path else 0)
+        return '(GLab.tabK_node _ _ _ %s %s)' % (tabpf(path + '0'), tabpf(path + '1'))
+
+    def runpf(a, b):
+        return 'GLab.Run.of_seg seg%d_ok' % a if a + 1 == b else 'GLab.Run.append (GLab.Run.of_seg seg%d_ok) (%s)' % (a, runpf(a + 1, b))
+
+    def run(X, a, b):
+        segs = ['seg%d' % i for i in range(a, b)]
+        return '''theorem gates%(X)s_eq : raw.gates%(X)s = %(cat)s := by
+  show List.flatten %(lst)s = _
+  simp only [List.flatten_cons, List.flatten_nil, List.append_nil]
+/-- the replay of phase %(X)s, composed from its kernel-checked segments -/
+theorem run%(X)s : GLab.Run par st%(a)d raw.gates%(X)s st%(b)d r%(X)s := by
+  rw [gates%(X)s_eq]
+  exact %(pf)s
+''' % dict(X=X, cat=nest(segs), lst=ll_(segs), a=a, b=b, pf=runpf(a, b))
+    lab = 'valid.lab S Ro cf hMA hMAi hMB hMBi reg hCc hreg'
+    txt = '''import Work.GCert.Labels.Main
+import %(base)s.%(name)sPrice
+import %(imps)s
+/-! # (key: gx-data) gcert/1 certificate `%(name)s` (h = %(h)d): THE LABEL HALF and its price
+GENERATED by checks/wht26/gx-data/py/gxgen.py from %(src)s.  No `sorry`.
+`v = %(v)d` ports, `R = %(R)d` slots, %(nc)d scratch copies of rank %(c)d, copy cost `cst = %(cst)d`, `N = %(N)d` unit moves.
+* `aux : GLab.Aux`, `valid : GLab.Valid raw aux`: frame table (`tab<i>_ok` composed by `tabK_node`), the replays of phase A
+  and phase B (`seg<i>_ok` composed by `Run.append`), the final climbs (`segF_ok`, `fin_ok`), the ends (`end_ok`);
+* `lab_cost`, `lab_price`: for ANY scalar half `S` the label half `valid.lab S ..` costs
+  `rcost rA + card C * bcost %(c)d + rcost rB + rcost rF` = `invPrice φ` = %(terms)s
+  (`Gen.%(name)sHist`, `%(name)sPrice`) when there are %(nc)d copies. -/
+set_option maxRecDepth 100000
+namespace GXD.%(name)s
+open GXD OAI.PowerSaving OAI.PowerSaving.CB
+theorem raw_h : raw.h = %(h)d := rfl
+theorem raw_v : raw.v = %(v)d := rfl
+theorem raw_R : raw.R = %(R)d := rfl
+/-- what the label checks read next to `raw` -/
+noncomputable def aux : GLab.Aux := ⟨par, st0, st%(a)d, st%(n)d, st%(f)d, fins, rA, rB, rF⟩
+theorem fin_ok : GLab.noAdds fins = true := by decide +kernel
+theorem cen_ok : (raw.ret.map Prod.fst).all (fun r => Nat.beq (GLab.cdim (par.tab.get (st%(a)d.get r))) %(c)d) = true := by
+  decide +kernel
+theorem tab_ok : GLab.tabK par.h par.tab = true := %(tabpf)s
+%(runA)s%(runB)s/-- **the kernel-checked label facts** of the certificate -/
+theorem valid : GLab.Valid raw aux where
+  hn := rfl
+  tab := tab_ok
+  runA := runA
+  runB := runB
+  runF := GLab.Run.of_seg segF_ok
+  fin := fin_ok
+  ends := end_ok
+theorem cen_dim (r : Nat) (hr : r ∈ raw.ret.map Prod.fst) : GLab.cdim (par.tab.get (st%(a)d.get r)) = %(c)d :=
+  Nat.eq_of_beq_eq_true (List.all_eq_true.mp cen_ok r hr)
+/-- price of the label half over ANY scalar half -/
+theorem lab_cost {C : Type} [Fintype C] [DecidableEq C] (S : GX.Scal (Fin raw.v) (Fin raw.R) C)
+    {emb : CR.L3 (Fin raw.v) (Fin raw.R) → Nat} {un : Nat → CR.L3 (Fin raw.v) (Fin raw.R)}
+    (Ro : GLab.Roles raw.v raw.R emb un) (cf : Co → SSC.Coef) (hMA) (hMAi) (hMB) (hMBi)
+    (reg : C → Nat) (hCc) (hreg) (φ : ℕ → ℝ) :
+    (%(lab)s).cost φ
+      = rcost φ rA + (Fintype.card C : ℝ) * bcost φ %(c)d + rcost φ rB + rcost φ rF := by
+  obtain ⟨h1, h2, h3, h4⟩ := valid.lab_ranks S Ro cf hMA hMAi hMB hMBi reg hCc hreg
+  have h5 : ∀ k, bcost φ (((%(lab)s).cen k).dim) = bcost φ %(c)d :=
+    fun k => by rw [h4 k]; exact congrArg (bcost φ) (cen_dim _ (hreg k))
+  rw [GX.Lab.cost, h1, h2, h3, Finset.sum_congr rfl (fun k _ => h5 k), Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
+  rfl
+/-- **the label half costs one invocation price** (`invPrice`, by rank, from the kernel counts) -/
+theorem lab_price {C : Type} [Fintype C] [DecidableEq C] (S : GX.Scal (Fin raw.v) (Fin raw.R) C)
+    {emb : CR.L3 (Fin raw.v) (Fin raw.R) → Nat} {un : Nat → CR.L3 (Fin raw.v) (Fin raw.R)}
+    (Ro : GLab.Roles raw.v raw.R emb un) (cf : Co → SSC.Coef) (hMA) (hMAi) (hMB) (hMBi)
+    (reg : C → Nat) (hCc) (hreg) (hC : Fintype.card C = %(nc)d) (φ : ℕ → ℝ) :
+    (%(lab)s).cost φ = invPrice φ := by
+  rw [lab_cost S Ro cf hMA hMAi hMB hMBi reg hCc hreg φ, hC, ← inv_cost φ]
+  norm_num [bcost]
+end GXD.%(name)s
+#print axioms GXD.%(name)s.valid
+#print axioms GXD.%(name)s.lab_price
+''' % dict(info, imps='\nimport '.join(c[0] for c in chk if 'Scal' not in c[0] and 'YChk' not in c[0]), c=cop[0], a=nA, n=ns,
+           f=ns + 1, tabpf=tabpf(''), runA=run('A', 0, nA), runB=run('B', nA, ns), lab=lab)
+    open('%s/%sCert.lean' % (out, name), 'w').write(txt)
+    return ['%s.%sCert' % (base, name), '%s.%sScalOK' % (base, name)]
+
+
+def ll_(xs):
+    return '[' + ', '.join(xs) + ']'
+
+
+def final_module(name, cname, d, dx, g, out):
+    """<Inst>.lean: the final theorems OVER ANY SCALAR HALF (compiles as soon as <C>Cert does): label half, price,
+    unit list, rate fact and gx-chain's `GX.wht_of_halves` are all closed; what is left is `S : GX.Scal` with its
+    gate matrices = the ordered products of the adds of `raw` (gx-scalar)."""
+    h, v, R, W, m, nc = g['h'], g['v'], g['R'], d['W'], d['m'], g['nc']
+    X = 'GXD.%s' % cname
+    args = 'S Ro cf hMA hMAi hMB hMBi reg hCc hreg'
+
+    def thm(nm, rate, A, B, s, ux):
+        return f'''
+/-- **Walsh-Hadamard transform, whole-block recursion on the bridged word B_2 with the gcert/1 circuit `{cname}`**,
+over any scalar half of `{X}.raw`: the statement `WHTGoal` of `WHTCheck/Challenge.lean` with `1 - 2/10^11`
+replaced by `1 - {A}/10^{B}`. -/
+theorem {nm} (Ro : GLab.Roles {X}.raw.v {X}.raw.R emb un)
+    (hMA : S.MA = SSC.matP un (GLab.mic cf ({X}.raw.gatesA.flatMap GXD.Gate.adds)))
+    (hMAi : S.MAi = SSC.matPinv un (GLab.mic cf ({X}.raw.gatesA.flatMap GXD.Gate.adds)))
+    (hMB : S.MB = SSC.matP un (GLab.mic cf ({X}.raw.gatesB.flatMap GXD.Gate.adds)))
+    (hMBi : S.MBi = SSC.matPinv un (GLab.mic cf ({X}.raw.gatesB.flatMap GXD.Gate.adds)))
+    (reg : C → Nat) (hCc : ∀ k (q : Fin {X}.raw.R), S.Cc k q ≠ 0 → reg k = 2 * {X}.raw.v + q.val)
+    (hreg : ∀ k, reg k ∈ {X}.raw.ret.map Prod.fst) (hC : Fintype.card C = {nc}) :
+    ∃ solve W, WHTProgram solve W ∧ BlockWHT.WHTTimeBoundsAt (1 - {A}/(10:ℝ)^{B}) W :=
+  GX.wht_of_halves S ({X}.valid.lab {args})
+    (by rw [Fintype.card_fin]; decide) {m} {W} (by rw [Fintype.card_fin]; rfl)
+    (by rw [Fintype.card_fin, Fintype.card_fin]; rfl) (by norm_num) BlockAccounting.blocks{rate}
+    (fun φ => by
+      rw [{X}.lab_price {args} hC φ, Fintype.card_fin, Fintype.card_fin]
+      exact BridgeRate.{name}.unit_cost{ux} φ)
+    {s} (1 - {A}/(10:ℝ)^{B}) (by norm_num) (by norm_num) BlockAccounting.fold_{rate}
+
+/-- the same from the FIVE scalar identities of gx-scalar's matrices of `{X}.raw` (`Work/GCert/Data/Glue.lean`) -/
+theorem {nm.replace('_of_scal', '_of_ids')} (hv : 0 < {X}.raw.v)
+    (hAi : GS.MAi {X}.raw hv * GS.MA {X}.raw hv = 1) (hBi : GS.MBi {X}.raw hv * GS.MB {X}.raw hv = 1)
+    (hx : ∀ (t : Fin {X}.raw.v) (j : GS.Rl {X}.raw), (GS.MB {X}.raw hv * CR.scatM (GS.Jrm {X}.raw * GS.Ccm {X}.raw)
+      * GS.MA {X}.raw hv) (CR.lX t) j = if CR.lX t = j then 1 else 0)
+    (hy : ∀ (i : GS.Rl {X}.raw) (t : Fin {X}.raw.v), (GS.MB {X}.raw hv * CR.scatM (GS.Jrm {X}.raw * GS.Ccm {X}.raw)
+      * GS.MA {X}.raw hv) i (CR.lY t) = if i = CR.lY t then 1 else 0)
+    (hid : ∀ S t : Fin {X}.raw.v, (GS.MB {X}.raw hv * CR.scatM (GS.Jrm {X}.raw * GS.Ccm {X}.raw)
+      * GS.MA {X}.raw hv) (CR.lY S) (CR.lX t) = if S = t then 1 else 0) :
+    ∃ solve W, WHTProgram solve W ∧ BlockWHT.WHTTimeBoundsAt (1 - {A}/(10:ℝ)^{B}) W :=
+  {nm} (GXD.scalOf {X}.raw hv hAi hBi hx hy hid) GS.toCoef (GXD.roles {X}.raw hv)
+    (GXD.scalOf_MA _ hv hAi hBi hx hy hid) (GXD.scalOf_MAi _ hv hAi hBi hx hy hid)
+    (GXD.scalOf_MB _ hv hAi hBi hx hy hid) (GXD.scalOf_MBi _ hv hAi hBi hx hy hid)
+    (GXD.reg {X}.raw) (GXD.reg_slot {X}.raw hv hAi hBi hx hy hid) (GXD.reg_mem {X}.raw)
+    (by rw [Fintype.card_fin]; decide)
+'''
+    txt = f'''import Work.GCert.Chain.End
+import Work.GCert.Data.Glue
+import Work.GCert.Data.{cname}Cert
+import Work.GCert.Data.Inst{name}
+
+/-!
+# (key: gx-data) Bridged word B_2 with the gcert/1 circuit `{cname}` (h = {h}, m = {m}): final theorems over a scalar half
+GENERATED by checks/wht26/gx-data/py/gxrate.py.  No `sorry`.
+`GX.wht_of_halves` (gx-chain) on the label half `{X}.valid` (kernel checks of `Work/GCert/Data/Gen/{cname}*`), its
+price `{X}.lab_price`, the unit list `BlockAccounting.blocks{name}` (`W = {W}` live roles) and the rate facts
+`fold_{name}` (`1 - {d['A']}/10^{d['B']}`), `fold_{name}x` (`1 - {dx['A']}/10^{dx['B']}`).
+The ONLY hypothesis left is the scalar half: `S : GX.Scal (Fin {v}) (Fin {R}) C` whose gate matrices are the ordered
+products of the adds of `raw` (`hMA .. hMBi`), with `{nc}` copies read from the retained registers (`reg`, `hCc`, `hreg`, `hC`);
+`wht_{name}_of_ids`, `wht_{name}x_of_ids`: the same from the five identities `hAi hBi hx hy hid` of gx-scalar's matrices
+`GS.MA GS.MAi GS.MB GS.MBi GS.Jrm GS.Ccm` of `raw` (the fields of the planned `GS.ScalOK`).
+-/
+
+namespace OAI
+namespace PowerSaving
+namespace WHT
+open BlockWHT
+
+variable {{C : Type}} [Fintype C] [DecidableEq C] (S : GX.Scal (Fin {X}.raw.v) (Fin {X}.raw.R) C)
+  {{emb : CR.L3 (Fin {X}.raw.v) (Fin {X}.raw.R) → Nat}} {{un : Nat → CR.L3 (Fin {X}.raw.v) (Fin {X}.raw.R)}}
+  (cf : GXD.Co → SSC.Coef)
+''' + thm('wht_' + name + '_of_scal', name, d['A'], d['B'], d['s'], '') + thm('wht_' + name + 'x_of_scal', name + 'x', dx['A'], dx['B'], dx['s'], '_x') + f'''
+end WHT
+end PowerSaving
+end OAI
+#print axioms OAI.PowerSaving.WHT.wht_{name}_of_scal
+#print axioms OAI.PowerSaving.WHT.wht_{name}x_of_scal
+#print axioms OAI.PowerSaving.WHT.wht_{name}_of_ids
+#print axioms OAI.PowerSaving.WHT.wht_{name}x_of_ids
+'''
+    open(out + '%s.lean' % name, 'w').write(txt)
+
+
+def main_module(name, cname, d, dx, out):
+    """<Inst>Main.lean: the CLOSED theorems (what the comparator solution cites).  WRITTEN against gx-scalar's planned
+    `GS.ScalOK` (fields / theorems hv hAi hBi hx hy hid, SCALAR v2); compiles when Work/GCert/Scalar/Main.lean exists."""
+    K = 'GXD.%s.scalOK' % cname
+
+    def thm(sfx, A, B):
+        return f'''
+/-- **Walsh-Hadamard transform in `O(n (log n)^(1 - {A}/10^{B}))`-type time bounds: whole-block recursion on the bridged
+word B_2 with the kernel-checked gcert/1 circuit `{cname}`.** -/
+theorem wht_main_{name}{sfx} :
+    ∃ solve W, WHTProgram solve W ∧ BlockWHT.WHTTimeBoundsAt (1 - {A}/(10:ℝ)^{B}) W :=
+  wht_{name}{sfx}_of_ids {K}.hv {K}.hAi {K}.hBi {K}.hx {K}.hy {K}.hid
+'''
+    txt = f'''import Work.GCert.Scalar.Main
+import Work.GCert.Data.{cname}ScalOK
+import Work.GCert.Data.{name}
+
+/-!
+# (key: gx-data) Bridged word B_2 with the gcert/1 circuit `{cname}`: THE CLOSED THEOREMS
+GENERATED by checks/wht26/gx-data/py/gxrate.py.  No `sorry`.
+`wht_{name}_of_ids` (label half, price, unit list, rate fact, gx-chain's end theorem) applied to the scalar half
+`{K} : GS.ScalOK GXD.{cname}.raw` (kernel checks `scal<i>_ok`, `ychk_ok`; soundness: gx-scalar).
+-/
+
+namespace OAI
+namespace PowerSaving
+namespace WHT
+open BlockWHT
+''' + thm('', d['A'], d['B']) + thm('x', dx['A'], dx['B']) + f'''
+end WHT
+end PowerSaving
+end OAI
+#print axioms OAI.PowerSaving.WHT.wht_main_{name}
+#print axioms OAI.PowerSaving.WHT.wht_main_{name}x
+'''
+    open(out + '%sMain.lean' % name, 'w').write(txt)
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/tools__gx__gxconv.py b/research/five-stage-source527-banks/inputs/complex/sources/tools__gx__gxconv.py
new file mode 100644
index 0000000000..49ebb65973
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/tools__gx__gxconv.py
@@ -0,0 +1,131 @@
+"""gxconv.py: gcert/0 -> gcert/1 for files whose in-place pair op SCALES its target
+(`op dest ctrl ca cb`: dest := ca dest + cb ctrl with ca not in {1}; #193 has ca = -2), questions.md D2.
+COPY of `convert` of gx.py with ONE change: a scaling is not a gate, it changes the
+UNIT in which the register is stored.  wanted(r) = lam[r] * stored(r), lam = 1 at the start;
+    op dest ctrl ca cb :  lam[dest] *= ca ;  stored(dest) += (cb lam[ctrl] / lam[dest]) stored(ctrl)
+    any other add t += k s :  stored(t) += (k lam[s] / lam[t]) stored(s)
+    scatter y_t += coef(t, k) total_k :  coefficient coef(t, k) * lam[slot of total k]   (a TABLE if some lam != 1)
+Only slots may be scaled (tested: lam = 1 on every x and y register, always), so the x / y rows and columns of
+the product (E5: hx, hy, hid) are those of the unscaled circuit; frames, blocks and N are untouched.
+The result is tested by the reference checker `gx.check1` (labels and scalars) before it is written.
+
+usage: gxconv.py <gcert0.json[.gz]> out=<gcert1.json[.gz]> [noext]
+"""
+import sys, time
+from fractions import Fraction as Fr
+import gxpaths
+import gx
+import gxrun
+need = gx.need
+
+
+def convert(c0):
+    need(c0["format"] == "gcert/0", "input is not gcert/0")
+    v, h, R = c0["v"], c0["h"], c0["R"]
+    ev = c0["events"]
+    cen = [n for n, e in enumerate(ev) if e[0] == "centre"]
+    need(cen == list(range(cen[0], cen[0] + len(cen))), "centres are not contiguous: no single scatter cut")
+    A, B, ret = [], [], []
+    neg, openq, lam, nsc = set(), {}, {}, 0
+    L = lambda r: lam.get(r, Fr(1))
+
+    def co(t, s, a, b=1):                       # stored coefficient of the wanted add t += (a/b) s
+        w = Fr(a, b) * L(s) / L(t)
+        return [w.numerator, w.denominator]
+
+    def o(f, s, tg, ex=()):
+        return ["out", f, s, [[t] + co(t, s, a, b) for t, a, b in tg], list(ex)]
+
+    def i(f, t, sr):
+        return ["in", f, t, [[s] + co(t, s, a, b) for s, a, b in sr]]
+    scat_lam = None
+    for n, e in enumerate(ev):
+        k = e[0]
+        out = A if n < cen[0] else B
+        if k in ("read0", "undo", "uninj"):
+            continue
+        if k == "inj":
+            need(e[2] not in neg, "inj reads a negated x register")
+            out.append(o(e[3], e[2], [[e[1], 1, 1]]))
+        elif k == "op":
+            if e[3] != 1:
+                need(e[3] != 0 and e[1] >= 2 * v, "op scales a register that is not a slot (or by 0)")
+                lam[e[1]] = L(e[1]) * e[3]
+                nsc += 1
+            out.append(o(e[5], e[2], [[e[1], e[4], 1]]))
+        elif k == "centre":
+            ret.append([e[2], e[1], e[3]])
+            if scat_lam is None:
+                scat_lam = dict(lam)
+        elif k == "gread":
+            hit = set(t for t, _, _ in e[3])
+            out.append(o(e[2], e[1], [[v + t, a, b] for t, a, b in e[3]], [v + t for t in e[4] if t not in hit]))
+        elif k == "rread":
+            out.append(o(e[2], e[1], [[v + t, a, b] for t, a, b in e[3]]))
+        elif k == "ysh":
+            out.append(o(e[4], e[2], [[e[1], e[3], 1]]))
+        elif k == "yw":
+            out.append(o(e[4], e[2], [[e[1], e[3][0], e[3][1]]]))
+        elif k == "kd":
+            out.append(o(e[3], e[2], [[e[1], -1 if e[2] in neg else 1, 1]]))
+        elif k == "ktr":
+            q = tuple(e[1])
+            need(e[2] == gx.KJ, "ktr matrix is not 1 - J/2")
+            x1, rest = q[0], list(q[1:])
+            if q not in openq:
+                out.append(i(e[3], x1, [[r, 1, 1] for r in rest]))
+                out.append(o(e[3], x1, [[r, -1, 2] for r in rest]))
+                out.append(i(e[3], x1, [[r, 1, 1] for r in rest]))
+                openq[q] = n
+                neg.add(x1)
+            else:
+                out.append(i(e[3], x1, [[r, -1, 1] for r in rest]))
+                out.append(o(e[3], x1, [[r, 1, 2] for r in rest]))
+                out.append(i(e[3], x1, [[r, -1, 1] for r in rest]))
+                del openq[q]
+                neg.discard(x1)
+        else:
+            raise AssertionError("unknown event kind %r" % k)
+        need(all(r >= 2 * v for r, x in lam.items() if x != 1), "an x or y register is scaled")
+    need(not openq and not neg, "a K block is not closed")
+    ret = sorted(ret)
+    need([z[0] for z in ret] == list(range(len(ret))), "retained totals are not 0 .. n-1")
+    sin, sout = Fr(*c0["scat"]["inside"]), Fr(*c0["scat"]["outside"])
+    sl = [(scat_lam or {}).get(z[1], Fr(1)) for z in ret]
+    if all(x == 1 for x in sl):
+        scat = dict(inside=c0["scat"]["inside"], outside=c0["scat"]["outside"])
+    else:
+        need(len(ret) == h, "star rule with a number of totals other than h")
+        scat = dict(table=[[[kk, (w * sl[kk]).numerator, (w * sl[kk]).denominator]
+                            for kk in range(h) for w in [sin if c0["ports"][t] >> kk & 1 else sout]] for t in range(v)])
+    ext = sorted([z[3], z[1]] for z in c0["gauge"] if z[5])
+    c1 = dict(format="gcert/1", derived_from=c0["name"] + " (gcert/0; %d scalings as units, gxconv.py)" % nsc, p=c0["p"],
+              h=h, v=v, R=R, cst=c0["cst"], N=c0["N"], ports=c0["ports"], frames=c0["frames"], start=c0["start"],
+              final=c0["final"], ext=ext, ret=ret, scat=scat, A=A, B=B, blocks=c0["blocks"])
+    return c1, nsc, sorted(set(str(x) for x in lam.values())), sum(1 for x in sl if x != 1)
+
+
+if __name__ == "__main__":
+    t0 = time.time()
+    opt = dict(a.split("=", 1) for a in sys.argv[2:] if "=" in a)
+    c0 = gxrun.load(sys.argv[1])
+    c1, nsc, lams, nsl = convert(c0)
+    c1["two_hosts"] = sorted(2 * c0["v"] + i for i, hs in enumerate(c0["hosts"]) if len(hs) > 1)
+    if "noext" in sys.argv and c1["ext"]:        # as gxrun.py: exterior gauges become first blocks
+        for r, f in c1["ext"]:
+            d = str(len(c1["frames"][f]))
+            c1["start"][r] = 0
+            c1["blocks"]["s"][d] = c1["blocks"]["s"].get(d, 0) + 1
+            c1["N"] += int(d)
+        c1["blocks"]["s"] = {k: c1["blocks"]["s"][k] for k in sorted(c1["blocks"]["s"], key=int)}
+        c1["derived_from"] += ", exterior gauges turned into first blocks"
+        c1["ext"] = []
+    st = {}
+    gx.check1(c1, scalar=True, stats=st)
+    print("ACCEPTED by gx.check1: %s h=%d v=%d R=%d N=%d gates A=%d B=%d ext=%d; scalings %d, final units %s, scaled totals %d, "
+          "scatter %s, stats %s (%.1fs)" % (c0["name"], c1["h"], c1["v"], c1["R"], c1["N"], len(c1["A"]), len(c1["B"]),
+                                           len(c1["ext"]), nsc, lams, nsl, "table" if "table" in c1["scat"] else "star", st,
+                                           time.time() - t0))
+    if "out" in opt:
+        gxrun.dump(c1, opt["out"])
+        print("written", opt["out"])
diff --git a/research/five-stage-source527-banks/inputs/complex/sources/tools__gx__gxcore.py b/research/five-stage-source527-banks/inputs/complex/sources/tools__gx__gxcore.py
new file mode 100644
index 0000000000..3e3f0a3076
--- /dev/null
+++ b/research/five-stage-source527-banks/inputs/complex/sources/tools__gx__gxcore.py
@@ -0,0 +1,375 @@
+"""gxcore.py: gcert/1 JSON -> normal data D, PYTHON MIRROR of the two Lean checks
+(label side: E0 frames, E1 pairs, E1/E2/E6 replay in segments; scalar side: E3 kinds, E4 bracket, E5 replay
+in source batches, E7), derived data (distinct climbs, cut states, block histogram) and kernel sizes.
+Own code; the reference is gx.py (check1), not imported.  Python 3, standard library only."""
+import json, gzip
+from collections import Counter
+from math import gcd
+
+
+class Reject(Exception):
+    pass
+
+
+def need(c, msg):
+    if not c:
+        raise Reject(msg)
+
+
+def load(path):
+    c = json.load(gzip.open(path, "rt") if path.endswith(".gz") else open(path))
+    need(c.get("format") == "gcert/1", "E0: format is not gcert/1")
+    return c
+
+
+def co(num, den):
+    need(den > 0, "E0: denominator")
+    return (num < 0, abs(num), den)
+
+
+def normal(c):
+    """the Lean-facing data: gates as tuples, coefficients (neg, num, den), ret by total"""
+    def gate(g):
+        if g[0] == "out":
+            return ("out", g[1], g[2], [(t, co(a, b)) for t, a, b in g[3]], list(g[4]))
+        need(g[0] == "in", "E0: unknown gate kind %r" % (g[0],))
+        return ("in", g[1], g[2], [(s, co(a, b)) for s, a, b in g[3]])
+    ret = sorted(c["ret"])
+    need([k for k, _, _ in ret] == list(range(len(ret))), "E2: retained totals numbered 0..")
+    sc = c["scat"]
+    scat = ("table", [[(k, co(a, b)) for k, a, b in row] for row in sc["table"]]) if "table" in sc else \
+        ("star", co(*sc["inside"]), co(*sc["outside"]))
+    return dict(h=c["h"], v=c["v"], R=c["R"], N=c["N"], cst=c["cst"], ports=list(c["ports"]),
+                frames=[list(f) for f in c["frames"]], start=list(c["start"]), final=list(c["final"]),
+                ext=[tuple(e) for e in c.get("ext", [])], ret=[(s, f) for _, s, f in ret], scat=scat,
+                A=[gate(g) for g in c["A"]], B=[gate(g) for g in c["B"]],
+                blocks={k: {int(r): n for r, n in c["blocks"][k].items()} for k in "xysc"})
+
+
+def regs(g):
+    return [g[2]] + [t for t, _ in g[3]] + (g[4] if g[0] == "out" else [])
+
+
+def adds(g):
+    """single adds (tgt, src, coef) in order"""
+    return [(t, g[2], k) for t, k in g[3]] if g[0] == "out" else [(g[2], s, k) for s, k in g[3]]
+
+
+def reduce(u, basis):
+    for b in basis:
+        if u >> (b.bit_length() - 1) & 1:
+            u ^= b
+    return u
+
+
+def echelon_ok(f):
+    piv = [b.bit_length() - 1 for b in f]
+    if any(b == 0 for b in f) or any(piv[i] <= piv[i + 1] for i in range(len(piv) - 1)):
+        return False
+    return all(not (b >> p & 1) for i, b in enumerate(f) for j, p in enumerate(piv) if i != j)
+
+
+def par(x):
+    return bin(x).count("1") & 1
+
+
+def cls(D, r):
+    return "x" if r < D["v"] else "y" if r < 2 * D["v"] else "s"
+
+
+# ---- label side ------------------------------------------------------------------------------------------
+def chk_frames(D):
+    """E0: ports; frame table reduced echelon, no repeats, zero first, full second"""
+    h, v, fr = D["h"], D["v"], D["frames"]
+    p = D["ports"]
+    need(len(p) == v == len(set(p)) and all(par(q) and 0 < q < 1 << h for q in p), "E0: ports")
+    need(len(fr) >= 2 and fr[0] == [] and fr[1] == [1 << i for i in range(h - 1, -1, -1)], "E0: frames 0 and 1")
+    need(all(echelon_ok(f) and all(b < 1 << h for b in f) for f in fr), "E0: frame table not reduced echelon")
+    need(len(set(map(tuple, fr))) == len(fr), "E0: frame table has repeats")
+    return [len(f) for f in fr]
+
+
+def derive(D):
+    """distinct climbs in order of first use (NOT a check: plain replay of the frame ids)"""
+    cur, seen, pairs = list(D["start"]), set(), []
+
+    def go(r, f):
+        o = cur[r]
+        if o != f:
+            if (o, f) not in seen:
+                seen.add((o, f))
+                pairs.append((o, f))
+            cur[r] = f
+    for g in D["A"] + D["B"]:
+        for r in regs(g):
+            go(r, g[1])
+    for r, f in enumerate(D["final"]):
+        go(r, f)
+    return pairs
+
+
+def chk_pairs(D, dim, pairs):
+    """E1, level 2: every listed pair is a nested pair of the table; returns the mask-operation count"""
+    fr, n, work = D["frames"], len(D["frames"]), 0
+    for a, b in pairs:
+        need(a < n and b < n and dim[a] < dim[b], "E1: pair (%d, %d) does not climb" % (a, b))
+        need(all(reduce(u, fr[b]) == 0 for u in fr[a]), "E1: pair (%d, %d) is not nested" % (a, b))
+        work += dim[a] * dim[b]
+    return work
+
+
+def cut_points(D, gmax, wmax=210000):
+    """replay segments: each phase (A, B) is cut into k pieces of equal gate count, k = the least number with at
+    most `gmax` gates AND at most `wmax` work per piece on average (work = sum over gates of the dimension of the
+    gate's frame; measured at h = 22: about 1 ms of kernel time per unit, NOTES.md).
+    Cut points = gate counts in A ++ B (always 0, |A| and the end)."""
+    nA, nB = len(D["A"]), len(D["B"])
+    cuts = [0]
+    for off, gs in ((0, D["A"]), (nA, D["B"])):
+        n = len(gs)
+        k = max(1, -(-n // gmax), -(-sum(len(D["frames"][g[1]]) for g in gs) // wmax))
+        cuts += [off + (n * i) // k for i in range(1, k + 1)]
+    return sorted(set(cuts))
+
+
+def label_data(D, dim, cuts):
+    """for gx-labels (gcert-interface.md LABELS v1): ordered block ranks per replay segment, the final climbs as
+    gates without adds (same final frame, at most 32 registers, only registers that still climb), their ranks,
+    the largest number of registers named by one gate.  Plain replay (the mirror has accepted the file)."""
+    cur, G = list(D["start"]), D["A"] + D["B"]
+    rs, most = [], 0
+    for a, b in zip(cuts, cuts[1:]):
+        r = []
+        for g in G[a:b]:
+            most = max(most, len(regs(g)))
+            for x in regs(g):
+                if cur[x] != g[1]:
+                    r.append(dim[g[1]] - dim[cur[x]])
+                    cur[x] = g[1]
+        rs.append(r)
+    by = {}
+    for x, f in enumerate(D["final"]):
+        if cur[x] != f:
+            by.setdefault(f, []).append(x)
+    fins, rF = [], []
+    for f in sorted(by):
+        for i in range(0, len(by[f]), 32):
+            xs = by[f][i:i + 32]
+            fins.append(("out", f, xs[0], [], xs[1:]))
+            rF += [dim[f] - dim[cur[x]] for x in xs]
+    return dict(rs=rs, fins=fins, rF=rF, most=max(most, 32 if fins else 0))
+
+
+def chk_replay(D, dim, pairs, cuts):
+    """E1 (labels), E2 (scatter cut), E6 (price), in the segments given by `cuts`.
+    Returns (states at the cuts, histogram by class, visits per segment)."""
+    h, v, R, fr = D["h"], D["v"], D["R"], D["frames"]
+    n = 2 * v + R
+    start, final, ext = D["start"], D["final"], dict(D["ext"])
+    need(len(start) == len(final) == n, "E0: start / final")
+    for t in range(v):
+        need(fr[start[t]] == [D["ports"][t]] and final[t] == 1, "E1: x start / final")
+        f = fr[final[v + t]]
+        need(start[v + t] == 0 and len(f) == h - 1 and all(par(u & D["ports"][t]) == 0 for u in f), "E1: y start / final")
+    for r in range(2 * v, n):
+        need(start[r] == ext.get(r, 0) and final[r] == 1, "E1: slot start / final")
+    ok = set(pairs)
+    need(len(ok) == len(pairs), "E1: pair list has repeats")
+    cur = list(start)
+    H = {k: Counter() for k in "xysc"}
+
+    def climb(r, f, where):
+        o = cur[r]
+        if o != f:
+            need((o, f) in ok, "E1: climb (%d, %d) is not a tested pair (%s, register %d)" % (o, f, where, r))
+            H[cls(D, r)][dim[f] - dim[o]] += 1
+            seg[dim[f] - dim[o]] += 1
+            cur[r] = f
+    seg, segs = Counter(), []
+    G, nA = D["A"] + D["B"], len(D["A"])
+    states, visits = [], []
+    need(cuts[0] == 0 and cuts[-1] == len(G) and cuts == sorted(cuts), "cut points")
+    for a, b in zip(cuts, cuts[1:]):
+        states.append(list(cur))
+        vs = 0
+        for i in range(a, b):
+            if i == nA:
+                scatter_cut(D, cur, dim, H, seg)
+            g = G[i]
+            rs = regs(g)
+            need(all(0 <= r < n for r in rs) and len(set(rs)) == len(rs), "E3: registers of gate %d" % i)
+            need(i >= nA or all(r < v or r >= 2 * v for r in rs), "E2: y role named in phase A (gate %d)" % i)
+            vs += len(rs)
+            for r in rs:
+                climb(r, g[1], "gate %d" % i)
+        visits.append(vs)
+        if b < len(G):
+            segs.append(seg)
+            seg = Counter()
+    if nA == len(G):
+        scatter_cut(D, cur, dim, H, seg)
+    states.append(list(cur))
+    for r in range(n):
+        climb(r, final[r], "final")
+    segs.append(seg)
+    Hb = {k: dict(sorted(w.items())) for k, w in H.items()}
+    need(Hb == D["blocks"], "E6: block histogram differs from `blocks`")
+    N = sum(r * c for w in H.values() for r, c in w.items())
+    need(N == D["N"] and N + sum(dim[f] for f in ext.values()) == R * h + 2 * v * (h - 1) + D["cst"], "E6: N")
+    need(sum(r * c for r, c in H["c"].items()) == D["cst"], "E6: cst")
+    return states, Hb, visits, [[w[r] for r in range(h + 1)] for w in segs]
+
+
+def scatter_cut(D, cur, dim, H, seg):
+    v = D["v"]
+    if D["scat"][0] == "table":
+        need(len(D["scat"][1]) == v and all(k < len(D["ret"]) for row in D["scat"][1] for k, _ in row), "E2: scatter table")
+    else:
+        need(len(D["ret"]) == D["h"], "E2: one retained total per coordinate (star rule)")
+    for s, f in D["ret"]:
+        need(s >= 2 * v and cur[s] == f, "E2: retained slot %d is not at its frame" % s)
+        H["c"][dim[f]] += 1
+        seg[dim[f]] += 1
+    need(all(cur[v + t] == 0 for t in range(v)), "E2: a y role left frame 0 before the scatter")
+
+
+# ---- scalar side -----------------------------------------------------------------------------------------
+SC = 6            # all scalar entries are kept as integers in units of 1/6
+KINDS = {("x", "s"), ("s", "s"), ("s", "y"), ("x", "y"), ("x", "x"), ("y", "y")}
+
+
+def chk_shape(D):
+    """E3 kinds, E4 bracket"""
+    ysum, piv, ytg = Counter(), set(), set()
+    for ph in "AB":
+        for i, g in enumerate(D[ph]):
+            for t, s, (ng, a, b) in adds(g):
+                kd = (cls(D, s), cls(D, t))
+                need(kd in KINDS and t != s, "E3: gate kind %s -> %s (gate %s%d)" % (kd + (ph, i)))
+                need(ph == "B" or kd in {("x", "s"), ("s", "s")}, "E3: bank gate before the scatter (gate A%d)" % i)
+                if kd == ("y", "y"):
+                    need(SC % b == 0, "E4: denominator beyond sixths")
+                    ysum[s, t] += (-a if ng else a) * (SC // b)
+                    piv.add(s)
+                    ytg.add(t)
+    need(not (piv & ytg), "E4: a pivot is the target of a gate reading y")
+    need(all(x == 0 for x in ysum.values()), "E4: y-reads of a (pivot, target) pair do not cancel")
+    return len(ysum)
+
+
+def scat_row(D, t):
+    sc = D["scat"]
+    if sc[0] == "table":
+        return sc[1][t]
+    return [(k, sc[1] if D["ports"][t] >> k & 1 else sc[2]) for k in range(D["h"])]
+
+
+
+def replay(D, init, count=None):
+    """exact replay through A, scatter, B of sparse vectors with entries in sixths; `init` = register -> {key: 6}"""
+    v = D["v"]
+    reg = [dict(init.get(r, ())) for r in range(2 * v + D["R"])]
+
+    def axpy(t, src, k):
+        ng, a, b = k
+        d = reg[t]
+        for key, u in src.items():
+            w = u * a
+            need(w % b == 0, "E5: an entry needs a denominator beyond sixths (register %d)" % t)
+            w = w // b
+            w = d.get(key, 0) + (-w if ng else w)
+            if count is not None:
+                count[key] += 1
+            if w:
+                d[key] = w
+            else:
+                d.pop(key, None)
+
+    def run(gates):
+        for g in gates:
+            if g[0] == "out":
+                src = dict(reg[g[2]])
+                for t, k in g[3]:
+                    axpy(t, src, k)
+            else:
+                for s, k in g[3]:
+                    axpy(g[2], reg[s], k)
+    run(D["A"])
+    afterA = sum(len(d) for d in reg)
+    tot = [dict(reg[s]) for s, _ in D["ret"]]
+    for t in range(v):
+        for k, cf in scat_row(D, t):
+            if tot[k]:
+                axpy(v + t, tot[k], cf)
+    run(D["B"])
+    return reg, afterA
+
+
+def chk_scalar(D, batches=1):
+    """E5 (+ E7 when `ext` is non-empty).  Returns stats: denominators and largest numerators per class in the
+    units of the spec (x halves, y sixths, slots integers), updates per source batch, entries after A."""
+    v = D["v"]
+    count = Counter()
+    reg, afterA = replay(D, {t: {t: SC} for t in range(v)}, count)
+    need(all(reg[t] == {t: SC} for t in range(v)), "E5: an x role is not restored (X3)")
+    need(all(reg[v + t] == {t: SC} for t in range(v)), "E5: a target does not receive exactly its source (hid)")
+    if D["ext"]:
+        rg, _ = replay(D, {r: {r: SC} for r, _ in D["ext"]})
+        need(all(not rg[v + t] for t in range(v)), "E7: the dirt of an exterior-gauged slot reaches a y role")
+    step = -(-v // batches)
+    per = [sum(count[k] for k in range(lo, min(lo + step, v))) for lo in range(0, v, step)]
+    return dict(updates=sum(count.values()), per_batch=per, afterA=afterA,
+                batches=[(lo, min(lo + step, v)) for lo in range(0, v, step)])
+
+
+def digits(D):
+    """largest |entry| met and denominators per class, by a second replay that watches every write (slow path
+    kept separate from the check): returns {class: (lcm of denominators, largest |numerator| in units 1/den)}"""
+    v = D["v"]
+    den = {"x": 1, "y": 1, "s": 1}
+    mx = {"x": 0, "y": 0, "s": 0}
+    reg = [({r: SC} if r < v else {}) for r in range(2 * v + D["R"])]
+
+    def axpy(t, src, k):
+        ng, a, b = k
+        d, kk = reg[t], cls(D, t)
+        for key, u in src.items():
+            w = d.get(key, 0) + (-(u * a // b) if ng else u * a // b)
+            if w:
+                d[key] = w
+                q = SC // gcd(w, SC)
+                if den[kk] % q:
+                    den[kk] = den[kk] * q // gcd(den[kk], q)
+                if abs(w) > mx[kk]:
+                    mx[kk] = abs(w)
+            else:
+                d.pop(key, None)
+    for g in D["A"]:
+        for t, s, k in adds(g):
+            axpy(t, reg[s], k)
+    tot = [dict(reg[s]) for s, _ in D["ret"]]
+    for t in range(v):
+        for k, cf in scat_row(D, t):
+            axpy(v + t, tot[k], cf)
+    for g in D["B"]:
+        for t, s, k in adds(g):
+            axpy(t, reg[s], k)
+    return {k: (den[k], mx[k] * den[k] // SC if (mx[k] * den[k]) % SC == 0 else "%d/6" % mx[k]) for k in "xys"}
+
+
+def mirror(D, gmax=12000, batches=1, pairs=None, cuts=None):
+    """both checks; returns everything the generator and the dry run need"""
+    dim = chk_frames(D)
+    pairs = derive(D) if pairs is None else pairs
+    work = chk_pairs(D, dim, pairs)
+    cuts = cut_points(D, gmax) if cuts is None else cuts
+    states, H, visits, segH = chk_replay(D, dim, pairs, cuts)
+    ysh = chk_shape(D)
+    S = chk_scalar(D, batches)
+    hist = Counter()
+    for k in "xysc":
+        for r, n in H[k].items():
+            hist[r] += n
+    return dict(dim=dim, pairs=pairs, pair_work=work, cuts=cuts, states=states, H=H, visits=visits, ysh=ysh,
+                scal=S, hist=dict(sorted(hist.items())), segH=segH,
+                lab=label_data(D, dim, cuts))
diff --git a/research/five-stage-source527-banks/loader/LICENSE b/research/five-stage-source527-banks/loader/LICENSE
new file mode 100644
index 0000000000..261eeb9e9f
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/LICENSE
@@ -0,0 +1,201 @@
+                                 Apache License
+                           Version 2.0, January 2004
+                        http://www.apache.org/licenses/
+
+   TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION
+
+   1. Definitions.
+
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+      and distribution as defined by Sections 1 through 9 of this document.
+
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+      outstanding shares, or (iii) beneficial ownership of such entity.
+
+      "You" (or "Your") shall mean an individual or Legal Entity
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+      the Work and Derivative Works thereof.
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+      the original version of the Work and any modifications or additions
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diff --git a/research/five-stage-source527-banks/loader/MANIFEST.json b/research/five-stage-source527-banks/loader/MANIFEST.json
new file mode 100644
index 0000000000..c9b7512456
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/MANIFEST.json
@@ -0,0 +1,60 @@
+{
+  "excluded_outputs": [
+    "check_development_equivalence.py",
+    "equivalence_result.json",
+    "preparation_result.json",
+    "raw_ledger_result.json",
+    "qualification_result.json"
+  ],
+  "files": {
+    "LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+    "NOTICE.md": "ee632728cc4d89565292d288f8289f483ac794b5d02cc6e42c2b2c43c43ff1d8",
+    "README.md": "4c36794b0f702ed521c0a7eaeacb1b7cc258760d59193cc8eb29be590e709dcd",
+    "SOURCE_PINS.json": "5a4fa413ea0ecc74050359329e90521723151e421cc0c19f774b92bd608a3ac9",
+    "expected_ledger.json": "4910004287e016f7f2db10f0a54eed24bf1912aa148b347b3cb4795fa3ac63f5",
+    "expected_semantics.json": "b7de3178436c824b6be6d99842270a82e1821b8008dda66d8b18519ad347da2d",
+    "raw_ledger.py": "89ba53114488a77a12b941db2787850c2c41612ad0332c47699a011c0c596b32",
+    "source_loader.py": "e5e19ee6d6899fce8273d0f06a49caf7cd8bfe940a2a9a103dbc9d3a62b6b291",
+    "test_public.py": "2103c953b37cf447d1a2bbb86c26f96fa5569a3ab018567e93884c12ebde8a2d"
+  },
+  "schema": "five-stage-focused-loader-package/1",
+  "scope": "Clean source preparation and raw path ledger only; no complete supplier or multiplication exponent claim",
+  "source_roots": {
+    "base_bit": {
+      "commit": "8d8d67bcf69c5ea67d3a29dbc64ba588156d6e8d",
+      "files": {
+        "LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+        "NOTICE": "c9850418622aea8d1382a230306c940947b5da818f4854b4b272ed60cf239f6c",
+        "research/paired-cube-diagonal-bit-168/bit/base_word.py": "9fe3a81b7561d2b09c086f63fac4626e88a32369ef8bc5579c94878de1218a89",
+        "research/paired-cube-diagonal-bit-168/bit/word.py": "85822e9cca5b1b2335c3704a3df2f3c6ef01e42d8328daacb33701d30cee80ab",
+        "research/paired-cube-diagonal-bit-168/references/pr168-v4/research/paired-cube-bit/check_paired_cube_bit.py": "4d7c86cd999a1519e0dfb6dcf33a8f26e442a353c3665a0f06881216a260fbf3",
+        "research/paired-cube-diagonal-bit-168/selected/bit/frames_p12.json.gz": "ad8e2705652b68476713d7647a8592505e222f2b3ccd1728dedce153aac6d462",
+        "research/paired-cube-diagonal-bit-168/selected/bit/graph_p12.json": "31a09a55920e64c8bfbac8b845dc01754c112aa6e207812e2fd1f4939d8f64cb",
+        "research/paired-cube-diagonal-bit-168/selected/bit/kchron_p12.json": "0fab548e84cf5ca7c04e8bd19b5810850463f36e0f92180efe73aeff17105e39",
+        "research/paired-cube-diagonal-bit-168/selected/bit/profile_p12.json": "c44ef8640cbcbcc9bac3a30e13373bd79efddde55d7cea514354ca2cdb60d66d",
+        "research/paired-cube-diagonal-bit-168/selected/bit/sinks.json": "c00ff70c564829d4b97740f92796fc6ce14fbcd5ca519663828650552b80249d",
+        "research/paired-cube-diagonal-bit-168/selected/bit/word_p12.json.gz": "1cb7e8ed45cf542e304fddc4b12247dbb248a0dc84dedbd832f336b2d172793a"
+      },
+      "repository": "CrocSwap/integer-mult-bounds"
+    },
+    "pr210": {
+      "commit": "df95878d11190518e45ef9717c9ee05011f88ace",
+      "files": {
+        "research/coordinated-crossover-pr200/LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+        "research/coordinated-crossover-pr200/NOTICE": "9fc5561c0995892dbdde777c713899486920c32d3562392f5f72377a4953bf98",
+        "research/coordinated-crossover-pr200/borrow/selection.json": "6932749407e14b199651f0a7deff02c6de74e2c1be0a1300263f1e164927fb8b",
+        "research/coordinated-crossover-pr200/frames/opframe-bases.json": "687496f3c15818e7cad1beb5fdf3812b8e38825ea302bfd60aea7ac23f3516b4",
+        "research/coordinated-crossover-pr200/gaugeb/selection.json": "b0f9f0e67f94a3c21069f393ea27f9483d315bff2f53f77270163065a6345d1c",
+        "research/coordinated-crossover-pr200/joint/joint-selection.json": "13f6420d6c341c22235c9d136d5f7c32896974bb165a893fecd7e583b790d133",
+        "research/coordinated-crossover-pr200/joint/joint_word.py": "de37cbed753fc6f13a7e5dc25c8346279cfee5f1868df8c12a5fcb6ca8f631c0",
+        "research/coordinated-crossover-pr200/joint/pairs.json": "da54b863601b5cbc21d27fc5cd623bf805f974a3e47b81ed29a85aa5bad0fa05",
+        "research/coordinated-crossover-pr200/newg/replay.py": "0cbec81a11e758156e9303f63dc8c617601da69523073fdf2a77394d76acf56a",
+        "research/coordinated-crossover-pr200/newg/schedule.json": "ab39d61fff7a2cfcb13e2c10d65f2c603e76a91013d197260d6c7b645ccb27a1",
+        "research/coordinated-crossover-pr200/newg/selection.json": "d94ef4e0b8ca98a7385b364cf245816a7cad08f850a60ce8b67aab4c3a89f3f4"
+      },
+      "repository": "eumemic/integer-mult-bounds"
+    },
+    "schema": "five-stage-public-source-loader/1"
+  },
+  "source_snapshot_bytes": 7928724
+}
diff --git a/research/five-stage-source527-banks/loader/NOTICE.md b/research/five-stage-source527-banks/loader/NOTICE.md
new file mode 100644
index 0000000000..47090d6e93
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/NOTICE.md
@@ -0,0 +1,11 @@
+# Attribution
+
+This focused preparation adapter is distributed under Apache-2.0. It preserves the original source files, headers and notices in the supplied source roots.
+
+- eumemic: the current PR210 operation frames, joint gauges, donor assignments, source borrowing and retimed source471 schedule at df95878d11190518e45ef9717c9ee05011f88ace; original OpenAI Codex assistance disclosure retained.
+- Chafik Boukhalfa (chafreaky): the PR200 physical bit construction and terminal interfaces, carried by the consumed PR202 base at 8d8d67bcf69c5ea67d3a29dbc64ba588156d6e8d.
+- Rohan Arun, Dugongue, evmckinney9, icekylinx, James Chang, eumemic, Douglas Colkitt, OpenAI, and preceding contributors retain the attribution and assistance disclosures recorded in the supplied original notices.
+
+The new contribution here is the explicit public-root binding, source-derived terminal records, exact semantic comparison and independent raw-path recount. It does not claim authorship of the inherited helper, frames, compiler or theorem interfaces, nor review or endorsement by the named contributors.
+
+Assisted with ChatGPT.
diff --git a/research/five-stage-source527-banks/loader/README.md b/research/five-stage-source527-banks/loader/README.md
new file mode 100644
index 0000000000..0bc303ed81
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/README.md
@@ -0,0 +1,48 @@
+# Minimal source471 preparation and raw-ledger adapter
+
+This focused adapter consumes the current public PR210 source and pinned base-bit files directly. It prepares the actual retimed helper context and recounts its physical paths. It does not run the all-column scalar proof, prime suite, five-stage global lowering, or outer assembly.
+
+## Inputs
+
+`SOURCE_PINS.json` lists 22 exact public files, including original licences and notices:
+
+- eumemic/integer-mult-bounds at `df95878d11190518e45ef9717c9ee05011f88ace`
+- CrocSwap/integer-mult-bounds at `8d8d67bcf69c5ea67d3a29dbc64ba588156d6e8d`
+
+Supply those source roots with their original relative paths. The supplied `source_inputs/` contains just the listed files for this qualification. Normal pinned upstream checkouts can be used instead. No network access or implicit downloading is performed by the adapter. Python 3.11 or newer with assertions enabled is required; no third-party Python package is used by this focused loader.
+
+## Targeted verification
+
+From this directory:
+
+    python3 -B test_public.py --pr210-root source_inputs/pr210 --base-bit-root source_inputs/base_bit
+
+To print the prepared-source fingerprint:
+
+    python3 -B source_loader.py --pr210-root source_inputs/pr210 --base-bit-root source_inputs/base_bit
+
+To print the independently rebuilt raw ledger:
+
+    python3 -B raw_ledger.py --pr210-root source_inputs/pr210 --base-bit-root source_inputs/base_bit
+
+These commands do not alter inputs or write result files. Redirect output to a new external path if desired. The tests distinguish executable checks from the compact deterministic fixtures `expected_semantics.json` and `expected_ledger.json`.
+
+## Programmatic interface
+
+`source_loader.prepare(pr210_root, base_bit_root)` returns the source preparation namespace. It contains actual `W`, `C`, `schedule`, `borrow`, `terminal`, `regs`, `adj`, `at`, `kpair`, `deliveries`, selections and `SOURCE_TEXT` for the pinned scalar source. The payload replay function is deliberately not loaded or invoked.
+
+The unmodified public joint candidate body is loaded with its base-bit root explicitly rebound. The unmodified current source471 preparation body is then executed, except for the source's historical input-list lines and terminal receipt lookup. Terminal records are reconstructed from pinned source root data and the pinned sink selectors. All input hashes are checked before importing source code.
+
+`raw_ledger.reconstruct(context)` recounts internal, copied-center, original-source and target paths, including actual compensation and terminal chronology. It pays all copied-center calls and independent exterior completions. It neither reads nor rescales a packed historical profile.
+
+The semantic fingerprint binds exact basis rows, original-source/root frames, operation frames, gauge order/read times, aliases, partner mix/delivery chains, retiming, terminal records, all borrowed and independent dirty roles, and target responses. Dynamically allocated frame numbers differ from older preparation code that registered unused historical frames; exact basis/chronology equality is the relevant comparison. No rank-only substitution is made.
+
+## Qualification and exclusions
+
+The one-time development comparison is `check_development_equivalence.py`. It intentionally loads the earlier audited preparation once to compare exact semantics. It is not a runtime dependency and is excluded from the minimal loader package. `equivalence_result.json`, `preparation_result.json`, `raw_ledger_result.json`, and test logs are qualification outputs, not proof inputs.
+
+The minimal loader package consists of source_loader.py, raw_ledger.py, test_public.py, SOURCE_PINS.json, expected_semantics.json, expected_ledger.json, README.md, NOTICE.md, LICENSE and MANIFEST.json, plus the supplied source inputs or equivalent externally supplied pinned roots. Do not include historical search code, whole archived checkouts, or qualification receipts in that runtime closure.
+
+A successful focused run establishes only the preparation/ledger scope above. A complete portable five-stage candidate still needs its global lowering, mathematical witness coverage, finite bills, conditional contract binding and exact assembly connected to these actual source objects.
+
+Assisted with ChatGPT.
diff --git a/research/five-stage-source527-banks/loader/SOURCE_PINS.json b/research/five-stage-source527-banks/loader/SOURCE_PINS.json
new file mode 100644
index 0000000000..c3e9ef5269
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/SOURCE_PINS.json
@@ -0,0 +1,37 @@
+{
+  "base_bit": {
+    "commit": "8d8d67bcf69c5ea67d3a29dbc64ba588156d6e8d",
+    "files": {
+      "LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+      "NOTICE": "c9850418622aea8d1382a230306c940947b5da818f4854b4b272ed60cf239f6c",
+      "research/paired-cube-diagonal-bit-168/bit/base_word.py": "9fe3a81b7561d2b09c086f63fac4626e88a32369ef8bc5579c94878de1218a89",
+      "research/paired-cube-diagonal-bit-168/bit/word.py": "85822e9cca5b1b2335c3704a3df2f3c6ef01e42d8328daacb33701d30cee80ab",
+      "research/paired-cube-diagonal-bit-168/references/pr168-v4/research/paired-cube-bit/check_paired_cube_bit.py": "4d7c86cd999a1519e0dfb6dcf33a8f26e442a353c3665a0f06881216a260fbf3",
+      "research/paired-cube-diagonal-bit-168/selected/bit/frames_p12.json.gz": "ad8e2705652b68476713d7647a8592505e222f2b3ccd1728dedce153aac6d462",
+      "research/paired-cube-diagonal-bit-168/selected/bit/graph_p12.json": "31a09a55920e64c8bfbac8b845dc01754c112aa6e207812e2fd1f4939d8f64cb",
+      "research/paired-cube-diagonal-bit-168/selected/bit/kchron_p12.json": "0fab548e84cf5ca7c04e8bd19b5810850463f36e0f92180efe73aeff17105e39",
+      "research/paired-cube-diagonal-bit-168/selected/bit/profile_p12.json": "c44ef8640cbcbcc9bac3a30e13373bd79efddde55d7cea514354ca2cdb60d66d",
+      "research/paired-cube-diagonal-bit-168/selected/bit/sinks.json": "c00ff70c564829d4b97740f92796fc6ce14fbcd5ca519663828650552b80249d",
+      "research/paired-cube-diagonal-bit-168/selected/bit/word_p12.json.gz": "1cb7e8ed45cf542e304fddc4b12247dbb248a0dc84dedbd832f336b2d172793a"
+    },
+    "repository": "CrocSwap/integer-mult-bounds"
+  },
+  "pr210": {
+    "commit": "df95878d11190518e45ef9717c9ee05011f88ace",
+    "files": {
+      "research/coordinated-crossover-pr200/LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+      "research/coordinated-crossover-pr200/NOTICE": "9fc5561c0995892dbdde777c713899486920c32d3562392f5f72377a4953bf98",
+      "research/coordinated-crossover-pr200/borrow/selection.json": "6932749407e14b199651f0a7deff02c6de74e2c1be0a1300263f1e164927fb8b",
+      "research/coordinated-crossover-pr200/frames/opframe-bases.json": "687496f3c15818e7cad1beb5fdf3812b8e38825ea302bfd60aea7ac23f3516b4",
+      "research/coordinated-crossover-pr200/gaugeb/selection.json": "b0f9f0e67f94a3c21069f393ea27f9483d315bff2f53f77270163065a6345d1c",
+      "research/coordinated-crossover-pr200/joint/joint-selection.json": "13f6420d6c341c22235c9d136d5f7c32896974bb165a893fecd7e583b790d133",
+      "research/coordinated-crossover-pr200/joint/joint_word.py": "de37cbed753fc6f13a7e5dc25c8346279cfee5f1868df8c12a5fcb6ca8f631c0",
+      "research/coordinated-crossover-pr200/joint/pairs.json": "da54b863601b5cbc21d27fc5cd623bf805f974a3e47b81ed29a85aa5bad0fa05",
+      "research/coordinated-crossover-pr200/newg/replay.py": "0cbec81a11e758156e9303f63dc8c617601da69523073fdf2a77394d76acf56a",
+      "research/coordinated-crossover-pr200/newg/schedule.json": "ab39d61fff7a2cfcb13e2c10d65f2c603e76a91013d197260d6c7b645ccb27a1",
+      "research/coordinated-crossover-pr200/newg/selection.json": "d94ef4e0b8ca98a7385b364cf245816a7cad08f850a60ce8b67aab4c3a89f3f4"
+    },
+    "repository": "eumemic/integer-mult-bounds"
+  },
+  "schema": "five-stage-public-source-loader/1"
+}
diff --git a/research/five-stage-source527-banks/loader/expected_ledger.json b/research/five-stage-source527-banks/loader/expected_ledger.json
new file mode 100644
index 0000000000..cb7130ed2f
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/expected_ledger.json
@@ -0,0 +1,142 @@
+{
+  "auxiliary_entrance_rank_histogram": {
+    "12": 18,
+    "13": 48,
+    "18": 13,
+    "20": 2200
+  },
+  "five_stage_profile": {
+    "W": 23683,
+    "calls": 495304,
+    "deficit": 4400,
+    "histogram": {
+      "1": 210470,
+      "10": 3855,
+      "100": 2200,
+      "11": 2275,
+      "12": 7535,
+      "13": 2890,
+      "14": 5460,
+      "15": 1495,
+      "16": 5070,
+      "17": 2435,
+      "18": 2340,
+      "19": 1930,
+      "2": 93420,
+      "20": 8580,
+      "21": 24270,
+      "22": 8765,
+      "23": 3520,
+      "3": 51045,
+      "4": 18105,
+      "46": 3520,
+      "5": 12420,
+      "50": 3520,
+      "6": 7580,
+      "60": 18,
+      "65": 48,
+      "7": 6220,
+      "8": 3255,
+      "9": 3050,
+      "90": 13
+    },
+    "idle_histogram": {
+      "23": 3520,
+      "4": 3520,
+      "46": 3520,
+      "50": 3520
+    },
+    "m": 120,
+    "maxchild": 100,
+    "rank_mass": 2837560
+  },
+  "one_stage_helper_histogram_including_copies": {
+    "1": 42094,
+    "10": 771,
+    "11": 455,
+    "12": 1507,
+    "13": 578,
+    "14": 1092,
+    "15": 299,
+    "16": 1014,
+    "17": 487,
+    "18": 468,
+    "19": 386,
+    "2": 18684,
+    "20": 1716,
+    "21": 4854,
+    "22": 1753,
+    "3": 10209,
+    "4": 2917,
+    "5": 2484,
+    "6": 1516,
+    "7": 1244,
+    "8": 651,
+    "9": 610
+  },
+  "paid_center_copy_histogram": {
+    "22": 24
+  },
+  "physical_internal_excluding_center_copies": {
+    "1": 35186,
+    "10": 771,
+    "11": 453,
+    "12": 1473,
+    "13": 530,
+    "14": 1073,
+    "15": 295,
+    "16": 1010,
+    "17": 480,
+    "18": 372,
+    "19": 274,
+    "2": 17625,
+    "21": 3787,
+    "22": 1320,
+    "3": 10176,
+    "4": 2893,
+    "5": 2466,
+    "6": 1506,
+    "7": 1220,
+    "8": 622,
+    "9": 605
+  },
+  "physical_source_histogram": {
+    "1": 2496,
+    "11": 2,
+    "12": 18,
+    "14": 2,
+    "15": 4,
+    "16": 1,
+    "17": 3,
+    "18": 67,
+    "2": 899,
+    "20": 880,
+    "21": 373,
+    "22": 409,
+    "3": 19,
+    "4": 20,
+    "5": 1,
+    "6": 1,
+    "8": 1
+  },
+  "physical_target_histogram": {
+    "1": 4412,
+    "12": 16,
+    "13": 48,
+    "14": 17,
+    "16": 3,
+    "17": 4,
+    "18": 29,
+    "19": 112,
+    "2": 160,
+    "20": 836,
+    "21": 694,
+    "3": 14,
+    "4": 4,
+    "5": 17,
+    "6": 9,
+    "7": 24,
+    "8": 28,
+    "9": 5
+  }
+}
diff --git a/research/five-stage-source527-banks/loader/expected_semantics.json b/research/five-stage-source527-banks/loader/expected_semantics.json
new file mode 100644
index 0000000000..6281d56ff4
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/expected_semantics.json
@@ -0,0 +1,39 @@
+{
+  "combined_sha256": "e51978b8c119a5ab690efe3faffa483865b8215e2100aa77f94dffadf79a71bf",
+  "formal_frame_id_equality_required": false,
+  "h": 24,
+  "hashes": {
+    "dirty_roles": "108c430514cc77b08b467e2a0e60ee845dac32a802e39304a1280a847bbfc119",
+    "donor_recipient_pairs": "5ad9b750daa9c4482f66f223111914f62ec6b7f3c47f4a922a1679d6adfd9dc4",
+    "full_basis": "f30f79ae778a658bfaa437e05d0e08306d5ef27ed1b268fac4cbc16177830387",
+    "gauge_read_order": "7f6fcb15a0b2207ebb43e39bb9bff59b7614779e34057be802da58e9c21cfdb3",
+    "gauge_readtimes": "b71ffaea3099cc5cef8604d90f82532adeb998b7c53084d255b41b5c1302f909",
+    "gauge_records": "05b6426ec2c9b8458919e0c21fb368cbb562015590d72a4a283b5f95b47cfa2c",
+    "gauge_selection": "4f17d9727532aee750b79f035807e2b0ed81cbce56ae29eb6e22bb491533fe3a",
+    "graph": "8622ea91c133bfaa23f31ff78fc0b24bf075864865918100e2adab26d7c9a032",
+    "live_gauge_reads": "b6a3e4016f1fb5b9e7f843597e7e9b565cadba5865f70eb33b3fd3fa358adc6d",
+    "operation_bases": "8b170d4f1aeee1325e4dcfc6f077e28a8ce9eab657a6f6695cc38fb766aafa16",
+    "operations": "6bbf0eb766e39d20cee98b2f91002200b29ba9073d7dad0872ab955835442f4f",
+    "partner_chronology": "93fbe42b76bdde82b178b6471cf0937fbd5d820f5656b2c53fb04076d57589e5",
+    "partner_lookup": "e1b35ae1e94a59f90e8ce5a719459150a2e8989709ad045c5f7812395a042660",
+    "phase1": "f67fc46eca63076db3247791692787b102d4b06d2d1d690a3b0be0808e789dfc",
+    "physical_role_map": "2c6dcdec5420adec6ca5575d6837f062211eb0b856b8fa668543e395908f096c",
+    "rest": "03e2888758f0a95abfab5582bf77d435c484c8f4d1737eeb7a8ab9752fccda7a",
+    "root_bases": "747757d16a1e9b5112a296bff94754ac43e047126849b50599be3ded16d1e36d",
+    "schedule": "73294d81e5d32745f6df91ebe874153a5791e3c774eae6f833b4ca919f255e67",
+    "source_bases": "ff2b49b40d245a593561561457ec9ce2b16b739b2e4c277ce96898dba42c2c62",
+    "source_loans": "8d42978f946bd21279dc7c84f8f360974702e3d8e63e7278a484ff40805ab216",
+    "source_selection": "bd9e2b029487ba5b9a794c7a1721c92b8e9c18fc39d8903d3c2df0dbf5ba28d8",
+    "target_response": "f7302a72deaebd599cfc671eab4690a6866c547cefd41dd71f313a10e27f4aaf",
+    "terminal": "c8edf8b469bc4886f8af4ad7535cb8f35452fd3ff2e3236e93aa6b44877d359d"
+  },
+  "independent_dirty_roles": 16643,
+  "local_independent_columns": 20163,
+  "logical_roles": 18908,
+  "retimed_operations": 168,
+  "schema": "source471-exact-semantic-fingerprint/1",
+  "source_head": "df95878d11190518e45ef9717c9ee05011f88ace",
+  "source_loans": 471,
+  "terminal_sinks": 34,
+  "v": 1760
+}
diff --git a/research/five-stage-source527-banks/loader/raw_ledger.py b/research/five-stage-source527-banks/loader/raw_ledger.py
new file mode 100644
index 0000000000..c5e44b07fe
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/raw_ledger.py
@@ -0,0 +1,127 @@
+#!/usr/bin/env python3
+"""Fresh raw source471 path ledger from the clean public loader, no receipts.
+
+Reconstructs actual retimed source, target, auxiliary and copied-center paths.
+No packed histogram, historical search, scalar replay or network input is used.
+Assisted with ChatGPT. Retain supplied source licences and notices.
+"""
+from collections import Counter, defaultdict
+from pathlib import Path
+import argparse, hashlib, json, sys
+import source_loader
+sys.dont_write_bytecode = True
+if not __debug__: raise SystemExit('refusing optimized Python')
+
+
+def clean(hist):
+    return {str(k):v for k,v in sorted(hist.items()) if v}
+
+
+def reconstruct(context):
+    w, c = context['W'], context['C']
+    kp = {e['passive']:e for e in w.k['entries']}
+    borrow = context['borrow']; omitted = context['omitted']
+    recipient = {d:b for b,d in w.pairs}
+    roots = defaultdict(list)
+    for j, role in enumerate(w.w['rootroles']):
+        roots[role].append(w.w['root_frame'][j])
+    def chain(role):
+        out = [w.opframe[i] for i in w.role_ops[role] if i not in omitted]+roots[role]
+        if role in recipient:
+            r=recipient[role]
+            out += [w.gauge[r]['frame']]+[w.opframe[i] for i in w.role_ops[r] if i not in omitted]+roots[r]
+        return out+[w.w['full_frame']]
+    inclusion_cache={}
+    def included(a,b):
+        if (a,b) not in inclusion_cache: inclusion_cache[a,b]=c.sub(a,b)
+        return inclusion_cache[a,b]
+    def hist(start,path):
+        prev=start;rank=0 if start is None else c.dimf[start];result=Counter()
+        for frame in path:
+            assert prev is None or included(prev,frame), ('nonnested actual path',prev,frame)
+            delta=c.dimf[frame]-rank;assert delta>=0
+            if delta:result[delta]+=1
+            prev=frame;rank=c.dimf[frame]
+        return result
+    start={r:w.w['source_frame'][n] for n,r in w.source.items()}
+    start.update({r:g['frame'] for r,g in w.gauge.items()})
+    internal=Counter()
+    for r in context['regs']:
+        hh=hist(start.get(r),chain(r))
+        if r in w.source.values(): hh[1]+=1
+        internal.update(hh)
+    copies=Counter(c.dimf[w.w['root_frame'][j]] for j,r in enumerate(w.g['roots']) if r['kind']=='center')
+    by_source={r['source']:r for r in context['selection']+context['gauge_selection']}
+    source=Counter();source_paths=[]
+    for n,e in kp.items():
+        source.update(hist(e['carrier_chain'][0],e['carrier_chain'][1:]))
+        if n not in by_source:
+            source.update(hist(e['passive_chain'][0],e['passive_chain'][1:]));continue
+        r=by_source[n];role=r['role'];path=chain(role)
+        if 'early' in r:
+            assert all(i>=r['early'] for i in w.role_ops[role] if i not in omitted)
+            path=[e['mix_frame']]+path
+        else:
+            f=w.register(r['gauge_basis']) if 'gauge_basis' in r else w.gauge[role]['frame']
+            path=[e['mix_frame'],f]+path
+        hh=hist(w.w['source_frame'][n],path);source.update(hh)
+        source_paths.append(dict(source=n,role=role,rank_path=[1]+[c.dimf[f] for f in path],histogram=clean(hh)))
+    # Exact target chronology from the source preparation, including compensated
+    # gauge reads, terminal writes/posts, side-root reads and partner deliveries.
+    target=[[] for _ in range(w.v)]
+    def read(t,frame): target[t].append(frame)
+    for j,i in enumerate(w.rest):
+        for role in context['at'][j]:
+            for t in context['adj'][role]: read(t,w.gauge[role]['frame'])
+        if i in context['bywrite']:
+            e=context['bywrite'][i];read(e['pivot'],w.opframe[i])
+        if i in context['afterwrite']:
+            e=context['afterwrite'][i]
+            for t in e['targets']:read(t,e['root_frame'])
+    for role in context['at'][len(w.rest)]:
+        for t in context['adj'][role]:read(t,w.gauge[role]['frame'])
+    for j,r in enumerate(w.g['roots']):
+        if r['kind']=='side' and j not in context['deletedroots']:
+            for t in r['targets']:read(t,w.w['root_frame'][j])
+        for e in context['deliveries'][j]:
+            for t in e['receivers']:read(t,e['deliver_frame'])
+    targets=Counter()
+    for t,path in enumerate(target):
+        assert path
+        assert all(all(w.module.dot(c.cov[t],row)==0 for row in c.B[f]) for f in set(path))
+        targets.update(hist(None,path));gap=w.h-1-c.dimf[path[-1]];assert gap>=0
+        if gap:targets[gap]+=1
+    entrances=Counter(g['dim'] for r,g in w.gauge.items() if r not in w.donor and r not in borrow)
+    helper=internal+copies+source+targets
+    R=len(context['regs']);v=w.v;h=w.h
+    assert copies=={22:24} and entrances=={20:2200,18:13,12:18,13:48}
+    assert len(source_paths)==len(borrow)==471
+    assert sum(r*n for r,n in source.items())==sum(r*n for r,n in targets.items())==v*(h-1)
+    assert sum(r*n for r,n in helper.items())==h*R+2*v*(h-1)+sum(r*n for r,n in copies.items())-sum(r*n for r,n in entrances.items())
+    idle=Counter({r:2*v for r in (2*h-2,h-1,2*h+2,4)})
+    five=Counter({r:5*n for r,n in helper.items()});five.update(idle)
+    for a,n in entrances.items():five[5*a]+=n
+    W=4*v+R;mass=sum(r*n for r,n in five.items())
+    assert (sum(helper.values()),sum(five.values()),mass,5*h*W-mass,max(five))==(95789,495304,2837560,4400,100)
+    return dict(schema='source471-clean-raw-ledger/1',source_head='df95878d11190518e45ef9717c9ee05011f88ace',
+                h=h,v=v,physical_R=R,source_aliases=len(borrow),
+                terminal_sinks=len(context['terminal']),physical_donor_recipient_identifications=len(w.pairs),
+                physical_internal_excluding_center_copies=clean(internal),
+                paid_center_copy_histogram=clean(copies),physical_source_histogram=clean(source),
+                physical_target_histogram=clean(targets),one_stage_helper_histogram_including_copies=clean(helper),
+                auxiliary_entrance_rank_histogram=clean(entrances),
+                auxiliary_entrance_count=sum(entrances.values()),helper_rank_mass=sum(r*n for r,n in helper.items()),
+                five_stage_profile=dict(m=5*h,W=W,histogram=clean(five),idle_histogram=clean(idle),calls=sum(five.values()),rank_mass=mass,deficit=5*h*W-mass,maxchild=max(five)),
+                exact_nested_path_pairs_checked=len(inclusion_cache),
+                source_paths=source_paths,
+                scope='Fresh exact retimed physical-path/rank ledger; no scalar all-column replay or five-stage global-word admission.')
+
+
+def main():
+    ap=argparse.ArgumentParser(description=__doc__)
+    ap.add_argument('--pr210-root',type=Path,required=True)
+    ap.add_argument('--base-bit-root',type=Path,required=True)
+    args=ap.parse_args()
+    print(json.dumps(reconstruct(source_loader.prepare(args.pr210_root,args.base_bit_root)),sort_keys=True,indent=2))
+
+if __name__=='__main__':main()
diff --git a/research/five-stage-source527-banks/loader/source_inputs/base_bit/LICENSE b/research/five-stage-source527-banks/loader/source_inputs/base_bit/LICENSE
new file mode 100644
index 0000000000..261eeb9e9f
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/source_inputs/base_bit/LICENSE
@@ -0,0 +1,201 @@
+                                 Apache License
+                           Version 2.0, January 2004
+                        http://www.apache.org/licenses/
+
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+
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+   Licensed under the Apache License, Version 2.0 (the "License");
+   you may not use this file except in compliance with the License.
+   You may obtain a copy of the License at
+
+       http://www.apache.org/licenses/LICENSE-2.0
+
+   Unless required by applicable law or agreed to in writing, software
+   distributed under the License is distributed on an "AS IS" BASIS,
+   WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+   See the License for the specific language governing permissions and
+   limitations under the License.
diff --git a/research/five-stage-source527-banks/loader/source_inputs/base_bit/NOTICE b/research/five-stage-source527-banks/loader/source_inputs/base_bit/NOTICE
new file mode 100644
index 0000000000..ba5ea80dd7
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/source_inputs/base_bit/NOTICE
@@ -0,0 +1,399 @@
+A sharper exponent for integer multiplication
+Copyright 2026 Douglas Colkitt
+
+This project includes unmodified source files from OpenAI's manuscript
+"Integer multiplication below n log n" and its accompanying repository:
+https://github.com/openai/math
+Commit: adc7f1241b42e322a6451854ab7e4b4c146bf78a
+
+The original manuscript identifies OpenAI as its author. Its Apache-2.0
+license is retained at upstream/LICENSE. The exact source URLs and hashes
+are recorded in upstream/manifest.json. Proposed modifications are supplied
+as separate files under patches/; upstream source files remain unchanged.
+
+The original local research note, scripts, certificates, patches, and
+documentation were prepared by Douglas Colkitt with assistance from OpenAI
+Codex. The community release incorporates work by the contributors identified
+in the notices below; it is not solely Douglas Colkitt's work.
+This project is not an official OpenAI release or an OpenAI endorsement.
+
+Related contributions (not imported in this checkpoint):
+Zhihao Chen (jacklightChen), PR #7, supplied an earlier ternary five-subset
+motif, finite-alphabet transfer and a different compressed producer:
+https://github.com/CrocSwap/integer-mult-bounds/pull/7
+The present checkpoint claims no priority for that motif. Related compressed
+complex-network work was submitted by eumemic in PR #3:
+https://github.com/CrocSwap/integer-mult-bounds/pull/3
+The pending contributions and their stronger bounds require their own review;
+this notice does not represent adoption or verification of their claims.
+
+Additional community work, including parallel and superseded submissions, is
+acknowledged in CONTRIBUTORS.md, with pinned submission metadata in
+docs/research/contribution-snapshot.json. These acknowledgements do not imply
+joint authorship of the checkpoint, incorporation of pending code, or verification
+of submitted mathematical claims. Original source-specific notices remain in force.
+
+Community release (accepted by conditional maintainer audit):
+The following original contributor notices apply to the imported PR #39 chain.
+The preceding non-import statements describe the preserved 2^-30 checkpoint.
+
+Ternary five-subset construction and common-four-point resynthesis
+Copyright 2026 Zhihao Chen (jacklightChen)
+
+This is Zhihao Chen's first submission of this specific construction to this
+repository. Research, proof development, code and verification were prepared
+with substantial assistance from GPT-6 Astra (OpenAI Codex), as identified by
+the contributor. This is not independent review or an OpenAI endorsement.
+Future research building on this contribution should explicitly acknowledge
+Zhihao Chen (jacklightChen), cite the accompanying note and pull request, and
+retain the preceding contributors' attribution. This notice records research
+provenance; the repository's Apache-2.0 license remains unchanged.
+
+Dependencies retained from eumemic's Claude-assisted PRs #3 and #5:
+https://github.com/CrocSwap/integer-mult-bounds/pull/3
+https://github.com/CrocSwap/integer-mult-bounds/pull/5
+PR #5 commit: d3d370c34937503bd3fa63c83d505a05e8e50893
+These include complex_circuit.py, complex_network.py, fast_gaussian.py,
+their generators, construction notes and certificates. The Gaussian input
+scaling proof is corrected in this derivative; see the modification notice below.
+Aligned pairing also appears in Bortlesboat's PR #2. Retained totals and bank
+sharing appear in dleen's PR #4 and cheaper retained centers in eumemic's PR #6.
+Their results and ideas are credited separately from this new construction.
+
+Bulk recursion and controlled tensor basis, 8 October 2026
+Copyright 2026 icekylinx
+
+The new projector batching, controlled simultaneous basis, mixed-width tape
+recursions, whole-residual complex recursion, and dependency-path precision
+guard in the batched-23 proof package are contributed by icekylinx, developed
+with substantial OpenAI GPT-6 Astra assistance. Review corrections and
+repository integration were prepared with OpenAI Codex assistance. They build on the retained
+PR #7 finite producers and the preceding contributions credited above.
+The new package retains the repository's Apache-2.0 license and the original
+authors' attribution. It is not an official OpenAI release or endorsement.
+
+Modification notice, 8 October 2026 (icekylinx): the retained fast Gaussian
+proof now bounds shifted approximation error by F=2^ceil(1.14 alpha^2).
+The P=34p precision budget and asymptotic exponent are unchanged. New batching
+checks are integrated into make verify. Existing attribution and licenses
+are retained.
+
+Partial-swap extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The partial-swap framing, binary retained-total triple construction,
+dependency-preserving positive labels and carrier reuse, unequal (25,23,57)
+common basis, improved complex batching, and exact assembly are contributed
+by icekylinx with substantial OpenAI GPT-6 Astra and Codex assistance.
+This extension directly inherits PR #10, commit
+62691e395a0458ce089a1c7b5d89e74291e95e29. The portable producer adapters
+modify the retained paired-exclusion and shared-point implementations credited
+above. The original licenses, attribution, and Gaussian correction are retained.
+
+Endpoint-gauge extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The complementary auxiliary endpoint gauges, (32,30,40) controlled basis,
+general nondegenerate binary phase residuals, retained-center complex producer,
+updated path guard and exact assembly are contributed by icekylinx with
+substantial OpenAI GPT-6 Astra and Codex assistance. This extension inherits
+f2ab41aebad47861caf6316282c1793e5513845e and its PR #10 ancestry. New portable
+adapters retain the paired-exclusion and shared-point contributions credited
+above. Original licenses, notices and the Gaussian correction are retained.
+
+Structured-basis and semantic-bulk extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The selected (32,30,36) bit construction, A1/A5 block refinements, fixed
+middle-axis I+J profiles with bounded-minor CRT certificates, mixed dyadic
+complex centers, complete data-front batching and circuit-specific assembly
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex
+assistance. The immediate parent is ed90fd940279c336ebc45968631bdebfb087b505.
+
+The semantic/bulk compatibility argument, literal scalar charge, product row
+stock and parameterization adapt Zhihao Chen (jacklightChen)'s PR #23,
+commit 661ebabc1076c9f525d7ce4967c876b203fa9827. Translated complex endpoints
+are adopted from the same contributor's PR #21,
+commit 5ba6cf0bfb68f2be8d15610e7972207c50254d6a, following eumemic's source-frame
+work and the earlier nested-basis contributions.
+
+Completed-child semantic precision, arbitrary-coordinate routing, phase-cell
+Gaussian inversion and bulk tape transfer are credited to the RaD project
+(hipotures), commit 45d9b60355872041f9275f77c057514def0d45bc, exposed through
+PR #20 at 4f8d6c8272b5ff307a0da51df545ec3cd96a8b6e. Their selected proof
+sources retain their separate CC0 license and original NOTICE under
+references/semantic-bulk/rad20. The PR #23 source retains its recorded OpenAI
+Codex assistance; RaD retains its recorded Codex and GPT-6.1-Sol Ultra
+assistance. These are source attributions, not claims about this session's
+runtime model. Related contributions by Rohan Arun and Aurel Prosz remain
+acknowledged in the imported notices. No exclusive priority or additional
+license condition is asserted.
+
+Copied retained-center extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The copied retained-center schedule in both orientations, its complex-phase
+transfer, the Wm-N+L rank budget, and the selected (25,23)/(28,28) composition
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex
+assistance. This extension inherits PR #32 at
+0ef3aeb61f55cc0b321ce6a0ef00acee25cefe52 and the preceding PR #10/#18/#24 work.
+
+The two-stage topology and paid copied-stream endpoint correction are due
+to Aurel Prosz (Paureel), c82d09eb4781b68435e3fa3eb6beea72fa900fab,
+through Zhihao Chen (jacklightChen)'s PR #29 unequal-axis composition,
+9d963275075fa98f1da821e27757b238dafd6b3c. That source also credits Swapnil
+Jain's two-stage development at ae405eb474d1486b2d8aef90139869f927f4836a.
+
+The contiguous corner method, incidence factorization and exact partition
+certificates are due to Rohan Arun (PR #31), method commit
+02f68f95369c39955c5ffcfddbd75cd83e26b305. The selected (25,23) certificate
+specializes Dominik Scholz's parameterization (PR #33),
+a499a345c040aa15b68e984a0904eadc485c7a7e. Their source notices and proof
+snapshots are preserved in references/copied-centers. Historical OpenAI
+Codex, GPT-6 Astra and Anthropic Claude Opus 5.5 assistance is credited to
+the respective imported sources. Existing PR #21/#23 and RaD attribution,
+licenses and the Gaussian correction remain in force.
+
+Copied centers with reversed (23,25) data corners, 2026-10-08
+Rohan Arun with substantial OpenAI Codex assistance supplies the composition,
+exact dimension23 controls and cost/assembly integration. James Chang's PR34
+supplies the reversed boundary family; icekylinx's PR36 supplies copied retained
+centers and the selected complex graph. The balanced assembly remains attributed
+to RaD/hipotures and its PR34 specialization. Retain Dominik Scholz PR33, Rohan
+Arun PR31, Zhihao Chen PR29/21/23, Paureel/Aurel Prosz, Swapnil Jain, icekylinx,
+eumemic, Douglas Colkitt, OpenAI and all earlier notices and hypotheses.
+This is not a claim of formal verification, new foundational rank identities,
+global optimality or measured runtime speedup.
+
+Fixed middle basis with copied reversed corners (2026-10-08):
+Rohan Arun, with substantial OpenAI Codex assistance. Composition and
+simultaneous-basis proof retain icekylinx PR32/36, James Chang PR34,
+Dominik Scholz PR35/38, Zhihao Chen, Paureel, Swapnil Jain, RaD/hipotures,
+eumemic and all preceding attribution and licenses. See research/copied-fixed-reversed.
+
+Changed-graph composition with copied centers and fixed bases, 8 October 2026
+Prepared with OpenAI Codex assistance.
+The new composition uses icekylinx PR36 copied centers and PR32 fixed-I+J
+projectors/profiler, Dominik Scholz PR35 two fixed factors and CRT argument,
+Zhihao Chen (jacklightChen) PR29/23 interfaces, Aurel Prosz (Paureel) and
+Swapnil Jain's two-stage development, Rohan Arun PR25/31 corner methods,
+Dominik Scholz PR33/38 dimension/composition work; James Chang (jamesyc)
+PR34 reversed geometry and balanced transfer; Rohan Arun PR37/39/40
+reversed integration, modular data tests and both-fixed baseline;
+RaD (hipotures) PR41 changed scalar DAGs, alternating point order and
+independent role compiler; RaD balanced routing/bulk interfaces, and all
+retained predecessors.
+Complete attribution and the conditional proof boundary are recorded in
+research/copied-fixed/PROOF.md; imported originals and source pins are retained.
+Original Apache-2.0 and CC0 notices remain applicable.
+
+Community follow-up integration (PRs #49, #45 and #26), 8 October 2026:
+Rohan Arun contributes weighted carrier matching and climbed summand/leave-one-out
+orders (PRs #44/#47/#49), with Anthropic Claude assistance. The final construction
+composes Chafik Boukhalfa's changed-graph checks, exact rational data recovery and
+reordered exclusion sums (PRs #43/#46/#48, OpenAI Codex assistance) with RaD /
+hipotures's alternating pair order and independent physical role compiler (#41).
+All predecessor attribution and conditional proof interfaces remain in force.
+
+Alejandro Zarzuelo Urdiales contributes Gaussian parity and canonical arithmetic
+proofs, finite tensor execution, profile precision accounting and mixed-center
+reference checks (#45), with substantial OpenAI Codex assistance. See
+research/gaussian-parity-synthesis/PROVENANCE.md for his full attribution.
+Ryan S / princezuda contributes historical Lean certificate arithmetic, algebraic
+frame/movement/guard lemmas and independent paired-circuit checks (#26). Original
+source notices and historical snapshots are preserved. Neither formal package
+claims a full formal proof of the multiplication algorithm or a new kappa.
+
+Joint frame compilation of skip-prefix strips
+
+The local compiler and exact finite witnesses were prepared by eumemic with
+substantial OpenAI Codex assistance. Avi Eisenberg / ikeboy with Anthropic
+Claude assistance supplied the unmodified PR #53 skip-prefix producer.
+The pinned PR #48 pipeline is Chafik Boukhalfa's contribution, incorporating
+Rohan Arun, RaD / hipotures, icekylinx, Dominik Scholz, James Chang,
+Zhihao Chen, Aurel Prosz / Paureel, Swapnil Jain, eumemic, Douglas Colkitt,
+OpenAI, David Harvey, Joris van der Hoeven and the earlier contributors
+credited in the retained sources. See references/frame-compiler/pr48/NOTICE
+and references/frame-compiler/pr53/NOTICE for their original attribution.
+No authorship or review by those predecessors is claimed for this compiler.
+
+Joint dual-skip/frame-compiler composition (2026-10-08)
+---------------------------------------------------
+Rohan Gupta / gupt1156 with Anthropic Claude assistance: PR55 dual-skip
+scalar graph; original LICENSE and NOTICE are retained under
+references/frame-compiler/pr55. eumemic with substantial OpenAI Codex
+assistance: PR57 joint reversible frame compiler, serialized physical-word
+replay and profile reconstruction. Chafik Boukhalfa with OpenAI Codex
+assistance: this composition, independent integration and exact certificate.
+Avi Eisenberg / ikeboy, Rohan Arun, RaD / hipotures, icekylinx, and all
+predecessor authors retain the notices above and in pinned source files.
+No original copyright or Apache-2.0 notice is removed.
+
+Ranked reclamation after the PR58 joint-dual composition (2026-10-08)
+----------------------------------------------------------------
+Chafik Boukhalfa with OpenAI Codex assistance: prioritize retired physical
+slots by descending current frame rank, retaining slot ID as the tie-breaker;
+complete replay, exact profiles, comparison and adversarial controls.
+The dedicated joint_dual_reclaim_compiler.py is derived from eumemic's PR57
+compiler. Its original shared compiler remains byte-identical. The PR55
+producer and all preceding attribution remain unchanged. The complete PR58
+comparison certificate and its original source/proof provenance are retained
+under references/frame-compiler/pr58.
+
+Skip-prefix strip producer, 8 October 2026
+By Avi Eisenberg (ikeboy), with Anthropic Claude (Claude Opus 5.5)
+assistance. The skip-prefix leave-one-out strip layout in
+research/skip-strips/skip_graph.py is new. Everything else reuses Chafik
+Boukhalfa's PR #43/#46/#48 research/copied-fixed pipeline, its verification
+code, exact data recovery and support-order sorting, together with:
+- Rohan Arun's PR #44 weighted carrier matching optimizer;
+- RaD (hipotures) PR #41 alternating point order;
+- every predecessor credited above.
+Complete attribution and the conditional proof boundary are recorded in
+research/skip-strips/PROOF.md. Original Apache-2.0 and CC0 notices remain
+applicable.
+
+Local split-pair skip-prefix research continuation (2026-10-08)
+Prepared with substantial OpenAI Codex assistance at the user's request.
+New contribution: singleton/pair recursive partition refinement, finite summand
+order and moment search, and verified composition with inherited paid clones.
+Avi Eisenberg/ikeboy with Anthropic Claude assistance: PR53 skip-prefix strips.
+Chafik Boukhalfa with OpenAI Codex assistance: PR54 original-envelope clone
+specialization and PR43/46/48 checking interfaces. RaD/hipotures: PR51
+whole-chain/source-partition cloning and earlier producer/interfaces.
+Rohan Arun: PR44 weighted matching and earlier fixed/reversed work, with all
+stated historical AI assistance. All predecessor notices remain applicable.
+
+Original-envelope whole-chain clones on skip-prefix strips (research/skip-clones):
+Chafik Boukhalfa, prepared with OpenAI Codex assistance. The complete-chain
+source-partition clone operation is adapted from RaD / hipotures, PR #51,
+with original envelopes retained here. The scalar base is Avi Eisenberg's
+PR #53 skip-prefix layout, prepared with Anthropic Claude assistance.
+All credited PR #43/#46/#48, PR #44 and earlier code/proof notices remain
+in force. This extension is provided under the repository's Apache-2.0 terms.
+
+Community review checkpoint #50–#61, 8 October 2026
+Maintainer integration and independent arithmetic: Douglas Colkitt, with OpenAI Codex.
+Avi Eisenberg / ikeboy: skip-prefix strip framework (#53), with Anthropic Claude.
+Rohan Gupta / gupt1156: parallel order search (#50, stated Antigravity assistance)
+and dual-suffix strips (#55, Anthropic Claude assistance).
+Eumemic: equal-frame joint compilation and paid signal reclamation (#57), with Codex.
+Chafik Boukhalfa: original-envelope paid clones (#54), composition (#58) and ranked
+reclamation (#60), with Codex. Rohan Arun: order search (#52) and positive-frame
+composition (#56), with Anthropic Claude and stated Codex validation assistance.
+RaD / hipotures: enlarged frames, source partitions and paid whole-chain clones
+(#51), with Codex. Rohan Garg / rohangar1: singleton/pair recursion and paid-clone
+composition (#59), with Codex. Alejandro Zarzuelo Urdiales: exact parameter
+refinement, scoped Lean arithmetic, matrix/parity and search tools (#61), with
+his disclosed multi-tool and Codex assistance.
+Parallel #50/#52 records are reviewed and credited without replacing the earlier
+climbed-48 implementation; their exact commit history is retained. The separate
+#51 witness is retained as research; its source-graph/negative-basis data proof
+has not received the same full replay as #56's consumed compiler machinery.
+See docs/research/community-round2-review.md for precise validation boundaries.
+All earlier licenses, contributions and notices remain applicable.
+
+Interval strips and core-aware pair assembly
+
+The interval leave-one-out strips and the core-aware group-pair assembly in
+research/pair-assembly were prepared by Avi Eisenberg / ikeboy with Anthropic
+Claude assistance. They build on PR #53's skip-prefix pipeline and its
+inherited PR #48 sources. The stacked witness in research/pair-assembly/frame
+uses eumemic's PR #57 joint frame compiler unchanged; its driver and verifier
+are adaptations of PR #57's skip_frame_compiler.py, skip_frame_compose.py and
+verify_skip_frame.py. Original notices are retained.
+
+Stopped product-ring extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The opposite-bank generic projector factorization, componentwise atom-ring
+adapters, stopped recurrence and outer ordinary-interchange conversion,
+aligned pair-star producer with rational centers, exact denominator-21
+grid and selected (23,23)/(24,24) composition are contributed by icekylinx
+with substantial OpenAI GPT-6 Astra and Codex assistance. This extension
+builds on merged PR #36, 11817ccacb564bb7f98789c20dc11d3fece207e3, and is
+integrated on upstream main 56b66d58297deca1d7dd130247d720e960f77a37.
+
+The retained ordinary leaf uses the preceding projector, corner and
+copied-center proofs. The selected two-stage topology retains Aurel Prosz
+(Paureel), Swapnil Jain and Zhihao Chen's credits; the semantic assembly
+retains PR #21/#23 and RaD's attribution. Paired producers, carrier matching,
+original source notices and each imported license remain unchanged.
+
+Partial-gauge extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The pair-first cube assembly, compatible backward frame enlargement,
+partial source-gauge schedule and stopped whole-projector composition are
+contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex
+assistance, extending PR #104 at 948ce1510df750f4c18b96bdaef436a86f8bf834.
+
+The physical deferred bit word, reflected event accounting, fan-order
+clarification and signed two-stage endpoint are adopted from Zhihao Chen
+(jacklightChen), PR #97, e15350b66e4bf796153a68c240a6cc1b788abe3e, with
+Codex assistance. Its deferred readouts, V leaves, lifted frames and frozen
+witness originate in Swapnil Jain's 741e7aa078392553815df7926ee17ac5e25a8c38,
+with the original Claude disclosure retained. Cyclic interval and core-aware
+pair primitives build on Avi Eisenberg (ikeboy), PR #62,
+ad0f25ff7b23cff7f08ad237c2254e6ecf74257e, with Anthropic Claude assistance.
+Prior two-stage, copied-center, ordinary-leaf, semantic/bulk, routing and
+precision credits remain in force. Imported source notices and licenses
+are preserved under references/partial-gauge/; see SOURCES.json.
+
+Three-stage cover extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The arbitrary-subspace Clifford frame theorem, three-stage Cayley geometry,
+weighted q-adic GL cover, unit-lower weighted compiler, charged local-ring
+fallback, uniform batching, internal row borrowing and exact composition
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra assistance. This
+repository integration uses Codex assistance and extends
+PR #115, cf60442983c153a3a0f5272ea4713f99eba13c86.
+
+The selected local complex DAG and support/label replay are adopted from
+Copyright 2026 eumemic, PR #117, cbb05ce504d571546d9b7794c186a613c659c3bf,
+with its original Apache-2.0 terms and Claude assistance disclosure.
+The unchanged bit physical word retains Zhihao Chen (jacklightChen),
+PR #97, and Swapnil Jain's underlying witness and assistance notices.
+The prior opposite-bank, stopped ordinary-leaf, copied-center, topology,
+semantic/bulk, precision and analytic credits remain applicable.
+Original source notices and license bytes are preserved in
+references/three-stage-cover/pr117/ and references/partial-gauge/.
+
+Paired-cube and shared-core extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The paired-coordinate triple-cube identity, signed overlap channels,
+original-source involution, coordinate-star centers, concrete frame/carrier
+compiler, chronological source gauges, PR97 gauge-subset selection,
+orthogonal cross-stage routing and exact supplier/assembly integration
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra assistance.
+Repository integration uses Codex assistance and extends PR #130 at
+6a9970a530119174507904e23592fd59ede19a5d.
+
+The substantial completed-core workspace-sharing principle is credited to
+an664, PR #128, 530588a019b4a74f09180680c9e3961bf649ec89, with its reported
+OpenAI Codex assistance. No source code, partition or serialized witness
+from that PR is imported. The new routing and paired-motif integration
+are described separately and claim no exclusive priority for broad sharing,
+paired-coordinate, Fourier or carrier-matching ideas.
+
+Selected coarse queries restrict the retained eumemic PR #117 positive DAG;
+its original Apache-2.0 and Claude assistance notices remain unchanged.
+The bit word and original frames retain Zhihao Chen's PR #97 and Swapnil
+Jain's frozen witness, notices and historical assistance disclosures.
+Earlier generalized Clifford, three-stage, uniform weighted recursion,
+ordinary-leaf, semantic/bulk, precision and analytic credits remain in force.
+Source-specific license bytes are preserved under references/.
+
+Source-assisted common-frame extension (2026-10-09)
+Copyright 2026 icekylinx. Apache-2.0.
+Research construction and local certificates: OpenAI GPT-6 Astra assistance.
+Repository integration and independent complex scalar replay: Codex assistance.
+Substantial dependency: eumemic PR168, fd25adb7fbaa12ee761d02c733c54d1d2a7687ee;
+its byte-preserved sources, licenses and original Claude assistance disclosure
+are retained in research/source-assisted/public/pr168_fd25adb7.tar.gz.
+Additional inherited contributions are recorded in research/source-assisted/CREDITS.md.
+The new work concerns source-assisted invertible common-frame lifts, controlled
+affine cleanup and compatible zero-fresh reuse; no exclusive claim is made for
+broad register reuse, paired cubes, frame theory or ordinary linear elimination.
diff --git a/research/five-stage-source527-banks/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/bit/base_word.py b/research/five-stage-source527-banks/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/bit/base_word.py
new file mode 100644
index 0000000000..044f278fbe
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/bit/base_word.py
@@ -0,0 +1,202 @@
+#!/usr/bin/env python3
+"""Reusable independent formal checks for a physical partner-pair bit word.
+
+The supplied frozen inputs are source-pinned. This never runs their producer.
+The Z check is of the defining integer decoder; only its F2 reduction is I.
+Prepared with OpenAI Codex assistance; Apache-2.0.
+"""
+from collections import Counter, defaultdict
+import importlib.util
+import json
+from pathlib import Path
+import sys
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[2]
+BASE = HERE.parent / 'references/pr168-v4/research/paired-cube-bit'
+P = 12
+if sys.flags.optimize: raise SystemExit('refusing -O')
+
+
+def need(ok, msg):
+    if not ok:
+        raise ValueError(msg)
+
+class Candidate:
+    def exact_frames(self):
+        C = self.C
+        C.decoder(); C.geometry()
+        for i in self.changed_frames:
+            f = self.opframe[i]; x = self.ops[i][2]; bits = C.sup[x]
+            need(C.nondeg(f), 'replacement operation frame nondegenerate')
+            while bits:
+                low = bits&-bits; s=low.bit_length()-1; bits-=low
+                need(C.in_frame(C.chi[s],f), 'value span in replacement operation frame')
+        for b, d in self.pairs:
+            need(C.sub(self.endframe[d], self.gauge[b]['frame']), 'exact Q donor-to-birth containment')
+            need(C.nondeg(self.endframe[d]) and C.nondeg(self.gauge[b]['frame']), 'handoff endpoints nondegenerate')
+
+    def register(self, rows):
+        C = self.C
+        B,_ = self.module.reduce_rows(rows,self.h if hasattr(self,'h') else C.h)
+        B = tuple(sorted(map(tuple,B),key=lambda r:next(i for i,x in enumerate(r) if x)))
+        if not hasattr(self,'frame_ids'):
+            self.frame_ids = {}
+        if B in self.frame_ids: return self.frame_ids[B]
+        f=max(C.B)+1; A,_=self.module.kernel(B,C.h)
+        C.B[f],C.A[f],C.dimf[f]=B,A,len(B);self.frame_ids[B]=f
+        return f
+
+
+    def row(self):
+        C,h,v=self.C,self.h,self.v
+        _,baseline_Y,src,_=C.chains()
+        seq=defaultdict(list)
+        for i,(a,b,x) in enumerate(self.ops):
+            seq[a].append(self.opframe[i]);seq[b].append(self.opframe[i])
+        for j,s in enumerate(self.w['rootroles']):seq[s].append(self.w['root_frame'][j])
+        start={b:self.w['source_frame'][x] for x,b in self.source.items()}
+        start.update({b:z['frame'] for b,z in self.gauge.items()})
+        recipient={d:b for b,d in self.pairs}
+        H=Counter()
+        for s in range(self.R):
+            if s in self.donor:continue
+            chain=list(seq[s])
+            if s in recipient:
+                b=recipient[s];chain += [self.gauge[b]['frame']]+seq[b]
+            chain.append(self.w['full_frame'])
+            prev=start.get(s);dp=0 if prev is None else C.dimf[prev]
+            if s in self.source.values():H[1]+=1
+            for f in chain:
+                need(prev is None or C.sub(prev,f),'physical role chain nested')
+                d=C.dimf[f]
+                if d>dp:H[d-dp]+=1
+                prev,dp=f,d
+        for j,r in enumerate(self.g['roots']):
+            if r['kind']=='center':H[C.dimf[self.w['root_frame'][j]]]+=1
+        Y=Counter();current=[None]*v
+        def read(t,f):
+            old=current[t];d=0 if old is None else C.dimf[old]
+            need(old is None or C.sub(old,f),'actual target frame chronology')
+            if C.dimf[f]>d:Y[C.dimf[f]-d]+=1
+            current[t]=f
+        for b in sorted(self.order,key=lambda b:self.readtime[b]):
+            for t in self.gauge[b]['targets']:read(t,self.gauge[b]['frame'])
+        deliveries=defaultdict(list)
+        for e in self.k['entries']:deliveries[e['deliver_after_root']].append(e)
+        for j,r in enumerate(self.g['roots']):
+            if r['kind']!='center':
+                for t in r['targets']:read(t,self.w['root_frame'][j])
+            for e in deliveries[j]:
+                for t in e['receivers']:read(t,e['deliver_frame'])
+        for t,f in enumerate(current):
+            need(f is not None and all(self.module.dot(C.cov[t],b)==0 for b in C.B[f]),'final target cap')
+            gap=h-1-C.dimf[f]
+            if gap:Y[gap]+=1
+        need(sum(r*n for r,n in Y.items())==v*(h-1),'target rank mass')
+        hist=Counter()
+        for part in (H,Y,src):
+            for r,n in part.items():
+                if r and n:hist[r]+=3*n
+        for b,z in self.gauge.items():
+            if b not in self.donor:hist[3*z['dim']]+=1
+        hist[2]+=2*v
+        hist={r:n for r,n in sorted(hist.items()) if n}
+        need(all(0<r<3*h and n>0 for r,n in hist.items()),'positive proper child bins')
+        R=self.R-len(self.pairs);W=2*v+R;mass=sum(r*n for r,n in hist.items());m=3*h
+        need(W*m-mass==C.prof['deficit_per_vertex'],'reuse preserves telescoping deficit')
+        return dict(h=h,v=v,R=R,virtual_R=self.R,reused_registers=len(self.pairs),W_per_vertex=W,m=m,
+            loss=C.prof['loss'],rank_per_vertex=mass,deficit_per_vertex=W*m-mass,
+            child_histogram=hist,maxchild=max(hist),selected_roles=len(self.gauge)-len(self.pairs),
+            selected_rank_histogram=dict(Counter(z['dim'] for b,z in self.gauge.items() if b not in self.donor)),
+            changed_operation_frames=len(self.changed_frames),
+            frames_not_contained_in_original=sum(not C.sub(self.opframe[i],self.original_opframe[i]) for i in self.changed_frames),
+            physical_target_histogram=dict(Y),physical_internal_histogram=dict(H),
+            foreign_producer_replays=0)
+
+    def adjoint(self):
+        adj = [dict() for _ in range(self.R)]
+        for r,s in zip(self.g['roots'], self.w['rootroles']):
+            for t in r['targets']: adj[s][t] = adj[s].get(t,0)+1
+        for a,b,_ in reversed(self.ops):
+            for t,c in adj[a].items(): adj[b][t] = adj[b].get(t,0)+c
+        return adj
+
+    def formal(self, ring, tamper=None):
+        """All source, target and dirty-register columns; no random sampling."""
+        v, phys = self.v, self.phys
+        regs = sorted(set(phys.values())); index = {r:2*v+j for j,r in enumerate(regs)}
+        if ring == 2:
+            unit = lambda i: 1<<i
+            def accum(dst,val,c): return dst ^ val if c&1 else dst
+        else:
+            unit = lambda i: {i:1}
+            def accum(dst,val,c):
+                out = dict(dst)
+                for i,z in val.items():
+                    k = out.get(i,0)+c*z
+                    if k: out[i]=k
+                    else: out.pop(i,None)
+                return out
+        x = [unit(s) for s in range(v)]; y = [unit(v+t) for t in range(v)]
+        a = {r:unit(index[r]) for r in regs}
+        adj = self.adjoint()
+        at = defaultdict(list)
+        for b in self.order: at[self.readtime[b]].append(b)
+        done = []
+        stale_used = False
+        def read(b):
+            nonlocal stale_used
+            if tamper == 'omit_compensation' and b == self.order[0]: return
+            value = a[phys[b]]
+            if tamper == 'stale' and b in self.donor and not stale_used:
+                value = unit(index[phys[b]]); stale_used = True
+            for t,c in adj[b].items(): y[t] = accum(y[t],value,-c)
+        def op(i,sign):
+            b,d,_ = self.ops[i]
+            a[phys[b]] = accum(a[phys[b]],a[phys[d]],sign)
+        for b in range(self.R):
+            if b not in self.gauge: read(b)
+        for s,b in self.source.items(): a[phys[b]] = accum(a[phys[b]],x[s],1)
+        for i in self.phase1: op(i,1); done.append(i)
+        for r,b in zip(self.g['roots'],self.w['rootroles']):
+            if r['kind']=='center':
+                for t in r['targets']:y[t]=accum(y[t],a[phys[b]],1)
+        for j,i in enumerate(self.rest):
+            for b in at[j]: read(b)
+            op(i,1); done.append(i)
+        for b in at[len(self.rest)]: read(b)
+        deliveries=defaultdict(list)
+        for e in self.k['entries']:deliveries[e['deliver_after_root']].append(e)
+        broken=False
+        for j,(r,b) in enumerate(zip(self.g['roots'],self.w['rootroles'])):
+            if r['kind']!='center':
+                for t in r['targets']: y[t] = accum(y[t],a[phys[b]],1)
+            for e in deliveries[j]:
+                c,d = e['carrier'],e['passive']
+                x[c] = accum(x[c],x[d],1)
+                for t in e['receivers']:
+                    if tamper == 'missing_partner' and not broken: broken=True;continue
+                    y[t] = accum(y[t],x[c],1)
+        for e in self.k['entries']:
+            x[e['carrier']] = accum(x[e['carrier']],x[e['passive']],-1)
+        for i in reversed(done): op(i,-1)
+        for s,b in self.source.items(): a[phys[b]] = accum(a[phys[b]],x[s],-1)
+        need(all(a[r] == unit(index[r]) for r in regs), 'all dirty register columns restored')
+        need(all(x[s] == unit(s) for s in range(v)), 'all source columns restored')
+        if ring == 2:
+            want = [accum(unit(v+t),unit(t),1) for t in range(v)]
+        else:
+            # Integer lift: the exact defining decoder, whose reduction is I over F2.
+            support = [{s:1} for s in range(v)]
+            for aa,bb in self.g['args'][v:]: support.append(accum(support[aa],support[bb],1))
+            want = [unit(v+t) for t in range(v)]
+            for r in self.g['roots']:
+                for t in r['targets']: want[t] = accum(want[t],support[r['node']],1)
+            for e in self.k['entries']:
+                for t in e['receivers']:
+                    want[t] = accum(want[t],unit(e['carrier']),1)
+                    want[t] = accum(want[t],unit(e['passive']),1)
+        need(y == want, 'all target columns equal defining decoder')
+        return dict(ring=str(ring),formal_variables=2*v+len(regs),
+            target_contract='F2 identity' if ring==2 else 'integer defining decoder',
+            identity=ring==2,defining_decoder=True,all_dirty_and_source_columns_restored=True)
diff --git a/research/five-stage-source527-banks/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/bit/word.py b/research/five-stage-source527-banks/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/bit/word.py
new file mode 100644
index 0000000000..6b8d96043a
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/bit/word.py
@@ -0,0 +1,104 @@
+"""Independent complete formal admission of the PR168-v4 physical bit word with our own physical layer (endpoint descent and reuse pairs) and terminal sinks.
+
+Retains eumemic's exact integer linear algebra and original decoder/geometry
+predicates. Physical chains, chronological formal columns and paid moments
+use our separate verifier. No producer or foreign aggregate checker runs.
+"""
+from pathlib import Path
+from collections import Counter,defaultdict
+from hashlib import sha256
+import gzip,importlib.util,json,sys,time
+HERE=Path(__file__).resolve().parent
+SOURCE=HERE.parent/'references/pr168-v4'
+PKG=HERE.parent
+def module(name,path):
+    spec=importlib.util.spec_from_file_location(name,path);m=importlib.util.module_from_spec(spec);spec.loader.exec_module(m);return m
+def read(p):
+    raw=p.read_bytes();return json.loads(gzip.decompress(raw) if p.suffix=='.gz' else raw)
+old=module('own_frozen_word',PKG/'bit/base_word.py');need=old.need
+
+class Candidate(old.Candidate):
+    def __init__(self):
+        self.module=backend=module('exact_integer_backend',SOURCE/'research/paired-cube-bit/check_paired_cube_bit.py')
+        C=backend.Checker.__new__(backend.Checker);self.C=C;C.mut=None
+        B=F=PKG/'selected/bit'
+        self.input_paths=[B/'graph_p12.json',B/'profile_p12.json',F/'word_p12.json.gz',F/'frames_p12.json.gz',F/'kchron_p12.json']
+        C.g,C.prof,C.w,C.fr,C.k=map(read,self.input_paths)
+        C.h=C.g['h'];C.v=C.g['v'];C.frames()
+        self.h,self.v,self.R=C.h,C.v,C.prof['R'];self.w,self.g,self.k=C.w,C.g,C.k
+        self.ops=self.w['ops'];self.nf={int(n):f for n,f in self.w['node_frame'].items()}
+        self.source={int(n):s for n,s in self.w['sources'].items()};self.gauge={z['role']:z for z in self.w['gauges']}
+        need(len(self.gauge)==len(self.w['gauges']),'unique gauge roles')
+        self.original_opframe=tuple(self.nf[n] for a,b,n in self.ops);self.opframe=self.w['op_frame'][:]
+        need(len(self.opframe)==len(self.ops),'complete operation frame vector')
+        self.changed_frames=[i for i,(a,b) in enumerate(zip(self.original_opframe,self.opframe)) if a!=b]
+        self.phase1=sorted(self.w['phase1']);phase=set(self.phase1)
+        self.rest=[i for i in range(len(self.ops)) if i not in phase];order=self.phase1+self.rest
+        self.position={i:p for p,i in enumerate(order)};pos={i:p for p,i in enumerate(self.rest)}
+        self.order=[z['role'] for z in reversed(self.w['gauges'])]
+        self.role_ops=defaultdict(list)
+        for i in order:
+            for s in self.ops[i][:2]:self.role_ops[s].append(i)
+        need(all(xs==sorted(xs) for xs in self.role_ops.values()),'execution order preserves every role chronology')
+        self.first={s:pos.get(xs[0],-1) for s,xs in self.role_ops.items()}
+        self.last={s:pos.get(xs[-1],-1) for s,xs in self.role_ops.items()}
+        self.endframe={s:self.opframe[xs[-1]] for s,xs in self.role_ops.items()}
+        self.rootroles=set(self.w['rootroles']);self.rootkind={s:r['kind'] for s,r in zip(self.w['rootroles'],self.g['roots'])}
+        need(len(self.rootroles)==len(self.w['rootroles']),'unique root roles')
+        reads={int(s):p for s,p in self.w['reads'].items()}
+        need(set(reads)<=set(self.gauge),'only gauges have explicit reads')
+        self.readtime={s:reads.get(s,len(self.phase1))-len(self.phase1) for s in self.order}
+        need(all(0<=self.readtime[s]<=self.first[s] for s in self.order),'all gauge reads after centre phase and before first use')
+        self.pairs=[[b,a] for a,b in self.w['pairs']];self.donor=dict(self.pairs)
+        need(len(self.donor)==len(self.pairs)==len(set(self.donor.values())),'one-to-one aliases')
+        need(not set(self.donor)&set(self.donor.values()),'disjoint donor recipient sets')
+        for b,d in self.pairs:
+            need(d not in self.gauge and d not in self.rootroles and b in self.gauge,'ungauged non-root donor and gauged recipient')
+            need(self.last[d]<self.readtime[b],'donor dead before recipient read')
+        self.phys={s:self.donor.get(s,s) for s in range(self.R)}
+        need(len(set(self.phys.values()))==self.R-len(self.pairs),'physical slot count')
+        need(all(self.phys[a]!=self.phys[b] for a,b,n in self.ops),'distinct physical gate ports')
+        C.chains=self.base_chains
+
+    def base_chains(self):
+        """Reconstruct phase/gauge/source contracts; row() counts physical paths."""
+        C,w,h,v=self.C,self.w,self.h,self.v
+        order=self.phase1+self.rest
+        roleprev={};pred=[]
+        for i,(a,b,n) in enumerate(self.ops):
+            pred.append((roleprev.get(a,-1),roleprev.get(b,-1)));roleprev[a]=roleprev[b]=i
+        pending=[roleprev[s] for s in self.rootroles if self.rootkind[s]=='center' and s in roleprev];closure=set()
+        while pending:
+            i=pending.pop()
+            if i not in closure:closure.add(i);pending.extend(j for j in pred[i] if j>=0)
+        need(closure==set(self.phase1),'phase one is exactly centre closure')
+        content=[0]*self.R
+        for n,s in self.source.items():content[s]=1<<n
+        for i in order:
+            a,b,n=self.ops[i];args=self.g['args'][n]
+            if content[a]:need(args is not None and {content[a],content[b]}=={C.sup[t] for t in args},'actual addition operands')
+            else:need(content[b]==C.sup[n],'actual copied node value')
+            content[a]|=content[b]
+        co=[set() for _ in range(self.R)]
+        for r,s in zip(self.g['roots'],w['rootroles']):co[s].update(r['targets'])
+        for i in reversed(order):
+            a,b,n=self.ops[i];co[b].update(co[a])
+        touched={s for i in self.phase1 for s in self.ops[i][:2]}
+        for s,z in self.gauge.items():
+            need(s not in self.source.values() and s not in touched,'gauge untouched in phase one and source injection')
+            need(set(z['targets'])==co[s] and C.dimf[z['frame']]==z['dim']>0,'gauge dimension and complete response support')
+        src=Counter();members=Counter();labels=list(map(set,self.g['labels']))
+        for e in self.k['entries']:
+            c,d,f,cap=e['carrier'],e['passive'],e['mix_frame'],e['deliver_frame']
+            need(len(labels[c]&labels[d])==1,'orthogonal partner pair')
+            need(C.dimf[f]==2 and C.in_frame(C.chi[c],f) and C.in_frame(C.chi[d],f),'partner mix span')
+            need(C.sub(f,cap) and all(self.module.dot(C.cov[t],u)==0 for t in e['receivers'] for u in C.B[cap]),'partner mix delivery cap')
+            need(e['carrier_chain']==[w['source_frame'][c],f,cap,w['full_frame']] and e['passive_chain']==[w['source_frame'][d],f,w['full_frame']],'exact partner chains')
+            root=self.g['roots'][e['deliver_after_root']]
+            need(root['kind']=='side' and sorted(e['receivers'])==sorted(root['targets']),'partner delivery at own receiver root')
+            for chain in (e['carrier_chain'],e['passive_chain']):
+                for a,b in zip(chain,chain[1:]):
+                    need(C.sub(a,b),'nested source chronology');src[C.dimf[b]-C.dimf[a]]+=1
+            members[c]+=1;members[d]+=1
+        need(all(members[s]==1 for s in range(v)),'every source in one partner pair')
+        return Counter(),Counter(),src,self.R
diff --git a/research/five-stage-source527-banks/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/references/pr168-v4/research/paired-cube-bit/check_paired_cube_bit.py b/research/five-stage-source527-banks/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/references/pr168-v4/research/paired-cube-bit/check_paired_cube_bit.py
new file mode 100644
index 0000000000..69d535a6c2
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/references/pr168-v4/research/paired-cube-bit/check_paired_cube_bit.py
@@ -0,0 +1,465 @@
+#!/usr/bin/env python3
+"""Standalone checker for subset-port bit words with broadcast roots (paired-cube bit word with partner-pair
+source mixing).  Standard library only; independent of the generator (own exact integer linear algebra, own
+register simulation).  Refuses to run under python -O.
+
+Inputs (plain JSON written by paired_cube_bit_word.py): graph_pP.json, word_pP.json, frames_pP.json,
+kchron_pP.json, profile_pP.json.  Checks:
+  1. exact frames: every frame record (integer annihilator rows 'a' or integer basis rows 'b') is completed to
+     the other representation by exact integer elimination; A.B^T = 0 exactly and rank A + rank B = h;
+  2. the mod-2 decoder: supports of all additions are disjoint; for every target the deliveries (star roots,
+     side roots, partner singles, partner-pair sums) XOR to {T}; stars read are exactly the c in T and the
+     side part is exactly sum_{|S cap T|=1} x_S;
+  3. geometry: source frames are the lines <chi_S>; side root frames lie in every receiver's cap
+     ker(3 chi_T - 1) and equal the common cap of the group; centre frames are the star spans; the frame of
+     every addition contains its operands' frames; every root node's frame lies in its root frame;
+  4. every role chain (start -> each op frame -> root frame -> full) is nested, including gauged starts
+     (sigma inside the first op frame, gauged roles untouched in phase one); every target chain (gauge reads
+     ascending, side root caps in root order, partner-pair delivery at the level-2 cap, final cap) is nested;
+     the partner-pair chronology: mix frame span{chi_c,chi_d} with H0-Gram I_2, contained in the receivers'
+     level-2 cap, carrier chain [<chi_c>, mix, cap, full] and passive chain [<chi_d>, mix, full] nested;
+  5. G = I - J/9 nondegenerate on every frame used (exact integer determinant test);
+  6. literal F2 replay with arbitrary dirty scratch: y = y0 + x, scratch and sources restored;
+  7. independent ledger recount from the role chains equals the saved child histogram (PR #144 bit ledger);
+  8. mutation controls (must all be REJECTED): broken partner mix, delivery outside the shared cap (full
+     frame), delivery below the level (face cap without span{chi_c,chi_d}), dropped star, non-nested role chain.
+Usage: python3 check_paired_cube_bit.py --dir out --p 12
+"""
+import argparse, json, random, sys
+from collections import Counter, defaultdict
+from math import gcd
+from operator import mul
+from pathlib import Path
+
+if sys.flags.optimize:
+    raise SystemExit('refusing to run under python -O: assertions are part of the check')
+
+
+class Fail(Exception):
+    pass
+
+
+def need(cond, msg):
+    if not cond:
+        raise Fail(msg)
+
+
+# ------------------------------------------------------------------ exact integer linear algebra
+def prim(v):
+    g = 0
+    for x in v:
+        g = gcd(g, abs(x))
+    if g > 1:
+        v = [x // g for x in v]
+    f = next((x for x in v if x), 0)
+    return tuple(-x for x in v) if f < 0 else tuple(v)
+
+
+def reduce_rows(rows, h):
+    """Integer reduced echelon form: returns (rows, pivot columns); each pivot column is zero outside its row."""
+    rows = [list(r) for r in rows]
+    out, pivs = [], []
+    for r in rows:
+        for pr, pc in zip(out, pivs):
+            if r[pc]:
+                a, b = pr[pc], r[pc]
+                r = [x * a - y * b for x, y in zip(r, pr)]
+        r = list(prim(r))
+        c = next((j for j in range(h) if r[j]), None)
+        if c is None:
+            continue
+        for i, (pr, pc) in enumerate(zip(out, pivs)):
+            if pr[c]:
+                a, b = r[c], pr[c]
+                out[i] = list(prim([x * a - y * b for x, y in zip(pr, r)]))
+        out.append(r); pivs.append(c)
+    return out, pivs
+
+
+def kernel(rows, h):
+    red, pivs = reduce_rows(rows, h)
+    ps = set(pivs)
+    basis = []
+    for f in range(h):
+        if f in ps:
+            continue
+        D = 1
+        for r, c in zip(red, pivs):
+            if r[f]:
+                D = D * abs(r[c]) // gcd(D, abs(r[c]))
+        v = [0] * h
+        v[f] = D
+        for r, c in zip(red, pivs):
+            if r[f]:
+                v[c] = -r[f] * D // r[c]
+        basis.append(prim(v))
+    return basis, len(red)
+
+
+def rank(rows, h):
+    return len(reduce_rows(rows, h)[0])
+
+
+def dot(a, b):
+    return sum(map(mul, a, b))
+
+
+# ------------------------------------------------------------------ checker
+class Checker:
+    def __init__(self, d, p, mutate=None):
+        self.mut = mutate
+        rd = lambda name: json.loads((d / ('%s_p%d.json' % (name, p))).read_text())
+        self.g, self.w, self.fr, self.k, self.prof = rd('graph'), rd('word'), rd('frames'), rd('kchron'), rd('profile')
+        self.h, self.v = self.g['h'], self.g['v']
+        if mutate == 'dropped_star':
+            j = next(i for i, r in enumerate(self.g['roots']) if r['kind'] == 'center')
+            self.g['roots'][j]['targets'] = []
+        if mutate == 'broken_mix':
+            self.k['entries'][0]['skip_mix'] = True
+        if mutate == 'outside_cap':
+            # raise the mixed carrier past the receivers' shared cap (to the full frame) and deliver there
+            e = self.k['entries'][0]
+            e['deliver_frame'] = self.w['full_frame']
+            e['carrier_chain'][2] = e['deliver_frame']
+        if mutate == 'below_level':
+            # deliver at the receivers' level-1 (face) cap, which does not contain span{chi_c, chi_d}
+            e = self.k['entries'][0]
+            face = next(j for j, r in enumerate(self.g['roots']) if r['kind'] == 'side' and set(e['receivers']) < set(r['targets']))
+            e['deliver_frame'] = self.w['root_frame'][face]
+            e['carrier_chain'][2] = e['deliver_frame']
+        if mutate == 'non_nested':
+            # move one addition's frame down to the frame of its smallest operand's source line
+            x = next(int(k) for k, f in sorted(self.w['node_frame'].items(), key=lambda kv: int(kv[0])) if int(k) >= self.v and str(int(k)) not in set(map(str, self.w['plain'])))
+            self.w['node_frame'][str(x)] = self.w['source_frame'][0]
+
+    # ---------------- frames
+    def frames(self):
+        h = self.h
+        self.A, self.B, self.dimf = {}, {}, {}
+        for key, rec in self.fr['frames'].items():
+            f = int(key)
+            if 'a' in rec:
+                A = [tuple(r) for r in rec['a']]
+                need(rank(A, h) == len(A), 'independent annihilator rows %d' % f)
+                B, _ = kernel(A, h)
+            else:
+                B = [tuple(r) for r in rec['b']]
+                need(rank(B, h) == len(B), 'independent basis rows %d' % f)
+                A, _ = kernel(B, h)
+            need(len(A) + len(B) == h and len(B) == rec['dim'], 'dimensions of frame %d' % f)
+            need(all(dot(a, b) == 0 for a in A for b in B), 'A.B = 0 for frame %d' % f)
+            self.A[f], self.B[f], self.dimf[f] = A, B, len(B)
+        self.cc = {}
+        return len(self.A)
+
+    def sub(self, f1, f2):
+        """frame f1 contained in frame f2 (exact)"""
+        if f1 == f2:
+            return True
+        key = (f1, f2)
+        r = self.cc.get(key)
+        if r is None:
+            B1, A2 = self.B[f1], self.A[f2]
+            r = len(B1) <= len(self.B[f2]) and all(dot(a, b) == 0 for a in A2 for b in B1)
+            self.cc[key] = r
+        return r
+
+    def in_frame(self, vec, f):
+        return all(dot(a, vec) == 0 for a in self.A[f])
+
+    def nondeg(self, f):
+        h = self.h
+        B, A = self.B[f], self.A[f]
+        if not B or len(B) == h:
+            return True
+        if len(B) <= len(A):
+            s = [sum(b) for b in B]
+            M = [[9 * dot(B[i], B[j]) - s[i] * s[j] for j in range(len(B))] for i in range(len(B))]
+            return rank(M, len(B)) == len(B)
+        s = [sum(a) for a in A]
+        M = [[(9 - h) * dot(A[i], A[j]) + s[i] * s[j] for j in range(len(A))] for i in range(len(A))]
+        return rank(M, len(A)) == len(A)
+
+    # ---------------- decoder
+    def decoder(self):
+        g, v = self.g, self.v
+        args = g['args']
+        lab = [frozenset(l) for l in g['labels']]
+        sup = [1 << i for i in range(v)] + [0] * (len(args) - v)
+        for x in range(v, len(args)):
+            a, b = args[x]
+            need(not sup[a] & sup[b], 'support-disjoint addition %d' % x)
+            sup[x] = sup[a] | sup[b]
+        self.sup = sup
+        acc, stars, side = [0] * v, [set() for _ in range(v)], [0] * v
+        for r in g['roots']:
+            need(r.get('coefficient', 1) == 1, 'mod-2 unit coefficients')
+            for t in r['targets']:
+                acc[t] ^= sup[r['node']]
+                if r['kind'] == 'center':
+                    stars[t].add(r['coordinate'])
+                else:
+                    side[t] ^= sup[r['node']]
+        for e in self.k['entries']:
+            for t in e['receivers']:
+                acc[t] ^= (1 << e['carrier']) | (1 << e['passive'])
+                side[t] ^= (1 << e['carrier']) | (1 << e['passive'])
+        for t in range(v):
+            need(acc[t] == 1 << t, 'decoder identity at target %d' % t)
+            need(stars[t] == set(lab[t]), 'stars read by target %d' % t)
+            orth = sum(1 << s for s in range(v) if len(lab[s] & lab[t]) == 1)
+            need(side[t] == orth, 'side part = sum over |S cap T| = 1 at target %d' % t)
+        for r in g['roots']:
+            if r['kind'] == 'center':
+                c = r['coordinate']
+                need(sup[r['node']] == sum(1 << s for s in range(v) if c in lab[s]), 'star support %d' % c)
+
+    # ---------------- geometry and chains
+    def geometry(self):
+        g, w, h, v = self.g, self.w, self.h, self.v
+        lab = g['labels']
+        chi = [tuple(1 if j in l else 0 for j in range(h)) for l in lab]
+        cov = [tuple((3 if j in l else 0) - 1 for j in range(h)) for l in lab]
+        nf = {int(k): f for k, f in w['node_frame'].items()}
+        rf = w['root_frame']
+        full = w['full_frame']
+        need(self.dimf[full] == h, 'full frame')
+        for s in range(v):
+            f = w['source_frame'][s]
+            need(self.dimf[f] == 1 and self.in_frame(chi[s], f), 'source line <chi_S>')
+        args = g['args']
+        for x, f in nf.items():
+            if x < v:
+                need(self.in_frame(chi[x], f), 'leaf label inside its frame')
+            else:
+                for y in args[x]:
+                    need(self.sub(nf[y], f), 'operand frame inside node frame (%d <- %d)' % (x, y))
+        loss = 0
+        for j, r in enumerate(g['roots']):
+            f = rf[j]
+            need(self.sub(nf[r['node']], f), 'root node frame inside root frame %d' % j)
+            if r['kind'] == 'center':
+                c = r['coordinate']
+                S = [chi[s] for s in range(v) if c in lab[s]]
+                need(all(self.in_frame(x, f) for x in S) and rank(S, h) == self.dimf[f], 'centre frame = star span')
+                loss += self.dimf[f]
+            else:
+                need(all(dot(cov[t], b) == 0 for t in r['targets'] for b in self.B[f]), 'root frame inside caps')
+                need(self.dimf[f] == h - rank([cov[t] for t in r['targets']], h), 'root frame = common cap')
+        need(loss == self.prof['loss'], 'copied-centre loss')
+        self.chi, self.cov, self.nf, self.rf = chi, cov, nf, rf
+
+    def chains(self):
+        """Role chains of the physical word and target chains; returns the recounted child histogram."""
+        g, w, h, v = self.g, self.w, self.h, self.v
+        nf, rf, full = self.nf, self.rf, w['full_frame']
+        ops, rootroles = w['ops'], w['rootroles']
+        R = 1 + max(max(a, b) for a, b, _ in ops)
+        R = max(R, 1 + max(rootroles), 1 + max(w['sources'].values()))
+        need(R == self.prof['R'], 'role count')
+        start = {}
+        for leaf, s in w['sources'].items():
+            start[s] = w['source_frame'][int(leaf)]
+        phase1 = set(w['phase1'])
+        touched_p1 = set()
+        for i in phase1:
+            touched_p1.update(ops[i][:2])
+        # phase one = closure of the centre roots' last ops under per-role precedence (recomputed)
+        prev_op, pred = {}, []
+        for i, (a, b, _) in enumerate(ops):
+            pred.append((prev_op.get(a, -1), prev_op.get(b, -1)))
+            prev_op[a] = prev_op[b] = i
+        stack = [prev_op[s] for r, s in zip(g['roots'], rootroles) if r['kind'] == 'center' and s in prev_op]
+        closure = set()
+        while stack:
+            i = stack.pop()
+            if i not in closure:
+                closure.add(i); stack.extend(j for j in pred[i] if j >= 0)
+        need(closure == phase1, 'phase one is the centre closure')
+        # conservative response support of every role (union incidence through the XOR word)
+        co = [0] * R
+        for j, (r, s) in enumerate(zip(g['roots'], rootroles)):
+            for t in r['targets']:
+                co[s] |= 1 << t
+        for a, b, _ in reversed(ops):
+            co[b] |= co[a]
+        gauge_of = {}
+        for z in w['gauges']:
+            need(sum(1 << t for t in z['targets']) == co[z['role']], 'gauge targets = response support of its role')
+            s = z['role']
+            need(s not in start, 'gauged role is not a source role')
+            need(s not in touched_p1, 'gauged role untouched in phase one')
+            need(self.dimf[z['frame']] == z['dim'] and z['dim'] > 0, 'gauge dimension')
+            need(self.nondeg(z['frame']), 'gauge frame nondegenerate')
+            start[s] = z['frame']; gauge_of[s] = z
+        seq = defaultdict(list)
+        for i, (a, b, x) in enumerate(ops):
+            f = nf[x]
+            seq[a].append(f); seq[b].append(f)
+        for j, s in enumerate(rootroles):
+            seq[s].append(rf[j])
+        H = Counter()
+        for s in range(R):
+            chain = [start.get(s)] + seq[s] + [full]
+            prev = chain[0]
+            dprev = 0 if prev is None else self.dimf[prev]
+            if prev is not None and s not in gauge_of:
+                H[1] += 1          # source role: injection frame <chi_S> (entrance child of width 1)
+            for f in chain[1:]:
+                if prev is not None:
+                    need(self.sub(prev, f), 'nested role chain (role %d)' % s)
+                d = self.dimf[f]
+                need(d >= dprev, 'ascending role chain (role %d)' % s)
+                if d > dprev:
+                    H[d - dprev] += 1
+                prev, dprev = f, d
+        for j, r in enumerate(g['roots']):
+            if r['kind'] == 'center':
+                H[self.dimf[rf[j]]] += 1   # copied-centre transform
+        # partner-pair chronology
+        src = Counter()
+        lab = [set(l) for l in g['labels']]
+        for e in self.k['entries']:
+            c, d = e['carrier'], e['passive']
+            need(len(lab[c] & lab[d]) == 1, 'carrier and passive orthogonal (H0-Gram I_2)')
+            mf, cf = e['mix_frame'], e['deliver_frame']
+            need(self.dimf[mf] == 2 and self.in_frame(self.chi[c], mf) and self.in_frame(self.chi[d], mf), 'mix frame = span{chi_c, chi_d}')
+            need(self.nondeg(mf), 'mix frame nondegenerate')
+            need(self.sub(mf, cf), 'mix frame inside the receivers shared cap')
+            need(all(dot(self.cov[t], b) == 0 for t in e['receivers'] for b in self.B[cf]), 'delivery frame inside receiver caps')
+            need(e['carrier_chain'][0] == w['source_frame'][c] and e['passive_chain'][0] == w['source_frame'][d], 'chains start at source lines')
+            need(e['carrier_chain'][1:] == [mf, cf, w['full_frame']] and e['passive_chain'][1:] == [mf, w['full_frame']], 'chain shape')
+            for ch in (e['carrier_chain'], e['passive_chain']):
+                for f1, f2 in zip(ch, ch[1:]):
+                    need(self.sub(f1, f2), 'nested source-register chain')
+                for f1, f2 in zip(ch, ch[1:]):
+                    src[self.dimf[f2] - self.dimf[f1]] += 1
+        # target chains: gauge reads (time order = reverse selection order), side caps in root order,
+        # partner-pair delivery at the level-2 cap right after its root, final cap
+        events = defaultdict(list)
+        for z in reversed(w['gauges']):
+            for t in z['targets']:
+                events[t].append(z['frame'])
+        deliveries = defaultdict(list)
+        for e in self.k['entries']:
+            deliveries[e['deliver_after_root']].append(e)
+        for j, r in enumerate(g['roots']):
+            if r['kind'] == 'center':
+                continue
+            for t in r['targets']:
+                events[t].append(rf[j])
+            for e in deliveries.get(j, []):
+                need(sorted(e['receivers']) == sorted(r['targets']), 'delivery after the receivers own level-2 root')
+                for t in e['receivers']:
+                    events[t].append(e['deliver_frame'])
+        Y = Counter()
+        cap = {}
+        for t in range(v):
+            capf = None
+            prev, dprev = None, 0
+            for f in events[t]:
+                if prev is not None:
+                    need(self.sub(prev, f), 'nested target chain (target %d)' % t)
+                need(all(dot(self.cov[t], b) == 0 for b in self.B[f]), 'target frame inside cap (target %d)' % t)
+                if self.dimf[f] > dprev:
+                    Y[self.dimf[f] - dprev] += 1
+                prev, dprev = f, self.dimf[f]
+            need(dprev <= h - 1, 'target chain below the cap')
+            Y[h - 1 - dprev] += 1 if h - 1 - dprev else 0
+        # every source register is in exactly one mixing pair
+        members = Counter()
+        for e in self.k['entries']:
+            members[e['carrier']] += 1; members[e['passive']] += 1
+        need(all(members[s] == 1 for s in range(v)), 'each source in exactly one partner pair')
+        for f in set(self.nf.values()) | set(self.rf):
+            need(self.nondeg(f), 'G-nondegenerate frame %d' % f)
+        return H, Y, src, R
+
+    # ---------------- replay
+    def replay(self, seeds=3):
+        g, w, v = self.g, self.w, self.v
+        ops, rootroles = w['ops'], w['rootroles']
+        R = self.prof['R']
+        srcmap = {int(k): s for k, s in w['sources'].items()}
+        roots = g['roots']
+        for seed in range(seeds):
+            rng = random.Random(1000 + seed)
+            x = [rng.getrandbits(1) for _ in range(v)]
+            y0 = [rng.getrandbits(1) for _ in range(v)]
+            z0 = [rng.getrandbits(1) for _ in range(R)]
+            X, Yv, z = x[:], y0[:], z0[:]
+            # y <- y - J M z (old-value response of the dirty scratch)
+            zd = z0[:]
+            for a, b, _ in ops:
+                zd[a] ^= zd[b]
+            for j, r in enumerate(roots):
+                for t in r['targets']:
+                    Yv[t] ^= zd[rootroles[j]]
+            for leaf, s in srcmap.items():      # z <- z + V x  (at the rank-one source lines)
+                z[s] ^= X[leaf]
+            for a, b, _ in ops:                  # z <- M z
+                z[a] ^= z[b]
+            for j, r in enumerate(roots):        # y <- y + J z
+                for t in r['targets']:
+                    Yv[t] ^= z[rootroles[j]]
+            for e in self.k['entries']:          # partner-pair sums on the source registers
+                if not e.get('skip_mix'):
+                    X[e['carrier']] ^= X[e['passive']]
+                for t in e['receivers']:
+                    Yv[t] ^= X[e['carrier']]
+            for e in self.k['entries']:          # undo at the full frame
+                if not e.get('skip_mix'):
+                    X[e['carrier']] ^= X[e['passive']]
+            for a, b, _ in reversed(ops):        # z <- M^-1 z
+                z[a] ^= z[b]
+            for leaf, s in srcmap.items():       # z <- z - V x
+                z[s] ^= X[leaf]
+            need(all(Yv[t] == y0[t] ^ x[t] for t in range(v)), 'replay: y = y0 + x (seed %d)' % seed)
+            need(z == z0 and X == x, 'replay: scratch and sources restored (seed %d)' % seed)
+
+    def run(self):
+        nfr = self.frames()
+        self.decoder()
+        self.geometry()
+        H, Y, src, R = self.chains()
+        self.replay()
+        h, v = self.h, self.v
+        m, W = 3 * h, 2 * v + R
+        C = Counter()
+        for part in (H, Y, src):
+            for r, c in part.items():
+                if r and c:
+                    C[r] += 3 * c
+        for z in self.w['gauges']:
+            C[3 * z['dim']] += 1
+        C[2] += 2 * v
+        saved = {int(k): c for k, c in self.prof['child_histogram'].items()}
+        need(dict(C) == saved, 'independent recount equals the saved child histogram')
+        mass = sum(r * c for r, c in C.items())
+        need(W == self.prof['W_per_vertex'] and W * m - mass == 2 * v - 3 * self.prof['loss'] == self.prof['deficit_per_vertex'],
+             'W = 2v + R and deficit 2v - 3 ell')
+        return dict(frames=nfr, roles=R, W=W, m=m, deficit=W * m - mass, children=sum(C.values()))
+
+
+def main():
+    ap = argparse.ArgumentParser(description=__doc__)
+    ap.add_argument('--dir', type=Path, default=Path(__file__).resolve().parent / 'out')
+    ap.add_argument('--p', type=int, default=12)
+    ap.add_argument('--no-controls', action='store_true')
+    a = ap.parse_args()
+    res = Checker(a.dir, a.p).run()
+    print('PASS p=%d: %s' % (a.p, res))
+    if a.no_controls:
+        return
+    for m in ('broken_mix', 'outside_cap', 'below_level', 'dropped_star', 'non_nested'):
+        try:
+            Checker(a.dir, a.p, mutate=m).run()
+        except Fail as e:
+            print('control %-13s REJECTED: %s' % (m, e))
+            continue
+        raise SystemExit('FAIL: control %s was accepted' % m)
+    print('PASS: all 5 mutation controls rejected')
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/five-stage-source527-banks/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/selected/bit/frames_p12.json.gz b/research/five-stage-source527-banks/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/selected/bit/frames_p12.json.gz
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diff --git a/research/five-stage-source527-banks/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/selected/bit/graph_p12.json b/research/five-stage-source527-banks/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/selected/bit/graph_p12.json
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+++ b/research/five-stage-source527-banks/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/selected/bit/graph_p12.json
@@ -0,0 +1 @@
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48],[18211,25849],[25850,25851],[18221,25852],[25853,25854],[25855,25856],[25857,25858],[25860,25861],[25859,25862],[25863,25864],[3679,3919],[4129,4309],[4669,4729],[4759,4999],[5539,5659],[5749,5809],[5839,6049],[7459,7579],[7669,7729],[7759,7879],[7969,8029],[8059,8149],[18595,25866],[18656,25867],[25868,25869],[18829,25870],[25871,25872],[18540,18543],[25873,25874],[25875,25876],[18526,25877],[18580,18665],[25878,25879],[25880,25881],[18631,25882],[18653,25883],[25884,25885],[18663,25886],[25887,25888],[25889,25890],[25891,25892],[25894,25895],[25893,25896],[25897,25898]],"decoder":"mod 2: Y_T += sum of root nodes whose targets contain T (star roots: T reads star(c) for c in T; side roots; partner singles) + x_carrier + x_passive for each partner_mix entry receiving T; equals x_T = sum_{c in T} star(c) + sum_{|S cap T|=1} 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diff --git a/research/five-stage-source527-banks/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/selected/bit/kchron_p12.json b/research/five-stage-source527-banks/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/selected/bit/kchron_p12.json
new file mode 100644
index 0000000000..9c00fdf366
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/selected/bit/kchron_p12.json
@@ -0,0 +1 @@
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\ No newline at end of file
diff --git a/research/five-stage-source527-banks/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/selected/bit/word_p12.json.gz b/research/five-stage-source527-banks/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/selected/bit/word_p12.json.gz
new file mode 100644
index 0000000000..d21cc9b2e7
Binary files /dev/null and b/research/five-stage-source527-banks/loader/source_inputs/base_bit/research/paired-cube-diagonal-bit-168/selected/bit/word_p12.json.gz differ
diff --git a/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/LICENSE b/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/LICENSE
new file mode 100644
index 0000000000..261eeb9e9f
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/LICENSE
@@ -0,0 +1,201 @@
+                                 Apache License
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diff --git a/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/NOTICE b/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/NOTICE
new file mode 100644
index 0000000000..d08cb66ac7
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/NOTICE
@@ -0,0 +1,431 @@
+A sharper exponent for integer multiplication
+Copyright 2026 Douglas Colkitt
+
+This project includes unmodified source files from OpenAI's manuscript
+"Integer multiplication below n log n" and its accompanying repository:
+https://github.com/openai/math
+Commit: adc7f1241b42e322a6451854ab7e4b4c146bf78a
+
+The original manuscript identifies OpenAI as its author. Its Apache-2.0
+license is retained at upstream/LICENSE. The exact source URLs and hashes
+are recorded in upstream/manifest.json. Proposed modifications are supplied
+as separate files under patches/; upstream source files remain unchanged.
+
+The original local research note, scripts, certificates, patches, and
+documentation were prepared by Douglas Colkitt with assistance from OpenAI
+Codex. The community release incorporates work by the contributors identified
+in the notices below; it is not solely Douglas Colkitt's work.
+This project is not an official OpenAI release or an OpenAI endorsement.
+
+Related contributions (not imported in this checkpoint):
+Zhihao Chen (jacklightChen), PR #7, supplied an earlier ternary five-subset
+motif, finite-alphabet transfer and a different compressed producer:
+https://github.com/CrocSwap/integer-mult-bounds/pull/7
+The present checkpoint claims no priority for that motif. Related compressed
+complex-network work was submitted by eumemic in PR #3:
+https://github.com/CrocSwap/integer-mult-bounds/pull/3
+The pending contributions and their stronger bounds require their own review;
+this notice does not represent adoption or verification of their claims.
+
+Additional community work, including parallel and superseded submissions, is
+acknowledged in CONTRIBUTORS.md, with pinned submission metadata in
+docs/research/contribution-snapshot.json. These acknowledgements do not imply
+joint authorship of the checkpoint, incorporation of pending code, or verification
+of submitted mathematical claims. Original source-specific notices remain in force.
+
+Community release (accepted by conditional maintainer audit):
+The following original contributor notices apply to the imported PR #39 chain.
+The preceding non-import statements describe the preserved 2^-30 checkpoint.
+
+Ternary five-subset construction and common-four-point resynthesis
+Copyright 2026 Zhihao Chen (jacklightChen)
+
+This is Zhihao Chen's first submission of this specific construction to this
+repository. Research, proof development, code and verification were prepared
+with substantial assistance from GPT-6 Astra (OpenAI Codex), as identified by
+the contributor. This is not independent review or an OpenAI endorsement.
+Future research building on this contribution should explicitly acknowledge
+Zhihao Chen (jacklightChen), cite the accompanying note and pull request, and
+retain the preceding contributors' attribution. This notice records research
+provenance; the repository's Apache-2.0 license remains unchanged.
+
+Dependencies retained from eumemic's Claude-assisted PRs #3 and #5:
+https://github.com/CrocSwap/integer-mult-bounds/pull/3
+https://github.com/CrocSwap/integer-mult-bounds/pull/5
+PR #5 commit: d3d370c34937503bd3fa63c83d505a05e8e50893
+These include complex_circuit.py, complex_network.py, fast_gaussian.py,
+their generators, construction notes and certificates. The Gaussian input
+scaling proof is corrected in this derivative; see the modification notice below.
+Aligned pairing also appears in Bortlesboat's PR #2. Retained totals and bank
+sharing appear in dleen's PR #4 and cheaper retained centers in eumemic's PR #6.
+Their results and ideas are credited separately from this new construction.
+
+Bulk recursion and controlled tensor basis, 8 October 2026
+Copyright 2026 icekylinx
+
+The new projector batching, controlled simultaneous basis, mixed-width tape
+recursions, whole-residual complex recursion, and dependency-path precision
+guard in the batched-23 proof package are contributed by icekylinx, developed
+with substantial OpenAI GPT-6 Astra assistance. Review corrections and
+repository integration were prepared with OpenAI Codex assistance. They build on the retained
+PR #7 finite producers and the preceding contributions credited above.
+The new package retains the repository's Apache-2.0 license and the original
+authors' attribution. It is not an official OpenAI release or endorsement.
+
+Modification notice, 8 October 2026 (icekylinx): the retained fast Gaussian
+proof now bounds shifted approximation error by F=2^ceil(1.14 alpha^2).
+The P=34p precision budget and asymptotic exponent are unchanged. New batching
+checks are integrated into make verify. Existing attribution and licenses
+are retained.
+
+Partial-swap extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The partial-swap framing, binary retained-total triple construction,
+dependency-preserving positive labels and carrier reuse, unequal (25,23,57)
+common basis, improved complex batching, and exact assembly are contributed
+by icekylinx with substantial OpenAI GPT-6 Astra and Codex assistance.
+This extension directly inherits PR #10, commit
+62691e395a0458ce089a1c7b5d89e74291e95e29. The portable producer adapters
+modify the retained paired-exclusion and shared-point implementations credited
+above. The original licenses, attribution, and Gaussian correction are retained.
+
+Endpoint-gauge extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The complementary auxiliary endpoint gauges, (32,30,40) controlled basis,
+general nondegenerate binary phase residuals, retained-center complex producer,
+updated path guard and exact assembly are contributed by icekylinx with
+substantial OpenAI GPT-6 Astra and Codex assistance. This extension inherits
+f2ab41aebad47861caf6316282c1793e5513845e and its PR #10 ancestry. New portable
+adapters retain the paired-exclusion and shared-point contributions credited
+above. Original licenses, notices and the Gaussian correction are retained.
+
+Structured-basis and semantic-bulk extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The selected (32,30,36) bit construction, A1/A5 block refinements, fixed
+middle-axis I+J profiles with bounded-minor CRT certificates, mixed dyadic
+complex centers, complete data-front batching and circuit-specific assembly
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex
+assistance. The immediate parent is ed90fd940279c336ebc45968631bdebfb087b505.
+
+The semantic/bulk compatibility argument, literal scalar charge, product row
+stock and parameterization adapt Zhihao Chen (jacklightChen)'s PR #23,
+commit 661ebabc1076c9f525d7ce4967c876b203fa9827. Translated complex endpoints
+are adopted from the same contributor's PR #21,
+commit 5ba6cf0bfb68f2be8d15610e7972207c50254d6a, following eumemic's source-frame
+work and the earlier nested-basis contributions.
+
+Completed-child semantic precision, arbitrary-coordinate routing, phase-cell
+Gaussian inversion and bulk tape transfer are credited to the RaD project
+(hipotures), commit 45d9b60355872041f9275f77c057514def0d45bc, exposed through
+PR #20 at 4f8d6c8272b5ff307a0da51df545ec3cd96a8b6e. Their selected proof
+sources retain their separate CC0 license and original NOTICE under
+references/semantic-bulk/rad20. The PR #23 source retains its recorded OpenAI
+Codex assistance; RaD retains its recorded Codex and GPT-6.1-Sol Ultra
+assistance. These are source attributions, not claims about this session's
+runtime model. Related contributions by Rohan Arun and Aurel Prosz remain
+acknowledged in the imported notices. No exclusive priority or additional
+license condition is asserted.
+
+Copied retained-center extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The copied retained-center schedule in both orientations, its complex-phase
+transfer, the Wm-N+L rank budget, and the selected (25,23)/(28,28) composition
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex
+assistance. This extension inherits PR #32 at
+0ef3aeb61f55cc0b321ce6a0ef00acee25cefe52 and the preceding PR #10/#18/#24 work.
+
+The two-stage topology and paid copied-stream endpoint correction are due
+to Aurel Prosz (Paureel), c82d09eb4781b68435e3fa3eb6beea72fa900fab,
+through Zhihao Chen (jacklightChen)'s PR #29 unequal-axis composition,
+9d963275075fa98f1da821e27757b238dafd6b3c. That source also credits Swapnil
+Jain's two-stage development at ae405eb474d1486b2d8aef90139869f927f4836a.
+
+The contiguous corner method, incidence factorization and exact partition
+certificates are due to Rohan Arun (PR #31), method commit
+02f68f95369c39955c5ffcfddbd75cd83e26b305. The selected (25,23) certificate
+specializes Dominik Scholz's parameterization (PR #33),
+a499a345c040aa15b68e984a0904eadc485c7a7e. Their source notices and proof
+snapshots are preserved in references/copied-centers. Historical OpenAI
+Codex, GPT-6 Astra and Anthropic Claude Opus 5.5 assistance is credited to
+the respective imported sources. Existing PR #21/#23 and RaD attribution,
+licenses and the Gaussian correction remain in force.
+
+Copied centers with reversed (23,25) data corners, 2026-10-08
+Rohan Arun with substantial OpenAI Codex assistance supplies the composition,
+exact dimension23 controls and cost/assembly integration. James Chang's PR34
+supplies the reversed boundary family; icekylinx's PR36 supplies copied retained
+centers and the selected complex graph. The balanced assembly remains attributed
+to RaD/hipotures and its PR34 specialization. Retain Dominik Scholz PR33, Rohan
+Arun PR31, Zhihao Chen PR29/21/23, Paureel/Aurel Prosz, Swapnil Jain, icekylinx,
+eumemic, Douglas Colkitt, OpenAI and all earlier notices and hypotheses.
+This is not a claim of formal verification, new foundational rank identities,
+global optimality or measured runtime speedup.
+
+Fixed middle basis with copied reversed corners (2026-10-08):
+Rohan Arun, with substantial OpenAI Codex assistance. Composition and
+simultaneous-basis proof retain icekylinx PR32/36, James Chang PR34,
+Dominik Scholz PR35/38, Zhihao Chen, Paureel, Swapnil Jain, RaD/hipotures,
+eumemic and all preceding attribution and licenses. See research/copied-fixed-reversed.
+
+Changed-graph composition with copied centers and fixed bases, 8 October 2026
+Prepared with OpenAI Codex assistance.
+The new composition uses icekylinx PR36 copied centers and PR32 fixed-I+J
+projectors/profiler, Dominik Scholz PR35 two fixed factors and CRT argument,
+Zhihao Chen (jacklightChen) PR29/23 interfaces, Aurel Prosz (Paureel) and
+Swapnil Jain's two-stage development, Rohan Arun PR25/31 corner methods,
+Dominik Scholz PR33/38 dimension/composition work; James Chang (jamesyc)
+PR34 reversed geometry and balanced transfer; Rohan Arun PR37/39/40
+reversed integration, modular data tests and both-fixed baseline;
+RaD (hipotures) PR41 changed scalar DAGs, alternating point order and
+independent role compiler; RaD balanced routing/bulk interfaces, and all
+retained predecessors.
+Complete attribution and the conditional proof boundary are recorded in
+research/copied-fixed/PROOF.md; imported originals and source pins are retained.
+Original Apache-2.0 and CC0 notices remain applicable.
+
+Community follow-up integration (PRs #49, #45 and #26), 8 October 2026:
+Rohan Arun contributes weighted carrier matching and climbed summand/leave-one-out
+orders (PRs #44/#47/#49), with Anthropic Claude assistance. The final construction
+composes Chafik Boukhalfa's changed-graph checks, exact rational data recovery and
+reordered exclusion sums (PRs #43/#46/#48, OpenAI Codex assistance) with RaD /
+hipotures's alternating pair order and independent physical role compiler (#41).
+All predecessor attribution and conditional proof interfaces remain in force.
+
+Alejandro Zarzuelo Urdiales contributes Gaussian parity and canonical arithmetic
+proofs, finite tensor execution, profile precision accounting and mixed-center
+reference checks (#45), with substantial OpenAI Codex assistance. See
+research/gaussian-parity-synthesis/PROVENANCE.md for his full attribution.
+Ryan S / princezuda contributes historical Lean certificate arithmetic, algebraic
+frame/movement/guard lemmas and independent paired-circuit checks (#26). Original
+source notices and historical snapshots are preserved. Neither formal package
+claims a full formal proof of the multiplication algorithm or a new kappa.
+
+Joint frame compilation of skip-prefix strips
+
+The local compiler and exact finite witnesses were prepared by eumemic with
+substantial OpenAI Codex assistance. Avi Eisenberg / ikeboy with Anthropic
+Claude assistance supplied the unmodified PR #53 skip-prefix producer.
+The pinned PR #48 pipeline is Chafik Boukhalfa's contribution, incorporating
+Rohan Arun, RaD / hipotures, icekylinx, Dominik Scholz, James Chang,
+Zhihao Chen, Aurel Prosz / Paureel, Swapnil Jain, eumemic, Douglas Colkitt,
+OpenAI, David Harvey, Joris van der Hoeven and the earlier contributors
+credited in the retained sources. See references/frame-compiler/pr48/NOTICE
+and references/frame-compiler/pr53/NOTICE for their original attribution.
+No authorship or review by those predecessors is claimed for this compiler.
+
+Joint dual-skip/frame-compiler composition (2026-10-08)
+---------------------------------------------------
+Rohan Gupta / gupt1156 with Anthropic Claude assistance: PR55 dual-skip
+scalar graph; original LICENSE and NOTICE are retained under
+references/frame-compiler/pr55. eumemic with substantial OpenAI Codex
+assistance: PR57 joint reversible frame compiler, serialized physical-word
+replay and profile reconstruction. Chafik Boukhalfa with OpenAI Codex
+assistance: this composition, independent integration and exact certificate.
+Avi Eisenberg / ikeboy, Rohan Arun, RaD / hipotures, icekylinx, and all
+predecessor authors retain the notices above and in pinned source files.
+No original copyright or Apache-2.0 notice is removed.
+
+Ranked reclamation after the PR58 joint-dual composition (2026-10-08)
+----------------------------------------------------------------
+Chafik Boukhalfa with OpenAI Codex assistance: prioritize retired physical
+slots by descending current frame rank, retaining slot ID as the tie-breaker;
+complete replay, exact profiles, comparison and adversarial controls.
+The dedicated joint_dual_reclaim_compiler.py is derived from eumemic's PR57
+compiler. Its original shared compiler remains byte-identical. The PR55
+producer and all preceding attribution remain unchanged. The complete PR58
+comparison certificate and its original source/proof provenance are retained
+under references/frame-compiler/pr58.
+
+Skip-prefix strip producer, 8 October 2026
+By Avi Eisenberg (ikeboy), with Anthropic Claude (Claude Opus 5.5)
+assistance. The skip-prefix leave-one-out strip layout in
+research/skip-strips/skip_graph.py is new. Everything else reuses Chafik
+Boukhalfa's PR #43/#46/#48 research/copied-fixed pipeline, its verification
+code, exact data recovery and support-order sorting, together with:
+- Rohan Arun's PR #44 weighted carrier matching optimizer;
+- RaD (hipotures) PR #41 alternating point order;
+- every predecessor credited above.
+Complete attribution and the conditional proof boundary are recorded in
+research/skip-strips/PROOF.md. Original Apache-2.0 and CC0 notices remain
+applicable.
+
+Local split-pair skip-prefix research continuation (2026-10-08)
+Prepared with substantial OpenAI Codex assistance at the user's request.
+New contribution: singleton/pair recursive partition refinement, finite summand
+order and moment search, and verified composition with inherited paid clones.
+Avi Eisenberg/ikeboy with Anthropic Claude assistance: PR53 skip-prefix strips.
+Chafik Boukhalfa with OpenAI Codex assistance: PR54 original-envelope clone
+specialization and PR43/46/48 checking interfaces. RaD/hipotures: PR51
+whole-chain/source-partition cloning and earlier producer/interfaces.
+Rohan Arun: PR44 weighted matching and earlier fixed/reversed work, with all
+stated historical AI assistance. All predecessor notices remain applicable.
+
+Original-envelope whole-chain clones on skip-prefix strips (research/skip-clones):
+Chafik Boukhalfa, prepared with OpenAI Codex assistance. The complete-chain
+source-partition clone operation is adapted from RaD / hipotures, PR #51,
+with original envelopes retained here. The scalar base is Avi Eisenberg's
+PR #53 skip-prefix layout, prepared with Anthropic Claude assistance.
+All credited PR #43/#46/#48, PR #44 and earlier code/proof notices remain
+in force. This extension is provided under the repository's Apache-2.0 terms.
+
+Community review checkpoint #50–#61, 8 October 2026
+Maintainer integration and independent arithmetic: Douglas Colkitt, with OpenAI Codex.
+Avi Eisenberg / ikeboy: skip-prefix strip framework (#53), with Anthropic Claude.
+Rohan Gupta / gupt1156: parallel order search (#50, stated Antigravity assistance)
+and dual-suffix strips (#55, Anthropic Claude assistance).
+Eumemic: equal-frame joint compilation and paid signal reclamation (#57), with Codex.
+Chafik Boukhalfa: original-envelope paid clones (#54), composition (#58) and ranked
+reclamation (#60), with Codex. Rohan Arun: order search (#52) and positive-frame
+composition (#56), with Anthropic Claude and stated Codex validation assistance.
+RaD / hipotures: enlarged frames, source partitions and paid whole-chain clones
+(#51), with Codex. Rohan Garg / rohangar1: singleton/pair recursion and paid-clone
+composition (#59), with Codex. Alejandro Zarzuelo Urdiales: exact parameter
+refinement, scoped Lean arithmetic, matrix/parity and search tools (#61), with
+his disclosed multi-tool and Codex assistance.
+Parallel #50/#52 records are reviewed and credited without replacing the earlier
+climbed-48 implementation; their exact commit history is retained. The separate
+#51 witness is retained as research; its source-graph/negative-basis data proof
+has not received the same full replay as #56's consumed compiler machinery.
+See docs/research/community-round2-review.md for precise validation boundaries.
+All earlier licenses, contributions and notices remain applicable.
+
+Interval strips and core-aware pair assembly
+
+The interval leave-one-out strips and the core-aware group-pair assembly in
+research/pair-assembly were prepared by Avi Eisenberg / ikeboy with Anthropic
+Claude assistance. They build on PR #53's skip-prefix pipeline and its
+inherited PR #48 sources. The stacked witness in research/pair-assembly/frame
+uses eumemic's PR #57 joint frame compiler unchanged; its driver and verifier
+are adaptations of PR #57's skip_frame_compiler.py, skip_frame_compose.py and
+verify_skip_frame.py. Original notices are retained.
+
+Stopped product-ring extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The opposite-bank generic projector factorization, componentwise atom-ring
+adapters, stopped recurrence and outer ordinary-interchange conversion,
+aligned pair-star producer with rational centers, exact denominator-21
+grid and selected (23,23)/(24,24) composition are contributed by icekylinx
+with substantial OpenAI GPT-6 Astra and Codex assistance. This extension
+builds on merged PR #36, 11817ccacb564bb7f98789c20dc11d3fece207e3, and is
+integrated on upstream main 56b66d58297deca1d7dd130247d720e960f77a37.
+
+The retained ordinary leaf uses the preceding projector, corner and
+copied-center proofs. The selected two-stage topology retains Aurel Prosz
+(Paureel), Swapnil Jain and Zhihao Chen's credits; the semantic assembly
+retains PR #21/#23 and RaD's attribution. Paired producers, carrier matching,
+original source notices and each imported license remain unchanged.
+
+Partial-gauge extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The pair-first cube assembly, compatible backward frame enlargement,
+partial source-gauge schedule and stopped whole-projector composition are
+contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex
+assistance, extending PR #104 at 948ce1510df750f4c18b96bdaef436a86f8bf834.
+
+The physical deferred bit word, reflected event accounting, fan-order
+clarification and signed two-stage endpoint are adopted from Zhihao Chen
+(jacklightChen), PR #97, e15350b66e4bf796153a68c240a6cc1b788abe3e, with
+Codex assistance. Its deferred readouts, V leaves, lifted frames and frozen
+witness originate in Swapnil Jain's 741e7aa078392553815df7926ee17ac5e25a8c38,
+with the original Claude disclosure retained. Cyclic interval and core-aware
+pair primitives build on Avi Eisenberg (ikeboy), PR #62,
+ad0f25ff7b23cff7f08ad237c2254e6ecf74257e, with Anthropic Claude assistance.
+Prior two-stage, copied-center, ordinary-leaf, semantic/bulk, routing and
+precision credits remain in force. Imported source notices and licenses
+are preserved under references/partial-gauge/; see SOURCES.json.
+
+Three-stage cover extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The arbitrary-subspace Clifford frame theorem, three-stage Cayley geometry,
+weighted q-adic GL cover, unit-lower weighted compiler, charged local-ring
+fallback, uniform batching, internal row borrowing and exact composition
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra assistance. This
+repository integration uses Codex assistance and extends
+PR #115, cf60442983c153a3a0f5272ea4713f99eba13c86.
+
+The selected local complex DAG and support/label replay are adopted from
+Copyright 2026 eumemic, PR #117, cbb05ce504d571546d9b7794c186a613c659c3bf,
+with its original Apache-2.0 terms and Claude assistance disclosure.
+The unchanged bit physical word retains Zhihao Chen (jacklightChen),
+PR #97, and Swapnil Jain's underlying witness and assistance notices.
+The prior opposite-bank, stopped ordinary-leaf, copied-center, topology,
+semantic/bulk, precision and analytic credits remain applicable.
+Original source notices and license bytes are preserved in
+references/three-stage-cover/pr117/ and references/partial-gauge/.
+
+Paired-cube and shared-core extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The paired-coordinate triple-cube identity, signed overlap channels,
+original-source involution, coordinate-star centers, concrete frame/carrier
+compiler, chronological source gauges, PR97 gauge-subset selection,
+orthogonal cross-stage routing and exact supplier/assembly integration
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra assistance.
+Repository integration uses Codex assistance and extends PR #130 at
+6a9970a530119174507904e23592fd59ede19a5d.
+
+The substantial completed-core workspace-sharing principle is credited to
+an664, PR #128, 530588a019b4a74f09180680c9e3961bf649ec89, with its reported
+OpenAI Codex assistance. No source code, partition or serialized witness
+from that PR is imported. The new routing and paired-motif integration
+are described separately and claim no exclusive priority for broad sharing,
+paired-coordinate, Fourier or carrier-matching ideas.
+
+Selected coarse queries restrict the retained eumemic PR #117 positive DAG;
+its original Apache-2.0 and Claude assistance notices remain unchanged.
+The bit word and original frames retain Zhihao Chen's PR #97 and Swapnil
+Jain's frozen witness, notices and historical assistance disclosures.
+Earlier generalized Clifford, three-stage, uniform weighted recursion,
+ordinary-leaf, semantic/bulk, precision and analytic credits remain in force.
+Source-specific license bytes are preserved under references/.
+
+Source-assisted common-frame extension (2026-10-09)
+Copyright 2026 icekylinx. Apache-2.0.
+Research construction and local certificates: OpenAI GPT-6 Astra assistance.
+Repository integration and independent complex scalar replay: Codex assistance.
+Substantial dependency: eumemic PR168, fd25adb7fbaa12ee761d02c733c54d1d2a7687ee;
+its byte-preserved sources, licenses and original Claude assistance disclosure
+are retained in research/source-assisted/public/pr168_fd25adb7.tar.gz.
+Additional inherited contributions are recorded in research/source-assisted/CREDITS.md.
+The new work concerns source-assisted invertible common-frame lifts, controlled
+affine cleanup and compatible zero-fresh reuse; no exclusive claim is made for
+broad register reuse, paired cubes, frame theory or ordinary linear elimination.
+
+Fourth finite bootstrap refinement (2026-10-09)
+Prepared by eumemic with substantial OpenAI Codex assistance.
+This arithmetic refinement builds on PR207 at
+cd14825023b75af4f5919a30e5e6d548b6ade5bc by adding one finite ordinary-leaf
+level. The 302 frame overrides, completed banks, scalar words and upstream
+assistance disclosures are inherited, not newly attributed to this work.
+The Apache-2.0 license and all preceding notices remain applicable.
+
+PR211 plateau frame refinement (2026-10-09)
+Prepared by eumemic with substantial OpenAI Codex assistance.
+The additional seven equal-frame component moves build directly on Rohan
+Arun's PR211 cascade witness at c63e50a5dde96fe1459d6b29e55e47f42f104347,
+prepared with Anthropic Claude assistance. Its original notice is preserved
+byte-for-byte in search/PR211-NOTICE. The offline certificate package and
+independent stage-private bank admission derive from Dugongue's PR207 at
+cd14825023b75af4f5919a30e5e6d548b6ade5bc. Original contributor notices,
+licenses, source pins and all upstream AI disclosures remain applicable.
+
+The final refinement also includes seeded bidirectional cascade search after
+the seven initial component moves. Exact canonical subspace comparison
+records 274 changed operations beyond PR211. Search, integration and replay
+were prepared with substantial OpenAI Codex assistance.
+
+Shared rank-18 gauges and two donor substitutions (2026-10-09)
+Prepared by eumemic with substantial OpenAI Codex assistance.
+The new actual gauge/read chronology, gauged-donor handoff, shared target
+transitions, rank6 residual banks and donor substitutions extend the same
+PR200/PR211 word and PR207/PR197 completed-bank framework. All original
+licenses, notices, authorship and upstream AI disclosures remain in force.
+
+Nested-gauge extension: three rank-19 physical entrances compose with thirteen shared rank-18 entrances; all target transitions and mixed4/5/6/24 banks are explicit. Prepared by eumemic with substantial OpenAI Codex assistance. All retained authorship, licenses and AI disclosures remain applicable.
diff --git a/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/borrow/selection.json b/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/borrow/selection.json
new file mode 100644
index 0000000000..80d0a76822
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/borrow/selection.json
@@ -0,0 +1,5282 @@
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diff --git a/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/frames/opframe-bases.json b/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/frames/opframe-bases.json
new file mode 100644
index 0000000000..b4d6b4509f
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/frames/opframe-bases.json
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new file mode 100644
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--- /dev/null
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diff --git a/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/joint/joint-selection.json b/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/joint/joint-selection.json
new file mode 100644
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+      [
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+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        -1
+      ]
+    ],
+    "role_targets": {
+      "14545": [
+        904,
+        906,
+        908,
+        910
+      ],
+      "14551": [
+        905,
+        907,
+        909,
+        911
+      ],
+      "14620": [
+        905,
+        907,
+        908,
+        910
+      ],
+      "14621": [
+        904,
+        906,
+        909,
+        911
+      ],
+      "14800": [
+        904,
+        906,
+        908,
+        910
+      ],
+      "14805": [
+        905,
+        907,
+        909,
+        911
+      ],
+      "14902": [
+        905,
+        906,
+        909,
+        910
+      ],
+      "14903": [
+        904,
+        907,
+        908,
+        911
+      ]
+    }
+  },
+  {
+    "roles": [
+      14557,
+      14563,
+      14622,
+      14623,
+      14810,
+      14815,
+      14904,
+      14905
+    ],
+    "gauge_basis": [
+      [
+        1,
+        1,
+        1,
+        1,
+        1,
+        1,
+        1,
+        1,
+        1,
+        1,
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+        0,
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+      ],
+      [
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+      [
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+      [
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+      [
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+      [
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+      [
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+        1,
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+        1,
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+      ]
+    ],
+    "role_targets": {
+      "14557": [
+        912,
+        914,
+        916,
+        918
+      ],
+      "14563": [
+        913,
+        915,
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+      ],
+      "14622": [
+        913,
+        915,
+        916,
+        918
+      ],
+      "14623": [
+        912,
+        914,
+        917,
+        919
+      ],
+      "14810": [
+        912,
+        914,
+        916,
+        918
+      ],
+      "14815": [
+        913,
+        915,
+        917,
+        919
+      ],
+      "14904": [
+        913,
+        914,
+        917,
+        918
+      ],
+      "14905": [
+        912,
+        915,
+        916,
+        919
+      ]
+    }
+  }
+]
diff --git a/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/joint/joint_word.py b/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/joint/joint_word.py
new file mode 100644
index 0000000000..52be474a77
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/joint/joint_word.py
@@ -0,0 +1,50 @@
+"""Actual joint entrance gauges on the frozen relaxed PR211 frame word.
+Experimental adapter, prepared with OpenAI Codex assistance; Apache-2.0.
+Pinned supplier files are never modified.
+"""
+from pathlib import Path
+from copy import deepcopy
+from collections import defaultdict
+import sys,json,hashlib
+sys.dont_write_bytecode=True
+D=Path(__file__).resolve().parent
+PACKAGE=D.parent;ROOT=PACKAGE.parents[1]
+P=ROOT/'research/paired-cube-diagonal-bit-168'
+sys.path.insert(0,str(P/'bit'))
+from word import Candidate as Original
+class Candidate(Original):
+ def __init__(self):
+  super().__init__()
+  self.original_w=deepcopy(self.w)
+  pairs=json.loads((D/'pairs.json').read_text())
+  assert len(pairs)==1760 and set(b for b,d in pairs)==set(self.donor)
+  assert len(set(d for b,d in pairs))==1760
+  old=dict(self.pairs);changed=[(b,d) for b,d in pairs if old[b]!=d]
+  assert changed==[(18892,8292),(18894,8293)]
+  assert all(d not in self.source.values() and d not in self.gauge and d not in self.rootroles for b,d in changed)
+  assert all(self.last[d]<self.readtime[b] for b,d in pairs)
+  self.pairs=pairs;self.donor=dict(pairs);self.w['pairs']=[[d,b] for b,d in pairs]
+  self.phys={s:self.donor.get(s,s) for s in range(self.R)}
+  assert all(self.phys[a]!=self.phys[b] for a,b,n in self.ops)
+  self.framepath=PACKAGE/'frames/opframe-bases.json'
+  for i,rows in json.loads(self.framepath.read_text()):self.opframe[i]=self.register(rows)
+  self.changed_frames=[i for i,(a,b) in enumerate(zip(self.original_opframe,self.opframe))if a!=b]
+  self.endframe={s:self.opframe[xs[-1]]for s,xs in self.role_ops.items()}
+  groups=json.loads((D/'joint-selection.json').read_text())
+  oldorder=self.order[:];touched={s for i in self.phase1 for s in self.ops[i][:2]};sources=set(self.source.values())
+  co=[set()for _ in range(self.R)]
+  for r,s in zip(self.g['roots'],self.w['rootroles']):co[s].update(r['targets'])
+  for a,b,n in reversed(self.ops):co[b].update(co[a])
+  newroles=[]
+  for group in groups:
+   f=self.register(group['gauge_basis']);assert self.C.dimf[f] in (12,13,18,19) and self.C.nondeg(f)
+   for s in group['roles']:
+    assert s not in self.gauge and s not in sources and s not in touched and s not in self.donor
+    assert self.C.sub(f,self.opframe[self.role_ops[s][0]])
+    targets=sorted(co[s]);assert targets==group['role_targets'][str(s)]
+    z=dict(role=s,frame=f,dim=self.C.dimf[f],targets=targets)
+    self.gauge[s]=z;self.w['gauges'].append(z);self.w['reads'][str(s)]=len(self.phase1);self.readtime[s]=0;newroles.append(s)
+  self.order=sorted(newroles,key=lambda s:self.gauge[s]['dim'])+oldorder
+  assert len(newroles)==82 and len(set(newroles))==82
+  self.newroles=newroles
+  assert self.original_w==json.loads(__import__('gzip').decompress(self.input_paths[2].read_bytes()))
diff --git a/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/joint/pairs.json b/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/joint/pairs.json
new file mode 100644
index 0000000000..33b56aa01d
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/joint/pairs.json
@@ -0,0 +1 @@
+[[17148, 9362], [17149, 1440], [17150, 9364], [17151, 1456], [17152, 14699], [17153, 12619], [17154, 14704], [17155, 12656], [17156, 486], [17157, 483], [17158, 506], [17159, 499], [17160, 8867], [17161, 576], [17162, 8869], [17163, 592], [17164, 461], [17165, 454], [17166, 476], [17167, 474], [17168, 14078], [17169, 12639], [17170, 14087], [17171, 12676], [17172, 451], [17173, 429], [17174, 467], [17175, 444], [17176, 14079], [17177, 14077], [17178, 14088], [17179, 14086], [17180, 608], [17181, 9326], [17182, 624], [17183, 9328], [17184, 12627], [17185, 9255], [17186, 12664], [17187, 9257], [17188, 422], [17189, 419], [17190, 442], [17191, 435], [17192, 14696], [17193, 9161], [17194, 14701], [17195, 9163], [17196, 326], [17197, 355], [17198, 346], [17199, 371], [17200, 14075], [17201, 9327], [17202, 14084], [17203, 9329], [17204, 397], [17205, 14073], [17206, 412], [17207, 14082], [17208, 13631], [17209, 1152], [17210, 13636], [17211, 1168], [17212, 390], [17213, 387], [17214, 410], [17215, 403], [17216, 13883], [17217, 13629], [17218, 13891], [17219, 13634], [17220, 365], [17221, 8734], [17222, 380], [17223, 8736], [17224, 14697], [17225, 12615], [17226, 14702], [17227, 12652], [17228, 1536], [17229, 12637], [17230, 1552], [17231, 12674], [17232, 9123], [17233, 9084], [17234, 9125], [17235, 9086], [17236, 269], [17237, 198], [17238, 284], [17239, 218], [17240, 12622], [17241, 704], [17242, 12659], [17243, 720], [17244, 1280], [17245, 9102], [17246, 1296], [17247, 9105], [17248, 12620], [17249, 8826], [17250, 12657], [17251, 8829], [17252, 294], [17253, 8771], [17254, 314], [17255, 8773], [17256, 1472], [17257, 640], [17258, 1488], [17259, 656], [17260, 230], [17261, 259], [17262, 250], [17263, 275], [17264, 8848], [17265, 8753], [17266, 8851], [17267, 8755], [17268, 291], [17269, 8885], [17270, 307], [17271, 8887], [17272, 14424], [17273, 8846], [17274, 14430], [17275, 8849], [17276, 928], [17277, 12609], [17278, 944], [17279, 12646], [17280, 14422], [17281, 13884], [17282, 14428], [17283, 13892], [17284, 301], [17285, 109], [17286, 316], [17287, 124], [17288, 8847], [17289, 8385], [17290, 8850], [17291, 8387], [17292, 166], [17293, 8384], [17294, 186], [17295, 8386], [17296, 14074], [17297, 8828], [17298, 14083], [17299, 8831], [17300, 8697], [17301, 12608], [17302, 8699], [17303, 12645], [17304, 8605], [17305, 8421], [17306, 8607], [17307, 8423], [17308, 358], [17309, 8902], [17310, 378], [17311, 8904], [17312, 14700], [17313, 8567], [17314, 14705], [17315, 8569], [17316, 205], [17317, 8456], [17318, 220], [17319, 8458], [17320, 8641], [17321, 8622], [17322, 8643], [17323, 8624], [17324, 13881], [17325, 12643], [17326, 13889], [17327, 12680], [17328, 12623], [17329, 8604], [17330, 12660], [17331, 8606], [17332, 8492], [17333, 1728], [17334, 8494], [17335, 1744], [17336, 8623], [17337, 8366], [17338, 8625], [17339, 8368], [17340, 13880], [17341, 8678], [17342, 13888], [17343, 8680], [17344, 8640], [17345, 8367], [17346, 8642], [17347, 8369], [17348, 3], [17349, 8529], [17350, 19], [17351, 8531], [17352, 12618], [17353, 8974], [17354, 12655], [17355, 8976], [17356, 195], [17357, 13628], [17358, 211], [17359, 13633], [17360, 8679], [17361, 8584], [17362, 8681], [17363, 8587], [17364, 32], [17365, 672], [17366, 48], [17367, 688], [17368, 1312], [17369, 832], [17370, 1328], [17371, 848], [17372, 333], [17373, 12640], [17374, 348], [17375, 12677], [17376, 12611], [17377, 1568], [17378, 12648], [17379, 1584], [17380, 35], [17381, 262], [17382, 51], [17383, 282], [17384, 8993], [17385, 512], [17386, 8995], [17387, 528], [17388, 323], [17389, 237], [17390, 339], [17391, 252], [17392, 9344], [17393, 9047], [17394, 9346], [17395, 9049], [17396, 448], [17397, 9029], [17398, 464], [17399, 9031], [17400, 12626], [17401, 1088], [17402, 12663], [17403, 1104], [17404, 13], [17405, 134], [17406, 28], [17407, 154], [17408, 12616], [17409, 8420], [17410, 12653], [17411, 8422], [17412, 227], [17413, 6], [17414, 243], [17415, 26], [17416, 12629], [17417, 9066], [17418, 12666], [17419, 9069], [17420, 12642], [17421, 8770], [17422, 12679], [17423, 8772], [17424, 8903], [17425, 8716], [17426, 8905], [17427, 8719], [17428, 768], [17429, 12636], [17430, 784], [17431, 12673], [17432, 8884], [17433, 8402], [17434, 8886], [17435, 8404], [17436, 131], [17437, 12625], [17438, 147], [17439, 12662], [17440, 1344], [17441, 1056], [17442, 1360], [17443, 1072], [17444, 12614], [17445, 13882], [17446, 12651], [17447, 13890], [17448, 8921], [17449, 8475], [17450, 8923], [17451, 8477], [17452, 141], [17453, 12632], [17454, 156], [17455, 12669], [17456, 9308], [17457, 8975], [17458, 9310], [17459, 8977], [17460, 9290], [17461, 14420], [17462, 9292], [17463, 14426], [17464, 8957], [17465, 8510], [17466, 8959], [17467, 8512], [17468, 96], [17469, 8511], [17470, 112], [17471, 8513], [17472, 14421], [17473, 1664], [17474, 14427], [17475, 1680], [17476, 173], [17477, 163], [17478, 188], [17479, 179], [17480, 14080], [17481, 9217], 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7700], [18650, 12040], [18651, 7701], [18652, 7960], [18653, 10458], [18654, 7961], [18655, 10460], [18656, 12983], [18657, 12096], [18658, 13020], [18659, 12098], [18660, 7974], [18661, 7708], [18662, 7975], [18663, 7709], [18664, 13965], [18665, 9015], [18666, 13973], [18667, 9017], [18668, 8034], [18669, 13005], [18670, 8035], [18671, 13042], [18672, 9088], [18673, 7712], [18674, 9090], [18675, 7713], [18676, 12213], [18677, 11996], [18678, 12215], [18679, 11998], [18680, 12157], [18681, 12137], [18682, 12159], [18683, 12139], [18684, 7944], [18685, 13000], [18686, 7945], [18687, 13037], [18688, 11972], [18689, 9032], [18690, 11975], [18691, 9034], [18692, 9436], [18693, 7188], [18694, 9439], [18695, 7189], [18696, 13014], [18697, 10586], [18698, 13051], [18699, 10588], [18700, 12078], [18701, 11928], [18702, 12081], [18703, 11931], [18704, 7356], [18705, 7196], [18706, 7357], [18707, 7197], [18708, 13003], [18709, 7500], [18710, 13040], [18711, 7501], [18712, 12195], [18713, 9051], [18714, 12197], [18715, 9053], [18716, 7180], [18717, 11020], [18718, 7181], [18719, 11022], [18720, 14162], [18721, 12991], [18722, 14171], [18723, 13028], [18724, 12980], [18725, 11929], [18726, 13017], [18727, 11932], [18728, 13966], [18729, 12038], [18730, 13974], [18731, 12041], [18732, 7612], [18733, 6144], [18734, 7613], [18735, 6160], [18736, 12036], [18737, 9602], [18738, 12039], [18739, 9604], [18740, 12115], [18741, 5920], [18742, 12117], [18743, 5936], [18744, 13680], [18745, 11952], [18746, 13685], [18747, 11954], [18748, 8036], [18749, 13079], [18750, 8037], [18751, 13116], [18752, 9182], [18753, 7732], [18754, 9184], [18755, 7733], [18756, 12417], [18757, 12000], [18758, 12419], [18759, 12002], [18760, 12357], [18761, 12337], [18762, 12359], [18763, 12339], [18764, 13077], [18765, 13074], [18766, 13114], [18767, 13111], [18768, 11978], [18769, 9126], [18770, 11981], [18771, 9128], [18772, 7728], [18773, 7208], [18774, 7729], [18775, 7209], [18776, 13088], [18777, 10590], [18778, 13125], [18779, 10592], [18780, 9165], [18781, 12276], [18782, 9167], [18783, 12279], [18784, 8078], [18785, 7216], [18786, 8079], [18787, 7217], [18788, 7200], [18789, 8150], [18790, 7201], [18791, 8151], [18792, 12397], [18793, 9147], [18794, 12399], [18795, 9149], [18796, 7962], [18797, 11038], [18798, 7963], [18799, 11040], [18800, 14180], [18801, 13065], [18802, 14189], [18803, 13102], [18804, 13054], [18805, 11935], [18806, 13091], [18807, 11938], [18808, 13982], [18809, 12232], [18810, 13990], [18811, 12235], [18812, 9456], [18813, 8206], [18814, 9459], [18815, 8207], [18816, 12230], [18817, 9620], [18818, 12233], [18819, 9622], [18820, 12319], [18821, 7780], [18822, 12321], [18823, 7781], [18824, 13690], [18825, 11956], [18826, 13695], [18827, 11958], [18828, 12282], [18829, 9241], [18830, 12285], [18831, 9243], [18832, 8080], [18833, 7232], [18834, 8081], [18835, 7233], [18836, 8152], [18837, 13151], [18838, 8153], [18839, 13188], [18840, 12507], [18841, 9221], [18842, 12509], [18843, 9223], [18844, 11056], [18845, 9476], [18846, 11058], [18847, 9479], [18848, 14198], [18849, 13139], [18850, 14207], [18851, 13176], [18852, 13128], [18853, 11941], [18854, 13165], [18855, 11944], [18856, 13998], [18857, 12238], [18858, 14006], [18859, 12241], [18860, 7964], [18861, 8208], [18862, 7965], [18863, 8209], [18864, 12236], [18865, 9638], [18866, 12239], [18867, 9640], [18868, 8290], [18869, 12435], [18870, 8291], [18871, 12437], [18872, 13700], [18873, 12042], [18874, 13705], [18875, 12044], [18876, 13202], [18877, 11961], [18878, 13239], [18879, 11964], [18880, 14014], [18881, 12244], [18882, 14022], [18883, 12247], [18884, 7966], [18885, 8210], [18886, 7967], [18887, 8211], [18888, 12242], [18889, 9658], [18890, 12245], [18891, 9660], [18892, 8292], [18893, 12439], [18894, 8293], [18895, 12441], [18896, 13710], [18897, 12046], [18898, 13715], [18899, 12048], [18900, 8212], [18901, 12443], [18902, 8213], [18903, 12445], [18904, 13720], [18905, 12066], [18906, 13725], [18907, 12068]]
diff --git a/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/newg/replay.py b/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/newg/replay.py
new file mode 100644
index 0000000000..a0af83c245
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/newg/replay.py
@@ -0,0 +1,208 @@
+"""Scalar prototype: source471 with168 frame-preserving commuting operations retimed after centers.
+Prepared with OpenAI Codex assistance; Apache-2.0.
+"""
+from pathlib import Path
+from collections import Counter,defaultdict
+import json,sys,importlib.util,time,gc,hashlib
+sys.dont_write_bytecode=True
+HERE=Path(__file__).resolve().parent;D=HERE.parent/'joint';spec=importlib.util.spec_from_file_location('source82',D/'joint_word.py');m=importlib.util.module_from_spec(spec);spec.loader.exec_module(m)
+W=m.Candidate();C=W.C
+W.changed_frames=[i for i,(a,b)in enumerate(zip(W.original_opframe,W.opframe))if a!=b]
+W.endframe={s:W.opframe[xs[-1]]for s,xs in W.role_ops.items()}
+C.decoder();C.geometry();selection=json.loads((HERE.parent/'borrow/selection.json').read_text());new_gauge_selection=json.loads((HERE/'selection.json').read_text());new_gauge_selection=new_gauge_selection['selection']if isinstance(new_gauge_selection,dict)else new_gauge_selection;gauge_selection=json.loads((HERE.parent/'gaugeb/selection.json').read_text())+new_gauge_selection
+schedule=json.loads((HERE/'schedule.json').read_text())
+original_phase1=W.phase1[:];original_rest=W.rest[:];omitted_for_schedule={r['omit_copy']for r in selection};deferred=schedule['deferred_ops'];assert len(deferred)==168
+W.phase1=schedule['new_phase1'];W.rest=deferred+original_rest
+assert set(W.phase1)|set(W.rest)|omitted_for_schedule==set(range(len(W.ops)))
+assert not(set(W.phase1)&set(W.rest))and not(set(W.phase1+W.rest)&omitted_for_schedule)
+W.role_ops=defaultdict(list)
+for i in W.phase1+W.rest:
+ for role in W.ops[i][:2]:W.role_ops[role].append(i)
+pos={i:j for j,i in enumerate(W.rest)};W.first={role:pos.get(ii[0],-1)for role,ii in W.role_ops.items()};W.last={role:pos.get(ii[-1],-1)for role,ii in W.role_ops.items()};W.position={i:j for j,i in enumerate(W.phase1+W.rest)}
+W.endframe={role:W.opframe[ii[-1]]for role,ii in W.role_ops.items()}
+W.readtime={role:t+168 for role,t in W.readtime.items()};new_phase_roles={8584,9424}
+assert new_phase_roles<=set(schedule['new_untouched_roles'])
+for role,t in W.readtime.items():W.w['reads'][str(role)]=len(W.phase1)+t
+knew={e['passive']:e for e in W.k['entries']};raw_response=W.adjoint();newroles=[];target_frames={};touched={s for i in W.phase1 for s in W.ops[i][:2]}
+for r in new_gauge_selection:
+ role,source=r['role'],r['source'];mix=knew[source]['mix_frame'];f=W.register(r['gauge_basis'])if 'gauge_basis'in r else mix;targets=sorted(raw_response[role]);assert targets==r['targets'] and mix==r['mix_frame'] and C.sub(mix,f)
+ assert role not in W.gauge and role not in W.donor and role not in W.source.values() and role not in touched
+ assert C.dimf[f]in(2,4) and C.nondeg(f) and C.sub(f,W.opframe[W.role_ops[role][0]])
+ assert all(t not in target_frames or C.sub(target_frames[t],f) and C.sub(f,target_frames[t])for t in targets);target_frames.update({t:f for t in targets})
+ assert all(all(W.module.dot(C.cov[t],b)==0 for b in C.B[f])for t in targets)
+ z=dict(role=role,frame=f,dim=C.dimf[f],targets=targets);W.gauge[role]=z;W.w['gauges'].append(z);W.readtime[role]=0 if role in new_phase_roles else 168;W.w['reads'][str(role)]=len(W.phase1)+W.readtime[role];newroles.append(role)
+W.order=newroles+W.order
+W.exact_frames()
+assert all(0<=W.readtime[s]<=W.first[s]for s in W.gauge)
+assert all(W.last[d]<W.readtime[b]and C.sub(W.endframe[d],W.gauge[b]['frame'])for b,d in W.pairs)
+assert all(not(set(W.gauge)&set(W.ops[i][:2]))for i in W.phase1)
+print('PASS scalar schedule timing, all gauge/alias readtimes, new source prefix inclusions; independent target/reflected ledger required',flush=True)
+borrow={r['role']:r['source']for r in selection};by_source={r['source']:r for r in selection};omitted={r['omit_copy']for r in selection};early={r['early']:r for r in selection};assert len(early)==len(borrow)==len(selection)==440
+gauge_borrow={r['role']:r['source']for r in gauge_selection};kpair={e['passive']:e for e in W.k['entries']};used_sources=set(by_source)|{r['partner']for r in selection}
+assert len(gauge_borrow)==len(gauge_selection) and len(set(gauge_borrow.values()))==len(gauge_selection)
+for r in gauge_selection:
+ assert r['source']in kpair and r['partner']==kpair[r['source']]['carrier']
+ assert r['role']in W.gauge and r['role']not in W.donor and r['role']not in borrow
+ assert r['source']not in used_sources and r['partner']not in used_sources
+ used_sources.update([r['source'],r['partner']])
+borrow.update(gauge_borrow);by_source.update({r['source']:r for r in gauge_selection})
+input_paths=[*W.input_paths,W.framepath,D/'joint_word.py',D/'joint-selection.json',D/'pairs.json',Path(W.module.__file__),D/'joint-replay.json',(HERE/'selection.json'),(HERE.parent/'gaugeb/selection.json'),Path(__file__),HERE/'schedule.json',(HERE.parent/'borrow/selection.json')]
+input_pins={str(p):hashlib.sha256(p.read_bytes()).hexdigest()for p in input_paths}
+adj=[{}for _ in range(W.R)]
+for r,s in zip(W.g['roots'],W.w['rootroles']):
+ for t in r['targets']:adj[s][t]=adj[s].get(t,0)+1
+for i in reversed(W.phase1+W.rest):
+ if i in omitted:continue
+ a,b,_=W.ops[i]
+ for t,c in adj[a].items():adj[b][t]=adj[b].get(t,0)+c
+oldadj=W.adjoint()
+assert all(all(0<c<=oldadj[s].get(t,0)for t,c in row.items())for s,row in enumerate(adj))
+assert all(adj[s]==oldadj[s]for s in W.gauge), 'all gauge reads and paid target supports retained exactly'
+assert len(by_source)==440+len(gauge_selection) and all(r['early'] in r['windows']for r in selection)
+# The formal column layout excludes each actual borrowed dirty slot, retaining
+# distinct original X columns and every other arbitrary dirty input.
+terminal=json.loads((D/'joint-replay.json').read_text())['selected'];removed={e['role']for e in terminal};deletedroots={e['root']for e in terminal};bywrite={i:e for e in terminal for i in e['writes']};afterwrite={e['writes'][-1]:e for e in terminal};assert not(set(early)&set(bywrite)) and not(set(borrow)&removed)
+regs=sorted(set(W.phys.values())-set(borrow)-removed);idx={r:2*W.v+j for j,r in enumerate(regs)}
+assert not set(borrow)&set(W.donor)
+at=defaultdict(list)
+for b in W.order:at[W.readtime[b]].append(b)
+deliveries=defaultdict(list)
+for e in W.k['entries']:deliveries[e['deliver_after_root']].append(e)
+
+def replay(mode='F2',direction=1,bits=24,mutation=None):
+ major=mode=='bound';binary=mode=='F2';v=W.v
+ unit=(lambda i:1)if major else(lambda i:1<<i)if binary else(lambda i:1<<(bits*i))
+ def add(x,c,y):return x+abs(c)*y if major else x^y if binary and c&1 else x if binary else x+c*y
+ target_frames=[None]*v;target_paths=[[]for _ in range(v)];center_reads=[];target_hist=Counter()
+ def target_move(t,f):
+  previous=target_frames[t];assert previous is None or C.sub(previous,f),('actual target chronology',t,previous,f)
+  assert all(W.module.dot(C.cov[t],b)==0 for b in C.B[f]),('actual target cap',t,f)
+  d=C.dimf[f]-(0 if previous is None else C.dimf[previous]);assert d>=0
+  if d:target_hist[d]+=1
+  target_frames[t]=f;target_paths[t].append(f)
+ X0=[unit(i)for i in range(v)];Y0=[unit(v+i)for i in range(v)];Z0={r:unit(idx[r])for r in regs};x=X0[:];y=Y0[:];z=Z0.copy();top=1;done=[]
+ def track(value):
+  nonlocal top
+  if major:top=max(top,value)
+  return value
+ def value(s):
+  s=W.phys[s];return x[borrow[s]]if s in borrow else z[s]
+ def assign(s,val):
+  s=W.phys[s]
+  if s in borrow:x[borrow[s]]=track(val)
+  else:z[s]=track(val)
+ responses=W.adjoint() if mutation=='old_adjoint' else adj
+ def read(s):
+  if mutation=='omit_gauge_compensation' and s==new_gauge_selection[-1]['role']:return
+  for t,c in responses[s].items():
+   if s in W.gauge:target_move(t,W.gauge[s]['frame'])
+   else:assert target_frames[t]is None,'initial correction leaves target atD0'
+   y[t]=track(add(y[t],-direction*c,value(s)))
+ def gate(i,sign):
+  a,b,_=W.ops[i];assign(a,add(value(a),sign,value(b)))
+ for s in range(W.R):
+  if s not in W.gauge and s not in borrow and s not in removed:read(s)
+ for n,s in W.source.items():assign(s,add(value(s),1,x[n]))
+ for r in gauge_selection:
+  if mutation!='omit_gauge_mix' or r is not new_gauge_selection[-1]:x[r['partner']]=track(add(x[r['partner']],1,x[r['source']]))
+ def forward(i):
+  if i not in omitted:gate(i,1);done.append(i)
+ for k in schedule['center_prefix_events']:
+  event=schedule['events'][k];i=event['op']
+  if event['kind']=='early_mix':
+   r=early[i];a,b=r['partner'],r['source']
+   if mutation!='omit_early_mix' or r is not selection[0]:x[a]=track(add(x[a],1,x[b]))
+  else:forward(i)
+ for r,s in zip(W.g['roots'],W.w['rootroles']):
+  if r['kind']=='center':
+   for t in r['targets']:
+    assert all(f is None for f in target_frames),'copied centers scatter before any nonzero target movement'
+    center_reads.append((s,t));y[t]=track(add(y[t],direction,value(s)))
+ for e in terminal:
+  assert all(target_frames[t]is None for t in e['targets']),'terminal setup afterD0center scatter'
+  for t in e['targets']:
+   if t!=e['pivot']:y[t]=track(add(y[t],-1,y[e['pivot']]))
+ for j,i in enumerate(W.rest):
+  for s in at[j]:
+   if mutation!='late_phase1_gauge' or s not in new_phase_roles:read(s)
+  if mutation=='late_phase1_gauge'and j==168:
+   for s in sorted(new_phase_roles):read(s)
+  if i in bywrite:
+   e=bywrite[i];a,b,n=W.ops[i];target_move(e['pivot'],W.opframe[i]);y[e['pivot']]=track(add(y[e['pivot']],direction,value(b)))
+  else:forward(i)
+  if i in afterwrite:
+   e=afterwrite[i]
+   for t in e['targets']:
+    target_move(t,e['root_frame'])
+    if t!=e['pivot']:y[t]=track(add(y[t],1,y[e['pivot']]))
+ for s in at[len(W.rest)]:read(s)
+ for j,(r,s)in enumerate(zip(W.g['roots'],W.w['rootroles'])):
+  if r['kind']=='side' and j not in deletedroots:
+   for t in r['targets']:target_move(t,W.w['root_frame'][j]);y[t]=track(add(y[t],direction,value(s)))
+  for e in deliveries[j]:
+   a,b=e['carrier'],e['passive']
+   if b not in by_source:x[a]=track(add(x[a],1,x[b]))
+   for t in e['receivers']:target_move(t,e['deliver_frame']);y[t]=track(add(y[t],direction,x[a]))
+ for e in W.k['entries']:
+  if e['passive']not in by_source:x[e['carrier']]=track(add(x[e['carrier']],-1,x[e['passive']]))
+ if mutation=='undo_before_restore':
+  for e in W.k['entries']:
+   if e['passive']in by_source:x[e['carrier']]=track(add(x[e['carrier']],-1,x[e['passive']]))
+ for i in reversed(done):gate(i,-1)
+ if mutation!='undo_before_restore':
+  for e in W.k['entries']:
+   if e['passive']in by_source:x[e['carrier']]=track(add(x[e['carrier']],-1,x[e['passive']]))
+ for n,s in W.source.items():assign(s,add(value(s),-1,x[n]))
+ if binary:want=[Y0[t]^X0[t]for t in range(v)]
+ else:
+  want=Y0[:];cache={}
+  for root in W.g['roots']:
+   n=root['node']
+   if n not in cache:
+    mask=C.sup[n];val=0
+    while mask:low=mask&-mask;mask-=low;val=add(val,1,X0[low.bit_length()-1])
+    cache[n]=val
+   for t in root['targets']:want[t]=add(want[t],direction,cache[n])
+  for e in W.k['entries']:
+   val=add(X0[e['carrier']],1,X0[e['passive']])
+   for t in e['receivers']:want[t]=add(want[t],direction,val)
+ # Finish every local target at its codimension-one cap, then reverse the exact
+ # annihilator stream. All copied-center scatters lie before these movements,
+ # hence atD0 forward and atD1 after the complementary movements reverse.
+ reflected=Counter()
+ for t,chain in enumerate(target_paths):
+  assert chain
+  d=W.h-1-C.dimf[chain[-1]];assert d>=0
+  if d:target_hist[d]+=1
+  previous=[C.cov[t]];dimension=1
+  for f in reversed(chain):
+   assert all(W.module.dot(a,b)==0 for a in previous for b in C.B[f])
+   basis=C.A[f];step=len(basis)-dimension;assert step>=0
+   if step:reflected[step]+=1
+   previous=basis;dimension=len(basis)
+  if W.h>dimension:reflected[W.h-dimension]+=1
+ assert reflected==target_hist
+ assert len(center_reads)==sum(len(r['targets'])for r in W.g['roots']if r['kind']=='center')
+ assert sum(k*n for k,n in target_hist.items())==W.v*(W.h-1)
+ if major:return max(top,max(a+b for a,b in zip(y,want)),max(z[r]+Z0[r]for r in regs),max(a+b for a,b in zip(x,X0)))
+ assert y==want,('wrong targets',mode,[t for t in range(v)if y[t]!=want[t]][:10])
+ assert x==X0,('source restoration',mode,[s for s in range(v)if x[s]!=X0[s]][:10])
+ assert z==Z0,('dirty restoration',mode)
+ return dict(mode=mode,direction=direction,formal_columns=2*v+len(regs),sources=v,targets=v,dirty=len(regs),all_targets=True,all_source_and_dirty_restored=True,copied_center_target_reads=len(center_reads),all_forward_centers_at_D0=True,all_reflected_centers_at_D1=True,actual_target_histogram={str(k):n for k,n in sorted(target_hist.items())},actual_reflected_target_histogram={str(k):n for k,n in sorted(reflected.items())})
+if __name__=='__main__':
+ start=time.monotonic();out=dict(status='PASS_SCALAR_SOURCE471_RETIMED_PHASE1_GAUGES',selection=selection,gauge_selection=gauge_selection,new_gauge_selection=new_gauge_selection,geometry_and_paid_admission=False)
+ out['F2']=replay();print('PASS F2',out['F2'],'seconds',time.monotonic()-start,flush=True)
+ bound=replay('bound');bits=8*((bound.bit_length()+2+7)//8);out['bound']=bound;out['bits']=bits;out['terminal_sinks']=len(terminal);print('bound',bound,'bits',bits,flush=True)
+ out['integer']=[]
+ if True:
+  for direction in(1,-1):out['integer'].append(replay('Z',direction,bits));gc.collect();print('PASS integer',direction,'seconds',time.monotonic()-start,flush=True)
+ out['controls']={}
+ for mutation in ['omit_early_mix','old_adjoint','undo_before_restore','omit_gauge_mix','omit_gauge_compensation','late_phase1_gauge']:
+  try:replay(mutation=mutation)
+  except AssertionError as error:out['controls'][mutation]=dict(status='REJECTED',error=str(error))
+  else:raise AssertionError('Invalid mutation accepted: '+mutation)
+ assert (1<<bits)>2*bound
+ assert input_pins=={str(p):hashlib.sha256(p.read_bytes()).hexdigest()for p in input_paths}
+ out['input_pins']=input_pins
+ (HERE/'replay.json').write_text(json.dumps(out,indent=2)+'\n')
+ print('PASS actual471 source word, both integer signs, all columns and six controls',flush=True)
diff --git a/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/newg/schedule.json b/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/newg/schedule.json
new file mode 100644
index 0000000000..51c2c8df5a
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/newg/schedule.json
@@ -0,0 +1 @@
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diff --git a/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/newg/selection.json b/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/newg/selection.json
new file mode 100644
index 0000000000..3a0ad4bcfb
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/source_inputs/pr210/research/coordinated-crossover-pr200/newg/selection.json
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+    "mix_frame": 125501,
+    "targets": [
+      24,
+      25,
+      26,
+      27
+    ],
+    "gauge_basis": [
+      [
+        1,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        0,
+        0
+      ],
+      [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        -1,
+        0,
+        0,
+        0
+      ],
+      [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        -1,
+        0,
+        0
+      ],
+      [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        -1,
+        0,
+        0,
+        0,
+        0,
+        -1,
+        1,
+        0,
+        0
+      ]
+    ]
+  },
+  {
+    "role": 13088,
+    "source": 1634,
+    "partner": 1633,
+    "mix_frame": 122675,
+    "targets": [
+      1624,
+      1625,
+      1626,
+      1627
+    ]
+  },
+  {
+    "role": 8584,
+    "source": 98,
+    "partner": 97,
+    "mix_frame": 126131,
+    "targets": [
+      208,
+      209,
+      210,
+      211
+    ]
+  },
+  {
+    "role": 9424,
+    "source": 458,
+    "partner": 457,
+    "mix_frame": 125321,
+    "targets": [
+      568,
+      569,
+      570,
+      571
+    ]
+  }
+]
diff --git a/research/five-stage-source527-banks/loader/source_loader.py b/research/five-stage-source527-banks/loader/source_loader.py
new file mode 100644
index 0000000000..5665769ab2
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/source_loader.py
@@ -0,0 +1,207 @@
+#!/usr/bin/env python3
+"""Source-bound current PR210 preparation with no historical search imports.
+
+This is a focused preparation adapter, not a full scalar or supplier verifier.
+The pinned public joint and source471 preparation bodies are retained literally;
+only the explicit base source root and terminal-selector source are rebound.
+Original source-specific notices remain beside the supplied public inputs.
+New path adapter and semantic fingerprints: assisted with ChatGPT.
+"""
+from collections import Counter, defaultdict
+from copy import deepcopy
+from pathlib import Path
+import argparse
+import contextlib
+import gzip
+import hashlib
+import importlib.util
+import io
+import json
+import sys
+
+sys.dont_write_bytecode = True
+if not __debug__:
+    raise SystemExit('refusing optimized Python: assertions are required')
+HERE = Path(__file__).resolve().parent
+BASE_PACKAGE = 'research/paired-cube-diagonal-bit-168'
+PR210_PACKAGE = 'research/coordinated-crossover-pr200'
+
+
+def sha(data):
+    return hashlib.sha256(data).hexdigest()
+
+
+def canonical(value):
+    return json.dumps(value, sort_keys=True, separators=(',', ':')).encode()
+
+
+def check_sources(pr210_root, base_bit_root):
+    """Validate all inputs before importing any source implementation."""
+    roots = {'pr210': Path(pr210_root).resolve(), 'base_bit': Path(base_bit_root).resolve()}
+    pins = json.loads((HERE/'SOURCE_PINS.json').read_text())
+    checked = {}
+    for group, root in roots.items():
+        checked[group] = {}
+        for relative, expected in pins[group]['files'].items():
+            path = (root/relative).resolve()
+            if not path.is_relative_to(root):
+                raise ValueError('source path escapes supplied root')
+            got = sha(path.read_bytes())
+            if got != expected:
+                raise ValueError('source hash mismatch: '+group+'/'+relative)
+            checked[group][relative] = got
+    return roots, pins, checked
+
+
+def import_file(name, path):
+    spec = importlib.util.spec_from_file_location(name, path)
+    module = importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(module)
+    return module
+
+
+def terminal_records(word, selectors):
+    """Derive terminal data from pinned roots and selectors, never a receipt."""
+    writes = defaultdict(list)
+    for i, (a, b, node) in enumerate(word.ops):
+        writes[a].append(i)
+    result = []
+    for selected in selectors:
+        root = selected['root']; pivot = selected['pivot']
+        role = word.w['rootroles'][root]
+        entry = word.g['roots'][root]
+        assert entry['kind'] == 'side' and pivot in entry['targets']
+        assert role not in word.source.values() and role not in word.gauge
+        result.append(dict(root=root, pivot=pivot, role=role,
+                           targets=entry['targets'], root_frame=word.w['root_frame'][root],
+                           writes=writes[role]))
+    assert len(result) == 34
+    assert len({r['role'] for r in result}) == 34
+    return result
+
+
+def prepare(pr210_root, base_bit_root):
+    roots, pins, checked = check_sources(pr210_root, base_bit_root)
+    p = roots['base_bit']/BASE_PACKAGE
+    package = roots['pr210']/PR210_PACKAGE
+    base = import_file('five_stage_public_base_word', p/'bit/word.py')
+    joint_path = package/'joint/joint_word.py'
+    joint = joint_path.read_text()
+    old = "P=ROOT/'research/paired-cube-diagonal-bit-168'\nsys.path.insert(0,str(P/'bit'))\nfrom word import Candidate as Original"
+    assert joint.count(old) == 1, 'joint import boundary changed'
+    joint = joint.replace(old, 'P=__base_package__\nOriginal=__base_candidate__', 1)
+    js = {'__name__': 'five_stage_public_joint', '__file__': str(joint_path),
+          '__base_package__': p, '__base_candidate__': base.Candidate}
+    exec(compile(joint, str(joint_path)+' [explicit base-root binding]', 'exec'), js)
+    word = js['Candidate']()
+    c = word.C
+    # These are decoder/geometry checks, not all-column payload replay.
+    c.decoder(); c.geometry()
+    word.changed_frames = [i for i, (a, b) in enumerate(zip(word.original_opframe, word.opframe)) if a != b]
+    word.endframe = {r: word.opframe[ii[-1]] for r, ii in word.role_ops.items()}
+    selection = json.loads((package/'borrow/selection.json').read_text())
+    gauges = json.loads((package/'gaugeb/selection.json').read_text())
+    current = json.loads((package/'newg/selection.json').read_text())
+    if isinstance(current, dict):
+        current = current['selection']
+    terminal = terminal_records(word, json.loads((p/'selected/bit/sinks.json').read_text()))
+    source_path = package/'newg/replay.py'
+    source = source_path.read_text()
+    start = source.index('schedule=json.loads')
+    stop = source.index('\ndef replay(')
+    preparation = source[start:stop]
+    # Original input_paths names an output-only joint replay receipt. All actual
+    # code/data inputs have already been checked against SOURCE_PINS.json.
+    lines = preparation.splitlines()
+    removed = [line for line in lines if line.startswith(('input_paths=', 'input_pins='))]
+    assert len(removed) == 2, 'source input-list boundary changed'
+    preparation = '\n'.join(line for line in lines if line not in removed)+'\n'
+    old = "terminal=json.loads((D/'joint-replay.json').read_text())['selected'];"
+    assert preparation.count(old) == 1, 'terminal source boundary changed'
+    preparation = preparation.replace(old, 'terminal=__terminal_records__;', 1)
+    namespace = {'__name__': 'five_stage_public_source471', '__file__': str(source_path),
+                 'Path': Path, 'Counter': Counter, 'defaultdict': defaultdict,
+                 'json': json, 'sys': sys, 'hashlib': hashlib,
+                 'HERE': package/'newg', 'D': package/'joint',
+                 'W': word, 'C': c, 'selection': selection,
+                 'new_gauge_selection': current, 'gauge_selection': gauges+current,
+                 '__terminal_records__': terminal}
+    with contextlib.redirect_stdout(io.StringIO()):
+        exec(compile(preparation, str(source_path)+' [preparation only; terminal selectors rebound]', 'exec'), namespace)
+    assert 'replay' not in namespace, 'payload replay must not be executed or loaded by preparation'
+    namespace['SOURCE_TEXT'] = source
+    namespace['source_pins'] = checked
+    namespace['source_revisions'] = {key: pins[key]['commit'] for key in roots}
+    namespace['source_preparation_sha256'] = sha(source[start:stop].encode())
+    namespace['loader_scope'] = 'Pinned source preparation and exact geometry only; no all-column payload replay, five-stage supplier admission, or final exponent.'
+    return namespace
+
+
+def semantic_fingerprint(context):
+    """Exact-basis semantics; generated frame IDs need not match legacy order."""
+    w, c = context['W'], context['C']
+    basis_cache = {}
+    def basis(f):
+        if f not in basis_cache:
+            basis_cache[f] = sha(canonical([list(r) for r in c.B[f]]))
+        return basis_cache[f]
+    def framed(entry):
+        out = dict(entry)
+        for key in ('frame', 'root_frame', 'mix_frame', 'deliver_frame'):
+            if key in out:
+                out[key] = basis(out[key])
+        for key in ('carrier_chain', 'passive_chain'):
+            if key in out:
+                out[key] = [basis(f) for f in out[key]]
+        return out
+    parts = dict(
+        partner_chronology=[framed(e) for e in w.k['entries']],
+        partner_lookup=[(n,framed(e)) for n,e in sorted(context['kpair'].items())],
+        graph=w.g,
+        operations=w.ops,
+        operation_bases=[basis(f) for f in w.opframe],
+        source_bases=[basis(f) for f in w.w['source_frame']],
+        root_bases=[basis(f) for f in w.w['root_frame']],
+        full_basis=basis(w.w['full_frame']),
+        gauge_records=[framed(w.gauge[r]) for r in sorted(w.gauge)],
+        gauge_read_order=w.order,
+        gauge_readtimes=sorted(w.readtime.items()),
+        donor_recipient_pairs=w.pairs,
+        physical_role_map=sorted(w.phys.items()),
+        phase1=w.phase1,
+        rest=w.rest,
+        schedule=context['schedule'],
+        source_loans=sorted(context['borrow'].items()),
+        terminal=[framed(e) for e in context['terminal']],
+        dirty_roles=context['regs'],
+        target_response=context['adj'],
+        live_gauge_reads=sorted(context['at'].items()),
+        source_selection=context['selection'],
+        gauge_selection=context['gauge_selection'],
+    )
+    hashes = {key: sha(canonical(value)) for key, value in parts.items()}
+    return dict(schema='source471-exact-semantic-fingerprint/1',
+                hashes=hashes, combined_sha256=sha(canonical(hashes)),
+                source_head='df95878d11190518e45ef9717c9ee05011f88ace',
+                h=w.h, v=w.v, logical_roles=w.R,
+                independent_dirty_roles=len(context['regs']),
+                local_independent_columns=2*w.v+len(context['regs']),
+                source_loans=len(context['borrow']),
+                terminal_sinks=len(context['terminal']),
+                retimed_operations=len(context['deferred']),
+                formal_frame_id_equality_required=False)
+
+
+def main():
+    ap = argparse.ArgumentParser(description=__doc__)
+    ap.add_argument('--pr210-root', type=Path, required=True)
+    ap.add_argument('--base-bit-root', type=Path, required=True)
+    args = ap.parse_args()
+    ctx = prepare(args.pr210_root, args.base_bit_root)
+    print(json.dumps(dict(status='PASS_PINNED_PUBLIC_SOURCE_PREPARATION',
+                         scope=ctx['loader_scope'],
+                         fingerprint=semantic_fingerprint(ctx),
+                         source_revisions=ctx['source_revisions']), indent=2, sort_keys=True))
+
+if __name__ == '__main__':
+    main()
diff --git a/research/five-stage-source527-banks/loader/test_public.py b/research/five-stage-source527-banks/loader/test_public.py
new file mode 100644
index 0000000000..255fce4f7d
--- /dev/null
+++ b/research/five-stage-source527-banks/loader/test_public.py
@@ -0,0 +1,88 @@
+#!/usr/bin/env python3
+"""Targeted preparation/ledger tests. No all-column scalar replay is invoked."""
+import argparse
+import json
+from pathlib import Path
+import subprocess
+import sys
+import unittest
+from unittest.mock import patch
+import source_loader
+import raw_ledger
+
+HERE=Path(__file__).resolve().parent
+PR210=HERE/'source_inputs/pr210'
+BASE=HERE/'source_inputs/base_bit'
+
+class IntegrityControls(unittest.TestCase):
+    def test_all_public_input_hashes(self):
+        _,_,pins=source_loader.check_sources(PR210,BASE)
+        self.assertEqual(sum(map(len,pins.values())),22)
+    def test_corrupt_input_rejected_before_import(self):
+        target=(PR210/source_loader.PR210_PACKAGE/'frames/opframe-bases.json').resolve()
+        original=Path.read_bytes
+        def corrupt(path):
+            data=original(path)
+            return data+b'\n' if path.resolve()==target else data
+        with patch.object(Path,'read_bytes',corrupt):
+            with self.assertRaisesRegex(ValueError,'source hash mismatch'):
+                source_loader.check_sources(PR210,BASE)
+    def test_missing_input_rejected(self):
+        target=(PR210/source_loader.PR210_PACKAGE/'newg/schedule.json').resolve()
+        original=Path.read_bytes
+        def missing(path):
+            if path.resolve()==target:raise FileNotFoundError(str(path))
+            return original(path)
+        with patch.object(Path,'read_bytes',missing):
+            with self.assertRaises(FileNotFoundError):source_loader.check_sources(PR210,BASE)
+    def test_optimized_execution_rejected(self):
+        proc=subprocess.run([sys.executable,'-O','-B',str(HERE/'source_loader.py'),'--help'],capture_output=True,text=True)
+        self.assertNotEqual(proc.returncode,0)
+        self.assertIn('refusing optimized',proc.stderr)
+
+class SemanticControls(unittest.TestCase):
+    @classmethod
+    def setUpClass(cls):
+        cls.context=source_loader.prepare(PR210,BASE)
+        cls.fingerprint=source_loader.semantic_fingerprint(cls.context)
+        cls.expected=json.loads((HERE/'expected_semantics.json').read_text())
+        cls.ledger=raw_ledger.reconstruct(cls.context)
+    def test_exact_semantic_fixture(self):
+        self.assertEqual(self.fingerprint,self.expected)
+    def test_no_scalar_replay_loaded(self):
+        self.assertNotIn('replay',self.context)
+        self.assertIn('SOURCE_TEXT',self.context)
+    def test_wrong_gauge_order_detected(self):
+        w=self.context['W'];before=w.order[:]
+        try:
+            w.order[0],w.order[1]=w.order[1],w.order[0]
+            self.assertNotEqual(source_loader.semantic_fingerprint(self.context)['hashes']['gauge_read_order'],self.expected['hashes']['gauge_read_order'])
+        finally:w.order[:]=before
+    def test_wrong_partner_delivery_detected(self):
+        e=self.context['W'].k['entries'][0];before=e['deliver_after_root']
+        try:
+            e['deliver_after_root']=before+1
+            changed=source_loader.semantic_fingerprint(self.context)['hashes']
+            self.assertNotEqual(changed['partner_chronology'],self.expected['hashes']['partner_chronology'])
+            self.assertNotEqual(changed['partner_lookup'],self.expected['hashes']['partner_lookup'])
+        finally:e['deliver_after_root']=before
+    def test_raw_ledger_fixture(self):
+        expected=json.loads((HERE/'expected_ledger.json').read_text())
+        for key,value in expected.items():self.assertEqual(self.ledger[key],value,key)
+    def test_all_center_and_completion_calls_paid(self):
+        self.assertEqual(self.ledger['paid_center_copy_histogram'],{'22':24})
+        profile=self.ledger['five_stage_profile']
+        self.assertEqual((profile['calls'],profile['rank_mass'],profile['deficit'],profile['maxchild']),(495304,2837560,4400,100))
+        self.assertEqual({r:profile['histogram'][r] for r in ['60','65','90','100']},{'60':18,'65':48,'90':13,'100':2200})
+
+if __name__=='__main__':
+    parser=argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--pr210-root',type=Path,default=PR210)
+    parser.add_argument('--base-bit-root',type=Path,default=BASE)
+    parser.add_argument('--pins-only',action='store_true')
+    args=parser.parse_args();PR210=args.pr210_root;BASE=args.base_bit_root
+    suite=unittest.TestSuite()
+    suite.addTests(unittest.defaultTestLoader.loadTestsFromTestCase(IntegrityControls))
+    if not args.pins_only:suite.addTests(unittest.defaultTestLoader.loadTestsFromTestCase(SemanticControls))
+    result=unittest.TextTestRunner(verbosity=2).run(suite)
+    raise SystemExit(not result.wasSuccessful())
diff --git a/research/five-stage-source527-banks/math_check.py b/research/five-stage-source527-banks/math_check.py
new file mode 100644
index 0000000000..d687bea3cb
--- /dev/null
+++ b/research/five-stage-source527-banks/math_check.py
@@ -0,0 +1,57 @@
+"""Source-bound banked five-stage moments and all47outerconstraints.
+Prepared with substantial OpenAI Codex assistance; Apache-2.0.
+Uses PR234 five-stage architecture and complex baseline, our527 helper and
+PR197-derived actual five-stage entrance banks. Inputs must be live verifier outputs.
+"""
+from pathlib import Path
+from fractions import Fraction as Q
+from collections import Counter
+import importlib.util
+HERE=Path(__file__).resolve().parent
+
+def load(name):
+ p=HERE/(name+'.py');s=importlib.util.spec_from_file_location('banked527_'+name,p);m=importlib.util.module_from_spec(s);s.loader.exec_module(m);return m
+
+def serial(x):
+ if isinstance(x,Q):return str(x)
+ if isinstance(x,dict):return {str(k):serial(v)for k,v in x.items()}
+ if isinstance(x,(list,tuple)):return list(map(serial,x))
+ return x
+
+def hist(x):return Counter({int(k):v for k,v in x.items()if v})
+
+def run(raw,complex_histogram,banks,global_result):
+ cost=load('moment');other=load('base_two_moment');outer=load('outer')
+ assert raw['source_aliases']==527 and raw['physical_R']==16587 and raw['h']==24 and raw['v']==1760
+ H=hist(raw['five_stage_profile']['histogram']);assert H==hist(global_result['paid_histogram'])
+ assert global_result['paid_rank_mass']==sum(k*n for k,n in H.items())==2830840
+ gauges=hist(raw['auxiliary_entrance_rank_histogram']);assert gauges=={20:2200,18:13,12:18,13:48}
+ assert banks['assignments']==4976100 and banks['source527_roles']==16587 and banks['physical_replicas']==60
+ assert banks['literal_stock']==1304935 and banks['banks_total']==882535 and banks['charts']==231
+ for a,n in gauges.items():assert H.pop(5*a)==n
+ assert len(H)>0 and min(H.values())>0 and max(H)==50
+ literal={k:n*60 for k,n in H.items()};literal_mass=sum(k*n for k,n in literal.items());literal_W=banks['literal_stock']
+ assert 120*literal_W-literal_mass==60*4400
+ # Divide the literal profile by5 to an integer normalization. The executed
+ # realization is still60replicas, and finite accounting uses its literalstock.
+ assert literal_W%5==0 and all(n%5==0 for n in literal.values())
+ normalized={k:n//5 for k,n in literal.items()};W=literal_W//5;m=120;mass=sum(k*n for k,n in normalized.items())
+ bp=dict(m=m,W=W,histogram=normalized,calls=sum(normalized.values()),rank_mass=mass,deficit=m*W-mass,maxchild=max(normalized),normalization=12,physical_replicas=60,literal_stock=literal_W,literal_children=sum(literal.values()),literal_rank_mass=literal_mass)
+ root=cost.certify(normalized,m,W,True);c=Q(int(Q(root['lower'])*10**18),10**18);bm=cost.moment(normalized,m,W,c,True);nextbm=cost.moment(normalized,m,W,c+Q(1,10**18),True);assert bm[1]<1<nextbm[0]
+ fallback=32*m*m*sum(normalized.values())
+ _,upper=other.moment(m,W,list(normalized.items()),c);_,bad=other.moment(m,W,[(1,fallback)],c)
+ lower,_=other.moment(m,W,list(normalized.items()),c+Q(1,10**18));badlower,_=other.moment(m,W,[(1,fallback)],c+Q(1,10**18));assert upper+Q(1,10**16)*bad<1<lower+Q(1,10**16)*badlower
+ CH=hist(complex_histogram);assert sum(CH.values())==361405 and sum(k*n for k,n in CH.items())==1613040
+ cp=dict(m=110,W=14692,histogram=dict(CH),calls=sum(CH.values()),rank_mass=sum(k*n for k,n in CH.items()),deficit=3080,maxchild=max(CH));b=Q(747454944651775,10**18);cm=cost.moment(dict(CH),110,14692,b,False);assert cm[1]<1
+ chain=[Q(384599,10**10)]
+ for _ in range(3):
+  a=(1-c)*c+c*chain[-1];assert chain[-1]<a<c<1-a;chain.append(a)
+ bit=chain[-1];eta=Q(1,10**12);beta=Q(1,10**9);assert bit<(1-beta)*b
+ bridge=dict(proof='PROOF.md',representation='Exact powers with source-bound finite overcharges',semantic=dict(G=dict(base=2,exponent=30000),E=dict(base=2,exponent=100000),B_upper=dict(base=2,exponent=100001),C0=dict(base=2,exponent=210000),C1=1,strict_literal_gap=1,induction_gap_lower=1),rows=dict(coefficient=20161,degree=10**6,suffix_slope=4*10**6,degree_gap=Q(10**6)-Q(51*20161,25)))
+ assert bridge['rows']['degree_gap']>0
+ q=bit*(1-2*eta);minimum=(1-eta)*q/(1+q);ticks=minimum*10**18;k=Q((ticks.numerator-1)//ticks.denominator,10**18)
+ assembly=outer.assembly(bit,b,bridge,k,eta=eta,beta=beta);assert len(assembly['strict_constraints'])==47 and len(assembly['margins'])==7 and min(assembly['strict_constraints'].values())>0
+ try:outer.assembly(bit,b,bridge,k+Q(1,10**18),eta=eta,beta=beta)
+ except AssertionError:pass
+ else:raise AssertionError('Adjacentkappaadmitted')
+ return serial(dict(status='PASS_BANKED_FIVE_STAGE527_EXACT_MOMENTS_AND_OUTER47',mathematics=dict(bit_profile=bp,complex_profile=cp,bit_coarse=c,complex_coarse=b,bit_moment_interval=bm,complex_moment_interval=cm,next_bit_grid_excluded=nextbm,independent_base_two_pass=True,ordinary_bootstrap_chain=chain,finite_bridge=bridge,assembly=assembly,kappa=k,kappa_decimal=cost.decimal(k),kappa_scientific=format(float(k),'.15e'),binding='bit'),adjacent_grid_point_rejected=True))
diff --git a/research/five-stage-source527-banks/moment.py b/research/five-stage-source527-banks/moment.py
new file mode 100644
index 0000000000..1e43aaeaca
--- /dev/null
+++ b/research/five-stage-source527-banks/moment.py
@@ -0,0 +1,114 @@
+"""Independent prospective ordinary five-stage cost, never supplier admission.
+
+Consumes a raw one-invocation ledger made from the actual source/internal/target
+paths.  No packed profile is used as an input.  Missing physical/finite/compiler
+obligations must remain explicit in the accompanying report.
+"""
+from collections import Counter
+from decimal import Decimal, localcontext, ROUND_FLOOR
+from fractions import Fraction as F
+from functools import lru_cache
+from math import factorial
+import hashlib, json, sys
+from pathlib import Path
+
+DYAD = 1 << 224
+
+def floor(x): return F(x.numerator*DYAD//x.denominator, DYAD)
+def ceil(x): return -floor(-x)
+
+@lru_cache(None)
+def logarithm(x):
+    """Mercator bounds after binary reduction; ln(2) via two Mercator sums.
+
+    For 1 <= y < 2, log(y) is an alternating series in z=y-1.
+    We instead reduce to 1 <= y < 3/2, and bound log(3/2) and
+    log(2) by the same alternating series at z=1/2 and z=1/3,
+    using log(2)=log(3/2)+log(4/3). All sums are exact rationals.
+    """
+    assert x >= 1
+    k = 0
+    while x >= 2: x /= 2; k += 1
+    j = int(x >= F(3, 2))
+    if j: x /= F(3, 2)
+    def mercator(y):
+        z=y-1
+        assert 0 <= z <= F(1,2)
+        # An even partial sum is below log; the next term bounds its tail.
+        t=z; s=F(0)
+        for n in range(1, 225):
+            s += (t if n & 1 else -t)/n
+            t *= z
+        return s, s+t/225
+    a,b=mercator(x); c,d=mercator(F(3,2)); e,f=mercator(F(4,3))
+    return floor(a+(k+j)*c+k*e), ceil(b+(k+j)*d+k*f)
+
+def exponential(l,u):
+    assert 0 <= l <= u < F(1,10)
+    def poly(x):
+        s=t=F(1)
+        for j in range(1,25): t=t*x/j; s+=t
+        return s,t
+    lo,_=poly(l);hi,last=poly(u)
+    return floor(lo),ceil(hi+last*u/25/(1-u/26))
+
+def moment(H,m,W,a,bad=False):
+    lo=hi=F(0)
+    for r,n in H.items():
+        assert 0 < r < m and n > 0
+        l,u=logarithm(F(m,r)); e,f=exponential(a*l,a*u)
+        p=F(r*n,m*W);lo+=p*e;hi+=p*f
+    if bad:
+        # Existing envelope shape, only conditionally portable to the new cover.
+        l,u=logarithm(F(m));e,f=exponential(a*l,a*u)
+        p=F(32*m*sum(H.values()),10**16*W)
+        lo+=p*e;hi+=p*f
+    return floor(lo),ceil(hi)
+
+def decimal(x):
+    with localcontext() as c:
+        c.prec=70
+        return str(Decimal(x.numerator)/Decimal(x.denominator))
+
+def certify(H,m,W,bad=False):
+    with localcontext() as c:
+        c.prec=70
+        terms=[(Decimal(r*n)/Decimal(m*W),(Decimal(m)/Decimal(r)).ln()) for r,n in H.items()]
+        fallback=Decimal(32*m*sum(H.values()))/Decimal(10**16*W) if bad else Decimal(0)
+        lm=Decimal(m).ln();lo=Decimal(0);hi=Decimal('.003')
+        for _ in range(210):
+            a=(lo+hi)/2
+            if sum(p*(a*l).exp() for p,l in terms)+fallback*(a*lm).exp()<1:lo=a
+            else:hi=a
+        q=int(((lo+hi)/2*10**24).to_integral_value(rounding=ROUND_FLOOR))
+    lo=F(q,10**24);hi=lo+F(1,10**24)
+    lv=moment(H,m,W,lo,bad);hv=moment(H,m,W,hi,bad)
+    assert lv[1]<1<hv[0],(lo,hi,lv,hv)
+    k=F(694782719690783,10**18);required=k/(1-k)
+    at=moment(H,m,W,required,bad)
+    return dict(lower=decimal(lo),upper=decimal(hi),lower_endpoint_residual_upper=decimal(lv[1]-1),upper_endpoint_residual_lower=decimal(hv[0]-1),strict_rational_signs=True,moment_minus_one_at_current_k_required_rate=[decimal(x-1) for x in at],nominal_k_ceiling_interval=[decimal(lo/(1+lo)),decimal(hi/(1+hi))])
+
+def price(core,gauges,R,v=1760,h=24):
+    core=Counter({int(k):n for k,n in core.items() if n})
+    gauges=Counter({int(k):n for k,n in gauges.items() if n})
+    m=5*h;W=4*v+R;ell=24*22
+    assert sum(r*n for r,n in core.items())==h*R+2*v*(h-1)+ell-sum(r*n for r,n in gauges.items())
+    idle=Counter({r:2*v for r in (2*h-2,h-1,2*h+2,4)})
+    ans={}
+    for name,bundled in [('separate_rank_a_corrections',False),('bundled_rank_5a_corrections',True)]:
+        H=Counter({r:5*n for r,n in core.items()});H.update(idle)
+        for a,n in gauges.items():H[5*a if bundled else a]+=n if bundled else 5*n
+        mass=sum(r*n for r,n in H.items());assert m*W-mass==4*v-5*ell==4400
+        ans[name]=dict(histogram=dict(sorted(H.items())),calls=sum(H.values()),rank_mass=mass,deficit=m*W-mass,maxchild=max(H),root=certify(H,m,W),root_with_inherited_bad_envelope=certify(H,m,W,True))
+    return dict(status='PROSPECTIVE_COST_ONLY_NOT_PHYSICAL_OR_ASSEMBLY_ADMISSION',m=m,W=W,R=R,v=v,h=h,copy_calls_per_invocation=24,copy_rank=22,ell=ell,raw_core=dict(sorted(core.items())),auxiliary_entrances=dict(sorted(gauges.items())),idle_histogram=dict(sorted(idle.items())),profiles=ans,interval_method='Independent exact rational alternating Mercator log bounds (224 terms), 24-degree exponential with geometric tail, 224-bit outward dyadic rounding; Decimal only proposes endpoints.',bad_envelope_scope='10^-16 and 32*m^2 rank-one fallback calls per paid child are inherited assumptions, not a new m120 admission.')
+
+if __name__=='__main__':
+    path=Path(sys.argv[1]);raw=path.read_bytes();data=json.loads(raw)
+    core=data['one_stage_helper_histogram_including_copies']
+    gauges=data['auxiliary_entrance_rank_histogram']
+    result=price(core,gauges,data['physical_R'])
+    result['input_ledger_sha256']=hashlib.sha256(raw).hexdigest()
+    result['input_head']=data['head']
+    result['one_invocation_parts']={key:data[key] for key in ('physical_internal_excluding_center_copies','paid_center_copy_histogram','physical_source_histogram','physical_target_histogram')}
+    result['source_owned_gauges_not_banked']={19:3,2:26,4:2}
+    print(json.dumps(result,indent=2))
diff --git a/research/five-stage-source527-banks/outer.py b/research/five-stage-source527-banks/outer.py
new file mode 100644
index 0000000000..43007488ad
--- /dev/null
+++ b/research/five-stage-source527-banks/outer.py
@@ -0,0 +1,58 @@
+#!/usr/bin/env python3
+"""Exact semantic assembly with the balanced prefix, for the paired-cube network.
+
+Prepared by eumemic with Anthropic Claude assistance. Apache-2.0. The 47 strict constraints, the seven margins and
+the finite bridge are unchanged from icekylinx's scripts/structured_bulk_assembly.py (PR144, adapted from Zhihao
+Chen's PR23). The prefix is the RaD project's (hipotures) balanced positional transform, reviewed in
+references/semantic-bulk/rad20/reports/review-balanced-transform.md: the prefix margin becomes 1-eps instead of
+1-eps(1+c), and eps(1+c)<1 remains a strict geometric condition (K_geometry) whose gap need not exceed kappa.
+The parameters c=q+eta/4 and eps=(1-eta)/(1+q) are those of the reviewed balanced assembly in
+research/copied-fixed/balanced_assembly.py, with its backoff h equal to eta. Nothing else differs from
+structured_bulk_assembly.assembly.
+"""
+from fractions import Fraction as Q
+
+
+def assembly(bit_saving,complex_saving,bridge,kappa,eta=Q(1,10**8),beta=Q(1,4)):
+    a,b=bit_saving,complex_saving
+    tau,sigma=1-a,1-b
+    q=a*(1-2*eta); lp=1-q; lam=(tau+lp)/2
+    c=q+eta/4; eps=(1-eta)/(1+q)
+    minimum=eps*q; r=(minimum+1-eps)/2; delta=eta/8
+    internal=tau+(1-beta)*max(sigma-tau,Q(0));leaf=sigma+beta*(1-sigma)
+    margins=dict(balanced_prefix=1-eps,coordinate_movement=a,
+        compact_phase_layer=minimum,bulk_exposure=a,
+        Gaussian_arithmetic=min(1-eps-delta,r-delta),
+        scalar_work=1-eps-delta,dimension=eps)
+    slacks=dict(bit_positive=a,complex_above_bit=b-a,
+        complex_below_one_over32=Q(1,32)-b,beta_positive=beta,
+        beta_below_one=1-beta,leaf_saving_above_bit=(1-beta)*b-a,
+        q_positive=q,q_below_internal=1-internal-q,q_below_leaf=1-leaf-q,
+        c_positive=c,c_below_one=1-c,q_below_reservations=c-q,
+        lambda_above_tau=lam-tau,lambda_above_sigma=lam-sigma,
+        lambda_above_internal=lam-internal,lambda_prime_above_lambda=lp-lam,
+        compact_leaf=lp-leaf,compact_reservations=lp-(1-c),
+        lambda_prime_below_one=q,epsilon_positive=eps,epsilon_below_one=1-eps,
+        guard_width=1-eps,K_geometry=1-eps*(1+c),K_dominates_log=eps*c,
+        record_suffix=1-eps,phase_local=1-eps-delta,phase_boundary=r-delta,
+        gamma_sublinear=1-eps-r,cell_above_band=eps-(1-r)/2,
+        prime_interval_packing=1-eps,alpha_positive=r,alpha_below_one=1-r,
+        alpha_below_one_fourth=Q(1,4)-r,delta_positive=delta,
+        delta_below_one_eighth=Q(1,8)-delta,short_record_fallback=eps-a,
+        small_field_exposure=1-eps-minimum,
+        artificial_boundary=8-eps+r-delta-minimum,
+        literal_scalar_guard=Q(bridge['semantic']['strict_literal_gap']),
+        row_product_gap=bridge['rows']['degree_gap'])
+    slacks.update({name+'_above_kappa':val-kappa for name,val in margins.items()})
+    assert len(slacks)==47 and len(margins)==7
+    assert all(v>0 for v in slacks.values()),{k:str(v) for k,v in slacks.items() if v<=0}
+    assert min(margins.values())==minimum
+    assert margins['balanced_prefix']-minimum==eta and 1-eps-r==eta/2
+    assert 1-eps*(1+c)==eta-eps*eta/4
+    return dict(parameters=dict(a_bit=a,a_complex=b,tau=tau,sigma=sigma,
+            eta=eta,beta=beta,q=q,c=c,epsilon=eps,lambda_=lam,
+            lambda_prime=lp,alpha_squared_power=r,delta=delta,
+            C0=bridge['semantic']['C0'],C1=1,kappa=kappa),
+        strict_constraints=slacks,margins=margins,minimum_margin=minimum,
+        absorption_gap=minimum-kappa,
+        recurrence=dict(internal=internal,leaf=leaf,reservations=1-c))
diff --git a/research/five-stage-source527-banks/parity_transform.py b/research/five-stage-source527-banks/parity_transform.py
new file mode 100644
index 0000000000..11c218eafa
--- /dev/null
+++ b/research/five-stage-source527-banks/parity_transform.py
@@ -0,0 +1,143 @@
+"""F2 payload identity elision and exact nested address-MOVE fusion.
+The inherited signed source decoder is retained as producer evidence.
+All odd-prime address projectors and routes remain unchanged.
+Prepared with substantial OpenAI Codex assistance; Apache-2.0.
+"""
+from array import array
+from collections import Counter
+from pathlib import Path
+import gzip,hashlib,json,time
+if not __debug__:raise SystemExit('assertions required')
+
+def transform(records):
+    out = array('i')
+    pending = {}
+    chains = []
+    removed = Counter()
+    hist = Counter()
+    merged = 0
+    maxchain = 0
+    uses = Counter()
+
+    def flush(a):
+        nonlocal merged, maxchain
+        if a in pending:
+            old, new, rank, count, first, last = pending.pop(a)
+            if rank:
+                hist[rank] += 1
+            out.extend((0, a, old, new, rank, 0))
+            merged += count - 1
+            maxchain = max(maxchain, count)
+            if count > 1:
+                chains.append(dict(stream=a, old=old, new=new, rank=rank, moves=count, first=first, last=last))
+    for index in range(0, len(records), 6):
+        op, a, b, c, f, z = records[index:index + 6]
+        if op == 1 and c % 2 == 0:
+            removed[c] += 1
+            continue
+        if op == 0:
+            assert z == 0
+            if a in pending:
+                old, prev, rank, count, first, last = pending[a]
+                assert prev == b, ('disconnected movechain', a, prev, b)
+                pending[a] = (old, c, rank + f, count + 1, first, index // 6)
+            else:
+                pending[a] = (b, c, f, 1, index // 6, index // 6)
+            continue
+        assert op in (1, 2, 3)
+        flush(a)
+        flush(b)
+        if op == 1:
+            uses[a] += 1
+            uses[b] += 1
+        elif op == 2:
+            hist[z] += 1
+            uses[a] += 1
+            uses[b] += 1
+        out.extend((op, a, b, c, f, z))
+    for a in sorted(pending):
+        flush(a)
+    return (out, dict(removed_even_adds=dict(removed), removed_even_add_count=sum(removed.values()), merged_move_count=merged, max_chain=maxchain, fused_chains=len(chains), paid_histogram=dict(sorted(hist.items())), paid_calls=sum(hist.values()), paid_rank_mass=sum((r * n for r, n in hist.items())), surviving_additions=sum((out[k] == 1 for k in range(0, len(out), 6))), output_records=len(out) // 6, unused_streams=[i for i in range(20107) if not uses[i]], stream_use_counts=dict(uses)), chains)
+
+def run(producer,output_dir=None):
+    begun=time.monotonic();W,C=producer['W'],producer['C'];v=W.v;n=20107
+    old=producer['records'];out,stats,chains=transform(old);oldmeta=producer['physical']
+    assert stats['removed_even_add_count']==1783196 and stats['surviving_additions']==786621
+    assert stats['paid_rank_mass']==434502 and stats['merged_move_count']==3
+    initial=producer['initial_state'];ZERO,FULL=producer['ZERO'],producer['FULL']
+    # Independent required-frame census: ignore all original MOVE instructions.
+    # Use only surviving gate operands, copy sources and mathematical endpoints.
+    needs={i:[] for i in initial};copies=Counter();copy=None
+    for k in range(0,len(old),6):
+        op,a,b,c,f,z=old[k:k+6]
+        if op==1 and c%2:
+            needs[a].append(f)
+            if b!=n:needs[b].append(f)
+            else:assert copy is not None
+        elif op==2:
+            assert copy is None;copy=(a,b,c);needs[a].append(c);copies[z]+=1
+        elif op==3:assert copy==(a,b,c);copy=None
+    assert copy is None and copies=={22:24}
+    final={i:FULL for i in initial}
+    for t in range(v):final[v+t]=W.register(W.module.kernel([C.cov[t]],W.h)[0])
+    census=Counter(copies);per_role={};pairs=set()
+    for i,oldf in initial.items():
+        H=Counter()
+        for f in needs[i]+[final[i]]:
+            assert C.sub(oldf,f);rank=C.dimf[f]-C.dimf[oldf]
+            if rank:H[rank]+=1
+            pairs.add((oldf,f));oldf=f
+        per_role[i]=H;census.update(H)
+    assert census==Counter({int(r):c for r,c in stats['paid_histogram'].items()})
+    # Read back the actual annihilator rows as a separate reflected ledger.
+    # A(new) annihilating B(old) states new^perp <= old^perp exactly.
+    reflected_pairs=0;endpoint_frames={f for pair in pairs for f in pair};nondeg_bases=set()
+    for f in endpoint_frames:
+        assert len(C.B[f])==C.dimf[f] and len(C.A[f])+C.dimf[f]==24
+        key=tuple(map(tuple,C.B[f]))
+        if key not in nondeg_bases:
+            assert C.nondeg(f),'singular required endpoint frame'
+            nondeg_bases.add(key)
+    for oldf,newf in sorted(pairs):
+        assert len(C.A[newf])<=len(C.A[oldf])
+        assert all(W.module.dot(a,b)==0 for a in C.A[newf]for b in C.B[oldf])
+        reflected_pairs+=1
+    # Validate the actual transformed instruction stream against its own frames.
+    state=dict(initial);copy=None;used=set(initial.values());hist=Counter();cat=Counter();coeff=Counter();centers=[];count=0
+    scalar=hashlib.sha256();tagged=hashlib.sha256();category_names=oldmeta['category_names']
+    def event(d,row):d.update(json.dumps(row,separators=(',',':')).encode());d.update(b'\n')
+    for k in range(0,len(out),6):
+        op,a,b,c,f,z=out[k:k+6]
+        if op==0:
+            assert state[a]==b and C.sub(b,c) and C.dimf[c]-C.dimf[b]==f
+            # Complement reflection reverses containment and preserves the gap.
+            assert (24-C.dimf[b])-(24-C.dimf[c])==f
+            state[a]=c;used.update((b,c))
+            if f:hist[f]+=1
+        elif op==1:
+            assert c%2 and state[a]==state[b]==f and a!=b
+            kind=category_names[z];source=copy['source']if b==n else b
+            event(scalar,[a,source,c]);event(tagged,[a,b,c,f,kind]);count+=1;cat[kind]+=1;coeff[abs(c)]+=1;used.add(f)
+        elif op==2:
+            assert copy is None and b==n and state[a]==c and f==ZERO and z==22
+            state[b]=f;hist[z]+=1;copy=dict(source=a,frame=c,rank=z,temporary=b,first_event=count,copy_output_frame=f)
+        else:
+            assert op==3 and copy is not None and (a,b,c)==(copy['source'],copy['temporary'],copy['frame']) and state[b]==f==ZERO
+            copy.update(after_event=count,scatter_reads=count-copy['first_event']);assert copy['scatter_reads']==220
+            centers.append(copy);copy=None;del state[b]
+    assert copy is None and state==final and hist==census
+    assert count==786621 and coeff=={1:785301,3:1320} and len(centers)==24
+    delta=Counter(hist);delta.subtract(Counter({int(r):c for r,c in oldmeta['paid_histogram'].items()}));delta={r:c for r,c in delta.items()if c}
+    assert delta=={8:-3,12:-3,20:3}
+    sourceH=Counter();targetH=Counter();internalH=Counter(copies)
+    for i,H in per_role.items():(sourceH if i<v else targetH if i<2*v else internalH).update(H)
+    receipt=dict(status='PASS_FRESH_PARITY_FUSION_AND_INDEPENDENT_REQUIRED_FRAME_CENSUS',removed_even_add_count=stats['removed_even_add_count'],remaining_payload_additions=count,merged_moves=3,chains=chains,local_histogram_delta=delta,both_reflected_ledgers=True,unchanged_all_input_output_frames=True,unique_required_frame_pairs=len(pairs),explicit_reflected_annihilator_pairs=reflected_pairs,nondegenerate_endpoint_bases=len(nondeg_bases),source_histogram=dict(sourceH),target_histogram=dict(targetH),internal_histogram_including_copies=dict(internalH),copied_center_histogram=dict(copies),producer_scalar_sha256=oldmeta['scalar_projection_sha256'],producer_tagged_sha256=oldmeta['tagged_scalar_sha256'],transform_sha256=hashlib.sha256(Path(__file__).read_bytes()).hexdigest())
+    meta=dict(oldmeta);meta.update(status='PASS_PHYSICAL527_PARITY_FUSED_EXACT_FRAME_AND_SCALAR_PROJECTION',scalar_projection_sha256=scalar.hexdigest(),tagged_scalar_sha256=tagged.hexdigest(),weighted_scalar_events=count,coefficient_histogram=dict(coeff),categories=dict(cat),copied_center_blocks=centers,paid_histogram=dict(sorted(hist.items())),paid_rank_mass=sum(r*c for r,c in hist.items()),positive_rank_moves=sum(hist.values())-24,used_frames=[dict(frame_id=f,dimension=C.dimf[f],basis_sha256=hashlib.sha256(json.dumps(C.B[f],separators=(',',':')).encode()).hexdigest())for f in sorted(used)],parity_transform=receipt,producer_physical_emitter_sha256=oldmeta['physical_emitter_sha256'],physical_emitter_sha256=receipt['transform_sha256'])
+    result=dict(producer);result.update(records=out,physical=meta,result=meta,producer_physical=oldmeta,parity_census=receipt)
+    if output_dir is not None:
+        p=Path(output_dir);p.mkdir(parents=True,exist_ok=True)
+        (p/'records.bin.gz').write_bytes(gzip.compress(out.tobytes(),mtime=0))
+        (p/'physical.json').write_text(json.dumps(meta,indent=2)+'\n')
+        (p/'parity.json').write_text(json.dumps(receipt,indent=2)+'\n')
+    print('PASS parity fusion',count,'ADDs',len(out)//6,'records',flush=True)
+    return result
diff --git a/research/five-stage-source527-banks/portable_bit.py b/research/five-stage-source527-banks/portable_bit.py
new file mode 100644
index 0000000000..cd584216cc
--- /dev/null
+++ b/research/five-stage-source527-banks/portable_bit.py
@@ -0,0 +1,55 @@
+"""Fresh source527 local physical and complete five-stage global lowering.
+
+Inherited source-bound PR234 lowering, integrated with source527 using
+substantial OpenAI Codex assistance. Cover expansion is phase-major.
+"""
+from pathlib import Path
+import hashlib,importlib.util,sys,time,json
+sys.dont_write_bytecode=True
+if not __debug__:raise SystemExit('assertions required')
+HERE=Path(__file__).resolve().parent
+
+def load(name,path):
+ spec=importlib.util.spec_from_file_location(name,path);m=importlib.util.module_from_spec(spec);sys.modules[name]=m;spec.loader.exec_module(m);return m
+
+def phase_major_templates(lower):
+ """Expand all cover classes for each yielded phase before advancing."""
+ for stage in range(5):
+  yield('helper',stage,lambda s=stage:lower.iter_stage(s))
+  if stage in(1,2,4):
+   i={1:0,2:1,4:2}[stage]
+   yield('idle',i,lambda i=i:lower.iter_idle(i))
+   yield('bridge',i,lambda i=i:lower.iter_bridges(i))
+ yield('completion',0,lower.iter_completions)
+ yield('terminal_exchange',0,lower.iter_exchanges)
+
+def run(prepared_context=None,raw=None,output_dir=None,run_geometry=True,package_root=None):
+ begun=time.monotonic();package=Path(package_root or HERE).resolve()
+ if prepared_context is None:prepared_context=load('portable527_prepare',package/'prepare.py').prepare()
+ if raw is None:raw=load('portable527_raw_ledger',package/'raw_ledger.py').run(prepared_context)
+ physical_module=load('portable527_physical',package/'code/physical527.py');global_module=load('portable527_global',package/'code/global_lowering.py')
+ context=dict(prepared_context);source=context['SOURCE_TEXT']
+ producer_output=Path(output_dir)/'producer-bit' if output_dir is not None else None
+ producer=physical_module.run(context,source,output_dir=producer_output)
+ physical_run=load('portable527_parity',package/'parity_transform.py').run(producer,output_dir=output_dir)
+ raw=load('portable527_parity_raw',package/'raw_ledger.py').rebind_parity(raw,physical_run['parity_census'])
+ physical=physical_run['physical'];physical['source_head']=raw['source_head']
+ assert physical['source_heads']==raw['source_aliases']and physical['independent_dirty_registers']==raw['physical_R']
+ assert physical['paid_histogram']=={int(k):v for k,v in raw['one_stage_helper_histogram_including_copies'].items()}
+ assert physical['initial_independent_entrances']=={int(k):v for k,v in raw['auxiliary_entrance_rank_histogram'].items()}
+ lower=global_module.Lowerer(physical_run['records'],physical_run);global_result=global_module.verify(lower,raw)
+ assert global_result['paid_histogram']=={int(k):v for k,v in raw['five_stage_profile']['histogram'].items()}
+ global_result.update(source_head=raw['source_head'],scalar_projection_sha256=physical['scalar_projection_sha256'],local_tagged_sha256=physical['tagged_scalar_sha256'],weighted_stage_additions=global_result['opcode_counts'][global_module.ADD],bridge_additions=global_result['opcode_counts'][global_module.BRIDGE],total_weighted_additions=global_result['opcode_counts'][global_module.ADD]+global_result['opcode_counts'][global_module.BRIDGE],m=global_module.M,global_live=global_module.LIVE,external_work_family=global_module.WORK)
+ phases=[(name,index)for name,index,method in phase_major_templates(lower)]
+ assert phases==[('helper',0),('helper',1),('idle',0),('bridge',0),('helper',2),('idle',1),('bridge',1),('helper',3),('helper',4),('idle',2),('bridge',2),('completion',0),('terminal_exchange',0)]
+ geometry=None
+ if run_geometry:geometry=load('portable527_geometry',package/'code/geometry527.py').run(context,global_module)
+ result=dict(context=context,W=context['W'],C=context['C'],records=physical_run['records'],lower=lower,raw=raw,physical=physical,global_result=global_result,geometry=geometry,phase_major_schedule=phases,scalar_observer_result=physical_run['scalar_result'],seconds=time.monotonic()-begun)
+ if output_dir is not None:
+  out=Path(output_dir);out.mkdir(parents=True,exist_ok=True)
+  for name in('physical','global_result','geometry','raw'):
+   if result[name]is not None:(out/(name+'.json')).write_text(json.dumps(result[name],indent=2)+'\n')
+ return result
+
+def summary(result):
+ return dict(status='PASS_PORTABLE527_FRESH_PHYSICAL_GLOBAL_AND_GEOMETRY',source_head=result['raw']['source_head'],local_record_count=len(result['records'])//6,scalar_projection_sha256=result['physical']['scalar_projection_sha256'],physical_tagged_sha256=result['physical']['tagged_scalar_sha256'],global_result=result['global_result'],geometry=result['geometry'],phase_major_schedule=result['phase_major_schedule'],seconds=result['seconds'])
diff --git a/research/five-stage-source527-banks/portable_complex.py b/research/five-stage-source527-banks/portable_complex.py
new file mode 100644
index 0000000000..c068be3e1c
--- /dev/null
+++ b/research/five-stage-source527-banks/portable_complex.py
@@ -0,0 +1,151 @@
+#!/usr/bin/env python3
+"""Fresh complex label, scalar, splice and finite-guard checks from primary inputs.
+No saved PASS receipts, workstation paths, network or Lean rebuild are required.
+"""
+from collections import Counter
+from copy import deepcopy
+from fractions import Fraction as Q
+from hashlib import sha256
+from math import prod
+from pathlib import Path
+import gzip, importlib.util, json, sys
+sys.dont_write_bytecode = True
+
+PIN='f010392c923279e3dd59ef3aa5fedad23affc5fd'
+CHRONOLOGY=['stage0','stage1','climbA','bridgeA','stage2','climbB','bridgeB','stage3','stage4','final_climbs','bridgeC','terminal_shifts','signed_exchange']
+
+def need(ok, why):
+    if not ok: raise ValueError(why)
+
+def module(root, name):
+    spec=importlib.util.spec_from_file_location('portable_'+name,root/'code'/(name+'.py'))
+    result=importlib.util.module_from_spec(spec);spec.loader.exec_module(result)
+    return result
+
+def finite_guard(m, b, basis):
+    p=m['parameters'];l=m['ledger'];dim=p['m'];R=p['R'];h=p['h'];W=p['W_live']
+    need((dim,R,h,W)==(110,9412,22,14692),'complex family changed')
+    scalar=max(b[o]['totals']['real_component_primitive_steps'] for o in ['forward','backward'])
+    macros=max(b[o]['totals']['scalar_macros'] for o in ['forward','backward'])
+    scalar5=3*b['forward']['totals']['real_component_primitive_steps']+2*b['backward']['totals']['real_component_primitive_steps']
+    physical=W+R+h+1
+    adapters=8*(macros+m['checks']['invocation_calls']+p['invocation_live']+h)+32*dim
+    outside=8*(l['idle_calls']+6*p['v']+physical)+32*dim
+    compiler=64*(dim+1)**3
+    local=scalar+adapters*compiler
+    pervertex=scalar5+12*p['v']+(5*adapters+outside)*compiler+64*physical
+    group=(1<<(dim-1+(dim//2-1)**2))*prod((1<<(2*i))-1 for i in range(1,dim//2))
+    live_stock=group*W;physical_stock=group*physical;s=group*l['rank_mass']
+    K=group*pervertex+64;router=compiler*(K+1)*(physical_stock+1)**2
+    G,E,C0=1<<30000,1<<100000,1<<210000;B=s+E
+    literal=2*G*physical_stock**2+8*s+4*physical_stock+4+32*dim
+    need(local==213063694563472 and local<1<<48 and K<1<<6050,'fresh local/global bill')
+    need((group.bit_length(),live_stock.bit_length(),physical_stock.bit_length(),s.bit_length())==(5995,6009,6010,6016),'fresh cover/work stock')
+    need(router<G and literal<E and B<1<<100001,'fresh router/literal guard')
+    need(2*B*(dim-l['max_child'])-s-E==127*B,'completed-child induction')
+    need(32*dim*B*B<C0 and 2*B+18<C0 and 2*l['max_child']<dim,'semantic and half-shrink guard')
+    row=live_stock.bit_length()+9909+252;rowphysical=physical_stock.bit_length()+9909+252
+    need(row<=rowphysical<20161,'external rows')
+    gap=Q(10**6)-Q(51,25)*20161;need(gap>0,'row reserve gap')
+    need(2*(dim*(dim-1)+3*dim+dim+3*dim*(dim-1)//2+dim+1)+2*dim<compiler,'explicit quadratic-phase lowering')
+    for e in range(1,100001):need((e-1).bit_length()+1<=2*e,'grid-depth endpoint')
+    def validate(c):
+        need(c['physical']==c['live']+R+h+1,'missing work stock')
+        need(dim*c['live']-l['rank_mass']==l['deficit'],'wrong live normalization')
+        need(c['copy_calls']==5*h,'missing center child')
+        need(c['denominators']==[1,2,3,6],'unrecorded denominator')
+        need(physical_stock<1<<c['stock_exponent'],'false stock bound')
+    original=dict(physical=physical,live=W,copy_calls=5*h,denominators=[1,2,3,6],stock_exponent=6010)
+    validate(original);controls={}
+    for name,key,value in [('missing_tableau','physical',physical-R),('scratch_in_live','live',physical),('dropped_center','copy_calls',5*h-1),('unrecorded_divisor','denominators',[1,2,3,5,6]),('stale_stock_bound','stock_exponent',5208)]:
+        bad=dict(original);bad[key]=value
+        try:validate(bad)
+        except ValueError:controls[name]=True
+        else:raise ValueError('finite guard mutation accepted: '+name)
+    return dict(status='PASS_FRESH_EXACT_COMPLEX_GUARD',m=dim,live_per_vertex=W,physical_per_vertex=physical,
+        coefficient_denominators=[1,2,3,6],semantic_C1=1,complete_local_group_upper=local,
+        complete_local_group_bits=local.bit_length(),global_logical_groups_bits=K.bit_length(),
+        simple_global_logical_bound_exponent=6050,router_bits=router.bit_length(),G_exponent=30000,
+        literal_charge_bits=literal.bit_length(),E_exponent=100000,required_C0_bits=(32*dim*B*B).bit_length(),
+        C0_exponent=210000,induction_gap_multiple_of_B=127,group_bits=group.bit_length(),
+        live_stock_bits=live_stock.bit_length(),physical_stock_bits=physical_stock.bit_length(),
+        rank_mass_bits=s.bit_length(),live_row_coefficient=row,physical_row_overcharge_coefficient=rowphysical,
+        retained_row_coefficient=20161,retained_row_gap=str(gap),controls=controls,
+        binary_basis_count_tests=basis.gl_count_test(),
+        odd_denominator_grid='2^(-P)3^(-G(D+1)); no return rounding')
+
+def run(package_root=None, progress=lambda text:None):
+    root=Path(package_root).resolve() if package_root else Path(__file__).resolve().parent
+    labels=module(root,'complex_labels');scalar=module(root,'complex_scalars')
+    splice=module(root,'complex_splice');basis=module(root,'complex_basis')
+    pinpath=root/'inputs/complex/source-pins.json'
+    if not pinpath.is_file():pinpath=root/'inputs/complex-source-pins.json'
+    pins=json.loads(pinpath.read_text());verified={};by_upstream={}
+    for relative,spec in pins['files'].items():
+        path=root/relative;got=sha256(path.read_bytes()).hexdigest()
+        need(got==spec['sha256'],'primary proof input changed: '+relative)
+        verified[relative]=got;by_upstream[spec['upstream']]=(path,spec)
+    cp=pins['certificate'];z=(root/cp['path']).read_bytes()
+    need(sha256(z).hexdigest()==cp['compressed_sha256'],'compressed certificate binding')
+    raw=gzip.decompress(z);need(sha256(raw).hexdigest()==cp['uncompressed_sha256'],'expanded certificate binding')
+    verified[cp['path']]=sha256(z).hexdigest();c=json.loads(raw)
+    progress('Checking every complex source label and paid child.')
+    checks=labels.replay_label_counts(c)
+    H=Counter({int(k):5*n for k,n in checks['invocation_histogram'].items()})
+    v,R,h=c['v'],c['R'],c['h'];dim=5*h;W=4*v+R
+    for rank in [2*h-2,h-1,2*h+2,4]:H[rank]+=2*v
+    ledger=dict(histogram={str(k):n for k,n in sorted(H.items())},calls=sum(H.values()),
+        rank_mass=sum(k*n for k,n in H.items()),max_child=max(H),deficit=dim*W-sum(k*n for k,n in H.items()),
+        copy_calls=5*h,idle_calls=8*v)
+    need((ledger['calls'],ledger['rank_mass'],ledger['deficit'],ledger['max_child'])==(361405,1613040,3080,46),'fresh five-stage ledger')
+    source_path,source_spec=by_upstream['Work/GCert/Chain/NetDef.lean'];lines=source_path.read_text().splitlines();refs={}
+    for name in ['x01','y01','x12','y12','x23','y23','x34','y34','sfin','term']:
+        indices=[i for i,line in enumerate(lines) if name+' :' in line];need(indices,'missing exact matrix field '+name)
+        i=indices[-1] if name=='sfin' else indices[0]
+        text=' '.join(x.strip() for x in lines[i:i+(2 if name=='sfin' else 1)])
+        refs[name]=dict(source='Work/GCert/Chain/NetDef.lean',anchor=name+' :',line=i+1,declaration_line=text,
+            url='https://github.com/jacobalansussman/wht-power-saving-lean/blob/'+PIN+'/Work/GCert/Chain/NetDef.lean#L'+str(i+1))
+    m=dict(source_pin=PIN,parameters=dict(h=h,m=dim,v=v,R=R,W_live=W,invocation_live=2*v+R),
+        ledger=ledger,checks=checks,chronology=CHRONOLOGY,source_frame_equalities=refs)
+    progress('Deriving both complex scalar words and temporary lifetimes.')
+    A=scalar.flat(c['A']);B=scalar.flat(c['B']);b={};life={}
+    for orientation in ['forward','backward']:
+        b[orientation]=scalar.bill(scalar.inv_schedule(orientation,A,B,c))
+        life[orientation]=scalar.lifecycle(scalar.inv_schedule(orientation,A,B,c))
+    need((b['forward']['totals']['real_component_primitive_steps'],b['backward']['totals']['real_component_primitive_steps'])==(1492984,1549456),'fresh scalar expansion')
+    b['scratch_lifecycle_checks']=life;b['template_tests']=scalar.toy_checks()
+    b['five_invocation_scalar_primitives_per_vertex']=3*b['forward']['totals']['real_component_primitive_steps']+2*b['backward']['totals']['real_component_primitive_steps']
+    b['scratch']=dict(live_roles=W,physical_roles_per_cover=W+R+h+1,additional_recursive_calls=0,extra_external_row_coordinate=False)
+    progress('Merging scalar events, exact frames and all final climbs.')
+    programs={};controls=[]
+    for orientation in ['forward','backward']:
+        events=list(splice.emit(c,orientation));programs[orientation]=splice.validate(events,c,orientation,m,b)
+        controls.extend(splice.controls(events,c,orientation,m,b))
+    g=splice.global_contract(m);splice.validate_global(g,m)
+    mutations=[('wrong_helper_owner',lambda x:x['stage_recipe'][1].__setitem__('helper_owner','s1^(-1)(d)')),
+        ('rank_only_frame',lambda x:x['exact_frame_bindings']['x23'].__setitem__('declaration_line','same rank only')),
+        ('scratch_alias',lambda x:x['scratch_namespace'].__setitem__('centers',[14691,14713])),
+        ('wrong_live_normalization',lambda x:x.__setitem__('recursive_volume_denominator',24127)),
+        ('free_RAM',lambda x:x.__setitem__('primitive_policy','free_RAM')),
+        ('missing_tableau',lambda x:x.__setitem__('physical',14715))]
+    for name,fn in mutations:
+        bad=deepcopy(g);fn(bad)
+        try:splice.validate_global(bad,m)
+        except ValueError as error:controls.append(dict(name=name,rejected=True,reason=str(error)))
+        else:raise ValueError('global mutation accepted: '+name)
+    progress('Recomputing complex cover, router, precision and row inequalities.')
+    precision=finite_guard(m,b,basis)
+    need(labels.scalar_word([(1,0,1),(0,1,-1),(0,2,1),(3,1,-1),(1,0,1),(3,1,1),(0,2,-1),(2,3,-1),(3,2,1),(1,3,-1),(2,0,1)])==[[0,-1,0,0],[1,0,0,0],[0,0,0,-1],[0,0,1,0]],'global scalar signed exchange')
+    return dict(status='PASS_FRESH_COMPLEX_LABEL_SCALAR_SPLICE_PRECISION',five_stage_histogram=ledger['histogram'],
+        label_checks=checks,ledger=ledger,scalar_bill=b,
+        splice=dict(program=programs,global_contract=g,controls=controls,source_pin=PIN,certificate_sha256=cp['compressed_sha256']),
+        precision_guard=precision,input_pins=verified,
+        inherited_theorem_dependencies=[
+            'Pinned hAi,hBi,hx,hy,hid establish the full scalar identity; this is not a fresh Lean or full scalar-column replay.',
+            'Pinned Lab.inv dirty lifting, exact gwinB/G0..G4 geometry, owner bijections and NetCert composition are inherited source theorems.',
+            'Retained general Clifford synthesis, compact complete-stream tape primitives, common-grid precision and complete-row proofs remain analytic dependencies.',
+            'The finite operational recipe is regenerated and checked; no astronomical cover enumeration or universal tape-machine execution is claimed.'])
+
+if __name__=='__main__':
+    result=run(progress=lambda text:print(text,file=sys.stderr))
+    print(json.dumps(result,indent=2))
diff --git a/research/five-stage-source527-banks/prepare.py b/research/five-stage-source527-banks/prepare.py
new file mode 100644
index 0000000000..c7e814a70a
--- /dev/null
+++ b/research/five-stage-source527-banks/prepare.py
@@ -0,0 +1,135 @@
+"""Clean source527 preparation from pinned source471 inputs and explicit selectors.
+Preserves source notices, with substantial OpenAI Codex assistance.
+No old PASS receipt is consumed and no payload execution occurs here.
+"""
+from pathlib import Path
+from collections import Counter,defaultdict
+from types import SimpleNamespace
+import hashlib,importlib.util,json,sys
+sys.dont_write_bytecode=True
+HERE=Path(__file__).resolve().parent
+
+def prepare():
+ S=HERE/'source527'
+ p=HERE/'loader/source_loader.py';spec=importlib.util.spec_from_file_location('portable527_source471_loader',p);loader=importlib.util.module_from_spec(spec);spec.loader.exec_module(loader)
+ context=loader.prepare(HERE/'loader/source_inputs/pr210',HERE/'loader/source_inputs/base_bit')
+ m=SimpleNamespace(**context);W=m.W;C=m.C
+ # Apply the nineteen PR230 operation-frame improvements retained by source527.
+ # The active full witness differs from PR230's raw search output at retained
+ # predecessor choices; its source493 manifest and provenance pin both forms.
+ framepath=S/'pr230/active-opframe-bases.json'
+ assert hashlib.sha256(framepath.read_bytes()).hexdigest()=='9a2bba5c72aebc9ee4f21e59b10ce810b2da98a1fa1748fbc44c50a080e584ff'
+ changed=[]
+ for operation,rows in json.loads(framepath.read_text()):
+  target_frame=W.register(rows)
+  if C.B[W.opframe[operation]]!=C.B[target_frame]:changed.append(operation)
+  W.opframe[operation]=target_frame
+ assert changed==[18750,23544,23545,23591,23592,23638,23639,23685,23686,23732,23733,27900,27904,27908,27912,28052,28056,28060,28064],changed
+ W.changed_frames=[i for i,(a,b)in enumerate(zip(W.original_opframe,W.opframe))if a!=b]
+ W.endframe={r:W.opframe[ii[-1]]for r,ii in W.role_ops.items()}
+ W.exact_frames();m.pr230_operation_changes=changed
+ extra=json.loads((S/'extra/selection.json').read_text());oldborrow=m.borrow.copy();used={s for r in m.selection+m.gauge_selection for s in[r['source'],r['partner']]};assert len(oldborrow)==471
+ for r in extra:
+  role,n=r['role'],r['source'];e=m.kpair[n];assert e['carrier']==r['partner'];assert not({n,r['partner']}&used);used.update([n,r['partner']])
+  assert role not in m.borrow and role not in m.W.gauge and role not in m.W.donor and role not in m.removed and role not in m.W.source.values()
+  assert set(m.adj[role])==set(r['targets'])and set(m.adj[role].values())=={1}
+  assert role not in {s for i in m.W.phase1 for s in m.W.ops[i][:2]};f=m.W.register(r['split_gauge_basis'])if'split_gauge_basis'in r else e['mix_frame'];assert m.C.dimf[f]in(2,4) and m.C.nondeg(f)and m.C.sub(f,m.W.opframe[m.W.role_ops[role][0]])
+  assert all(all(m.W.module.dot(m.C.cov[t],b)==0 for b in m.C.B[f])for t in r['targets'])
+  z=dict(role=role,frame=f,dim=m.C.dimf[f],targets=r['targets']);m.W.gauge[role]=z;m.W.w['gauges'].append(z);m.W.readtime[role]=0;m.W.w['reads'][str(role)]=len(m.W.phase1)
+  m.borrow[role]=n;m.by_source[n]=r;m.gauge_borrow[role]=n;m.gauge_selection.append(r)
+ m.W.order=[r['role']for r in extra]+m.W.order
+ # The third ordered rank4 source entrance follows all retained new-source gauges.
+ m.W.readtime[14073]=168;m.W.w['reads']['14073']=len(m.W.phase1)+168
+ m.W.order.remove(14073);at_index=max(m.W.order.index(r['role'])for r in m.new_gauge_selection)+1;m.W.order.insert(at_index,14073)
+ m.at=defaultdict(list)
+ for role in m.W.order:m.at[m.W.readtime[role]].append(role)
+ m.regs=sorted(set(m.W.phys.values())-set(m.borrow)-m.removed);m.idx={r:2*m.W.v+j for j,r in enumerate(m.regs)};assert len(m.borrow)==493 and len(m.regs)==16621
+ assert all(0<=m.W.readtime[s]<=m.W.first[s]for s in m.W.gauge);m.W.exact_frames()
+ m.extra_roles={r['role']for r in extra};m.extra_selection=extra
+ 
+ groups=json.loads((S/'targetagg/selection.json').read_text())['groups'];agg=[];role_groups=defaultdict(list);owner={}
+ for k,g in enumerate(groups):
+  F=W.register(g['frame_basis']);ts=g['targets'];assert C.nondeg(F)and g['pivot']in ts
+  for t in ts:assert t not in owner;owner[t]=k
+  for s in g['roles']:
+   assert set(ts)<=set(m.adj[s])and len({m.adj[s][t]for t in ts})==1 and C.sub(W.gauge[s]['frame'],F);role_groups[s].append(k)
+  agg.append(dict(targets=ts,pivot=g['pivot'],roles=g['roles'],frame=F))
+ assert len(agg)==82 and len(owner)==316
+ byrole={r['role']:r for r in extra};rf={}
+ for k,g in enumerate(agg):
+  for role in g['roles']:
+   r=byrole.get(role);f=W.gauge[role]['frame']
+   if r and'split_gauge_basis'in r and r['early_role']not in g['roles']:f=r['mix_frame']
+   assert C.sub(f,g['frame']);rf[role,k]=f
+ for role,ks in role_groups.items():ks.sort(key=lambda k:C.dimf[rf[role,k]])
+ m.agg=agg;m.role_groups=role_groups;m.target_group=owner;m.read_frames=rf;m.extra_byrole=byrole
+ rankselection=json.loads((S/'rank/selection.json').read_text());rankgroups=[];rankowner={}
+ for k,g in enumerate(rankselection['groups']):
+  F=W.register(g['frame_basis']);dep={int(t):{int(p):c for p,c in cs.items()}for t,cs in g['dependent'].items()};assert not(set(dep)&{p for cs in dep.values()for p in cs})
+  for t in g['targets']:assert t not in owner and t not in rankowner;rankowner[t]=k;assert all(W.module.dot(C.cov[t],b)==0 for b in C.B[F])
+  for s in g['prefix_roles']:
+   assert C.sub(W.gauge[s]['frame'],F)
+   for t,cs in dep.items():assert m.adj[s].get(t,0)==sum(c*m.adj[s].get(p,0)for p,c in cs.items())
+  rankgroups.append(dict(targets=g['targets'],dependent=dep,roles=set(g['prefix_roles']),frame=F))
+ assert len(rankgroups)==7 and len(rankowner)==56 and sum(len(g['dependent'])for g in rankgroups)==27
+ m.rankgroups=rankgroups;m.rankowner=rankowner
+ 
+ echelonselection=json.loads((S/'echelon/selection.json').read_text());echelon=[];eowner={}
+ def addrow(a,c,b):
+  z=a.copy()
+  for s,v in b.items():
+   z[s]=z.get(s,0)+c*v
+   if not z[s]:del z[s]
+  return z
+ for k,r in enumerate(echelonselection['groups']):
+  F=W.register(r['frame_basis']);R={int(t):{int(s):c for s,c in a.items()}for t,a in r['transformed_response'].items()};moves=[tuple(v)for v in r['moves']]
+  A={t:{s:m.adj[s].get(t,0)for s in r['prefix_roles']if m.adj[s].get(t,0)}for t in r['targets']}
+  for t,p,c in moves:assert t!=p;A[t]=addrow(A[t],c,A[p])
+  assert A==R and C.nondeg(F)
+  for t in r['targets']:assert t not in owner and t not in rankowner and t not in eowner;eowner[t]=k;assert all(W.module.dot(C.cov[t],b)==0 for b in C.B[F])
+  for s in r['prefix_roles']:assert C.sub(W.gauge[s]['frame'],F)
+  echelon.append(dict(targets=r['targets'],roles=set(r['prefix_roles']),response=R,moves=moves,frame=F))
+ assert len(echelon)==4 and len(eowner)==20
+ m.echelon=echelon;m.eowner=eowner
+ 
+ selection=json.loads((S/'fresh/selection.json').read_text());chosen=[];fresh={}
+ used={r[k]for r in m.selection+m.gauge_selection for k in('source','partner')};newused=set()
+ for raw in selection:
+  r=dict(raw);role,n=r['role'],r['source'];entry=m.kpair[n]
+  assert entry['carrier']==r['partner']and not({n,r['partner']}&(used|newused));newused.update((n,r['partner']))
+  assert role not in W.gauge and role not in m.borrow and n not in m.by_source
+  f=entry['mix_frame'];q=W.register(r['mix_basis']);assert C.sub(f,q)and C.sub(q,f)and C.dimf[f]==2
+  assert W.role_ops[role][0]==r['first_operation']and C.sub(f,W.opframe[r['first_operation']])and C.nondeg(f)
+  r['mix_frame']=f;chosen.append(r);fresh[role]=r
+  z=dict(role=role,frame=f,dim=2,targets=sorted(m.adj[role]));W.gauge[role]=z;W.w['gauges'].append(z);W.readtime[role]=0;W.w['reads'][str(role)]=len(W.phase1)
+  m.borrow[role]=n;m.by_source[n]=r;m.gauge_borrow[role]=n;m.gauge_selection.append(r);m.extra_byrole[role]=r
+ assert len(chosen)==3 and len(newused)==6
+ W.order=list(fresh)+W.order;m.at=defaultdict(list)
+ for role in W.order:m.at[W.readtime[role]].append(role)
+ m.regs=sorted(set(W.phys.values())-set(m.borrow)-m.removed);m.idx={r:2*W.v+j for j,r in enumerate(m.regs)}
+ assert len(m.borrow)==496 and len(m.regs)==16618 and len(m.extra_byrole)==25
+ W.exact_frames();m.fresh=fresh
+ 
+ chosen=json.loads((S/'paired-selection.json').read_text());pair_byalias={r['aliased_role']:r for r in chosen};pair_bydirty={int(a):r for r in chosen for a in r['retained_dirty_coefficients']}
+ for r in chosen:
+  b,n=r['aliased_role'],r['source'];f=m.kpair[n]['mix_frame'];r['mix_frame']=f;anchors={int(a):c for a,c in r['retained_dirty_coefficients'].items()};r['retained_dirty_coefficients']=anchors
+  response=Counter(m.adj[b])
+  for a,c in anchors.items():
+   for t,v in m.adj[a].items():response[t]+=c*v
+  assert not any(response.values()),('nonzero combined response',r)
+  assert b not in W.gauge and b not in m.borrow and n not in m.by_source
+  assert all(a not in W.gauge and a not in m.borrow for a in anchors)
+  canonical=W.register(r['mix_basis']);assert m.kpair[n]['carrier']==r['partner'] and C.dimf[f]==2 and C.sub(f,canonical)and C.sub(canonical,f)
+  assert all(C.sub(f,W.opframe[W.role_ops[a][0]])for a in [b]+list(anchors))and C.nondeg(f)
+  z=dict(role=b,frame=f,dim=2,targets=[]);W.gauge[b]=z;W.w['gauges'].append(z);W.readtime[b]=0;W.w['reads'][str(b)]=len(W.phase1)
+  q=dict(r,role=b);m.borrow[b]=n;m.by_source[n]=q;m.gauge_borrow[b]=n;m.gauge_selection.append(q);m.extra_byrole[b]=q
+ W.order=list(pair_byalias)+W.order;m.at=defaultdict(list)
+ for role in W.order:m.at[W.readtime[role]].append(role)
+ m.regs=sorted(set(W.phys.values())-set(m.borrow)-m.removed);m.idx={r:2*W.v+j for j,r in enumerate(m.regs)};assert len(m.borrow)==527 and len(m.regs)==16587
+ W.exact_frames();m.pair_byalias=pair_byalias;m.pair_bydirty=pair_bydirty;m.pair_selection=chosen;m.zero=W.register([])
+
+ m.SOURCE_TEXT=(S/'word.py').read_text()
+ assert hashlib.sha256(m.SOURCE_TEXT.encode()).hexdigest()=='e675d4eb9d90ff0b44279fae0b46f1a8b39f65421cc38454b70a6b75d0e10b42'
+ m.input_pins={}
+ m.portable_source_pins={str(p.relative_to(HERE)):hashlib.sha256(p.read_bytes()).hexdigest()for p in S.rglob('*')if p.is_file()}
+ return dict(m.__dict__)
diff --git a/research/five-stage-source527-banks/prime_check.py b/research/five-stage-source527-banks/prime_check.py
new file mode 100644
index 0000000000..c0f9ed28d5
--- /dev/null
+++ b/research/five-stage-source527-banks/prime_check.py
@@ -0,0 +1,67 @@
+"""Recompute exact determinants for every physically consumed source527 basis.
+Based on PR234's prime coverage check; substantial OpenAI Codex assistance.
+"""
+from collections import Counter
+from pathlib import Path
+import hashlib,importlib.util,json,time
+HERE=Path(__file__).resolve().parent
+
+def basis_hash(basis):return hashlib.sha256(json.dumps(basis,separators=(',',':')).encode()).hexdigest()
+
+def validate_record(row,dimension,determinant,allowed_primes):
+    assert type(row['dimension'])is int and type(row['determinant'])is int and type(row['residual'])is int,'exact integer witness fields'
+    assert row['dimension']==dimension and row['determinant']==determinant!=0,'exact determinant mismatch or singular basis'
+    assert 0<row['residual']<2**80,'retained prime lower bound'
+    product=row['residual']
+    for prime,exponent in row['factors'].items():
+        p=int(prime);assert p in allowed_primes and type(exponent)is int and exponent>0,'factor shape';product*=p**exponent
+    assert product==abs(determinant),'factor identity'
+
+def run(context,physical,progress=lambda text:None):
+    started=time.monotonic();w,c=context['W'],context['C']
+    p=HERE/'prime_witnesses.py';spec=importlib.util.spec_from_file_location('source527_exact_prime_validator',p);module=importlib.util.module_from_spec(spec);spec.loader.exec_module(module)
+    required=set(w.opframe)|set(w.w['source_frame'])|set(w.w['root_frame'])|{w.w['full_frame'],w.register([])}
+    required.update(g['frame']for g in w.gauge.values())
+    for e in w.k['entries']:required.update((e['mix_frame'],e['deliver_frame']))
+    for t in range(w.v):required.add(w.register(w.module.kernel([c.cov[t]],w.h)[0]))
+    required.update(g['frame']for g in context['agg']+context['rankgroups']+context['echelon'])
+    required.update(r['mix_frame']for r in context['pair_selection'])
+    used=set();groups={}
+    for row in physical['used_frames']:
+        f=row['frame_id'];assert f not in used;used.add(f)
+        h=basis_hash(c.B[f]);assert h==row['basis_sha256'] and len(c.B[f])==c.dimf[f]==row['dimension'],'fresh physical basis identity'
+        groups.setdefault(h,f)
+    assert {basis_hash(c.B[f])for f in required}<=set(groups),'base and added transform coverage'
+    determinants={};largest=0;prime_factors=set();witnesses={}
+    for j,(h,f)in enumerate(sorted(groups.items())):
+        B=c.B[f];s=list(map(sum,B))
+        gram=[[9*sum(a*b for a,b in zip(x,y))-s[i]*s[k]for k,y in enumerate(B)]for i,x in enumerate(B)]
+        det=module.det(gram);powers,residual=module.factor_witness(det)
+        row=dict(dimension=len(B),determinant=det,residual=residual,factors={p:e for p,e in powers.items()if e})
+        validate_record(row,len(B),det,module.PRIMES);prime_factors.update(map(int,row['factors']));witnesses[h]=row
+        determinants[h]=det;largest=max(largest,abs(det).bit_length())
+        if j and j%5000==0:progress('Validated '+str(j)+' exact basis determinants')
+    ranks=Counter();bundle_bases=set()
+    for role,g in w.gauge.items():
+        if role in w.donor or role in context['borrow']:continue
+        a=g['dim'];h=basis_hash(c.B[g['frame']]);d=determinants[h]
+        assert 0<5*a<120 and d!=0;ranks[a]+=1;bundle_bases.add(h)
+        assert abs(d**5)==abs(d)**5
+    assert ranks=={12:18,13:48,18:13,20:2200}and len(bundle_bases)==231
+    assert 2*120**3*10**16<2**80
+    from copy import deepcopy
+    controls=[]
+    def reject(name,call):
+        try:call()
+        except AssertionError:controls.append(name)
+        else:raise AssertionError('invalid witness accepted: '+name)
+    first=next(iter(groups));row=witnesses[first];det=determinants[first]
+    missing=dict(witnesses);missing.pop(first)
+    def coverage(actual,wanted):assert set(actual)==set(wanted),'complete current basis coverage'
+    reject('missing current basis',lambda:coverage(groups,missing))
+    changed=deepcopy(row);changed['determinant']=det+1;reject('incorrect determinant',lambda:validate_record(changed,row['dimension'],det,module.PRIMES))
+    changed_factor=deepcopy(row);changed_factor['residual']+=1;reject('incorrect factor identity',lambda:validate_record(changed_factor,row['dimension'],det,module.PRIMES))
+    singular=deepcopy(row);singular['determinant']=0;reject('singular basis',lambda:validate_record(singular,row['dimension'],module.det([[1,2],[1,2]]),module.PRIMES))
+    huge=deepcopy(row);huge['residual']=2**80;reject('residual at prime lower bound',lambda:validate_record(huge,row['dimension'],det,module.PRIMES))
+    noninteger=deepcopy(row);noninteger['residual']=float(row['residual']);reject('noninteger residual',lambda:validate_record(noninteger,row['dimension'],det,module.PRIMES))
+    return dict(status='PASS_FRESH_ALL_ACTUAL527_BASIS_DETERMINANTS_AND_FIVEFOLD_BUNDLES',unique_bases=len(groups),current_frame_ids=len(used),basis_inventory_sha256=hashlib.sha256(json.dumps(sorted(groups),separators=(',',':')).encode()).hexdigest(),maximum_determinant_bits=largest,prime_factors=sorted(prime_factors),all_remaining_factors_below_2_power_80=True,independent_entrances=sum(ranks.values()),bundled_unique_bases=len(bundle_bases),entrance_rank_counts=dict(ranks),controls=controls,physical_inventory_bound=True,seconds=time.monotonic()-started,scope='Every actual physical basis is freshly evaluated over exact integers; all527 source frames, target transformations and caps are covered. Five disjoint copies inherit nondegeneracy; compiler and prime supply remain theorem dependencies.')
diff --git a/research/five-stage-source527-banks/prime_witnesses.py b/research/five-stage-source527-banks/prime_witnesses.py
new file mode 100644
index 0000000000..7f16d93030
--- /dev/null
+++ b/research/five-stage-source527-banks/prime_witnesses.py
@@ -0,0 +1,99 @@
+"""Exact finite prime exclusions for the selected new rational bit frames.
+
+For integer basis B and G=I-J/9, det(9 B G B^t) is a nonzero integer.
+Excluding its prime divisors and 2,3,5 makes the reduced frame nondegenerate
+over every odd local ring Z/q^w. Projector denominators divide this determinant
+times fixed powers of 3. Retained frames retain their old exclusion witnesses.
+The selected q is an arbitrarily large fixed prime chosen after this finite
+set; this is not verification at a preselected numeric q.
+"""
+from pathlib import Path
+import argparse,gzip,hashlib,json,sys
+
+HERE=Path(__file__).resolve().parent
+
+def det(A):
+    A=[list(r) for r in A];n=len(A)
+    if not n:return 1
+    sign=1;previous=1
+    for k in range(n-1):
+        pivot=next((i for i in range(k,n) if A[i][k]),None)
+        if pivot is None:return 0
+        if pivot!=k:A[k],A[pivot]=A[pivot],A[k];sign=-sign
+        p=A[k][k]
+        for i in range(k+1,n):
+            for j in range(k+1,n):
+                v=A[i][j]*p-A[i][k]*A[k][j]
+                assert v%previous==0
+                A[i][j]=v//previous
+            A[i][k]=0
+        previous=p
+    return sign*A[-1][-1]
+
+PRIMES=(2,3,5,7,11,13,17,19,23,29,31)
+
+def factor_witness(determinant):
+    assert type(determinant) is int and determinant != 0
+    residual=abs(determinant);powers={}
+    for prime in PRIMES:
+        power=0
+        while residual%prime==0:residual//=prime;power+=1
+        powers[str(prime)]=power
+    validate_factor(determinant,powers,residual)
+    return powers,residual
+
+def validate_factor(determinant,powers,residual):
+    assert type(determinant) is int and determinant != 0
+    assert set(powers)==set(map(str,PRIMES))
+    assert all(type(x) is int and x>=0 for x in powers.values())
+    assert type(residual) is int and 0<residual<2**80
+    reconstructed=residual
+    for p in PRIMES:reconstructed*=p**powers[str(p)]
+    assert reconstructed==abs(determinant)
+
+def certificate(word=None):
+    if word is None:
+        from word import Candidate
+        word=Candidate()
+    C=word.C;h=word.h
+    used=set(word.opframe)|set(word.w['source_frame'])|set(word.w['root_frame'])|{word.w['full_frame']}
+    used.update(z['frame'] for z in word.w['gauges'])
+    for e in word.k['entries']:used.update((e['mix_frame'],e['deliver_frame']))
+    groups={}
+    for frame in sorted(used):
+        B=C.B[frame]
+        key=json.dumps(B,separators=(',',':'));groups.setdefault(key,[]).append(frame)
+    records=[]
+    for key,frames in sorted(groups.items()):
+        B=json.loads(key);s=list(map(sum,B))
+        gram=[[9*sum(a*b for a,b in zip(x,y))-s[i]*s[j] for j,y in enumerate(B)] for i,x in enumerate(B)]
+        determinant=det(gram);powers,residual=factor_witness(determinant)
+        records.append(dict(basis_sha256=hashlib.sha256(key.encode()).hexdigest(),
+            frame_ids=frames,dimension=len(B),cleared_gram_determinant=determinant,
+            small_prime_powers=powers,remaining_factor=residual,gram_denominator_power_of_9=len(B)))
+    controls=[];zeros={str(p):0 for p in PRIMES}
+    for name,args in (('zero determinant',(0,zeros,1)),('incorrect factor identity',(15,zeros,1)),
+        ('residual above retained prime lower bound',(2**80+7,zeros,2**80+7))):
+        try:validate_factor(*args)
+        except AssertionError:controls.append(name)
+        else:raise ValueError('Invalid prime witness accepted: '+name)
+    return dict(status='PASS exact prime witnesses for every used physical frame',h=h,
+        source_commit='4a3c769e5c5430e7114c4d3e099ff34664677f17',
+        input_sha256={str(p.relative_to(HERE.parents[2])):hashlib.sha256(p.read_bytes()).hexdigest() for p in word.input_paths},
+        script_sha256=hashlib.sha256(Path(__file__).read_bytes()).hexdigest(),
+        stripped_primes=list(PRIMES),frame_witnesses=records,unique_used_bases=len(records),
+        total_used_frames=len(used),total_operation_frames=len(word.changed_frames),
+        maximum_determinant_bits=max(abs(z['cleared_gram_determinant']).bit_length() for z in records),
+        all_remaining_factors_below_retained_prime_lower_bound=True,adverse_controls=controls,
+        rule='Every q>2^80 avoids every nonzero used-frame determinant, so the original prime range is unchanged.',
+        scope='Exact integer cleared-Gram factors for all physical frames including every changed operation, gauge, source, root, partner-mix, delivery and ambient frame. Uniform split-rank and fallback arguments remain the retained local-ring interface.')
+
+def main():
+    assert not sys.flags.optimize
+    ap=argparse.ArgumentParser();ap.add_argument('--write',action='store_true');args=ap.parse_args()
+    record=certificate();target=HERE/'prime-witnesses.json.gz'
+    if args.write:target.write_bytes(gzip.compress((json.dumps(record,sort_keys=True,separators=(',',':'))+'\n').encode(),mtime=0))
+    else:assert record==json.loads(gzip.decompress(target.read_bytes())),'Exact prime witnesses changed'
+    print(json.dumps({k:v for k,v in record.items() if k!='frame_witnesses'}))
+
+if __name__=='__main__':main()
diff --git a/research/five-stage-source527-banks/proof/FINITE_BIT_ACCOUNTING.md b/research/five-stage-source527-banks/proof/FINITE_BIT_ACCOUNTING.md
new file mode 100644
index 0000000000..42a585d742
--- /dev/null
+++ b/research/five-stage-source527-banks/proof/FINITE_BIT_ACCOUNTING.md
@@ -0,0 +1,126 @@
+# Portable five-stage BIT finite bill
+
+The executable `finite_check.run` accepts the freshly reconstructed source
+ledger, physical tags, independently checked scalar projection, exact current
+basis-prime result, exact moment calculation and optional global lowering
+result. It reads no saved execution receipt. Source expectations are checked
+against those computed objects, not substituted for them. The complete
+package runner is responsible for generating its inputs before this call.
+
+## Quantifiers and inherited primitive hypotheses
+
+Fix this candidate and an eligible odd prime q>2^80 before the input widths
+n,w are chosen. All represented determinant and route exclusions lie in
+{2,3,5,7}. The primitive constant C may depend on q, the fixed source recipe,
+the finite bootstrap level and the retained primitive implementations. It
+does not depend on n,w, ancestor values or unrelated spectator widths except
+through logical volume V.
+
+The three byte-pinned public sources supplied here are
+`three-stage-cover-bit.tex`, `three-stage-cover-rows.tex`, and
+`copied-centers-lemma.tex`. Their exact-tape, completed ordinary-leaf and
+complete-stream hypotheses are retained. All high matrix fibers are batched;
+prefix descriptor preparation has fixed polynomial cost in w and is paid
+over the complete q^w target fiber. Unrelated spectators use ripple counters.
+Fresh child role fields never enter the ancestor weight descriptor. Finite
+low-type tables, initialization and sequential scans are charged in C.
+There is no unit-cost random-access oracle or new numerical Turing-machine
+constant requirement.
+
+## Actual finite recipe
+
+Let m=120, W=23,683, E=495,304, v=1,760, R=16,643, d=m^2 and
+J=20v+10R+4v=208,670. The last 4v=7,040 families pay the explicit final
+complete-stream exchange; they are not equal-frame payload shears.
+
+The check derives from the fresh word 12,858,690 weighted additions or
+21,787,770 unit additions, including all six bridge families. Its
+cancellation-free prefix lift is
+64*37,623^3*9,430,154^2=303,094,329,841,208,644,962,256,113,408<2^98.
+No old defining-integer endpoint is imposed on the F2 supplier.
+
+The actual rational 48-dimensional route is reconstructed, eliminated and
+inversely replayed: 195 additions, 44 swaps and 27 scalings, with denominator
+set {1,2,3,6}. A missing addition fails reconstruction. Coordinate
+permutations and embedding give the other ports and stages.
+
+Per route, pay at most 23,707*|GL_120(F_q)| low-role transpositions and
+16*(d+1)^2 high affine factors. Per good edge, pay at most 8m^2+8 elementary
+normalizer wrappers. The same compiler pays all exterior completions.
+There are 120 copied-center episodes. Their recursive rank-22 calls and
+their scalar reads are already included in E and the scalar count; their
+copy/erase and parking passes are separately paid by the complete-stream
+primitive, using one additional work stream.
+
+For a bad edge with N=2m=240, the exact inherited Gaussian fallback uses at
+most 6N(N-1)+3N+6(N-1)=346,314<32m^2=460,800 children. Keep the entire
+larger allowance in the moment. Coefficient preparation uses only unit
+pivot inversions. Allocate 128N^3 ring operations per edge per low class:
+four 8N^3 elimination/inversion allocations and sixteen 2N^3 matrix products
+give 64N^3; the remaining 64N^3 pays extraction, copying, generic tests,
+factor output and the bad-branch elimination. Matrix bit costs and descriptor
+arithmetic are paid by the inherited primitive theorem. Invertibility testing
+of each candidate low matrix additionally gets 16m^3 operations.
+
+Overcharging every finite recipe item over every low matrix gives
+
+    B(q) <= [unit+J*16(d+1)^2+J*(W+24)+E*(8m^2+8)
+             +E*128(2m)^3+16m^3+120+E+1]*q^14400
+         = 1,568,901,757,016,837*q^14400 < q^14401.
+
+These are explicit ring/recipe counts, not single-step machine costs. All
+fixed-tape, table, setup and selector constants stay in C_primitive, giving
+the normalized toll C*n+C*w^(1-a_old) after increasing one finite C.
+No recursive child is absorbed into this nonrecursive constant.
+
+## Weighted induction and rows
+
+Set c=704197288590873/10^18 and tau=1-c. Exact interval recomputation,
+adding the entire 10^-16*32m^2 fallback envelope per edge, proves
+
+    1-mu_tau > 774/10^21,
+    1-mu_1 = 107421874442783/69383789062500000 > 0.
+
+For the retained recurrence
+T(n,w)<=sum_r c_r*T(r*floor(n/120),w)+C*n+C*w^(1-a_old),
+take A=C*(1+1/(1-mu_1)+1/(1-mu_tau)). Induction bounds T by
+A*(n+n^tau*w^(1-a_old)): the linear and stopped gaps separately pay the
+two tolls; the leaves follow A>=C. Floors only decrease the child sizes.
+
+The maximum ratio is 5/6, so four levels halve. The depth bound
+4*ceil(log2(max(2,n))) changes stock, not the one-level moment. Paying all
+24 temporary roles gives per-level log_q stock below 14400w+1. Add the
+inherited old-leaf reserve. The new contribution is at most
+57,604w*ceil(log2(max(2,n))). Borrow and restore these complete high digits;
+successive residuals and the two-call ordinary wrapper reuse the reserve.
+One outer row padding factor is below two; binary-radix padding costs less
+than q^2 in logical volume. The temporary roles never enter the denominator W.
+
+## Finite ordinary bootstrap and cutoff
+
+Starting with completed ordinary a0=384599/10^10, set beta=c and repeat
+a_(j+1)=(1-c)c+c*a_j exactly three times. The finite result is
+
+    a3=704197288358392787581160456097840009769165126434639147437106139267359/10^72.
+
+At each stage, the paid exponents are 1-c, 1-a_(j+1), c with a logarithm,
+and c(1-a_j). The exact positive gaps are c-a_(j+1),
+1-a_(j+1)-c, 1-a_(j+1)-c(1-a_j), and 1-c (for row fitting).
+
+Choose a finite integer C>=1 dominating that stage's already fixed primitive,
+recurrence, wrapper, row, setup, padding and previous ordinary constants.
+Let delta be the minimum positive gap and
+
+    L=ceil(max(1,36/delta^2,2*(4+ceil(log2 C))/delta)), e0=2^L.
+
+For e>=e0, C(1+log2 e)e^(-delta)<=1/16. Indeed for t=log2 e,
+log2(1+t)<=3sqrt(t)<=delta*t/2 and log2 C+4<=delta*t/2.
+Thus every lower-order toll is covered and the borrowed high part fits.
+Below e0 use the previously completed ordinary algorithm. A finite next
+constant is C_large+C_old*2^(L*(a_(j+1)-a_j)). This choice is sequential,
+not circular, and uniform in the input and complete spectators.
+
+This proves the finite-accounting and ordinary-bootstrap step at the same
+conditional theorem level as the retained public baseline. It does not claim
+an independent proof of those inherited primitives, a measured universal
+machine constant, or execution at every input width.
diff --git a/research/five-stage-source527-banks/proof/INTEGRATED_PROOF.md b/research/five-stage-source527-banks/proof/INTEGRATED_PROOF.md
new file mode 100644
index 0000000000..c624da6c5c
--- /dev/null
+++ b/research/five-stage-source527-banks/proof/INTEGRATED_PROOF.md
@@ -0,0 +1,87 @@
+# Source-bound five-stage compositional candidate
+
+9 October 2026. This is an isolated research admission branch. Its executable checks immutable finite receipts and recomputes exact scalar moments, a finite ordinary bootstrap, and all 47 balanced outer inequalities. It additionally binds fresh public-source bit preparation, raw-ledger construction, and the complete globally lowered physical event stream. It does not repeat the exhaustive F2 column proof, run the Lean kernel or universal tape machine, or claim Mac qualification or publication.
+
+## 1. Meaning of admission here
+
+The branch is an explicit conjunction of constructive source theorems and finite source-bound evidence. It does not replace any proof with a numerical root or infer source correctness from a PASS string. The new family names are `new5stageF2` and `baseline5stageComplex`; the old dimension-72 entry point and written bounds are kept byte-identical. The branch has no general-purpose dimension override.
+
+The executable's semantic guards reject a wrong source, omitted copied centers, a rank-only complex frame contract, a work/live denominator substitution, omitted final climbs, incorrect reverse-copy lowering, and a one-level bit-halving substitution. The source lock identifies the exact evidence, code, and primary proof files consumed. The more detailed emitted-program verifiers exercise local chronology and addresses, beyond these top-level schema controls.
+
+## 2. Raw source471 provenance
+
+The saved raw helper ledger pins an older screen.py. The older screen bytes are unavailable; no byte-delta assertion is made. The rebind verifier reruns the current exact geometry/ledger construction with all writes suppressed and a hash-verified cached certificate in place of its redundant network fetch. Every semantic field, all 471 source paths, and all 1,760 target paths agree. The saved ledger is not rewritten.
+
+The apparent target-prefix duplication is resolved by the actual chronology: screen.py removes the two latest loan prefixes for its predecessor positive controls; raw_ledger.py puts those two prefixes back. There are exactly eight restored target reads. Current-source dependency hashes and the canonical semantic-payload digest make this equivalence independently inspectable. The sole historical pin exception is specifically this screen.py field and is allowed only with the exact rebind witness.
+
+## 3. Local bit helper and global lowering
+
+The source is PR210 df95878d11190518e45ef9717c9ee05011f88ace. The independently recorded local scalar word has 2,569,626 weighted events. Its projection is fixed by SHA256 e74e43692685ad5f716b41b363acb8b443c2c90096634a55f0e3728980713944. Both literal forward and signed inverse pass all 20,163 F2 columns, including arbitrary targets and 16,643 independent dirty helpers. The 471 borrowed sources are source aliases, never extra independent dirty coordinates.
+
+The tagged physical word has digest 614f19da4288021cd5c10a575ec7759c069962a6030bf30edb047573ed76edf6 and realizes the telescope Dout M Din^-1. Its input frames are X:Pq, Y:0, Z:sigma and output frames X:H, Y:H-Pq, Z:H. Thus a completed invocation has no data/helper cross blocks and each independent helper residual is D_(H-sigma). This is a full arbitrary-input operator identity, not just zero-dirt testing.
+
+The five-stage compiler maps every local live index through the stage owner injection, maps every local temporary to external family 23,683, and applies the stage's data or helper cover bijection. The active source/target pairs are (X1,Y1), (Y1,X1), (X1,Y1), (Y2,X2), (X2,Y2). The unused pair stays parked. A data route is right multiplication by the exact rational rho_i,S; a helper route is the whole-block exchange tau_i. Both are bijections of the eligible GL cover. They are charged address movements, not dense payload arithmetic.
+
+Cover expansion is strictly phase-major: complete every cover class in stage i before starting the next routed stage or boundary phase. The same rule applies separately to idle, bridge, completion and terminal-exchange phases. Running all five stages for one class before proceeding to another class would be incorrect because rho and tau refer to other classes. The callable stage and boundary iterators specify the phases; a flattened one-class-per-phase template is not a class-major execution schedule. The admission contract rejects that alternative loop ordering.
+
+For backward stages the compiler reverses the complete local event order, negates scalar coefficients, and replaces local P by H-P. A copied-center block is lowered afresh: copy the original at H-U, transform the copy to H, perform reverse signed scatter, and erase it. It is forbidden to invert an erased temporary. Every original center also retains its complementary live climb. Copy/live index injectivity is checked for all stage orientations.
+
+The global lowering artifact must emit bridges, idle children, independent gauge completions, and the final signed bank exchange. That exchange is an explicitly paid complete-stream role permutation, not a same-frame shear. A stage-descriptor string alone does not meet this obligation.
+
+## 4. Rational five-window geometry and prime eligibility
+
+Take H=Q^24, G=I-J/9 and a triple line q*. Its projector has denominator 6; coordinate permutations supply the port isometries. The two-block involution rho_i,S fixes the two port lines and exchanges their complementary spaces. With Q_i the direct sum of the first i copied complements, the frame is D_[g rho_i,S(P direct_sum Q_i)rho_i,S^-1 g^-1]. The four elementary two-block projector identities imply all eight bank boundary equalities. Rational complements are I-P.
+
+The exact rational partial swap is D_P=[[I-P,P],[P,I-P]]. It is involutory. Every same-frame scalar gate telescopes exactly, all idle transitions are nested actual projectors, and D_I D_Pq=D_(I-Pq) gives the terminal all-data address endpoint after the scalar exchange. No Pauli or Fourier phase claim is imported from the complex compiler into this bit proof.
+
+For each independent physical helper, the five routed residuals occupy different H_i. Their supported projectors have pairwise zero products. Their product is D_(I-sum_i sigma_i), so one paid child D_(sum_i sigma_i) completes D_I. This is one correction for each helper, not a packing of unrelated dirty rows. The 2,279 independent entrances have ranks {12:18,13:48,18:13,20:2200}; the 31 source-owned gauge entrances receive no extra completion.
+
+All 26,750 used physical frame IDs (26,380 distinct bases) have exact determinant witnesses. Their factors lie in {2,3,5,7}; the 231 distinct gauge-bundle bases use {2,3,5}. Window-map denominators also use {2,3,5}. Hence every retained q>2^80 preserves these ranks. General GL charts use gPg^-1, not a recomputed Euclidean orthogonal projection.
+
+The weighted generic normalizer allows every proper rank, including 60,65,90,100 in dimension120. Its cleared pivot-polynomial degree is at most m r(r+1)/2 < m^3. Thus the bad fraction is at most2m^3/q<10^-16. Each bad edge retains the full invertible local-ring residual and is paid 32m^2=460,800 rank-one fallback children. No field elimination at nonunit pivots is substituted.
+
+## 5. Exact bit ledger and ordinary finite recurrence
+
+One invocation has 95,789 positive-rank children and rank mass435,846. Its24 rank22 center copies are included once, as are each original center's live climbs. With H this histogram,
+
+n5 = 5H + 3520(delta4+delta23+delta46+delta50)
+     +18delta60+48delta65+13delta90+2200delta100.
+
+Thus m=120, W=23,683, calls495,304, mass2,837,560, deficit4,400. Work streams do not enlarge the live moment denominator. Five invocations retain all120 rank22 centers.
+
+The counted finite-bit adapter pays the literal scalar word, six bridge families, weighted good-edge wrappers, bad-class residuals, copy/read/erase, class routing, selector arithmetic, and fixed-tape movement. Its retained primitive theorem fixes the eligible prime and finite source recipe before varying width, so its finite constant may depend on that prime and recipe. It cannot depend on width, ancestor payload values or unrelated spectators except through complete volume. This is the correct quantifier order; an invented numerical Turing-machine constant is unnecessary.
+
+The largest child ratio is5/6. Three levels do not halve and four do: (5/6)^4=625/1296<1/2. The same one-level weighted moment induction remains valid; the factor4 is used only in depth/descriptor/borrowed-row bounds. Even paying24 simultaneous temporaries gives physical stock23,707 per GL cover vertex and log_q Wphysical(w)<14400w+1. The resulting internal log_q row charge is at most4ceil(log2(max(2,n)))(14400w+1), plus the old-leaf reserve. High digits are borrowed and restored internally; this is not a new external q-adic group factor.
+
+The defining-integer lift need not equal the older supplier's decoder endpoint. A safe cancellation-free prefix bound is64*37623^3*9430154^2<2^98. It uses neither signed integer restoration nor commutation between stage-dependent decoder channels. F2 payload exactness does not require complex floating precision.
+
+Let c=704197288590873/10^18. The freshly enclosed full bit moment, including all bad fallback charges, has a strictly positive gap. Start from the retained completed ordinary leaf a0=384599/10^10. Three finite levels with atom exponent beta=c give
+
+a_(j+1)=(1-c)c+c a_j,
+a3=.000704197288358392787581160456097840009769165126434639147437106139267359.
+
+At every level a_j<a_(j+1)<c<1-a_(j+1). The actual finite-cutoff witness binds the positive moment and lower-order toll gaps to the new finite recipe. This is not an infinite limiting bootstrap or a packed-profile substitution.
+
+## 6. Complex source, complete splice and finite guard
+
+The complex source is f010392c923279e3dd59ef3aa5fedad23affc5fd. The published gcert/1 gzip and expanded JSON identities are checked directly. Its pinned scalar identities and GBridge2 matrix equalities are reused as source theorems, including phases; no same-rank substitute or changed-185 helper is admitted.
+
+The source-defined old-response and cleanup maps are realized by filtered, reversed and transposed literal helper words. Their scalar projections are checked against complete schedule hashes. Old-response uses a9,412-stream temporary tableau, cleared before all recursive children. The coefficient work stream is also cleared after use. The emitted label/scalar splice preserves13,798 phaseA transitions,46,937 phaseB transitions, and9,412 final climbs, totaling70,147 live transitions. It retains22 center children. Final climbs occur after B, followed by exact final-frame adapters and then cleanup.
+
+The five-stage source theorem supplies all ten exact matrix fields x01/y01 through x34/y34, sfin and term, the actual cover owner bijections, all idle banks and final signed exchange. The complete finite compiler uses the retained two-rank-zero-factor construction around each exact recursive C^r child. Rank-zero phase/permutation operations are charged selected-bit and complete-stream tape primitives, not free RAM lookups.
+
+The exact scalar primitive counts are1,492,984 forward and1,549,456 backward, totaling7,577,864 for three forward plus two backward invocations. All361,405 recursive children, including110 rank20 centers, remain charged. The profile has m110, W14692, mass1,613,040, deficit3,080 and maxchild46.
+
+Physical work stock is24,127|O(110,2)|, with6010 bits; live recursion stock has6009 bits. The proved local operation bill is below2^48 and the complete fixed-program bound below2^6050. Router and literal guards fit G=2^30000 and E=2^100000. Coefficients use only denominators1,2,3,6, so the exact common grid2^-P 3^-G(D+1) introduces no return rounding. Completed child semantics give C1=1, induction gap127B, and C0=2^210000>32mB^2. The complex half-shrink requirement is preserved:2*46<110.
+
+The physical row overcharge16,171 remains below20,161. Degree1,000,000 and suffix slope4,000,000 retain exact row gap23,971,789/25. These constants come from the changed stock and counted program, not from simply replacing72 by110 in the old guard.
+
+## 7. Fresh outer47 arithmetic and completion boundary
+
+Use complex saving747454944651775/10^18, strictly inside its freshly re-enclosed moment. The bit saving is the exact finite a3 above. Both moment gaps exceed7.7e-19. With eta=10^-12 and stopping fraction10^-9, the unchanged retained balanced assembly produces47 positive strict constraints and7 margins.
+
+The isolated candidate is kappa=703701743496697/10^18, or.000703701743496697. The minimum outer margin exceeds it by approximately7.3899e-19. This arithmetic value exceeds the retained frontier. Exact rational gaps, rather than printed decimals, are the evidence.
+
+The certificate remains a receipt-bound compositional research candidate with explicit source-theorem dependencies. The clean loader consumes 22 pinned public files and agrees in 23 exact semantic components with the audited context. Its fresh raw ledger and complete five-stage histogram agree; an empty-directory extraction passes all 10 focused tests. The clean-root global emitter regenerates all 12,858,690 weighted additions, 495,304 paid children, 120 fresh copies and erasures, 2,279 helper completions, and 3,520 final pair exchanges. Source-bound global lowering, exact h24 geometry, clean-loader qualification, and finite closure supplements are all mandatory before the entry point emits its PASS certificate.
+
+These fresh computations do not pretend that the inherited exhaustive F2 columns, exact determinant archive, or complex source theorems were independently reproved. Full portable packaging and Mac qualification remain separate delivery checks. The pinned comparison frontier is df95878's .000694782719690783; the subsequently reported public head31f4ccc1c16c538749845f9244b24747f1cf93dd claims .000695191451687307, which is a separate comparator pending independent validation and is not substituted into this source freeze.
diff --git a/research/five-stage-source527-banks/proof/PARITY-CONTRACT.md b/research/five-stage-source527-banks/proof/PARITY-CONTRACT.md
new file mode 100644
index 0000000000..fad6150b1f
--- /dev/null
+++ b/research/five-stage-source527-banks/proof/PARITY-CONTRACT.md
@@ -0,0 +1,45 @@
+# Parity-even payload gate removal: inherited contract audit
+
+Conclusion: removing an even-coefficient pointwise payload ADD is valid for the bit supplier. The odd-prime address arithmetic remains unchanged. A new paid profile still requires the actual filtered word and full frame schedule; this conclusion alone does not certify a new kappa value.
+
+## Primary source evidence
+
+The bundled, unmodified primary section [`primary/03-motifs.tex`](primary/03-motifs.tex) states the following. Its exact source, attribution, and file hashes are recorded in [`primary/PROVENANCE.md`](primary/PROVENANCE.md).
+
+* Line50: in the bit construction scalars lie in F2.
+* Lines149–177: replace a scalar by an array in R^Omega, with R the scalar ring of the network. Common R-linear frames give the exact operator `D_out S_Omega D_in^-1`.
+* Around line223: the scalar domain need not equal the label field. Bit labels are rational subspaces even though the payload scalars are binary.
+* Lines483–493: rational-address frames are F2-linear permutations of array entries; the bit network uses pointwise XOR gates.
+
+The byte-pinned [`copied-centers-lemma.tex`](copied-centers-lemma.tex) in this package opens with its scalar restoration identity explicitly over F2. Its copy contract preserves the complete role stream, controls and spectators; each copy's frame change and erasure remain paid.
+
+The retained PR234 [`INTEGRATED_PROOF.md`](INTEGRATED_PROOF.md), line56, explicitly says: "The defining-integer lift need not equal the older supplier's decoder endpoint" and the majorant "uses neither signed integer restoration nor commutation between stage-dependent decoder channels." The package's signed integer decoder checks are stronger supplier evidence; the inherited bit interface does not impose their particular integer endpoint on a newly synthesized F2-equivalent word.
+
+The retained weighted bit theorem, [`three-stage-cover-bit.tex`](three-stage-cover-bit.tex), uses the odd-prime local ring Z/q^w for address fields and weighted address interchanges. Its rational projectors, unit-pivot tests, route matrices and fallback coefficients are address geometry. They are not the F2 payload ADD coefficients.
+
+## Operator proof
+
+Fix the finite complete address set Omega and let one stream be a vector in F2^Omega. A payload gate is `x_a <- x_a + (c mod2) x_b`. For even c this is the identity on every input, including arbitrary correlated dirty streams and spectators. Removing any number of these identity gates leaves the complete scalar operator S_Omega unchanged. Negative odd coefficients and coefficient3 also reduce to1, though the simplest experimental witness can retain their original signed integer representatives.
+
+Retain each original input/output frame and every remaining gate's original common frame. The common-frame identity then gives the same `D_out S_Omega D_in^-1` for the filtered word. In particular, every independent helper still has its required partial-swap residual and the existing completed-bank proof still applies. No integer identity of the new lift is needed to conclude this F2 operator identity.
+
+## Safe frame compression
+
+Deleting an ADD removes its operand-frame alignment requirement. For a given stream, collect successive MOVE operations between surviving uses. A use includes any surviving ADD, COPY read/write, ERASE or required phase/endpoint boundary. No use may be crossed merely because another stream is idle.
+
+Suppose the original local frames form a nested chain U0 <= U1 <= ... <= Uk. Their exact address changes telescope to `D_Uk D_U0^-1`. Their orthogonal increments are disjoint commuting projectors, so this is one allowed residual of rank dim(Uk)-dim(U0). In the reflected orientation the chain is the reversed complementary chain; the same endpoint difference and rank follow exactly. The ambient stage background cancels. Rebuild both ledgers from the actual surviving uses and actual basis inclusions; do not infer this from summed ranks alone.
+
+Keep every physical input/output family and its endpoint, all stage boundaries, all data routes/bridges, and every copied-center lifetime. For source527 the120 center copies in the five-stage construction have odd scatter coefficients, so parity filtering does not itself remove those episodes. The shared work stream cannot be used before COPY or after ERASE. A COPY is a barrier even if its original register has no intervening ADD.
+
+## Required finite admission
+
+1. Filter actual emitted ADD records by the literal coefficient modulo2; retain all odd-coefficient records in their original order. Check all20107 formal F2 source, target and dirty columns against the admitted shear. The expected local event counts before any further resynthesis are1,783,196 removed and786,621 retained.
+2. Generate the actual complemented inverse schedule, with fresh reverse copies as before, and verify every formal F2 column again. Do not reverse an erasure.
+3. Compute cancellation-free prefix bounds for the new literal signed forward/inverse projections. Recompute the five-stage bound and finite bill. Failure to reproduce the previous integer decoder is not itself a contract failure; a missing F2 dirty restoration is.
+4. Regenerate exact monotone forward/reflected physical paths and the complete paid histogram. Every fused residual's actual endpoints must be checked. Keep all remaining recursive children, copied-center calls, fallback allowance, data idle edges and stock.
+5. Rebind the completed banks and final global address lowering to the new word and tags. Their actual rational address projectors, odd-q routes and normalizers must not be reduced modulo2.
+6. Rerun actual basis/prime coverage, finite ordinary-leaf assembly, both rational moment engines and all47 outer inequalities before claiming a result.
+
+For simple monotone MOVE fusion the paid rank mass and physical stock remain unchanged; combining positive increments can improve the concave moment while reducing the number of children. Any different deficit or stock must be explained by a separate exact endpoint change. The displayed bound is still conditional on the retained all-size compiler, selectors, common ancestor charts, restored rows, routing, prime and analytic interfaces.
+
+Prepared with OpenAI Codex assistance. This audit is independent of the physical parity-filter/fusion implementation owned by the complex research lane.
diff --git a/research/five-stage-source527-banks/proof/copied-centers-lemma.tex b/research/five-stage-source527-banks/proof/copied-centers-lemma.tex
new file mode 100644
index 0000000000..8933aaf508
--- /dev/null
+++ b/research/five-stage-source527-banks/proof/copied-centers-lemma.tex
@@ -0,0 +1,103 @@
+\subsection{Scalar, frame and temporary-stream contracts}
+The retained source-copy, reversible mixer and scatter maps satisfy
+$JLV=I$. Over $\mathbb F_2$ the chronological word
+\[
+ L,\ J,\ L^{-1},\ V,\ L,\ J,\ L^{-1},\ V
+\]
+adds the input to the target and restores every arbitrary auxiliary.
+Over Gaussian dyadics use its corresponding old-value subtraction and
+new-value addition. A designated retained output has a distinct terminal
+output-use carrier: a matching may feed that carrier, but cannot merge
+two designated terminals or append a later middle-producer consumer.
+Its remaining uses before cleanup are its scatter reads, all of which
+leave its scalar value unchanged.
+
+Within one tensor invocation suppress the fixed future line and put
+\[
+ D_0=B\otimes F_h,\quad D_1=A\otimes F_h,\quad
+ A=B\mathbin{\perp}P,\quad D_U=D_0\mathbin{\perp}(P\otimes U),
+\]
+where $P$ and the suppressed future factor are one-dimensional and
+$U$ is a nondegenerate retained-center space of dimension $r$.
+Thus a local change of rank $d$ has ambient rank $d$.
+Write $\mathcal F_Q$ for the address frame: it is the rational partial
+swap $D_Q$ in the bit construction, and the binary phase $C_Q$ in the
+complex construction. Changing a copy from frame $Q_1$ to $Q_2$ means
+applying $\mathcal F_{Q_2}\mathcal F_{Q_1}^{-1}$ to its physical stream.
+Orthogonal backgrounds cancel in this quotient.
+
+Copies use Paureel's retained copy/transform/read/discard interface
+\cite{paureel,pr29}. A copy has precisely the original complete role
+stream, including all control and spectator ranges, and volume $V/W$.
+It is processed while inactive original streams are parked on the fixed
+depth-first stack tapes. Copying, pointwise reads and erasure are linear
+in that stream's volume. Sequential processing needs one additional
+fixed work stream, not an independently varying row coordinate.
+
+\begin{lemma}[Copied retained-center schedule]
+In either invocation orientation a designated retained center's
+physical rank profile can be changed from $z^r+z^h$ to
+$z^r+z^{h-r}$, with the same scalar action, original role endpoints,
+and arbitrary-scratch restoration. The rank-$r$ transform on the copy
+is included in the recursive call list.
+\end{lemma}
+\begin{proof}
+In the forward orientation the original center has scalar value $w$
+at frame $D_U$ after the middle mixer. Its old sequence was
+$D_U\to D_0\to D_1$: scatter at $D_0$, then cleanup at $D_1$,
+with ranks $r,h$.
+Keep the original at $D_U$, copy its physical stream, and transform
+only the copy to $D_0$, paying rank $r$. Supply every scatter read
+from that transformed copy while its data targets remain at their
+unchanged common frame $D_0$, then erase the temporary. The reads
+are exactly those of $w$ in the old schedule. Since they do not change
+$w$, move the original directly from $D_U$ to $D_1$ before its first
+cleanup incidence; this costs $h-r$. Every subsequent original gate
+therefore receives the same scalar value at its required frame.
+
+In the reverse complementary orientation the old sequence was
+$D_0\to D_1\to D_{U^\perp}$, with ranks $h,r$ and scatter at $D_1$.
+After the preceding all-$D_0$ operations, advance the original directly
+to $D_{U^\perp}=D_0\mathbin{\perp}(P\otimes U^\perp)$, paying
+$h-r$. Copy it and transform the copy to $D_1$, paying $r$.
+Use that copy for all early scatter reads, then erase it. The original
+already has the exact frame and unchanged scalar value required by the
+following inverse mixer. Terminal output-use separation ensures no
+intervening producer consumer has been bypassed.
+
+For partial swaps either direction of the rank-$r$ change is the same
+involution. For complex phases the forward copy uses $C_U^{-1}$,
+the direct original growth uses $C_{U^\perp}$, and the reverse copy
+uses $C_FC_{U^\perp}^{-1}=C_U$. These are the same charged ranks,
+implemented by the retained forward/inverse residual interface.
+All tensor backgrounds cancel as stated. Every original scalar
+incidence, final source-copy cancellation and auxiliary source/sink
+operator is unchanged. Thus the scalar identity and all endpoints
+remain exact on arbitrary initial auxiliary values.
+\end{proof}
+
+For the selected centers $r=h-1$, and there are $h$ distinct retained
+output uses per invocation. Their old histogram has rank mass
+$hR_h+2\ell_h$, where $\ell_h=h(h-1)$. The new histogram is exactly
+\[
+ H'_{h,1}=H_{h,1}+h,\qquad H'_{h,h}=H_{h,h}-h,
+ \qquad H'_{h,r}=H_{h,r}\quad(r\ne1,h),
+\]
+and hence $\sum_r rH'_{h,r}=hR_h+\ell_h$.
+Every old rank-$(h-1)$ residual remains as a paid transform of a copy;
+its projector and ordinary bit pivot profile are unchanged. Only the
+full-rank original transition becomes the rank-one complement.
+The physical role count $R_h$ is unchanged.
+
+For two factors, put $N=v_av_b$, $B_a=v_bR_a$, $B_b=v_aR_b$,
+$W=2N+B_a+B_b$ and $L=v_b\ell_a+v_a\ell_b$.
+All exterior, data-front and endpoint-correction calls remain in place.
+The resulting complete rank mass is
+\[
+ s=Wm-N+L,\qquad Wm-s=N-L.
+\]
+This follows by subtracting exactly $L$ from the retained two-stage
+physical mass $Wm-N+2L$. The $N$ rank-one endpoint-copy calls are
+separate and remain fully charged. Because center copies retain the
+same complete role-volume fraction $1/W$, the existing complete-row,
+integer-width remainder and fixed-tape recurrence proofs apply.
diff --git a/research/five-stage-source527-banks/proof/primary/03-motifs.tex b/research/five-stage-source527-banks/proof/primary/03-motifs.tex
new file mode 100644
index 0000000000..cdb0ba527a
--- /dev/null
+++ b/research/five-stage-source527-banks/proof/primary/03-motifs.tex
@@ -0,0 +1,700 @@
+\section{Two finite networks with a rank saving}
+\label{sec:finite-networks}
+
+We construct two fixed linear networks that exchange two banks of values
+while restoring every auxiliary value.  When the values are replaced by
+arrays, a common representation on all wires at a gate lets that gate act
+pointwise without disturbing the representation.  We explain this invariant
+before constructing the subspace labels that make changes of representation
+cheap.  The first network uses bit values and rational subspaces; the second
+uses Gaussian dyadic values and binary subspaces.  The scalar domain specifies
+the values combined at a gate, while the subspace labels are used to construct operators
+on each wire's address-indexed array.
+
+Put
+\[
+ h=100,\qquad v=\binom h3=161700,\qquad
+ N=v^3=4227952113000000,\qquad m=h^3=1000000.
+\]
+Let $\mathcal T$ be the set of three-element subsets of $[h]$.
+There are two data banks $X_a,Y_a$, indexed by
+$a=(a_1,a_2,a_3)\in\mathcal T^3$.
+A wire carries one scalar.  A gate is a fixed linear map on a specified
+set of wires; untouched wires retain their values.  The vertices on a
+wire are its source, the gates that touch it in time order, and its sink;
+its edges join consecutive vertices.  We call the name of a wire its
+\emph{role}, so that we can distinguish a fixed place in the network
+from the value currently carried there.
+
+\subsection{Scalar operations and restored auxiliary values}
+
+There are three stages.  At stage $j$, fix the other two coordinates and
+operate on the $v$ values in each bank obtained by varying coordinate $j$.  Thus
+there are $v^2$ separate invocations at each stage.
+Each such invocation has its own auxiliary wires, distinct from all
+other invocations, including those in other stages.  There are
+\[
+ I=3v^2=78440670000
+\]
+invocations.  In a forward invocation, write $x_T$ for the data used as
+controls and $y_S$ for the data to be updated; $S,T\in\mathcal T$ are
+their varying $j$th coordinates.  These are the logical source and
+target banks of the invocation.  For each ordered pair $(S,T)$ that is
+neighboring under the rule below, there is an auxiliary wire $A_{ST}$.
+There are also central wires $C_i$, $i\in[h]$, and in the second
+construction a wire $C_\ast$.  We call all these auxiliary wires
+\emph{scratch wires}; their initial values may be arbitrary.
+
+The following maps specify the construction.
+\begin{itemize}
+\item In the bit construction, scalars lie in $\mathbb F_2$, and
+$S,T$ are neighbors when $|S\cap T|=1$.
+The copy map adds $x_T$ to $A_{ST}$ for each neighbor pair.
+The side injection adds $\sum_{T:\,|S\cap T|=1}A_{ST}$ to $y_S$.
+The gather adds $\sum_{T\ni i}x_T$ to $C_i$, and the scatter adds
+$\sum_{i\in S}C_i$ to $y_S$.
+\item In the complex construction, scalars lie in
+$\mathbb Z[i,1/2]$, and neighbors are distinct triples with even
+intersection.  Copy is unchanged.  Side injection adds
+\[
+ -\sum_{T\text{ neighboring }S}\frac{|S\cap T|-1}{2}A_{ST}
+\]
+to $y_S$.  Gather adds $\sum_{T\ni i}x_T$ to $C_i$ and
+$\sum_Tx_T$ to $C_\ast$.  Scatter adds
+$(\sum_{i\in S}C_i-C_\ast)/2$ to $y_S$.
+\end{itemize}
+Each update leaves its control wires unchanged.  The first two rows of
+the following schedule remove the effect of the old scratch values;
+the next four introduce the source values and add their effect to the
+target; the last two restore the scratch values.
+This cancellation of arbitrary initial scratch values is a linear
+instance of the restoration construction for transparent computation in
+Buhrman, Cleve, Kouck\'y, Loff and Speelman~\cite[Section~3]{BuhrmanCleveKouckyLoffSpeelman2014}.
+The forward schedule is
+\begin{equation}
+\begin{array}{c|l|l}
+\text{row}&\text{operation}&\text{gate grouping}\\ \hline
+0&\text{subtract side injection}&\text{one gate per target}\\
+1&\text{subtract scatter}&\text{one gate on targets and center}\\
+2&\text{copy}&\text{one gate per source}\\
+3&\text{gather}&\text{one gate on sources and center}\\
+4&\text{scatter}&\text{one gate on targets and center}\\
+5&\text{side injection}&\text{one gate per target}\\
+6&\text{undo gather}&\text{one gate on sources and center}\\
+7&\text{undo copy}&\text{one gate per source}.
+\end{array}
+\label{eq:motif-schedule}
+\end{equation}
+
+\begin{lemma}\label{lem:scalar-motifs}
+For arbitrary initial values of its auxiliary wires, a forward
+invocation changes $y_S$ to $y_S+x_S$ and leaves every other value
+unchanged.  Performing a forward invocation from $X$ to $Y$ at stage 1,
+the inverse forward schedule from $Y$ to $X$ at stage 2, and a forward
+invocation from $X$ to $Y$ at stage 3 sends
+\[
+ (X_a,Y_a)\longmapsto(-Y_a,X_a)
+\]
+and restores every auxiliary value.  In the bit construction this is the
+permutation exchanging $X_a$ and $Y_a$.
+\end{lemma}
+
+\begin{proof}
+Write $\mathcal V$ for copy, $\mathcal G$ for gather,
+$\mathcal J$ for side injection and $\mathcal R$ for scatter.
+If the initial side and central vectors are $a,c$, the first two rows
+subtract $\mathcal J a+\mathcal R c$ from $y$.  Rows 2 and 3 change
+them to $a+\mathcal Vx,c+\mathcal Gx$.  Rows 4 and 5 therefore add
+$\mathcal R(c+\mathcal Gx)+\mathcal J(a+\mathcal Vx)$.
+Rows 6 and 7 restore $c,a$.  The net target change is
+$(\mathcal J\mathcal V+\mathcal R\mathcal G)x$, independently of
+$a,c$.
+
+In the bit case, the central coefficient from $x_T$ to $y_S$ is
+$|S\cap T|\bmod2$.  The side contribution cancels exactly the
+intersection-one cases; distinct triples with intersection zero or two
+already have coefficient zero, and the self coefficient is one.
+In the complex case, the central coefficient is $(|S\cap T|-1)/2$.
+The side cancels it for distinct triples with intersection zero or two;
+intersection one contributes zero, and the self coefficient is one.
+Thus both net maps are $y\gets y+x$.
+
+The inverse schedule with logical source $Y$ and target $X$ effects
+$X\gets X-Y$.  Starting from $(x,y)$, the three stages give
+$(x,y+x)$, then $(-y,y+x)$, then $(-y,x)$.
+Each invocation restores its own auxiliary wires.
+\end{proof}
+
+A fixed triple has
+\[
+ z_{\rm b}=3\binom{97}{2}=13968,\qquad
+ z_{\rm c}=\binom{97}{3}+3\cdot97=147731
+\]
+neighbors in the two constructions, respectively.  The second count
+separates intersection zero and intersection two.  Ordered pairs must
+be counted: there are $vz$ side wires in each invocation, not $vz/2$.
+With $c_{\rm b}=100$ and $c_{\rm c}=101$ central wires per invocation,
+the total wire counts are
+\begin{align}
+ W_{\rm b}&=2N+I(vz_{\rm b}+c_{\rm b})
+          =177176569091445000000,\label{eq:bit-wire-count}\\
+ W_{\rm c}&=2N+I(vz_{\rm c}+c_{\rm c})
+          =1873807244643542670000.\label{eq:complex-wire-count}
+\end{align}
+All these numbers are fixed, independently of any eventual input length.
+
+\subsection{From scalar values to arrays}
+
+We record the representation principle before choosing the labels.  Let
+$R$ be the scalar ring of either network and let $\Omega$ be a finite address
+set common to all wires.  Replace each scalar by an array in $R^\Omega$, and
+apply each scalar gate pointwise at every address.  Write $\mathsf S_\Omega$
+for the resulting network on the vector of wire arrays.
+
+At each source, gate and sink $v$, assign an invertible $R$-linear operator
+$D_v$ on $R^\Omega$, called its \emph{frame}.  A frame acts on the values
+indexed by addresses within one wire's array and may mix them; a gate instead
+combines wire values at each fixed address.  All incoming and outgoing
+incidences of one gate use the same $D_v$.  On an edge from $u$ to $v$, apply
+$D_vD_u^{-1}$ to that wire's array.  Define the operators on all input and
+output roles by
+\[
+ D_{\rm in}=\bigoplus_w D_{\mathrm{source}(w)},\qquad
+ D_{\rm out}=\bigoplus_w D_{\mathrm{sink}(w)}.
+\]
+For a supplied physical input $f$, define its logical interpretation to be
+$g=D_{\rm in}^{-1}f$; this definition requires no physical preprocessing.
+At a vertex $v$, the invariant is that the stored array equals $D_v$ applied
+to the logical array there.  It holds at the sources by definition.  On an
+edge whose logical array is $\varphi$, the edge operator changes $D_u\varphi$
+to $D_v\varphi$.  At a gate with logical input arrays $\varphi_j$,
+$R$-linearity gives
+$\sum_j c_jD_v\varphi_j=D_v(\sum_j c_j\varphi_j)$ for each output combination, with
+$c_j\in R$.  Thus the common frame commutes with the pointwise gate, and
+the invariant continues to the sinks.  The implemented operator is therefore
+\begin{equation}
+ D_{\rm out}\,\mathsf S_\Omega\,D_{\rm in}^{-1}.
+ \label{eq:common-frame-identity}
+\end{equation}
+In particular, if the scalar network is a signed role permutation that
+sends input role $w$ to output role $\rho(w)$ with scalar sign
+$\varepsilon_w$, that route applies
+$\varepsilon_wD_{\mathrm{sink}(\rho(w))}
+D_{\mathrm{source}(w)}^{-1}$ to its input array.  All roles, including
+the auxiliary roles, enter this identity.  Once the scalar routing is fixed,
+the endpoint frames determine the array operator on each route; the
+intermediate frames can then be chosen to make each edge change inexpensive.
+
+Figure~\ref{fig:frame-compatibility} summarizes this invariant.
+\begin{figure}[ht]
+\centering
+\input{figures/proof-map.tex}
+\caption{An edge changes the representation of one role's logical array
+$g_w$. At the next gate $G$, all $t$ input arrays have the common frame
+$D_v$, so the pointwise linear gate commutes with that frame. The same
+$D_v$ acts on each output component.}
+\label{fig:frame-compatibility}
+\end{figure}
+
+\subsection{Subspaces attached to the gates}
+
+The scalar network already has the required signed exchange. We now
+choose its intermediate labels to make the changes of frame inexpensive.
+A label is a nondegenerate subspace of a fixed bilinear space. On every
+edge one label will contain the other. For such an edge, the
+\emph{orthogonal residual} is the orthogonal complement of the smaller label inside
+the larger one; its dimension is the absolute change in label dimension.
+The saving will come from keeping the total dimension lost on decreasing
+edges small relative to $N$.
+
+For the bit network, let $F=\mathbb Q^h$ with bilinear form
+\[
+ \langle x,y\rangle=x^{\mathsf T}(I-J/9)y,
+\]
+where $J$ is the all-one matrix.  This form has eigenvalues $1$ and
+$1-h/9=-91/9$, so it is nondegenerate.  For a triple $T$ let $t_T$ be
+its indicator vector.  Then
+\[
+ \langle t_S,t_T\rangle=|S\cap T|-1,
+ \qquad \langle t_T,t_T\rangle=2.
+\]
+For the complex network, take instead $F=\mathbb F_2^h$ with the
+ordinary dot product; here $t_T\cdot t_T=1$.
+Thus every triple line is nondegenerate, and neighboring triple lines
+are orthogonal, in the label field belonging to either construction.
+The scalar domain need not equal the label field.
+These degree-one intersection formulas specialize the construction
+described by Alon~\cite[Section~3, remark after Theorem~1.1]{Alon1998}
+to the prime $2$, with the ground-set size changed to $h=100$.
+
+The ambient label space is $\mathcal F=F^{\otimes3}$ with its tensor
+bilinear form.  It has dimension $m$ and is nondegenerate.  Write
+\[
+ u_a=t_{a_1}\otimes t_{a_2}\otimes t_{a_3},\qquad U_a=\langle u_a\rangle.
+\]
+The prescribed terminal labels are
+\[
+\begin{array}{c|cc}
+\text{role}&\text{source label}&\text{sink label}\\ \hline
+X_a&U_a&\mathcal F\\
+Y_a&0&U_a^\perp\\
+\text{each scratch role}&0&\mathcal F.
+\end{array}
+\]
+Write $W$ for the wire count in the construction under consideration.
+The source dimensions sum to $N$ and the sink dimensions to $Wm-N$.
+Thus the signed dimension changes will sum to $Wm-2N$, because the
+contributions at gates cancel. If decreasing edges lose a total
+dimension $L$, the sum of absolute changes is $Wm-2N+2L$.
+We will obtain $L<N/2$. The rational interface below will require
+an additional $N$ source ranks, so this inequality will leave a strict
+saving in both constructions.
+
+At stage $j$, fix an invocation.  Put
+\[
+ A=F^{\otimes(j-1)},\qquad
+ P=\langle t_{a_1}\otimes\cdots\otimes t_{a_{j-1}}\rangle,
+ \qquad B=P^\perp\subset A.
+\]
+An empty tensor product is the one-dimensional ground field; thus
+$B=0$ at stage 1.  Let
+$Q=\langle t_{a_{j+1}}\otimes\cdots\otimes t_{a_3}\rangle
+\subset F^{\otimes(3-j)}$, again using the empty tensor product at
+stage 3.  Since $P$ is nondegenerate,
+$A=B\mathbin{\perp}P$.  
+We prescribe the following transition for each data wire in the stage,
+where $t$ is the indicator of that wire's $j$th triple:
+\[
+\begin{array}{c|c|c}
+ &\text{incoming label}&\text{outgoing label}\\ \hline
+ X&A\otimes\langle t\rangle\otimes Q&(A\otimes F)\otimes Q\\
+ Y&B\otimes\langle t\rangle\otimes Q&
+ \bigl((B\otimes F)\mathbin{\perp}(P\otimes t^\perp)\bigr)\otimes Q.
+\end{array}
+\]
+The outgoing $Y$ label can also be written
+$(P\otimes\langle t\rangle)^\perp_{A\otimes F}\otimes Q$.
+These transitions expose one tensor factor at each stage. To see that
+they fit together, for $j<3$ write
+$Q=\langle t_{a_{j+1}}\rangle\otimes Q'$. The next stage has
+\[
+ A'=A\otimes F,\qquad
+ P'=P\otimes\langle t\rangle,\qquad
+ B'=(P')^\perp_{A'}
+    =(B\otimes F)\mathbin{\perp}(P\otimes t^\perp).
+\]
+Thus the prescribed outgoing labels are
+$A'\otimes\langle t_{a_{j+1}}\rangle\otimes Q'$ on $X$ and
+$B'\otimes\langle t_{a_{j+1}}\rangle\otimes Q'$ on $Y$, exactly
+the next incoming labels. At stage 1 they start at $U_a$ on $X_a$
+and zero on $Y_a$; at stage 3 they end at $\mathcal F$ on $X_a$
+and $U_a^\perp$ on $Y_a$, as prescribed at the terminals.
+
+The gate labels are organized by the decomposition
+\[
+ A\otimes F=(B\otimes F)\mathbin{\perp}(P\otimes F),
+\]
+with the common summand $B\otimes F$ and the remaining summand
+$P\otimes F$ of dimension $h$. Suppress the one-dimensional factor
+$Q$ for the moment. On a side wire joining neighboring physical triples
+with indicators $t_X,t_Y$, the common component grows to $B\otimes F$,
+while the remaining component passes through
+\[
+ P\otimes\langle t_X\rangle
+ \ \subset\ P\otimes t_Y^\perp
+ \ \subset\ P\otimes F.
+\]
+The first inclusion is possible because $t_X\perp t_Y$.
+For the central gates we assign $A\otimes F$ on the physical $X$
+bank and $B\otimes F$ on the physical $Y$ bank. A central wire
+returns once from the former label to the latter, losing exactly
+$P\otimes F$. The gate order below confines all decreases to this
+return, while the other edges realize the prescribed stage transitions.
+
+Number the successive times in an invocation by
+$r=0,\ldots,7$. At stages 1 and 3, time $r$ uses row $r$ of the scalar
+schedule~\eqref{eq:motif-schedule}. At stage 2 it uses the inverse of
+row $7-r$, with the logical banks exchanged. In either case the gates
+touch the physical wires in the order
+\[
+ Y\text{-side},\quad Y\text{-center},\quad X\text{-side},\quad
+ X\text{-center},\quad Y\text{-center},\quad Y\text{-side},\quad
+ X\text{-center},\quad X\text{-side}.
+\]
+A side wire still joins neighboring physical triples in stage 2 because
+the neighbor relation is symmetric. The following table assigns the
+common gate labels. It suppresses the tensor factor $Q$; for a gate
+indexed by a physical bank triple, $t$ denotes that triple's indicator.
+
+\begin{equation}
+\begin{array}{c|l|l}
+\text{time}&\text{physical wires}&\text{common gate label}\\ \hline
+0&Y,\ \text{its side wires}&B\otimes\langle t\rangle\\
+1&Y,\ \text{all center wires}&B\otimes F\\
+2&X,\ \text{its side wires}&(B\otimes F)\mathbin{\perp}(P\otimes\langle t\rangle)\\
+3&X,\ \text{all center wires}&A\otimes F\\
+4&Y,\ \text{all center wires}&B\otimes F\\
+5&Y,\ \text{its side wires}&(B\otimes F)\mathbin{\perp}(P\otimes t^\perp)\\
+6&X,\ \text{all center wires}&A\otimes F\\
+7&X,\ \text{its side wires}&A\otimes F.
+\end{array}
+\label{eq:motif-labels}
+\end{equation}
+All labels are tensored by $Q$ and viewed as subspaces of $\mathcal F$.
+Every displayed summand is nondegenerate.
+
+All scratch sources have label zero and all scratch sinks have label
+$\mathcal F$, as prescribed above. The next lemma verifies the stage
+boundaries and all intervening edge changes, including these scratch
+edges.
+
+\begin{lemma}\label{lem:motif-residuals}
+Every edge joins comparable nondegenerate labels.  The only decreasing
+edges are the central-wire edges from time 3 to time 4; their dimension
+loss is $h$.  Consequently, if $c$ is the number of central wires per
+invocation and $W$ the total number of wires, then
+\begin{equation}
+ L=Ich,\qquad
+ \sum_{e}\bigl|\dim U_{\mathrm{head}(e)}-
+                       \dim U_{\mathrm{tail}(e)}\bigr|
+ =Wm-2N+2L.                                      \label{eq:rank-budget-labels}
+\end{equation}
+For the binary label space, every nonzero orthogonal residual of an
+edge has an orthonormal basis.
+\end{lemma}
+
+\begin{proof}
+Write $a=\dim A=h^{j-1}$ and $f=h^{3-j}$.  In the table below,
+$\mathrm{in}$ denotes the preceding vertex on that data wire: its source
+at stage 1, and its last gate in the preceding stage otherwise.  The
+interstage calculation above gives its label.  The table lists the
+orthogonal residual and its dimension for every edge except the final
+data edges to their sinks, whose labels agree and whose residuals are zero.
+Tensor each residual by $Q$, except in the last row.  The symbols $t_X,t_Y$ are
+the indicators of the neighboring physical triples joined by a side wire.
+\[
+\begin{array}{l|l|r}
+\text{wire and edge}&\text{orthogonal residual}&\text{dimension}\\ \hline
+X:\ \mathrm{in}\to2&B\otimes t_X^\perp&(a-1)(h-1)\\
+X:\ 2\to3&P\otimes t_X^\perp&h-1\\
+X:\ 3\to6,\ 6\to7&0&0\\
+Y:\ \mathrm{in}\to0&0&0\\
+Y:\ 0\to1&B\otimes t_Y^\perp&(a-1)(h-1)\\
+Y:\ 1\to4&0&0\\
+Y:\ 4\to5&P\otimes t_Y^\perp&h-1\\
+\mathrm{center}:\ \mathrm{source}\to1&B\otimes F&(a-1)h\\
+\mathrm{center}:\ 1\to3&P\otimes F&h\\
+\mathrm{center}:\ 3\to4&P\otimes F&h\\
+\mathrm{center}:\ 4\to6&P\otimes F&h\\
+\mathrm{side}:\ \mathrm{source}\to0&B\otimes\langle t_Y\rangle&a-1\\
+\mathrm{side}:\ 0\to2&(B\otimes t_Y^\perp)\mathbin{\perp}(P\otimes\langle t_X\rangle)
+ &(a-1)(h-1)+1\\
+\mathrm{side}:\ 2\to5&P\otimes\langle t_X,t_Y\rangle^\perp&h-2\\
+\mathrm{side}:\ 5\to7&P\otimes\langle t_Y\rangle&1\\
+\mathrm{scratch}:\ \mathrm{last}\to\mathrm{sink}
+ &(A\otimes F)\otimes Q^\perp&ah(f-1).
+\end{array}
+\]
+The last gate on a central wire is at time 6, and the last gate on a side
+wire is at time 7.  Both have label $(A\otimes F)\otimes Q$.  Enlarging it
+to $\mathcal F$ at the sink gives the last residual in the table, with
+$Q^\perp$ taken in the entire future tensor space.
+The side edge $2\to5$ uses the neighbor relation
+essentially.  Since $t_X\perp t_Y$, we have
+$\langle t_X\rangle\subset t_Y^\perp$.  Hence its two labels satisfy
+\[
+ (B\otimes F)\mathbin{\perp}(P\otimes\langle t_X\rangle)
+ \ \subset\
+ (B\otimes F)\mathbin{\perp}(P\otimes t_Y^\perp).
+\]
+The complement of $\langle t_X\rangle$ inside $t_Y^\perp$ is
+$\langle t_X,t_Y\rangle^\perp$, of dimension $h-2$; tensoring by
+$P$ and then by $Q$ gives the residual in the table.  The other entries
+follow directly by splitting $A=B\mathbin{\perp}P$ and
+$F=\langle t\rangle\mathbin{\perp}t^\perp$.
+Label dimension is nondecreasing along every edge except center $3\to4$, which removes
+$P\otimes F\otimes Q$.  The table includes the interstage edges through
+$\mathrm{in}$, and the initial and final data labels match their terminals,
+so no other dimension changes occur.
+
+The signed sum of dimension changes telescopes at every gate: each
+touched wire has one incoming and one outgoing incidence with the same
+gate label.  The sum of source dimensions is $N$, while the sum of
+sink dimensions is $Wm-N$.  The signed increase is therefore $Wm-2N$.
+There are $Ic$ decreasing edges, each of loss $h$; replacing signed
+changes by absolute changes adds $2L$.
+
+Now work over $\mathbb F_2$.  A nondegenerate symmetric bilinear space
+is nonalternating precisely when it contains a vector of norm one.
+The lines $P,Q$ have such vectors.  If $B\ne0$, a coordinate unit
+outside the support of the earlier tensor vector belongs to $B$:
+that support has size $3^{j-1}<h^{j-1}$ when $j>1$.
+The same argument gives a unit vector in $Q^\perp$ when it is nonzero.
+A triple complement, and the complement of two neighboring triple
+lines, contain coordinate units outside supports of sizes at most
+three and six, respectively.  Full tensor spaces contain coordinate
+units as well.  In each nonzero tensor summand, tensoring these chosen
+norm-one vectors gives a norm-one vector; placing it in that orthogonal
+summand gives one in the whole residual.  Each residual is nondegenerate
+by the decompositions above.  Thus every nonzero residual is nonalternating.
+
+For completeness, every nondegenerate nonalternating binary symmetric
+space has an orthonormal basis.  Split off unit lines until the
+remaining space is alternating.  A nonzero nondegenerate alternating
+space splits into planes with bases $a,b$ satisfying
+$a\cdot a=b\cdot b=0$, $a\cdot b=1$; this follows by choosing a
+nonzero $a$, choosing $b$ with $a\cdot b=1$, and taking the orthogonal
+complement of their nondegenerate plane.  Retain one unit vector $w$
+from the lines already split off.  A plane orthogonal to $w$ can be
+absorbed into that line: the three vectors
+\[
+ w+a,\qquad w+b,\qquad w+a+b
+\]
+are independent, pairwise orthogonal, and have norm one.
+Use one of these new unit vectors in place of $w$ for the next plane;
+the remaining planes are still orthogonal to it.  Repeating this
+operation absorbs every alternating plane.
+\end{proof}
+
+The loss ratios have the exact values
+\[
+ \frac{L_{\rm b}}{N}=\frac{100}{539},\qquad
+ \frac{L_{\rm c}}{N}=\frac{101}{539}.
+\]
+In particular, both are smaller than $1/2$.  We now turn the dimension
+count into the two interfaces needed later.
+
+\subsection{The rational matrix interface}
+
+For a nondegenerate rational subspace $U\subset\mathcal F$, write
+$P_U$ for the projection onto $U$ along $U^\perp$.  Assign $P_U$ to
+each gate or sink with label $U$.  Assign the source matrix $-P_{U_a}$
+to $X_a$ and zero to every other source.
+These are fixed rational $m\times m$ matrices; write $M_v$ for the matrix
+at vertex $v$.  Define
+$\rho(X_a)=Y_a$, $\rho(Y_a)=X_a$, and let $\rho$ fix every auxiliary
+wire.
+
+In the tape application, two address chunks are written as vectors $H,D$
+of $m$ fields.  For a matrix $M$, the frame $\Phi_M$ moves the array entry
+at $(H,D)$ to $(H+MD,D)$, leaving the other address fields unchanged.
+The fields use a radix coprime to the fixed denominators, and addition is
+modulo their common range.  Hence $\Phi_M$ is an invertible
+$\mathbb F_2$-linear permutation of array entries, and
+$\Phi_{M'}\Phi_M=\Phi_{M'+M}$.  Using $D_v=\Phi_{M_v}$
+in \eqref{eq:common-frame-identity}, the endpoint matrix difference on each
+routed role is exactly the shear applied to that role.  The following identity
+makes it the same shear $H\gets H+D$ on every role;
+Section~\ref{sec:tape-interchange} turns that shear into a field interchange
+and implements an edge change with one smaller interchange per unit of its
+rational rank.
+
+\begin{proposition}\label{prop:bit-motif-interface}
+The bit network has $W_{\rm b}$ wires, uses only pointwise XOR gates,
+and sends its input vector to the permutation $\rho$ of that vector.
+Its rational matrices are common to all incidences of a gate and obey
+\[
+ M_{\mathrm{sink}(\rho(w))}-M_{\mathrm{source}(w)}=I_m
+ \quad\text{for every input role }w.
+\]
+The sum of the rational ranks of all edge differences is
+\begin{align}
+ s_{\rm b}
+ &=\sum_{e}\operatorname{rank}_{\mathbb Q}
+      (M_{\mathrm{head}(e)}-M_{\mathrm{tail}(e)})\\
+ &=W_{\rm b}m-N+2L_{\rm b}
+  =177176569088785861287000000<W_{\rm b}m.
+\end{align}
+\end{proposition}
+
+\begin{proof}
+Lemma~\ref{lem:scalar-motifs} gives the scalar permutation, including
+all auxiliary roles.  For $U\subset V$ nondegenerate,
+$P_V-P_U$ is the identity on $V\cap U^\perp$ and zero on its
+orthogonal complement.  Its rank is $\dim V-\dim U$; reversing the
+edge negates the matrix without changing its rank.
+The only exceptions are the $N$ negative source projections.
+At stage 1, $B=0$, and the first gate on $X_a$ has label exactly $U_a$.
+Its first edge matrix is therefore $2P_{U_a}$, of rational rank one,
+where the corresponding dimension change was zero.
+Lemma~\ref{lem:motif-residuals} consequently gives the stated rank sum.
+Its deficit from $W_{\rm b}m$ is $N-2L_{\rm b}>0$.
+
+For an input $X_a$ the endpoint difference is
+$P_{U_a^\perp}-(-P_{U_a})=I_m$.  Every other role has zero source
+matrix and identity sink matrix.  This proves the interface.
+\end{proof}
+
+\subsection{The binary phase interface}
+
+Use the complex scalar network and binary labels.  For a binary vector
+$x$, let $\operatorname{wt}(x)$ denote its integer Hamming weight, and
+write $[b]\in\{0,1\}$ for the integer representative of $b\in\mathbb F_2$.
+For a label $U$, let
+\[
+ q_U(x)=\operatorname{wt}(P_Ux)\pmod4.
+\]
+The projection here is over $\mathbb F_2$.
+Let
+\[
+ H=\frac1{\sqrt2}\begin{pmatrix}1&1\\1&-1\end{pmatrix},\quad
+ C=H\begin{pmatrix}1&0\\0&i\end{pmatrix}H
+   =aI+bX,\qquad a=\frac{1+i}{2},\quad b=\frac{1-i}{2},
+\]
+where $X$ interchanges the two coordinates.  On functions of
+$x\in\mathbb F_2^m$ define
+\[
+ \mathcal C_U=H^{\otimes m}\operatorname{diag}_x(i^{q_U(x)})H^{\otimes m}.
+\]
+The use of $H$ in this definition is an identity of matrices.  Each entry
+of $\mathcal C_U$ and its inverse is a Gaussian integer divided by $2^m$,
+so both preserve arrays over $\mathbb Z[i,1/2]$.  The kernels implemented
+below have Gaussian dyadic coefficients.
+
+To apply the scalar network to these functions, replace the scalar on
+each wire $w$ by an array
+$f_w:\mathbb F_2^m\to\mathbb Z[i,1/2]$.  At each address, a gate
+applies its scalar linear map to the values on the wires it touches.
+The frame in \eqref{eq:common-frame-identity} is $D_v=\mathcal C_U$ at a
+vertex with label $U$.  On an edge from label $U$ to label $V$, apply
+$\mathcal C_V\mathcal C_U^{-1}$ to that wire's array.
+All frames are diagonalized by the same matrix $H^{\otimes m}$.
+For an integer $k\ge1$, an address for $k$ columns is
+$(x^{(1)},\ldots,x^{(k)})\in(\mathbb F_2^m)^k$; every frame and edge
+operator is then the tensor product of its one-column version over
+these $k$ coordinates.  Each scalar gate is still applied once at each
+joint address, to the vector of values on the roles it touches.
+
+\begin{proposition}\label{prop:complex-motif-interface}
+For each edge of the complex network, write $U,V$ for its tail and
+head labels.  Its phase-frame difference
+$\mathcal C_V\mathcal C_U^{-1}$ is a product of one translation
+kernel $aI+bX_v$, or its inverse, per vector in an orthonormal basis
+of the edge residual.  Here $(X_vf)(x)=f(x+v)$.
+The sum of these basis lengths is
+\[
+ s_{\rm c}=W_{\rm c}m-2N+2L_{\rm c}
+ =1873807244636671267308000000<W_{\rm c}m.
+\]
+After source and sink diagonal corrections, and undoing the signed
+bank exchange, telescoping these phase frames gives $C^{\otimes m}$
+on every role, including auxiliary roles.  The same assertion holds
+on any positive number of columns by taking tensor products over columns.
+\end{proposition}
+
+\begin{proof}
+For binary vectors $z,w$,
+\[
+ \operatorname{wt}(z+w)
+ =\operatorname{wt}(z)+\operatorname{wt}(w)
+       -2|\operatorname{supp}(z)\cap\operatorname{supp}(w)|.
+\]
+If $z\cdot w=0$, the intersection cardinality is even, so weight is
+additive modulo four.  If $V=U\mathbin{\perp}E$ and $\mathcal B$
+is an orthonormal basis of $E$, it follows that
+\[
+ q_V(x)-q_U(x)=\sum_{v\in\mathcal B}
+                 \operatorname{wt}(v)[v\cdot x]\pmod4.
+\]
+For a decreasing edge the right side is negated.
+Each $v$ has norm one, so $\operatorname{wt}(v)$ is odd.  Its
+coefficient in the phase, including the sign for the edge direction,
+is therefore some $\varepsilon\in\{1,-1\}$ modulo four.  The function
+$i^{\varepsilon[v\cdot x]}$ equals
+\[
+ \frac{1+i^\varepsilon}{2}
+ +\frac{1-i^\varepsilon}{2}(-1)^{v\cdot x}.
+\]
+The identity $H\operatorname{diag}(1,-1)H=X$, tensored over the
+coordinates in $\operatorname{supp}(v)$, sends the diagonal character
+$(-1)^{v\cdot x}$ to translation $X_v$.  The displayed phase therefore
+becomes $aI+bX_v$ or its inverse.  The nonzero residuals have such
+bases by Lemma~\ref{lem:motif-residuals}; for a zero residual use the
+empty basis.  The same lemma gives the count $s_{\rm c}$.
+The deficit is $2N-2L_{\rm c}>0$.
+
+For an edge $e$, write $\mathcal B_e$ for its residual basis and
+$K_{e,v}$ for the forward or inverse translation kernel associated
+with $v\in\mathcal B_e$.  Taking tensor products preserves their
+product, so on $k$ columns
+\[
+ \bigl(\mathcal C_V\mathcal C_U^{-1}\bigr)^{\otimes k}
+ =\prod_{v\in\mathcal B_e}K_{e,v}^{\otimes k}.
+\]
+Thus each residual vector contributes one factor acting on all $k$
+columns, and there are $s_{\rm c}$ such factors over the network.
+
+Let $\rho$ denote the underlying permutation of the complex roles: it
+exchanges $X_a,Y_a$ and fixes each scratch role.
+By Lemma~\ref{lem:scalar-motifs}, the pointwise scalar network sends an input array at
+$X_a$ to $Y_a$ with sign $+1$, an input at $Y_a$ to $X_a$ with sign
+$-1$, and each scratch input to its own output with sign $+1$.
+Applying \eqref{eq:common-frame-identity} with $D_v=\mathcal C_{U_v}$,
+where $U_v$ is the label at vertex $v$, shows that an input role $w$ reaches
+output role $\rho(w)$ with that sign and with operator
+$\mathcal C_{U_{\mathrm{sink}(\rho(w))}}
+ \mathcal C_{U_{\mathrm{source}(w)}}^{-1}$.
+For $k$ columns, use $D_v=\mathcal C_{U_v}^{\otimes k}$ on the joint
+address set in the same identity.  The route operator is then the $k$th
+tensor power of the displayed address operator, multiplied by its scalar
+route sign once.
+The zero label has frame $I$, while the full label has frame
+$C^{\otimes m}$ because $q_{\mathcal F}(x)=\operatorname{wt}(x)$.
+Every pair except $X_a\to Y_a$ has these zero and full labels.
+For the remaining pair, write $u=u_a$.  Its binary norm is one and
+its weight is $3^3=27$.  Orthogonal weight additivity yields
+\begin{align*}
+ q_{U_a^\perp}(x)-q_{U_a}(x)
+ &=\operatorname{wt}(x)-2\operatorname{wt}(u)[u\cdot x]\\
+ &=\operatorname{wt}(x)+2[u\cdot x]\pmod4.
+\end{align*}
+Since $2[u\cdot x]\equiv2\sum_{j\in\operatorname{supp}(u)}x_j
+\pmod4$, the operator on $X_a\to Y_a$ is the tensor of $C^{-1}$ on the
+27 coordinates of $\operatorname{supp}(u)$ and $C$ elsewhere.
+Put $Z=\operatorname{diag}(1,-1)$.  Direct multiplication gives
+\[
+ C=iZC^{-1}Z,\qquad C^{-1}=-iZCZ.
+\]
+For $k$ columns, let $Z_{a,k}$ be the tensor product of $Z$ on
+those 27 coordinates in every column and the identity elsewhere.
+Apply $Z_{a,k}$ before the network at $X_a$, and apply
+$i^{27k}Z_{a,k}$ after the network at $Y_a$; the latter scalar is
+applied once to the whole role.  There are $27k$ inverse factors, so
+the identity $C=iZC^{-1}Z$ turns each of them into $C$.  The operator
+on this pair is therefore $C^{\otimes mk}$ as well.
+Only the physical $X$ outputs still carry the minus sign of the
+scalar exchange.  Negate those outputs and send output role $\rho(w)$
+back to role $w$.  Every role, including every scratch role, now has
+the required forward tensor product.
+\end{proof}
+
+\subsection{Explicit rational bounds for the two recurrences}
+
+The relative rank deficits are
+\[
+ \eta_{\rm b}=\frac{W_{\rm b}m-s_{\rm b}}{W_{\rm b}m}
+       =\frac{339}{22587335000000},\qquad
+ \eta_{\rm c}=\frac{W_{\rm c}m-s_{\rm c}}{W_{\rm c}m}
+       =\frac{73}{19906842167500}.
+\]
+For both recurrences we may use the fixed rational exponent
+\begin{equation}
+ \tau=\sigma=1-2^{-50}.
+ \label{eq:explicit-motif-exponents}
+\end{equation}
+Both exact deficits displayed above are greater than $20\cdot2^{-50}$.
+Since $m<2^{20}$ and $\log 2<1$, we have $\log m<20$, whence
+\[
+ m^{1-2^{-50}}
+ =m\exp(-2^{-50}\log m)
+ >m(1-20\cdot2^{-50})>m(1-\eta)
+\]
+for either $\eta=\eta_{\rm b}$ or $\eta=\eta_{\rm c}$.
+Consequently
+\[
+ \frac{s_{\rm b}}{W_{\rm b}}<m^\tau,
+ \qquad \frac{s_{\rm c}}{W_{\rm c}}<m^\sigma,
+ \qquad 0<\tau=\sigma<1.
+\]
+Here $\sigma$ is a rational upper bound for the logarithmic exponent
+of the complex recurrence.
diff --git a/research/five-stage-source527-banks/proof/primary/LICENSE b/research/five-stage-source527-banks/proof/primary/LICENSE
new file mode 100644
index 0000000000..261eeb9e9f
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+++ b/research/five-stage-source527-banks/proof/primary/LICENSE
@@ -0,0 +1,201 @@
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diff --git a/research/five-stage-source527-banks/proof/primary/PROVENANCE.md b/research/five-stage-source527-banks/proof/primary/PROVENANCE.md
new file mode 100644
index 0000000000..bebdb72910
--- /dev/null
+++ b/research/five-stage-source527-banks/proof/primary/PROVENANCE.md
@@ -0,0 +1,23 @@
+# Primary bit-scalar and common-frame contract
+
+The three unmodified source files in this directory are carried by Henry Grant / hcg890's PR234 snapshot:
+
+- Repository: https://github.com/hcg890/integer-mult-bounds
+- PR: https://github.com/CrocSwap/integer-mult-bounds/pull/234
+- Exact snapshot commit: `af3fe331ca60c936229e060681ffccbc1c208678`
+- Primary section path in that snapshot: `upstream/build/sections/03-motifs.tex`
+- Accompanying source paths: `upstream/README.md` and `upstream/LICENSE`
+
+The accompanying README identifies the primary work as **OpenAI, Integer multiplication below n log n, September 23, 2026** and records its original citation and publication link. It is retained as `UPSTREAM-README.md`. This is attribution to the primary source; the snapshot commit above belongs to the carrying repository and is not asserted to be an original OpenAI commit. These credits do not imply review or endorsement of this contribution.
+
+The accompanying Apache-2.0 license is retained without modification as `LICENSE`. These files have not been edited, excerpted, or reformatted. The exact content hashes are:
+
+| Bundled file | SHA256 |
+| --- | --- |
+| `03-motifs.tex` | `ed76761194988a01af77ead0ea6469cb5dcd29690ab985a0b5a4006b0f2b69bb` |
+| `UPSTREAM-README.md` | `8d1ceff898a4e0e681679f91c4def485820be9e99f04e4861f2ce601a364723e` |
+| `LICENSE` | `c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4` |
+
+`03-motifs.tex` supplies the bit scalar ring at line50, the common-frame identity at lines149–177, the distinction between scalar and label fields at line225, and the rational-address bit frame contract at lines483–493. It is bundled as theorem evidence, not as a standalone compilable document. The accompanying [`../PARITY-CONTRACT.md`](../PARITY-CONTRACT.md) proves the parity-filter adaptation and records the finite verification obligations. No theorem dependency in that parity argument requires a machine-local path.
+
+This provenance note and the parity contract were prepared with substantial OpenAI Codex assistance. Original source authorship and assistance disclosures remain separate from that disclosure.
diff --git a/research/five-stage-source527-banks/proof/primary/UPSTREAM-README.md b/research/five-stage-source527-banks/proof/primary/UPSTREAM-README.md
new file mode 100644
index 0000000000..c8cb3c3857
--- /dev/null
+++ b/research/five-stage-source527-banks/proof/primary/UPSTREAM-README.md
@@ -0,0 +1,17 @@
+# [Integer multiplication below n log n](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 23, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:Integer-multiplication-below-n-log-n-September-23-2026,
+  author = {{OpenAI}},
+  title = {{Integer multiplication below $n\log n$}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Integer-multiplication-below-n-log-n-September-23-2026/paper.pdf}{OAI:Integer-multiplication-below-n-log-n-September-23-2026}},
+  year = {2026}
+}
+```
diff --git a/research/five-stage-source527-banks/proof/three-stage-cover-bit.tex b/research/five-stage-source527-banks/proof/three-stage-cover-bit.tex
new file mode 100644
index 0000000000..f05225d21f
--- /dev/null
+++ b/research/five-stage-source527-banks/proof/three-stage-cover-bit.tex
@@ -0,0 +1,264 @@
+\subsection{Atom contract and common endpoint}
+Fix an odd prime $q$, chosen below, and put $R_w=\mathbb Z/q^w\mathbb Z$.
+An atom is a complete consecutive $w$-digit address field. Write $C_n$ for
+reversal of the $n$ atom positions, without reversing digits inside an atom.
+The recursive contract is the involution
+\[
+ F_A(H,D)=(AC_nD,C_nA^{-1}H),
+\]
+where $A$ is a diagonal array of units on the head atoms. Its entries may
+depend on exposed ancestor row fields and the atom index, but not on the
+fresh role fields of this invocation or on the head/data values. Only the
+polynomial-time descriptors generated by the construction are required;
+no constant-cost oracle for arbitrary functions of a prefix is assumed.
+Every call preserves all complete spectator fields and restores its row
+coordinate. The original unweighted call has $A=I$.
+
+For a split idempotent $P$ on the head coordinates set
+\[
+ D_P=\begin{pmatrix}I-P&PC_n\\C_nP&I-C_nPC_n\end{pmatrix},
+ \qquad F=D_I.
+\]
+These are involutions, and $D_P-I$ has split rank $\operatorname{rank}P$.
+Nested commuting-projector identities are polynomial identities and require
+no symmetry of $P$. For $n=mf$ and $G=g\otimes I_f$ define
+\[
+ Q_g=\operatorname{diag}(G,C_nGC_n).
+\]
+Then $Q_gF=FQ_g$ and $Q_gD_PQ_g^{-1}=D_{GPG^{-1}}$.
+Consequently conjugating every base frame by any $g\in\operatorname{GL}_m(R_w)$
+leaves the same full physical endpoint $F$.
+
+Put $Q_A=\operatorname{diag}(A,I)$. Conjugate the entire completed scalar
+word by $Q_A$. Scalar gates mix roles having the same address map, so they
+commute with $Q_A$. A current role routing also commutes with $Q_A$, since
+it changes only fresh fields. Thus only address residuals change, and the
+completed endpoint is $Q_AFQ_A^{-1}=F_A$. More explicitly, when a port
+changes charts from $g$ to $gR_s$, its local label changes from $M$ to
+$R_s^{-1}MR_s$, and
+\[
+ (gR_s)(R_s^{-1}MR_s)(gR_s)^{-1}=gMg^{-1}.
+\]
+This routing is a change of row/chart, not an uncharged dense address map.
+The adapted chart $Q_AQ_gQ_A^{-1}$ commutes with $F_A$. None of these
+conjugations is executed as an extra normalization pass.
+
+\subsection{Explicit rational three-stage geometry}
+Use the retained PR97 local producer with
+\[
+ h=23,\qquad v=1771,\qquad R_{\rm loc}=28866,\qquad \ell=506.
+\]
+Its physical projector ranks and scalar word are unchanged. On
+$\mathcal A=\mathbb Q^{23}$ take the form $H_0=(I-J/9)/2$.
+For a triple $S$ let $q_S=\chi_S$; then $q_S^tH_0q_S=1$.
+Fix $q_0=\chi_{\{1,2,3\}}$ and $K=q_0^\perp$. A rational basis is
+\[
+ k_i=e_i-(e_i^tH_0q_0)q_0,\qquad 2\le i\le23.
+\]
+Projection along $q_0$ is injective on the span of $e_2,\ldots,e_{23}$,
+since $q_0$ has nonzero first coordinate. Thus this is indeed a basis.
+Let $\mathcal B,\mathcal C$ be orthogonal copies of $K$ and put
+$\mathcal E=\mathcal A\perp\mathcal B\perp\mathcal C$, of dimension $m=67$.
+
+Choose a coordinate permutation $\pi_S$ taking $\{1,2,3\}$ to $S$.
+It preserves $H_0$. The rational isometry $R_{12,S}$ fixes $q_S$ and
+$\mathcal C$, and exchanges $\pi_Sk_i\in q_S^\perp\subset\mathcal A$
+with $k_i\in\mathcal B$. Define $R_{23,S}$ by exchanging the same vectors
+with their copies in $\mathcal C$, fixing $q_S$ and $\mathcal B$.
+Both maps are involutions. A hyperedge $(g,S)$ uses the invocations
+$gR_{12,S},g,gR_{23,S}$, with active spaces
+\[
+ g(\mathcal B+\mathbb Qq_S),\qquad g\mathcal A,
+ \qquad g(\mathcal C+\mathbb Qq_S).
+\]
+Their projectors commute; their pairwise and common intersection is the
+projector $P_{gq_S}$, and their sum is $I+2P_{gq_S}$.
+This follows from the orthogonal decomposition
+$\mathbb Qq_S\perp q_S^\perp\perp\mathcal B\perp\mathcal C$ before
+conjugation. These are the cumulative-projector three-stage transfer
+identities: the three scalar shears complete the interchange, with zero
+interstage connectors and no rank-one endpoint correction. Each port
+family visits the full cover bijectively, by right multiplication with
+its displayed involution. The physical ledger is therefore three copies
+of the local auxiliary, data and center histograms, and three copies of
+the exterior histogram $m-\text{nullity}$.
+
+\subsection{One weighted child on a generic residue class}
+Work first on one atom column. Let $g$ be its weighted residual involution,
+of split rank $r$ relative to $I$. If the cross block has generic northeast
+rank profile, unit lower row and column elimination, without normalizing
+pivots, gives
+\[
+ Lg_{XY}R=ED_pF_0^t,
+ \quad E=(e_1,\ldots,e_r),\quad
+ F_0=(e_m,\ldots,e_{m-r+1}),
+\]
+with invertible diagonal $D_p$. Conjugate by $\operatorname{diag}(L,R^{-1})$.
+Split rank and the unit pivot minor give a factorization
+\[
+ g-I=\binom EB(A_0,D_pF_0^t),\qquad
+ A_0E+D_pF_0^tB=-2I_r.
+\]
+For clarity, if $g-I=UV$, then $g^2=I$ gives
+$U(VU+2I)V=0$. The unit minor supplies a left inverse of $U$ and a right
+inverse of $V$, proving the identity over the local ring itself.
+Set
+\[
+ K=-(B+F_0D_p^{-1})E^t+F_0D_p^{-1}A_0(I-EE^t),
+ \qquad S=\begin{pmatrix}I&0\\K&I\end{pmatrix}.
+\]
+The identities $KE=-B-F_0D_p^{-1}$ and
+$D_pF_0^tK=A_0+E^t$ give
+\[
+ SgS^{-1}=\begin{pmatrix}
+ I-EE^t&ED_pF_0^t\\F_0D_p^{-1}E^t&I-F_0F_0^t
+ \end{pmatrix}.
+\]
+This is one weighted reversed interchange on the first $r$ head and last
+$r$ tail coordinates. All other factors are unit lower intrabank maps or
+lower cross-bank additive maps. Thus variable diagonal units remain in
+the child's weight array rather than becoming unsupported routing passes.
+
+For $n=mf$, group atoms by coarse coordinate $i$ and column $j$.
+If $L_P,R_P,D_P$ are unweighted normalizers for the current projector and
+$A_j=\operatorname{diag}(A[if+j])$, then
+$L_j=A_jL_PA_j^{-1}$ and $R_j=R_P$ are unit lower normalizers for
+$A_jPC_m$. The child's exact contiguous index rule is
+\[
+ A_{\rm child}[kf+j]=A[i_kf+j]D_P[k].
+\]
+The cross-bank atom pairing is reversed, and $C_r\otimes C_f=C_{rf}$.
+Hence the middle operation is exactly one $rf$-atom call of the same
+contract, with no interleaving or gathering. Its coefficients depend only
+on parent prefix fields, which become ancestor spectators at the child;
+the child's fresh roles come from disjoint unused row stock.
+
+All unit lower adapters require $O(n)$ atom passes. The retained ordered
+affine lemma implements each later-target shear in $O(V)$, including
+prefix-controlled offsets. The generated descriptors have polynomial cost
+in $w$: good-class evaluation follows one ancestral index per level;
+fallback evaluation branches by a constant per level. Since depth is
+$O(\log e)$ and $w=\lceil e^{1/1000}\rceil$, this costs $e^{O(1)}=w^{O(1)}$.
+Stored ancestor data use $O(w\log e)$ bits. The proof of the ordered affine
+lemma still applies with these controls: polynomial preparation per prefix
+is absorbed by the complete $q^w$ target fiber. Unrelated spectator counters
+are incremented by ripple counting rather than read in their entirety.
+
+\subsection{Weighted leaves and a genuine local-ring fallback}
+At one atom write $J_a=\left(\begin{smallmatrix}0&a\\a^{-1}&0\end{smallmatrix}\right)$,
+$U(a)=\left(\begin{smallmatrix}1&a\\0&1\end{smallmatrix}\right)$ and
+$L(b)=\left(\begin{smallmatrix}1&0\\b&1\end{smallmatrix}\right)$.
+Then
+\[
+ J_a=\operatorname{diag}(1,-1)U(a)L(-a^{-1})U(a),
+ \qquad U(a)=J_1L(a)J_1.
+\]
+Thus a weighted leaf uses four old ordinary atom interchanges, three
+prefix-controlled shifts and one fixed sign. With
+$a_{\rm old}=384599/10^{10}$ it costs $O(Vw^{1-a_{\rm old}})$.
+This constant occurs only at stopped leaves.
+
+A bad residue class cannot be treated by field Bruhat elimination:
+an entry zero modulo $q$ need not vanish modulo $q^w$. Instead Gaussian
+eliminate the full invertible residual of size $N_0=2m$ with unit pivots.
+Each invertible Schur complement has a unit in its next pivot column;
+row swaps and unnormalized additions clear that column. There are at most
+$N_0(N_0-1)$ additions and $N_0$ swaps. Pivot choices are determined by
+the unweighted low residue class: diagonal conjugation scales every
+Schur-complement row and column by units and preserves its zero pattern.
+The final diagonal has product a fixed sign, since the residual has
+determinant $(-1)^r$ and row swaps change only the sign.
+
+A lower addition is a paid additive adapter. An upper addition is
+conjugated into a lower addition by two ordinary coarse-field swaps.
+For a field of $f$ atoms, let $R_{\rm tail}$ reverse the later field.
+Then the ordinary weighted swap is
+\[
+ J_A=R_{\rm tail}F_AR_{\rm tail},\qquad
+ J_A(H,D)=(AD,A^{-1}H).
+\]
+Each $R_{\rm tail}$ is one unweighted reversed call on the first and last
+$\lfloor f/2\rfloor$ atoms, fixing a possible middle atom. Thus $J_A$
+costs at most three children of widths $f,\lfloor f/2\rfloor,
+\lfloor f/2\rfloor$. A paired diagonal scaling is $J_aJ_1$, at most six
+children. The full diagonal is a product of $N_0-1$ such pairs and a fixed
+sign. Across the two banks the reversed pairing already makes the swaps
+and diagonal pairs directly reversed calls; the preceding ordinary bounds
+remain conservative. Reindexing a tail weight by $j\mapsto f-1-j$ changes
+only its descriptor. Charging every addition as upper therefore gives
+\[
+ 6N_0(N_0-1)+3N_0+6(N_0-1)<32m^2.
+\]
+Every child has width at most $f$, hence ratio at most $1/m$. Coefficients
+are computed from constant-sized local-ring matrices and parent weight
+descriptors; only units are inverted. This proves the fallback cost,
+including its actual reversals, rather than assuming nonunit pivots away.
+
+\subsection{Finite covers, batching, and the strict moment}
+Take $q>2^{80}$ avoiding all fixed rational denominators and rank witnesses.
+At atom width $w$ use the finite cover $\operatorname{GL}_m(R_w)$ and its
+right-multiplication edges $g\mapsto gR_s$. Every vertex has the unique chart
+\[
+ g=g_0+qG_1,\quad g_0\in\operatorname{GL}_m(\mathbb F_q),
+ \quad G_1\in(\mathbb Z/q^{w-1}\mathbb Z)^{m^2},
+\]
+using fixed representatives for $g_0$. Right multiplication permutes $g_0$
+and acts affinely on $G_1$, with a carry determined by $g_0$. Its fixed
+dimensional rational linear part factors into fixed scalings, ordered
+shifts and a constant number of old width-$O(w)$ interchanges. Thus each
+fixed port routing costs $O(Vw^{1-a_{\rm old}})$.
+
+Let $W_0=|\operatorname{GL}_m(\mathbb F_q)|(2v+3R_{\rm loc})$.
+The full role count is $W(w)=W_0q^{m^2(w-1)}$. Split preceding rows into
+low role type, high matrix and unused stock. Batch \emph{all} high matrices
+for each low type in one stream. A frame edge is one child on this whole
+batch, of volume $V/W_0$, and routing is executed once per fixed operation
+type. Consequently neither the number of tapes nor the number of child
+programs grows with $w$. High cardinality cancels in the normalized moment,
+but its $m^2(w-1)$ digits are charged in the row reserve.
+
+For each fixed projector require the leading minors of sizes $1$ through
+$r$ to be units. Each cleared minor polynomial in $g_0$ is nonzero:
+similarity to $\operatorname{diag}(I_r,0)$ supplies a witness after the
+fixed rank is preserved modulo $q$. Their total degree is at most $m^3$.
+The polynomial zero bound and conditioning on invertibility (a factor at
+most two) give bad fraction at most $2m^3/q<10^{-16}$ for each edge type.
+No simultaneous union bound over edge types is needed, since their moments
+are averaged separately. Good classes use the whole-rank child; bad
+classes use the full $32m^2$-child fallback just proved.
+
+Let $H_r$ be the three-stage histogram described above and
+$W=2v+3R_{\rm loc}=90140$. The retained ledger gives
+\[
+ \sum_r rH_r=6037356,\qquad Wm-\sum_r rH_r=2024=2v-3\ell,
+ \qquad \max\{r:H_r>0\}=66.
+\]
+For $a=747/2000000$, $\tau=1-a$, and $b=10^{-16}$ the conservative moment is
+\[
+ \mu_\tau\le
+ \sum_r\frac{H_r}{W}(r/m)^\tau+
+ b\frac{32m^2\sum_rH_r}{W}(1/m)^\tau<1.
+\]
+The entire fallback is added; no displaced good contribution is subtracted.
+The exact script \path{scripts/three_stage_cover_network.py} and certificate
+\path{certificates/three-stage-cover-network.json}, from the retained axis
+\path{certificates/three-stage-cover-bit-input.json}, give
+$1-\mu_\tau>6.3397366396\cdot10^{-7}$.
+The certificate uses the 32-term positive atanh logarithm series with its
+geometric tail, $\exp(u)\le1+u+u^2/(2(1-u/3))$, and upward rounding to
+the grid $2^{-120}$. All inputs have $0\le u<3$; these are rational upper
+bounds, not floating-point root estimates. Since every child ratio is at
+most one and $\tau<1$, this same strict bound implies $\mu_1<1$.
+
+With $T(n,w)$ denoting time divided by logical volume, the fixed low-type
+profile $\{c_r\}$, including fallback classes, satisfies
+\[
+ T(n,w)\le\sum_rc_rT(r\lfloor n/m\rfloor,w)+Cn+Cw^{1-a_{\rm old}}
+ \quad(n\ge m),\qquad
+ T(n,w)\le Cw^{1-a_{\rm old}}\quad(n<m).
+\]
+Peeling fewer than $m$ leftover atoms uses the weighted leaf algorithm.
+The constants are independent of $w,n$ and ancestor values. Induction,
+using $\mu_1<1$ for the linear term and $\mu_\tau<1$ for the stopped term,
+gives $T(n,w)=O(n+n^\tau w^{1-a_{\rm old}})$. The maximum proper ratio
+$66/67$ has halving degree $47$, so recursion depth is $O(\log n)$.
diff --git a/research/five-stage-source527-banks/proof/three-stage-cover-rows.tex b/research/five-stage-source527-banks/proof/three-stage-cover-rows.tex
new file mode 100644
index 0000000000..590da4b892
--- /dev/null
+++ b/research/five-stage-source527-banks/proof/three-stage-cover-rows.tex
@@ -0,0 +1,95 @@
+\subsection{Internal row reserve and ordinary conversion}
+The finite cover requires a row divisor $W(w)$ at each coarse level.
+All children have ratio at most $66/67$ and depth
+$D(n)\le47\lceil\log_2\max(2,n)\rceil+O(1)$.
+A sufficient common divisor is $W(w)^{D(n)}$ times the inherited old-leaf
+reserve. The old supplier has its retained complete-row contract and
+logarithmic reserve. Therefore a sufficient integer $S(e,w)$ satisfies
+\[
+ \log_qS(e,w)=O(w\log e).
+\]
+Unused stock is divided only into fresh descendant role fields; exposed
+parent low/high fields are spectator prefixes. Their weights cannot depend
+on those fresh roles. Every completed call restores its row coordinate,
+so successive residuals and the outer wrapper reuse one reserve instead
+of multiplying reserves for sequential calls.
+
+The unweighted reversed endpoint $F_n(H,D)=(C_nD,C_nH)$ converts to an
+ordinary interchange by the chronological sequence
+\[
+ D\leftarrow D-H;\quad F_n;\quad D\leftarrow D+H;\quad
+ F_n;\quad D\leftarrow H-D.
+\]
+Indeed the successive values are $(H,D-H)$, $(C_n(D-H),C_nH)$,
+$(C_n(D-H),C_nD)$, $(D,D-H)$, and $(D,H)$.
+The three additive maps cost $O(Vn)$ in the atom model. The two calls
+reuse restored rows; their constant multiplicity is outside the recursion
+and does not multiply its characteristic moment.
+
+\subsection{Borrowing and restoring the stock}
+Set $\beta=1/1000$, $w=\lceil e^\beta\rceil$, and
+$k=\lceil\log_qS(e,w)\rceil=O(w\log e)$, using the depth bound for the
+original width $e$. For sufficiently large $e$, $k<e/2$. Write the two
+original complete radix-$q$ chunks as $H=(H_{\rm hi},H_{\rm body})$ and
+$D=(D_{\rm hi},D_{\rm body})$, both high parts having $k$ digits. The
+physical order, including arbitrary original complete spectators $Z$, is
+\[
+ H_{\rm hi},\ H_{\rm body},\ Z,\ D_{\rm hi},\ D_{\rm body}.
+\]
+The complete $q^k$ range of $H_{\rm hi}$ already precedes both bodies.
+Reinterpret it as a preceding row field; $D_{\rm hi}$ remains an intervening
+spectator. Pad only this row range to $\lceil q^k/S\rceil S$. As $q^k\ge S$,
+the padding factor is less than two.
+
+Execute the completed \emph{ordinary} body wrapper with this row reserve,
+then remove the padding. Each row is restored, so the genuine row range
+is invariant at the completed endpoint. The order of values is now
+\[
+ H_{\rm hi},\ D_{\rm body},\ Z,\ D_{\rm hi},\ H_{\rm body}.
+\]
+One inherited ordinary width-$k$ interchange of the two high parts,
+with the bodies and $Z$ as spectators, gives
+\[
+ D_{\rm hi},\ D_{\rm body},\ Z,\ H_{\rm hi},\ H_{\rm body}.
+\]
+This is precisely the original full-chunk interchange. Its extra cost is
+$O(Vk^{1-a_{\rm old}})\le O(Vw\log e)$, conservatively. No full body is
+moved once per borrowed digit. Padding occurs only in this outer wrapper;
+it contributes one volume constant, not a recursive branching factor.
+The wrapper does not rely on a raw reversed call preserving borrowed high
+positions. Small widths use the retained supplier directly.
+
+\subsection{Uniform saving and return to binary addresses}
+Select atom width from the original $e$. Since $k=o(e)$, the body width
+$e-k$ has the same asymptotic powers as $e$. Apply the ordinary wrapper to
+its complete atom body and the old ordinary supplier to fewer than $w$
+remaining digits. The stopped recurrence and the borrowing cost give
+\[
+ O\!\left(V\left[e^{1-\beta}+
+ e^{(1-\beta)(1-a)+\beta(1-a_{\rm old})}+e^\beta\log e\right]\right).
+\]
+Here $a=747/2000000$ and $a_{\rm old}=384599/10^{10}$. Set
+\[
+ a_{\rm bit}=(1-\beta)a+\beta a_{\rm old}
+ =\frac{3731649599}{10^{13}}=0.0003731649599.
+\]
+The inequalities $\beta>a_{\rm bit}$ and $\beta<1-a_{\rm bit}$ make both
+the atom-adapter and borrowing terms lower order. The remainder cost
+$O(Vw^{1-a_{\rm old}})$ is also covered. Thus the ordinary radix-$q$
+interchange has cost $O(Ve^{1-a_{\rm bit}})$, without an externally supplied
+superpolynomial stock of rows.
+
+For an original binary chunk range $[2^e]$, let
+$t=\lceil e/\log_2q\rceil$ and embed it numerically into $[q^t]$.
+Pad both exchanged ranges while retaining every spectator. The total
+logical-volume increase is below $q^2$. Execute the completed ordinary
+radix-$q$ interchange, then remove the padding. Its endpoint preserves the
+rectangular genuine set $[2^e]\times[2^e]$; intermediate maps may mix genuine
+and padded addresses, which does not affect this endpoint justification.
+As $q$ is fixed and $t=\Theta(e)$, the binary interchange has the same
+saving exponent. All schedules use a fixed number of finite-alphabet
+tapes; low residue tables and port types are fixed independently of $w$.
+
+The conclusion retains the original local-producer, ordered-affine,
+ordinary-leaf and multiplication-framework hypotheses. It adds no free
+unit-scaling, dense chart transformation, or external row-stock primitive.
diff --git a/research/five-stage-source527-banks/raw_ledger.py b/research/five-stage-source527-banks/raw_ledger.py
new file mode 100644
index 0000000000..0bac8c838e
--- /dev/null
+++ b/research/five-stage-source527-banks/raw_ledger.py
@@ -0,0 +1,188 @@
+"""Independent fresh source527 physical path census, including both reflections.
+Prepared with substantial OpenAI Codex assistance; Apache-2.0.
+No saved receipt, search script, or historical profile is consumed.
+"""
+from collections import Counter,defaultdict
+from types import SimpleNamespace
+
+def run(context):
+ m=SimpleNamespace(**context);W=m.W;C=m.C;chosen=m.pair_selection;fresh=m.fresh;pair_byalias=m.pair_byalias;pair_bydirty=m.pair_bydirty
+ compact=lambda h:{str(k):v for k,v in sorted(h.items())if v}
+ zero=W.register([]);compact=lambda h:{str(k):v for k,v in sorted(h.items())if v}
+ # Exact raw event sequence, with every actual extra source included.
+ events=[]
+ for j,op in enumerate(W.rest):
+  for s in m.at[j]:events.append(dict(kind='gauge',ref=s,frame=W.gauge[s]['frame'],targets=list(m.adj[s])))
+  if op in m.bywrite:events.append(dict(kind='terminalwrite',ref=op,frame=W.opframe[op],targets=[m.bywrite[op]['pivot']]))
+  if op in m.afterwrite:events.append(dict(kind='terminalread',ref=op,frame=m.afterwrite[op]['root_frame'],targets=m.afterwrite[op]['targets']))
+ for s in m.at[len(W.rest)]:events.append(dict(kind='gauge',ref=s,frame=W.gauge[s]['frame'],targets=list(m.adj[s])))
+ endgauge=len(events)
+ for j,r in enumerate(W.g['roots']):
+  if r['kind']=='side'and j not in m.deletedroots:events.append(dict(kind='root',ref=j,frame=W.w['root_frame'][j],targets=r['targets']))
+  for e in m.deliveries[j]:events.append(dict(kind='K',ref=e['passive'],frame=e['deliver_frame'],targets=e['receivers']))
+ for t in range(W.v):events.append(dict(kind='endpoint',ref=t,frame=W.register(W.module.kernel([C.cov[t]],W.h)[0]),targets=[t]))
+ 
+ def histogram(t,path):
+  h=Counter();rev=Counter();assert path[0]==zero and C.dimf[path[-1]]==W.h-1
+  for f in path:assert C.nondeg(f)and all(W.module.dot(C.cov[t],b)==0 for b in C.B[f]),('cap/nondeg',t,f)
+  for a,b in zip(path,path[1:]):
+   assert C.sub(a,b),('forward inclusion',t,a,b)
+   d=C.dimf[b]-C.dimf[a];assert d>=0
+   if d:h[d]+=1
+  for a,b in zip(reversed(path),list(reversed(path))[1:]):
+   assert all(W.module.dot(x,y)==0 for x in C.A[a]for y in C.B[b]),('reflected inclusion',t,a,b)
+   d=len(C.A[b])-len(C.A[a]);assert d>=0
+   if d:rev[d]+=1
+  assert h==rev and sum(k*v for k,v in h.items())==W.h-1
+  return h
+ 
+ def paths(agg,readframes,echelon,rankgroups):
+  rankowner={t:k for k,g in enumerate(rankgroups)for t in g['targets']}
+  out=[[zero]for _ in range(W.v)];owner={t:k for k,g in enumerate(agg)for t in g['targets']};roles=defaultdict(list)
+  for k,g in enumerate(agg):
+   for s in g['roles']:roles[s].append(k)
+  active=set(range(len(agg)));rankactive=set(range(len(rankgroups)));echactive=set(range(len(echelon)));seen=[set()for _ in echelon];restores=[]
+  for j,e in enumerate(events):
+   if j>=endgauge:assert not active and not rankactive and not echactive
+   s=e['ref'];T=set(e['targets'])
+   if e['kind']=='gauge':
+    if s in pair_byalias:continue
+    if s in fresh:
+     for t,c in fresh[s]['transformed_response'].items():
+      if c:out[int(t)].append(W.gauge[s]['frame'])
+     continue
+    skip=set()
+    for k,g in enumerate(echelon):
+     if k not in echactive:continue
+     if s not in g['roles']:
+      if T&set(g['targets']):
+       assert seen[k]==g['roles'];restores.append(dict(group=k,before_role=s,event=j))
+       for t in g['targets']:out[t].append(g['frame'])
+       echactive.remove(k)
+      continue
+     seen[k].add(s);skip.update(g['targets'])
+     for t,a in g['response'].items():
+      if a.get(s,0):out[t].append(W.gauge[s]['frame'])
+    for k in sorted({rankowner[t]for t in T if t in rankowner}):
+     if k in rankactive and s not in rankgroups[k]['roles']:
+      for t in rankgroups[k]['targets']:out[t].append(rankgroups[k]['frame'])
+      rankactive.remove(k)
+    for k in sorted({owner[t]for t in T if t in owner}):
+     if k in active and s not in agg[k]['roles']:
+      for t in agg[k]['targets']:out[t].append(agg[k]['frame'])
+      active.remove(k)
+    for k in roles[s]:
+     assert k in active;g=agg[k];out[g['pivot']].append(readframes[s,k]);skip.update(g['targets'])
+    for t in e['targets']:
+     if t in skip:continue
+     rk=rankowner.get(t)
+     if rk in rankactive and t in rankgroups[rk]['dependent']:continue
+     out[t].append(e['frame'])
+   else:
+    assert not(T&{t for k in active for t in agg[k]['targets']})
+    assert not(T&{t for k in echactive for t in echelon[k]['targets']})
+    for t in e['targets']:out[t].append(e['frame'])
+  H=Counter()
+  for t,p in enumerate(out):H.update(histogram(t,p))
+  return out,H,restores
+ 
+ newpaths,target,restores=paths(m.agg,m.read_frames,m.echelon,m.rankgroups)
+ # Recover raw literal adjoint independently, respecting all omitted copies.
+ adj=[{}for _ in range(W.R)]
+ for root,role in zip(W.g['roots'],W.w['rootroles']):
+  for t in root['targets']:adj[role][t]=adj[role].get(t,0)+1
+ for i in reversed(W.phase1+W.rest):
+  if i in m.omitted:continue
+  a,b,_=W.ops[i]
+  for t,c in adj[a].items():adj[b][t]=adj[b].get(t,0)+c
+ assert adj==m.adj
+ oldused={r[k]for r in m.selection+m.gauge_selection if r['role']not in pair_byalias for k in('source','partner')}
+ assert len({r[k]for r in chosen for k in('source','partner')})==62 and not oldused&{r[k]for r in chosen for k in('source','partner')}
+ for r in chosen:
+  b=r['aliased_role'];response=Counter(adj[b])
+  for a,c in r['retained_dirty_coefficients'].items():
+   for t,v in adj[a].items():response[t]+=c*v
+  assert not any(response.values())
+ # Complete physical source and auxiliary census, including all alias handoffs.
+ recipient={d:b for b,d in W.pairs};roots=defaultdict(list)
+ for j,s in enumerate(W.w['rootroles']):roots[s].append(W.w['root_frame'][j])
+ source_role={s:n for n,s in W.source.items()};plainborrow={r['role']:r for r in m.selection};gauges={r['role']:r for r in m.gauge_selection}
+ assert len(plainborrow)==440 and len(gauges)==87 and len(m.borrow)==527
+ seq={s:[('op',i,W.opframe[i])for i in W.role_ops[s]]+[('root',j,f)for j,f in enumerate(roots[s])]for s in range(W.R)}
+ def physical_path(s):
+  a=list(seq[s])
+  if s in pair_bydirty:a=[('paired-source-injection',s,pair_bydirty[s]['mix_frame'])]+a
+  if s in recipient:
+   b=recipient[s];a+=[('alias',b,W.gauge[b]['frame'])]+seq[b]
+  return a+[('full',s,W.w['full_frame'])]
+ count=0
+ 
+ def census_hist(start,events):
+  nonlocal count
+  count+=1;prev=start;dim=C.dimf[start]if start is not None else 0;out=Counter();chain=[]if start is None else[start]
+  for _,_,f in events:
+   assert C.nondeg(f)and(prev is None or C.sub(prev,f));d=C.dimf[f]-dim;assert d>=0
+   if d:out[d]+=1
+   prev=f;dim=C.dimf[f];chain.append(f)
+  assert dim==24;reflected=Counter()
+  for a,b in zip(reversed(chain),list(reversed(chain))[1:]):
+   assert all(W.module.dot(x,y)==0 for x in C.A[a]for y in C.B[b]);d=len(C.A[b])-len(C.A[a]);assert d>=0
+   if d:reflected[d]+=1
+  if start is None:
+   d=24-len(C.A[chain[0]])
+   if d:reflected[d]+=1
+  assert reflected==out
+  return out,chain
+ internal=Counter();sources=Counter();pathrows={};sourcepaths={};dirtypaths={}
+ for s in m.regs:
+  start=W.gauge[s]['frame']if s in W.gauge else W.w['source_frame'][source_role[s]]if s in source_role else None
+  H,chain=census_hist(start,physical_path(s))
+  if s in source_role:H[1]+=1
+  internal.update(H);pathrows['a'+str(s)]=compact(H)
+  if s in pair_bydirty:dirtypaths[str(s)]=[zero]+chain
+ for j,r in enumerate(W.g['roots']):
+  if r['kind']=='center':internal[C.dimf[W.w['root_frame'][j]]]+=1
+ for entry in W.k['entries']:
+  carrier,n=entry['carrier'],entry['passive'];H,chain=census_hist(entry['carrier_chain'][0],[('carrier',j,f)for j,f in enumerate(entry['carrier_chain'][1:])]);sources.update(H)
+  if n not in m.by_source:
+   H,chain=census_hist(entry['passive_chain'][0],[('passive',j,f)for j,f in enumerate(entry['passive_chain'][1:])]);sources.update(H);pathrows['s'+str(n)]=compact(H);continue
+  r=m.by_source[n];s=r['role'];events=physical_path(s)
+  if s in plainborrow:
+   r=plainborrow[s];events=[x for x in events if not(x[0]=='op'and x[1]==r['omit_copy'])];at=next(i for i,x in enumerate(events)if x[0]=='op'and x[1]==r['early']);events.insert(at,('mix',n,entry['mix_frame']))
+  else:
+   assert s in gauges;events=[('mix',n,entry['mix_frame']),('gauge',s,W.gauge[s]['frame'])]+events
+  H,chain=census_hist(W.w['source_frame'][n],events);sources.update(H);pathrows['s'+str(n)]=compact(H)
+  if s in m.extra_byrole:sourcepaths[str(s)]=chain
+ assert count==18347 and len(sourcepaths)==56 and len(dirtypaths)==48 and sum(k*v for k,v in sources.items())==1760*23
+ gauges=Counter(z['dim']for s,z in W.gauge.items()if s not in W.donor and s not in m.borrow)
+ assert gauges=={20:2200,18:13,12:18,13:48}
+ copies=Counter(C.dimf[W.w['root_frame'][j]]for j,r in enumerate(W.g['roots'])if r['kind']=='center');assert copies=={22:24}
+ withoutcopies=internal.copy();withoutcopies.subtract(copies);assert min(withoutcopies.values())>=0
+ helper=internal+sources+target;h=24;v=1760;R=len(m.regs);M=120;stock=4*v+R
+ assert R==16587 and len(m.borrow)==527
+ assert sum(k*n for k,n in helper.items())==h*R+2*v*(h-1)+sum(k*n for k,n in copies.items())-sum(k*n for k,n in gauges.items())
+ idle=Counter({r:2*v for r in (2*h-2,h-1,2*h+2,4)});five=Counter({r:5*n for r,n in helper.items()});five.update(idle)
+ for r,n in gauges.items():five[5*r]+=n
+ mass=sum(k*n for k,n in five.items());assert M*stock-mass==4400
+ raw=dict(schema='source527-fresh-raw-ledger/1',source_head=context.get('source_head','source527'),h=h,v=v,physical_R=R,source_aliases=len(m.borrow),terminal_sinks=len(m.terminal),physical_donor_recipient_identifications=len(W.pairs),physical_internal_excluding_center_copies=compact(withoutcopies),paid_center_copy_histogram=compact(copies),physical_source_histogram=compact(sources),physical_target_histogram=compact(target),one_stage_helper_histogram_including_copies=compact(helper),auxiliary_entrance_rank_histogram=compact(gauges),auxiliary_entrance_count=sum(gauges.values()),helper_rank_mass=sum(k*n for k,n in helper.items()),five_stage_profile=dict(m=M,W=stock,histogram=compact(five),idle_histogram=compact(idle),calls=sum(five.values()),rank_mass=mass,deficit=M*stock-mass,maxchild=max(five)),source_aux_paths=count,target_paths=1760,extra_source_paths=len(sourcepaths),paired_dirty_paths=len(dirtypaths),all_reflected_ledgers=True,all_zero_response_relations=True,scope='Fresh complete source527 raw physical source/internal/target and copied-center ledger. Both reflected chains checked; no saved execution receipt consumed.')
+ assert raw['five_stage_profile']['calls']==494994 and mass==2830840
+ return raw
+
+def rebind_parity(raw,receipt):
+ """Bind the fresh required-use census to the retained producer source ledger."""
+ from copy import deepcopy
+ def hist(x):return Counter({int(r):n for r,n in x.items()if n})
+ assert receipt['status']=='PASS_FRESH_PARITY_FUSION_AND_INDEPENDENT_REQUIRED_FRAME_CENSUS'
+ assert receipt['both_reflected_ledgers']and receipt['unchanged_all_input_output_frames']
+ source=hist(receipt['source_histogram']);target=hist(receipt['target_histogram']);internal=hist(receipt['internal_histogram_including_copies']);copies=hist(receipt['copied_center_histogram'])
+ assert source==hist(raw['physical_source_histogram'])
+ assert internal==hist(raw['physical_internal_excluding_center_copies'])+copies
+ delta=target.copy();delta.subtract(hist(raw['physical_target_histogram']));assert {r:n for r,n in delta.items()if n}=={8:-3,12:-3,20:3}
+ producer=deepcopy(raw);helper=source+target+internal;five=Counter({r:5*n for r,n in helper.items()});five.update(hist(raw['five_stage_profile']['idle_histogram']))
+ for a,n in hist(raw['auxiliary_entrance_rank_histogram']).items():five[5*a]+=n
+ assert sum(five.values())==494979 and sum(r*n for r,n in five.items())==2830840
+ raw['schema']='source527-parity-fused-fresh-raw-ledger/1';raw['producer_ledger']=producer
+ raw['physical_target_histogram']={str(r):n for r,n in sorted(target.items())};raw['one_stage_helper_histogram_including_copies']={str(r):n for r,n in sorted(helper.items())}
+ raw['five_stage_profile'].update(histogram={str(r):n for r,n in sorted(five.items())},calls=sum(five.values()))
+ raw['parity_transform']=receipt;raw['scope']='Fresh retained source527 producer ledger plus independently rebuilt actual surviving-use ledger after F2 payload identity elision and same-stream nested MOVE fusion. Both reflected endpoint inclusions checked. No address arithmetic reduced modulo2.'
+ return raw
diff --git a/research/five-stage-source527-banks/requirements.txt b/research/five-stage-source527-banks/requirements.txt
new file mode 100644
index 0000000000..1ca5fd0616
--- /dev/null
+++ b/research/five-stage-source527-banks/requirements.txt
@@ -0,0 +1 @@
+sympy==1.14.0
diff --git a/research/five-stage-source527-banks/scalar_check.py b/research/five-stage-source527-banks/scalar_check.py
new file mode 100644
index 0000000000..49702c9029
--- /dev/null
+++ b/research/five-stage-source527-banks/scalar_check.py
@@ -0,0 +1,76 @@
+"""Fresh F2 source proof, independent literal projection/inverse and controls.
+No saved execution receipt is accepted. Uses the actual clean source context
+and freshly emitted local physical records. Assisted with ChatGPT.
+"""
+from collections import Counter
+import hashlib,json,time
+
+EXPECTED_EVENT='aeeb17ee9162b8f6469e7205c2cb2abdd1a383e1b4876a68d791d26382160ded'
+
+
+def events(records,reverse=False):
+    center=None
+    for k in (range(len(records)-6,-1,-6) if reverse else range(0,len(records),6)):
+        op,a,b,c,f,z=records[k:k+6]
+        if op==1:
+            if b==20107:
+                assert center is not None
+                b=center
+            assert 0<=a<20107 and 0<=b<20107 and a!=b
+            yield a,b,-c if reverse else c
+        elif op==(3 if reverse else 2):
+            assert center is None and b==20107
+            center=a
+        elif op==(2 if reverse else 3):
+            assert center==a and b==20107
+            center=None
+    assert center is None
+
+
+def replay_projection(records,reverse=False,omit_event=None):
+    n=20107;v=1760
+    columns=[1<<i for i in range(n)];norms=[1]*n
+    wanted=columns[:]
+    for i in range(v):wanted[v+i]^=1<<i
+    largest=1;counts=Counter();digest=hashlib.sha256();count=0
+    for event_index,(a,b,c) in enumerate(events(records,reverse)):
+        if event_index==omit_event:continue
+        if c&1:columns[a]^=columns[b]
+        norms[a]+=abs(c)*norms[b];largest=max(largest,norms[a])
+        counts[abs(c)]+=1
+        digest.update(json.dumps([a,b,c],separators=(',',':')).encode());digest.update(b'\n');count+=1
+    assert columns==wanted,'independent literal all-column endpoint'
+    assert count==786621 and counts=={1:785301,3:1320}
+    if not reverse:assert digest.hexdigest()==EXPECTED_EVENT
+    return dict(all_formal_columns=n,all_source_and_dirty_restored=True,
+                arbitrary_target_contents_preserved=True,
+                event_sha256=digest.hexdigest(),weighted_additions=count,
+                coefficient_counts=dict(counts),literal_unit_additions=sum(c*n for c,n in counts.items()),
+                max_intermediate_row_l1=largest,final_source_row_l1=max(norms[:v]),
+                final_target_row_l1=max(norms[v:2*v]),final_dirty_row_l1=max(norms[2*v:]))
+
+
+
+def run(context,records,progress=lambda text:None):
+    import gc
+    started=time.monotonic();ns=dict(context);source=ns['SOURCE_TEXT']
+    assert hashlib.sha256(source.encode()).hexdigest()=='e675d4eb9d90ff0b44279fae0b46f1a8b39f65421cc38454b70a6b75d0e10b42'
+    exec(compile(source,'source527 exact scalar word','exec'),ns)
+    progress('Checking all20107 source, target and arbitrary dirty F2 columns')
+    exact=ns['replay']('F2');assert exact['formal_columns']==20107 and exact['all_targets']and exact['all_source_and_dirty_restored']
+    progress('Checking independent literal physical projection and inverse')
+    forward=replay_projection(records);inverse=replay_projection(records,True)
+    assert forward['max_intermediate_row_l1']==37631 and inverse['max_intermediate_row_l1']==3295796
+    try:replay_projection(records,omit_event=786620)
+    except AssertionError:filtered_control='omitted surviving odd payload ADD rejected'
+    else:raise AssertionError('filtered odd payload omission accepted')
+    bound=ns['replay']('bound');bits=8*((bound.bit_length()+2+7)//8);assert 1<<bits>2*bound
+    progress('Checking the signed integer source decoder and dirty restoration')
+    integer=[]
+    for direction in(1,-1):integer.append(ns['replay']('Z',direction,bits));gc.collect()
+    controls={}
+    for mutation in ['omit_sparse_anchor','wrong_sparse_sign','omit_pair_initialization','omit_pair_restoration','wrong_pair_restore_sign','subtract_paired_alias','omit_transformed_source_mix','omit_transformed_source_read','omit_echelon_setup','omit_echelon_inverse','flip_echelon_sign','omit_rank_setup','omit_rank_inverse','repeat_rank_read','omit_aggregation_setup','omit_aggregation_inverse','repeat_aggregation_read','omit_extra_mix','omit_extra_compensation','omit_early_mix','old_adjoint','undo_before_restore','omit_gauge_mix','omit_gauge_compensation','late_phase1_gauge']:
+        try:ns['replay']('Z'if mutation in('flip_echelon_sign','wrong_pair_restore_sign','wrong_sparse_sign')else'F2',bits=bits,mutation=mutation)
+        except AssertionError as error:controls[mutation]=dict(rejected=True,reason=str(error))
+        else:raise AssertionError('invalid source mutation accepted: '+mutation)
+    return dict(status='PASS_FRESH527_PRODUCER_SIGNED_DECODER_AND_PARITY_FUSED_F2_INVERSE_AND_CONTROLS',filtered_control=filtered_control,F2=exact,integer=integer,majorant=dict(residual_bound=bound,packing_bits=bits),forward=forward,inverse=inverse,controls=controls,source_sha256=hashlib.sha256(source.encode()).hexdigest(),seconds=time.monotonic()-started,scope='The unchanged source527 producer retains both signed defining-integer decoder proofs and25controls. The emitted parity-filtered word is independently checked on all20107formal F2columns and literalreverse, with its own cancellation-free signed-lift prefix bounds. Its integer endpoint is not asserted equal to the producer decoder; F2payload identity supplies the bit contract. Odd-prime address arithmetic is unchanged.')
diff --git a/research/five-stage-source527-banks/source527/echelon/selection.json b/research/five-stage-source527-banks/source527/echelon/selection.json
new file mode 100644
index 0000000000..291c1760f7
--- /dev/null
+++ b/research/five-stage-source527-banks/source527/echelon/selection.json
@@ -0,0 +1,2512 @@
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+        ],
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+        "undo_rank": 21,
+        "undo_frame_basis": [
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+          ],
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+          [
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+          ],
+          [
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+            0,
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+          ],
+          [
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+            0,
+            0,
+            0,
+            0,
+            1,
+            1,
+            1
+          ]
+        ]
+      }
+    ],
+    "early_role": 14432,
+    "late_role": 14092,
+    "split_gauge_basis": [
+      [
+        0,
+        0,
+        1,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
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+      ],
+      [
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+        0,
+        -1,
+        1,
+        0,
+        0,
+        0,
+        0
+      ],
+      [
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+        -1,
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+        0,
+        0,
+        0,
+        0
+      ],
+      [
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+        -1,
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+        0,
+        0,
+        0
+      ]
+    ]
+  },
+  {
+    "role": 14073,
+    "source": 90,
+    "partner": 89,
+    "targets": [
+      48,
+      49,
+      50,
+      51,
+      56,
+      57,
+      58,
+      59
+    ],
+    "first_operation": 24428,
+    "first_dimension": 14,
+    "before_target_event": 4,
+    "next_event": "newg",
+    "gauge_rank": 4,
+    "gauge_basis": [
+      [
+        1,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        0,
+        1,
+        0,
+        0,
+        0,
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+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ],
+      [
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+        0,
+        0,
+        1,
+        -1,
+        0,
+        0,
+        -1,
+        1,
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+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
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+      ],
+      [
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+        0,
+        0,
+        1,
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+        -1,
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+        0,
+        0,
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+        0,
+        0,
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+        0,
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+      ],
+      [
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+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ]
+    ],
+    "mix_frame": 126149,
+    "score": 0.09272955383185888,
+    "local_delta": {
+      "14": -1,
+      "22": -1,
+      "2": 5,
+      "10": 1,
+      "17": 8,
+      "19": -4,
+      "4": 4,
+      "21": -4
+    },
+    "split_gauge_basis": [
+      [
+        1,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        0,
+        1,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ],
+      [
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+        0,
+        0,
+        1,
+        -1,
+        0,
+        0,
+        -1,
+        1,
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+        0,
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+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ],
+      [
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+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        -1,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ],
+      [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        -1,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ]
+    ],
+    "early_role": 14075,
+    "late_role": 14073,
+    "new_target_groups": [
+      {
+        "targets": [
+          56,
+          57,
+          58,
+          59
+        ],
+        "pivot": 56
+      },
+      {
+        "targets": [
+          48,
+          49,
+          50,
+          51
+        ],
+        "pivot": 48
+      }
+    ],
+    "derive_closing_frames": true
+  }
+]
diff --git a/research/five-stage-source527-banks/source527/extra/upstream_setup.py b/research/five-stage-source527-banks/source527/extra/upstream_setup.py
new file mode 100644
index 0000000000..2edf47302d
--- /dev/null
+++ b/research/five-stage-source527-banks/source527/extra/upstream_setup.py
@@ -0,0 +1,28 @@
+"""Source493 role setup; actual word executed by targetagg/replay.py.
+Prepared with substantial OpenAI Codex assistance; Apache-2.0.
+The22 extra source mixes retime existing K mixes and unmixes.
+"""
+from pathlib import Path
+from collections import defaultdict
+import importlib.util,json,sys
+sys.dont_write_bytecode=True
+D=Path(__file__).resolve().parent;P=D.parent/'newg'
+sp=importlib.util.spec_from_file_location('source471base',P/'replay.py');m=importlib.util.module_from_spec(sp);sp.loader.exec_module(m)
+extra=json.loads((D/'selection.json').read_text());oldborrow=m.borrow.copy();used={s for r in m.selection+m.gauge_selection for s in[r['source'],r['partner']]};assert len(oldborrow)==471
+for r in extra:
+ role,n=r['role'],r['source'];e=m.kpair[n];assert e['carrier']==r['partner']and e['mix_frame']==r['mix_frame'];assert not({n,r['partner']}&used);used.update([n,r['partner']])
+ assert role not in m.borrow and role not in m.W.gauge and role not in m.W.donor and role not in m.removed and role not in m.W.source.values()
+ assert set(m.adj[role])==set(r['targets'])and set(m.adj[role].values())=={1}
+ assert role not in {s for i in m.W.phase1 for s in m.W.ops[i][:2]};f=m.W.register(r['split_gauge_basis'])if'split_gauge_basis'in r else e['mix_frame'];assert m.C.dimf[f]in(2,4) and m.C.nondeg(f)and m.C.sub(f,m.W.opframe[m.W.role_ops[role][0]])
+ assert all(all(m.W.module.dot(m.C.cov[t],b)==0 for b in m.C.B[f])for t in r['targets'])
+ z=dict(role=role,frame=f,dim=m.C.dimf[f],targets=r['targets']);m.W.gauge[role]=z;m.W.w['gauges'].append(z);m.W.readtime[role]=0;m.W.w['reads'][str(role)]=len(m.W.phase1)
+ m.borrow[role]=n;m.by_source[n]=r;m.gauge_borrow[role]=n;m.gauge_selection.append(r)
+m.W.order=[r['role']for r in extra]+m.W.order
+# The third ordered rank4 source entrance follows all retained new-source gauges.
+m.W.readtime[14073]=168;m.W.w['reads']['14073']=len(m.W.phase1)+168
+m.W.order.remove(14073);at_index=max(m.W.order.index(r['role'])for r in m.new_gauge_selection)+1;m.W.order.insert(at_index,14073)
+m.at=defaultdict(list)
+for role in m.W.order:m.at[m.W.readtime[role]].append(role)
+m.regs=sorted(set(m.W.phys.values())-set(m.borrow)-m.removed);m.idx={r:2*m.W.v+j for j,r in enumerate(m.regs)};assert len(m.borrow)==493 and len(m.regs)==16621
+assert all(0<=m.W.readtime[s]<=m.W.first[s]for s in m.W.gauge);m.W.exact_frames()
+m.extra_roles={r['role']for r in extra};m.extra_selection=extra
diff --git a/research/five-stage-source527-banks/source527/fresh/selection.json b/research/five-stage-source527-banks/source527/fresh/selection.json
new file mode 100644
index 0000000000..dc3a16d5e7
--- /dev/null
+++ b/research/five-stage-source527-banks/source527/fresh/selection.json
@@ -0,0 +1,191 @@
+[
+  {
+    "role": 14700,
+    "source": 58,
+    "partner": 57,
+    "mix_basis": [
+      [
+        1,
+        0,
+        0,
+        1,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        0,
+        0,
+        0,
+        0
+      ],
+      [
+        0,
+        0,
+        1,
+        -1,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        -1,
+        1,
+        0,
+        0,
+        0,
+        0
+      ]
+    ],
+    "first_operation": 28477,
+    "transformed_response": {
+      "160": 1
+    }
+  },
+  {
+    "role": 14710,
+    "source": 706,
+    "partner": 705,
+    "mix_basis": [
+      [
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+        1,
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+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
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+        0,
+        1,
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+      ],
+      [
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+        0,
+        0,
+        0,
+        0,
+        -1,
+        1,
+        0,
+        0
+      ]
+    ],
+    "first_operation": 28641,
+    "transformed_response": {
+      "520": 1
+    }
+  },
+  {
+    "role": 14720,
+    "source": 970,
+    "partner": 969,
+    "mix_basis": [
+      [
+        0,
+        0,
+        0,
+        0,
+        1,
+        0,
+        0,
+        0,
+        0,
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+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ],
+      [
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+        -1,
+        1,
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+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        0
+      ]
+    ],
+    "first_operation": 28805,
+    "transformed_response": {
+      "808": 1
+    }
+  }
+]
diff --git a/research/five-stage-source527-banks/source527/fresh/upstream_setup.py b/research/five-stage-source527-banks/source527/fresh/upstream_setup.py
new file mode 100644
index 0000000000..237b97a83d
--- /dev/null
+++ b/research/five-stage-source527-banks/source527/fresh/upstream_setup.py
@@ -0,0 +1,26 @@
+"""Three additional source aliases read through the existing target basis.
+Prepared by eumemic with substantial OpenAI Codex assistance; Apache-2.0.
+"""
+from pathlib import Path
+from collections import defaultdict
+import importlib.util,json,sys
+sys.dont_write_bytecode=True
+D=Path(__file__).resolve().parent;P=D.parent
+sp=importlib.util.spec_from_file_location('retained493_target_setup',P/'targetagg/setup.py');previous=importlib.util.module_from_spec(sp);sp.loader.exec_module(previous)
+m=previous.m;W=m.W;C=m.C;selection=json.loads((D/'selection.json').read_text());chosen=[];fresh={}
+used={r[k]for r in m.selection+m.gauge_selection for k in('source','partner')};newused=set()
+for raw in selection:
+ r=dict(raw);role,n=r['role'],r['source'];entry=m.kpair[n]
+ assert entry['carrier']==r['partner']and not({n,r['partner']}&(used|newused));newused.update((n,r['partner']))
+ assert role not in W.gauge and role not in m.borrow and n not in m.by_source
+ f=entry['mix_frame'];q=W.register(r['mix_basis']);assert C.sub(f,q)and C.sub(q,f)and C.dimf[f]==2
+ assert W.role_ops[role][0]==r['first_operation']and C.sub(f,W.opframe[r['first_operation']])and C.nondeg(f)
+ r['mix_frame']=f;chosen.append(r);fresh[role]=r
+ z=dict(role=role,frame=f,dim=2,targets=sorted(m.adj[role]));W.gauge[role]=z;W.w['gauges'].append(z);W.readtime[role]=0;W.w['reads'][str(role)]=len(W.phase1)
+ m.borrow[role]=n;m.by_source[n]=r;m.gauge_borrow[role]=n;m.gauge_selection.append(r);m.extra_byrole[role]=r
+assert len(chosen)==3 and len(newused)==6
+W.order=list(fresh)+W.order;m.at=defaultdict(list)
+for role in W.order:m.at[W.readtime[role]].append(role)
+m.regs=sorted(set(W.phys.values())-set(m.borrow)-m.removed);m.idx={r:2*W.v+j for j,r in enumerate(m.regs)}
+assert len(m.borrow)==496 and len(m.regs)==16618 and len(m.extra_byrole)==25
+W.exact_frames();m.fresh=fresh
diff --git a/research/five-stage-source527-banks/source527/paired-selection.json b/research/five-stage-source527-banks/source527/paired-selection.json
new file mode 100644
index 0000000000..ad93da9765
--- /dev/null
+++ b/research/five-stage-source527-banks/source527/paired-selection.json
@@ -0,0 +1,2479 @@
+[
+  {
+    "source": 738,
+    "partner": 737,
+    "retained_dirty_role": 14095,
+    "aliased_role": 14097,
+    "initialization_sign": -1,
+    "mix_frame": 124691,
+    "mix_basis": [
+      [
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+        -1,
+        1,
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+        0
+      ]
+    ],
+    "dirty_first_rank": 15,
+    "alias_first_rank": 12,
+    "dirty_first_op": 24571,
+    "alias_first_op": 24575,
+    "transformed_source_response": {},
+    "local_delta": {
+      "13": 1,
+      "15": -1,
+      "10": 1,
+      "12": -1,
+      "2": 1,
+      "22": -1
+    },
+    "score": -0.045791385257643924,
+    "initialization_restoration_unit_adds": 2,
+    "source_compensation_unit_adds": 0,
+    "retained_dirty_coefficients": {
+      "14095": -1
+    }
+  },
+  {
+    "source": 1218,
+    "partner": 1217,
+    "retained_dirty_role": 14131,
+    "aliased_role": 14133,
+    "initialization_sign": -1,
+    "mix_frame": 123611,
+    "mix_basis": [
+      [
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+      ],
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+      ]
+    ],
+    "dirty_first_rank": 15,
+    "alias_first_rank": 12,
+    "dirty_first_op": 24799,
+    "alias_first_op": 24803,
+    "transformed_source_response": {},
+    "local_delta": {
+      "13": 1,
+      "15": -1,
+      "10": 1,
+      "12": -1,
+      "2": 1,
+      "22": -1
+    },
+    "score": -0.045791385257643924,
+    "initialization_restoration_unit_adds": 2,
+    "source_compensation_unit_adds": 0,
+    "retained_dirty_coefficients": {
+      "14131": -1
+    }
+  },
+  {
+    "source": 1618,
+    "partner": 1617,
+    "retained_dirty_role": 14185,
+    "aliased_role": 14187,
+    "initialization_sign": -1,
+    "mix_frame": 122711,
+    "mix_basis": [
+      [
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+      [
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+      ]
+    ],
+    "dirty_first_rank": 15,
+    "alias_first_rank": 12,
+    "dirty_first_op": 25141,
+    "alias_first_op": 25145,
+    "transformed_source_response": {},
+    "local_delta": {
+      "13": 1,
+      "15": -1,
+      "10": 1,
+      "12": -1,
+      "2": 1,
+      "22": -1
+    },
+    "score": -0.045791385257643924,
+    "initialization_restoration_unit_adds": 2,
+    "source_compensation_unit_adds": 0,
+    "retained_dirty_coefficients": {
+      "14185": -1
+    }
+  },
+  {
+    "source": 1623,
+    "partner": 1620,
+    "retained_dirty_role": 14194,
+    "aliased_role": 14196,
+    "initialization_sign": -1,
+    "mix_frame": 122709,
+    "mix_basis": [
+      [
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+      ]
+    ],
+    "dirty_first_rank": 15,
+    "alias_first_rank": 12,
+    "dirty_first_op": 25198,
+    "alias_first_op": 25202,
+    "transformed_source_response": {},
+    "local_delta": {
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+      "15": -1,
+      "10": 1,
+      "12": -1,
+      "2": 1,
+      "22": -1
+    },
+    "score": -0.045791385257643924,
+    "initialization_restoration_unit_adds": 2,
+    "source_compensation_unit_adds": 0,
+    "retained_dirty_coefficients": {
+      "14194": -1
+    }
+  },
+  {
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+    "partner": 993,
+    "retained_dirty_role": 14113,
+    "aliased_role": 14115,
+    "initialization_sign": -1,
+    "mix_frame": 124115,
+    "mix_basis": [
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+    ],
+    "dirty_first_rank": 15,
+    "alias_first_rank": 12,
+    "dirty_first_op": 24685,
+    "alias_first_op": 24689,
+    "transformed_source_response": {},
+    "local_delta": {
+      "13": 1,
+      "15": -1,
+      "10": 1,
+      "12": -1,
+      "2": 1,
+      "22": -1
+    },
+    "score": -0.045791385257643924,
+    "initialization_restoration_unit_adds": 2,
+    "source_compensation_unit_adds": 0,
+    "retained_dirty_coefficients": {
+      "14113": -1
+    }
+  },
+  {
+    "source": 327,
+    "partner": 324,
+    "retained_dirty_role": 14086,
+    "aliased_role": 14088,
+    "initialization_sign": -1,
+    "mix_frame": 125625,
+    "mix_basis": [
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+    ],
+    "dirty_first_rank": 15,
+    "alias_first_rank": 12,
+    "dirty_first_op": 24514,
+    "alias_first_op": 24518,
+    "transformed_source_response": {},
+    "local_delta": {
+      "13": 1,
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+      "10": 1,
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+      "2": 1,
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+    },
+    "score": -0.045791385257643924,
+    "initialization_restoration_unit_adds": 2,
+    "source_compensation_unit_adds": 0,
+    "retained_dirty_coefficients": {
+      "14086": -1
+    }
+  },
+  {
+    "source": 687,
+    "partner": 684,
+    "retained_dirty_role": 14104,
+    "aliased_role": 14106,
+    "initialization_sign": -1,
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+    "mix_basis": [
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+    "alias_first_rank": 12,
+    "dirty_first_op": 24628,
+    "alias_first_op": 24632,
+    "transformed_source_response": {},
+    "local_delta": {
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+      "15": -1,
+      "10": 1,
+      "12": -1,
+      "2": 1,
+      "22": -1
+    },
+    "score": -0.045791385257643924,
+    "initialization_restoration_unit_adds": 2,
+    "source_compensation_unit_adds": 0,
+    "retained_dirty_coefficients": {
+      "14104": -1
+    }
+  },
+  {
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+    "partner": 972,
+    "retained_dirty_role": 14122,
+    "aliased_role": 14124,
+    "initialization_sign": -1,
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+    "mix_basis": [
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+    "alias_first_rank": 12,
+    "dirty_first_op": 24742,
+    "alias_first_op": 24746,
+    "transformed_source_response": {},
+    "local_delta": {
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+      "15": -1,
+      "10": 1,
+      "12": -1,
+      "2": 1,
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+    "score": -0.045791385257643924,
+    "initialization_restoration_unit_adds": 2,
+    "source_compensation_unit_adds": 0,
+    "retained_dirty_coefficients": {
+      "14122": -1
+    }
+  },
+  {
+    "source": 1199,
+    "partner": 1196,
+    "retained_dirty_role": 14140,
+    "aliased_role": 14142,
+    "initialization_sign": -1,
+    "mix_frame": 123663,
+    "mix_basis": [
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+    "dirty_first_rank": 15,
+    "alias_first_rank": 12,
+    "dirty_first_op": 24856,
+    "alias_first_op": 24860,
+    "transformed_source_response": {},
+    "local_delta": {
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+    "score": -0.045791385257643924,
+    "initialization_restoration_unit_adds": 2,
+    "source_compensation_unit_adds": 0,
+    "retained_dirty_coefficients": {
+      "14140": -1
+    }
+  },
+  {
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+    "partner": 1209,
+    "retained_dirty_role": 14461,
+    "aliased_role": 14730,
+    "initialization_sign": -1,
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+    "mix_basis": [
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+    ],
+    "dirty_first_rank": 16,
+    "alias_first_rank": 19,
+    "dirty_first_op": 26845,
+    "alias_first_op": 28969,
+    "transformed_source_response": {},
+    "local_delta": {
+      "14": 1,
+      "16": -1,
+      "17": 1,
+      "19": -1,
+      "2": 1,
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+    "score": -0.04344419638562158,
+    "initialization_restoration_unit_adds": 2,
+    "source_compensation_unit_adds": 0,
+    "retained_dirty_coefficients": {
+      "14461": -1
+    }
+  },
+  {
+    "source": 695,
+    "partner": 692,
+    "retained_dirty_role": 14443,
+    "aliased_role": 14715,
+    "initialization_sign": -1,
+    "mix_frame": 124797,
+    "mix_basis": [
+      [
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diff --git a/research/five-stage-source527-banks/source527/pr230/NOTICE b/research/five-stage-source527-banks/source527/pr230/NOTICE
new file mode 100644
index 0000000000..bc1cfc52a1
--- /dev/null
+++ b/research/five-stage-source527-banks/source527/pr230/NOTICE
@@ -0,0 +1,441 @@
+A sharper exponent for integer multiplication
+Copyright 2026 Douglas Colkitt
+
+This project includes unmodified source files from OpenAI's manuscript
+"Integer multiplication below n log n" and its accompanying repository:
+https://github.com/openai/math
+Commit: adc7f1241b42e322a6451854ab7e4b4c146bf78a
+
+The original manuscript identifies OpenAI as its author. Its Apache-2.0
+license is retained at upstream/LICENSE. The exact source URLs and hashes
+are recorded in upstream/manifest.json. Proposed modifications are supplied
+as separate files under patches/; upstream source files remain unchanged.
+
+The original local research note, scripts, certificates, patches, and
+documentation were prepared by Douglas Colkitt with assistance from OpenAI
+Codex. The community release incorporates work by the contributors identified
+in the notices below; it is not solely Douglas Colkitt's work.
+This project is not an official OpenAI release or an OpenAI endorsement.
+
+Related contributions (not imported in this checkpoint):
+Zhihao Chen (jacklightChen), PR #7, supplied an earlier ternary five-subset
+motif, finite-alphabet transfer and a different compressed producer:
+https://github.com/CrocSwap/integer-mult-bounds/pull/7
+The present checkpoint claims no priority for that motif. Related compressed
+complex-network work was submitted by eumemic in PR #3:
+https://github.com/CrocSwap/integer-mult-bounds/pull/3
+The pending contributions and their stronger bounds require their own review;
+this notice does not represent adoption or verification of their claims.
+
+Additional community work, including parallel and superseded submissions, is
+acknowledged in CONTRIBUTORS.md, with pinned submission metadata in
+docs/research/contribution-snapshot.json. These acknowledgements do not imply
+joint authorship of the checkpoint, incorporation of pending code, or verification
+of submitted mathematical claims. Original source-specific notices remain in force.
+
+Community release (accepted by conditional maintainer audit):
+The following original contributor notices apply to the imported PR #39 chain.
+The preceding non-import statements describe the preserved 2^-30 checkpoint.
+
+Ternary five-subset construction and common-four-point resynthesis
+Copyright 2026 Zhihao Chen (jacklightChen)
+
+This is Zhihao Chen's first submission of this specific construction to this
+repository. Research, proof development, code and verification were prepared
+with substantial assistance from GPT-6 Astra (OpenAI Codex), as identified by
+the contributor. This is not independent review or an OpenAI endorsement.
+Future research building on this contribution should explicitly acknowledge
+Zhihao Chen (jacklightChen), cite the accompanying note and pull request, and
+retain the preceding contributors' attribution. This notice records research
+provenance; the repository's Apache-2.0 license remains unchanged.
+
+Dependencies retained from eumemic's Claude-assisted PRs #3 and #5:
+https://github.com/CrocSwap/integer-mult-bounds/pull/3
+https://github.com/CrocSwap/integer-mult-bounds/pull/5
+PR #5 commit: d3d370c34937503bd3fa63c83d505a05e8e50893
+These include complex_circuit.py, complex_network.py, fast_gaussian.py,
+their generators, construction notes and certificates. The Gaussian input
+scaling proof is corrected in this derivative; see the modification notice below.
+Aligned pairing also appears in Bortlesboat's PR #2. Retained totals and bank
+sharing appear in dleen's PR #4 and cheaper retained centers in eumemic's PR #6.
+Their results and ideas are credited separately from this new construction.
+
+Bulk recursion and controlled tensor basis, 8 October 2026
+Copyright 2026 icekylinx
+
+The new projector batching, controlled simultaneous basis, mixed-width tape
+recursions, whole-residual complex recursion, and dependency-path precision
+guard in the batched-23 proof package are contributed by icekylinx, developed
+with substantial OpenAI GPT-6 Astra assistance. Review corrections and
+repository integration were prepared with OpenAI Codex assistance. They build on the retained
+PR #7 finite producers and the preceding contributions credited above.
+The new package retains the repository's Apache-2.0 license and the original
+authors' attribution. It is not an official OpenAI release or endorsement.
+
+Modification notice, 8 October 2026 (icekylinx): the retained fast Gaussian
+proof now bounds shifted approximation error by F=2^ceil(1.14 alpha^2).
+The P=34p precision budget and asymptotic exponent are unchanged. New batching
+checks are integrated into make verify. Existing attribution and licenses
+are retained.
+
+Partial-swap extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The partial-swap framing, binary retained-total triple construction,
+dependency-preserving positive labels and carrier reuse, unequal (25,23,57)
+common basis, improved complex batching, and exact assembly are contributed
+by icekylinx with substantial OpenAI GPT-6 Astra and Codex assistance.
+This extension directly inherits PR #10, commit
+62691e395a0458ce089a1c7b5d89e74291e95e29. The portable producer adapters
+modify the retained paired-exclusion and shared-point implementations credited
+above. The original licenses, attribution, and Gaussian correction are retained.
+
+Endpoint-gauge extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The complementary auxiliary endpoint gauges, (32,30,40) controlled basis,
+general nondegenerate binary phase residuals, retained-center complex producer,
+updated path guard and exact assembly are contributed by icekylinx with
+substantial OpenAI GPT-6 Astra and Codex assistance. This extension inherits
+f2ab41aebad47861caf6316282c1793e5513845e and its PR #10 ancestry. New portable
+adapters retain the paired-exclusion and shared-point contributions credited
+above. Original licenses, notices and the Gaussian correction are retained.
+
+Structured-basis and semantic-bulk extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The selected (32,30,36) bit construction, A1/A5 block refinements, fixed
+middle-axis I+J profiles with bounded-minor CRT certificates, mixed dyadic
+complex centers, complete data-front batching and circuit-specific assembly
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex
+assistance. The immediate parent is ed90fd940279c336ebc45968631bdebfb087b505.
+
+The semantic/bulk compatibility argument, literal scalar charge, product row
+stock and parameterization adapt Zhihao Chen (jacklightChen)'s PR #23,
+commit 661ebabc1076c9f525d7ce4967c876b203fa9827. Translated complex endpoints
+are adopted from the same contributor's PR #21,
+commit 5ba6cf0bfb68f2be8d15610e7972207c50254d6a, following eumemic's source-frame
+work and the earlier nested-basis contributions.
+
+Completed-child semantic precision, arbitrary-coordinate routing, phase-cell
+Gaussian inversion and bulk tape transfer are credited to the RaD project
+(hipotures), commit 45d9b60355872041f9275f77c057514def0d45bc, exposed through
+PR #20 at 4f8d6c8272b5ff307a0da51df545ec3cd96a8b6e. Their selected proof
+sources retain their separate CC0 license and original NOTICE under
+references/semantic-bulk/rad20. The PR #23 source retains its recorded OpenAI
+Codex assistance; RaD retains its recorded Codex and GPT-6.1-Sol Ultra
+assistance. These are source attributions, not claims about this session's
+runtime model. Related contributions by Rohan Arun and Aurel Prosz remain
+acknowledged in the imported notices. No exclusive priority or additional
+license condition is asserted.
+
+Copied retained-center extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The copied retained-center schedule in both orientations, its complex-phase
+transfer, the Wm-N+L rank budget, and the selected (25,23)/(28,28) composition
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex
+assistance. This extension inherits PR #32 at
+0ef3aeb61f55cc0b321ce6a0ef00acee25cefe52 and the preceding PR #10/#18/#24 work.
+
+The two-stage topology and paid copied-stream endpoint correction are due
+to Aurel Prosz (Paureel), c82d09eb4781b68435e3fa3eb6beea72fa900fab,
+through Zhihao Chen (jacklightChen)'s PR #29 unequal-axis composition,
+9d963275075fa98f1da821e27757b238dafd6b3c. That source also credits Swapnil
+Jain's two-stage development at ae405eb474d1486b2d8aef90139869f927f4836a.
+
+The contiguous corner method, incidence factorization and exact partition
+certificates are due to Rohan Arun (PR #31), method commit
+02f68f95369c39955c5ffcfddbd75cd83e26b305. The selected (25,23) certificate
+specializes Dominik Scholz's parameterization (PR #33),
+a499a345c040aa15b68e984a0904eadc485c7a7e. Their source notices and proof
+snapshots are preserved in references/copied-centers. Historical OpenAI
+Codex, GPT-6 Astra and Anthropic Claude Opus 5.5 assistance is credited to
+the respective imported sources. Existing PR #21/#23 and RaD attribution,
+licenses and the Gaussian correction remain in force.
+
+Copied centers with reversed (23,25) data corners, 2026-10-08
+Rohan Arun with substantial OpenAI Codex assistance supplies the composition,
+exact dimension23 controls and cost/assembly integration. James Chang's PR34
+supplies the reversed boundary family; icekylinx's PR36 supplies copied retained
+centers and the selected complex graph. The balanced assembly remains attributed
+to RaD/hipotures and its PR34 specialization. Retain Dominik Scholz PR33, Rohan
+Arun PR31, Zhihao Chen PR29/21/23, Paureel/Aurel Prosz, Swapnil Jain, icekylinx,
+eumemic, Douglas Colkitt, OpenAI and all earlier notices and hypotheses.
+This is not a claim of formal verification, new foundational rank identities,
+global optimality or measured runtime speedup.
+
+Fixed middle basis with copied reversed corners (2026-10-08):
+Rohan Arun, with substantial OpenAI Codex assistance. Composition and
+simultaneous-basis proof retain icekylinx PR32/36, James Chang PR34,
+Dominik Scholz PR35/38, Zhihao Chen, Paureel, Swapnil Jain, RaD/hipotures,
+eumemic and all preceding attribution and licenses. See research/copied-fixed-reversed.
+
+Changed-graph composition with copied centers and fixed bases, 8 October 2026
+Prepared with OpenAI Codex assistance.
+The new composition uses icekylinx PR36 copied centers and PR32 fixed-I+J
+projectors/profiler, Dominik Scholz PR35 two fixed factors and CRT argument,
+Zhihao Chen (jacklightChen) PR29/23 interfaces, Aurel Prosz (Paureel) and
+Swapnil Jain's two-stage development, Rohan Arun PR25/31 corner methods,
+Dominik Scholz PR33/38 dimension/composition work; James Chang (jamesyc)
+PR34 reversed geometry and balanced transfer; Rohan Arun PR37/39/40
+reversed integration, modular data tests and both-fixed baseline;
+RaD (hipotures) PR41 changed scalar DAGs, alternating point order and
+independent role compiler; RaD balanced routing/bulk interfaces, and all
+retained predecessors.
+Complete attribution and the conditional proof boundary are recorded in
+research/copied-fixed/PROOF.md; imported originals and source pins are retained.
+Original Apache-2.0 and CC0 notices remain applicable.
+
+Community follow-up integration (PRs #49, #45 and #26), 8 October 2026:
+Rohan Arun contributes weighted carrier matching and climbed summand/leave-one-out
+orders (PRs #44/#47/#49), with Anthropic Claude assistance. The final construction
+composes Chafik Boukhalfa's changed-graph checks, exact rational data recovery and
+reordered exclusion sums (PRs #43/#46/#48, OpenAI Codex assistance) with RaD /
+hipotures's alternating pair order and independent physical role compiler (#41).
+All predecessor attribution and conditional proof interfaces remain in force.
+
+Alejandro Zarzuelo Urdiales contributes Gaussian parity and canonical arithmetic
+proofs, finite tensor execution, profile precision accounting and mixed-center
+reference checks (#45), with substantial OpenAI Codex assistance. See
+research/gaussian-parity-synthesis/PROVENANCE.md for his full attribution.
+Ryan S / princezuda contributes historical Lean certificate arithmetic, algebraic
+frame/movement/guard lemmas and independent paired-circuit checks (#26). Original
+source notices and historical snapshots are preserved. Neither formal package
+claims a full formal proof of the multiplication algorithm or a new kappa.
+
+Joint frame compilation of skip-prefix strips
+
+The local compiler and exact finite witnesses were prepared by eumemic with
+substantial OpenAI Codex assistance. Avi Eisenberg / ikeboy with Anthropic
+Claude assistance supplied the unmodified PR #53 skip-prefix producer.
+The pinned PR #48 pipeline is Chafik Boukhalfa's contribution, incorporating
+Rohan Arun, RaD / hipotures, icekylinx, Dominik Scholz, James Chang,
+Zhihao Chen, Aurel Prosz / Paureel, Swapnil Jain, eumemic, Douglas Colkitt,
+OpenAI, David Harvey, Joris van der Hoeven and the earlier contributors
+credited in the retained sources. See references/frame-compiler/pr48/NOTICE
+and references/frame-compiler/pr53/NOTICE for their original attribution.
+No authorship or review by those predecessors is claimed for this compiler.
+
+Joint dual-skip/frame-compiler composition (2026-10-08)
+---------------------------------------------------
+Rohan Gupta / gupt1156 with Anthropic Claude assistance: PR55 dual-skip
+scalar graph; original LICENSE and NOTICE are retained under
+references/frame-compiler/pr55. eumemic with substantial OpenAI Codex
+assistance: PR57 joint reversible frame compiler, serialized physical-word
+replay and profile reconstruction. Chafik Boukhalfa with OpenAI Codex
+assistance: this composition, independent integration and exact certificate.
+Avi Eisenberg / ikeboy, Rohan Arun, RaD / hipotures, icekylinx, and all
+predecessor authors retain the notices above and in pinned source files.
+No original copyright or Apache-2.0 notice is removed.
+
+Ranked reclamation after the PR58 joint-dual composition (2026-10-08)
+----------------------------------------------------------------
+Chafik Boukhalfa with OpenAI Codex assistance: prioritize retired physical
+slots by descending current frame rank, retaining slot ID as the tie-breaker;
+complete replay, exact profiles, comparison and adversarial controls.
+The dedicated joint_dual_reclaim_compiler.py is derived from eumemic's PR57
+compiler. Its original shared compiler remains byte-identical. The PR55
+producer and all preceding attribution remain unchanged. The complete PR58
+comparison certificate and its original source/proof provenance are retained
+under references/frame-compiler/pr58.
+
+Skip-prefix strip producer, 8 October 2026
+By Avi Eisenberg (ikeboy), with Anthropic Claude (Claude Opus 5.5)
+assistance. The skip-prefix leave-one-out strip layout in
+research/skip-strips/skip_graph.py is new. Everything else reuses Chafik
+Boukhalfa's PR #43/#46/#48 research/copied-fixed pipeline, its verification
+code, exact data recovery and support-order sorting, together with:
+- Rohan Arun's PR #44 weighted carrier matching optimizer;
+- RaD (hipotures) PR #41 alternating point order;
+- every predecessor credited above.
+Complete attribution and the conditional proof boundary are recorded in
+research/skip-strips/PROOF.md. Original Apache-2.0 and CC0 notices remain
+applicable.
+
+Local split-pair skip-prefix research continuation (2026-10-08)
+Prepared with substantial OpenAI Codex assistance at the user's request.
+New contribution: singleton/pair recursive partition refinement, finite summand
+order and moment search, and verified composition with inherited paid clones.
+Avi Eisenberg/ikeboy with Anthropic Claude assistance: PR53 skip-prefix strips.
+Chafik Boukhalfa with OpenAI Codex assistance: PR54 original-envelope clone
+specialization and PR43/46/48 checking interfaces. RaD/hipotures: PR51
+whole-chain/source-partition cloning and earlier producer/interfaces.
+Rohan Arun: PR44 weighted matching and earlier fixed/reversed work, with all
+stated historical AI assistance. All predecessor notices remain applicable.
+
+Original-envelope whole-chain clones on skip-prefix strips (research/skip-clones):
+Chafik Boukhalfa, prepared with OpenAI Codex assistance. The complete-chain
+source-partition clone operation is adapted from RaD / hipotures, PR #51,
+with original envelopes retained here. The scalar base is Avi Eisenberg's
+PR #53 skip-prefix layout, prepared with Anthropic Claude assistance.
+All credited PR #43/#46/#48, PR #44 and earlier code/proof notices remain
+in force. This extension is provided under the repository's Apache-2.0 terms.
+
+Community review checkpoint #50–#61, 8 October 2026
+Maintainer integration and independent arithmetic: Douglas Colkitt, with OpenAI Codex.
+Avi Eisenberg / ikeboy: skip-prefix strip framework (#53), with Anthropic Claude.
+Rohan Gupta / gupt1156: parallel order search (#50, stated Antigravity assistance)
+and dual-suffix strips (#55, Anthropic Claude assistance).
+Eumemic: equal-frame joint compilation and paid signal reclamation (#57), with Codex.
+Chafik Boukhalfa: original-envelope paid clones (#54), composition (#58) and ranked
+reclamation (#60), with Codex. Rohan Arun: order search (#52) and positive-frame
+composition (#56), with Anthropic Claude and stated Codex validation assistance.
+RaD / hipotures: enlarged frames, source partitions and paid whole-chain clones
+(#51), with Codex. Rohan Garg / rohangar1: singleton/pair recursion and paid-clone
+composition (#59), with Codex. Alejandro Zarzuelo Urdiales: exact parameter
+refinement, scoped Lean arithmetic, matrix/parity and search tools (#61), with
+his disclosed multi-tool and Codex assistance.
+Parallel #50/#52 records are reviewed and credited without replacing the earlier
+climbed-48 implementation; their exact commit history is retained. The separate
+#51 witness is retained as research; its source-graph/negative-basis data proof
+has not received the same full replay as #56's consumed compiler machinery.
+See docs/research/community-round2-review.md for precise validation boundaries.
+All earlier licenses, contributions and notices remain applicable.
+
+Interval strips and core-aware pair assembly
+
+The interval leave-one-out strips and the core-aware group-pair assembly in
+research/pair-assembly were prepared by Avi Eisenberg / ikeboy with Anthropic
+Claude assistance. They build on PR #53's skip-prefix pipeline and its
+inherited PR #48 sources. The stacked witness in research/pair-assembly/frame
+uses eumemic's PR #57 joint frame compiler unchanged; its driver and verifier
+are adaptations of PR #57's skip_frame_compiler.py, skip_frame_compose.py and
+verify_skip_frame.py. Original notices are retained.
+
+Stopped product-ring extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The opposite-bank generic projector factorization, componentwise atom-ring
+adapters, stopped recurrence and outer ordinary-interchange conversion,
+aligned pair-star producer with rational centers, exact denominator-21
+grid and selected (23,23)/(24,24) composition are contributed by icekylinx
+with substantial OpenAI GPT-6 Astra and Codex assistance. This extension
+builds on merged PR #36, 11817ccacb564bb7f98789c20dc11d3fece207e3, and is
+integrated on upstream main 56b66d58297deca1d7dd130247d720e960f77a37.
+
+The retained ordinary leaf uses the preceding projector, corner and
+copied-center proofs. The selected two-stage topology retains Aurel Prosz
+(Paureel), Swapnil Jain and Zhihao Chen's credits; the semantic assembly
+retains PR #21/#23 and RaD's attribution. Paired producers, carrier matching,
+original source notices and each imported license remain unchanged.
+
+Partial-gauge extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The pair-first cube assembly, compatible backward frame enlargement,
+partial source-gauge schedule and stopped whole-projector composition are
+contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex
+assistance, extending PR #104 at 948ce1510df750f4c18b96bdaef436a86f8bf834.
+
+The physical deferred bit word, reflected event accounting, fan-order
+clarification and signed two-stage endpoint are adopted from Zhihao Chen
+(jacklightChen), PR #97, e15350b66e4bf796153a68c240a6cc1b788abe3e, with
+Codex assistance. Its deferred readouts, V leaves, lifted frames and frozen
+witness originate in Swapnil Jain's 741e7aa078392553815df7926ee17ac5e25a8c38,
+with the original Claude disclosure retained. Cyclic interval and core-aware
+pair primitives build on Avi Eisenberg (ikeboy), PR #62,
+ad0f25ff7b23cff7f08ad237c2254e6ecf74257e, with Anthropic Claude assistance.
+Prior two-stage, copied-center, ordinary-leaf, semantic/bulk, routing and
+precision credits remain in force. Imported source notices and licenses
+are preserved under references/partial-gauge/; see SOURCES.json.
+
+Three-stage cover extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The arbitrary-subspace Clifford frame theorem, three-stage Cayley geometry,
+weighted q-adic GL cover, unit-lower weighted compiler, charged local-ring
+fallback, uniform batching, internal row borrowing and exact composition
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra assistance. This
+repository integration uses Codex assistance and extends
+PR #115, cf60442983c153a3a0f5272ea4713f99eba13c86.
+
+The selected local complex DAG and support/label replay are adopted from
+Copyright 2026 eumemic, PR #117, cbb05ce504d571546d9b7794c186a613c659c3bf,
+with its original Apache-2.0 terms and Claude assistance disclosure.
+The unchanged bit physical word retains Zhihao Chen (jacklightChen),
+PR #97, and Swapnil Jain's underlying witness and assistance notices.
+The prior opposite-bank, stopped ordinary-leaf, copied-center, topology,
+semantic/bulk, precision and analytic credits remain applicable.
+Original source notices and license bytes are preserved in
+references/three-stage-cover/pr117/ and references/partial-gauge/.
+
+Paired-cube and shared-core extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The paired-coordinate triple-cube identity, signed overlap channels,
+original-source involution, coordinate-star centers, concrete frame/carrier
+compiler, chronological source gauges, PR97 gauge-subset selection,
+orthogonal cross-stage routing and exact supplier/assembly integration
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra assistance.
+Repository integration uses Codex assistance and extends PR #130 at
+6a9970a530119174507904e23592fd59ede19a5d.
+
+The substantial completed-core workspace-sharing principle is credited to
+an664, PR #128, 530588a019b4a74f09180680c9e3961bf649ec89, with its reported
+OpenAI Codex assistance. No source code, partition or serialized witness
+from that PR is imported. The new routing and paired-motif integration
+are described separately and claim no exclusive priority for broad sharing,
+paired-coordinate, Fourier or carrier-matching ideas.
+
+Selected coarse queries restrict the retained eumemic PR #117 positive DAG;
+its original Apache-2.0 and Claude assistance notices remain unchanged.
+The bit word and original frames retain Zhihao Chen's PR #97 and Swapnil
+Jain's frozen witness, notices and historical assistance disclosures.
+Earlier generalized Clifford, three-stage, uniform weighted recursion,
+ordinary-leaf, semantic/bulk, precision and analytic credits remain in force.
+Source-specific license bytes are preserved under references/.
+
+Source-assisted common-frame extension (2026-10-09)
+Copyright 2026 icekylinx. Apache-2.0.
+Research construction and local certificates: OpenAI GPT-6 Astra assistance.
+Repository integration and independent complex scalar replay: Codex assistance.
+Substantial dependency: eumemic PR168, fd25adb7fbaa12ee761d02c733c54d1d2a7687ee;
+its byte-preserved sources, licenses and original Claude assistance disclosure
+are retained in research/source-assisted/public/pr168_fd25adb7.tar.gz.
+Additional inherited contributions are recorded in research/source-assisted/CREDITS.md.
+The new work concerns source-assisted invertible common-frame lifts, controlled
+affine cleanup and compatible zero-fresh reuse; no exclusive claim is made for
+broad register reuse, paired cubes, frame theory or ordinary linear elimination.
+
+Fourth finite bootstrap refinement (2026-10-09)
+Prepared by eumemic with substantial OpenAI Codex assistance.
+This arithmetic refinement builds on PR207 at
+cd14825023b75af4f5919a30e5e6d548b6ade5bc by adding one finite ordinary-leaf
+level. The 302 frame overrides, completed banks, scalar words and upstream
+assistance disclosures are inherited, not newly attributed to this work.
+The Apache-2.0 license and all preceding notices remain applicable.
+
+PR211 plateau frame refinement (2026-10-09)
+Prepared by eumemic with substantial OpenAI Codex assistance.
+The additional seven equal-frame component moves build directly on Rohan
+Arun's PR211 cascade witness at c63e50a5dde96fe1459d6b29e55e47f42f104347,
+prepared with Anthropic Claude assistance. Its original notice is preserved
+byte-for-byte in search/PR211-NOTICE. The offline certificate package and
+independent stage-private bank admission derive from Dugongue's PR207 at
+cd14825023b75af4f5919a30e5e6d548b6ade5bc. Original contributor notices,
+licenses, source pins and all upstream AI disclosures remain applicable.
+
+The final refinement also includes seeded bidirectional cascade search after
+the seven initial component moves. Exact canonical subspace comparison
+records 274 changed operations beyond PR211. Search, integration and replay
+were prepared with substantial OpenAI Codex assistance.
+
+Shared rank-18 gauges and two donor substitutions (2026-10-09)
+Prepared by eumemic with substantial OpenAI Codex assistance.
+The new actual gauge/read chronology, gauged-donor handoff, shared target
+transitions, rank6 residual banks and donor substitutions extend the same
+PR200/PR211 word and PR207/PR197 completed-bank framework. All original
+licenses, notices, authorship and upstream AI disclosures remain in force.
+
+Nested-gauge extension: three rank-19 physical entrances compose with thirteen shared rank-18 entrances; all target transitions and mixed4/5/6/24 banks are explicit. Prepared by eumemic with substantial OpenAI Codex assistance. All retained authorship, licenses and AI disclosures remain applicable.
+
+Equal-frame plateau refinement on PR210's 471-source word (2026-10-10)
+Prepared by sennemmi with substantial OpenAI Codex assistance. The deterministic
+frame search, 19 operation retimings, exact price update and repository
+integration extend eumemic's PR210 head
+df95878d11190518e45ef9717c9ee05011f88ace. They do not claim the inherited
+source word, shared gauges, completed banks or upstream proof framework.
+All PR200, PR197, PR202/193, PR207, PR211, PR216 and PR210 attributions,
+licenses and AI disclosures remain in force; the repository's Apache-2.0
+license is unchanged.
diff --git a/research/five-stage-source527-banks/source527/pr230/PROVENANCE.json b/research/five-stage-source527-banks/source527/pr230/PROVENANCE.json
new file mode 100644
index 0000000000..ff39579387
--- /dev/null
+++ b/research/five-stage-source527-banks/source527/pr230/PROVENANCE.json
@@ -0,0 +1,27 @@
+{
+  "pull_request": "https://github.com/CrocSwap/integer-mult-bounds/pull/230",
+  "author": "sennemmi",
+  "assistance": "Substantial OpenAI Codex assistance, as disclosed by the original contributor.",
+  "imported_head": "fee4c05b91d4cdebc0b9d8cbb266a98069928a9a",
+  "scope": "Nineteen operation-frame changes inherited from the independently verified492-source composition and replayed with493sourcealiases,82equal-response groups,seven integer-response groups andfour signed echelon groups. Original search,result,framewitness andnotice remain byte-preserved.",
+  "original_files": {
+    "NOTICE": "7dfbad087ee49b74fd5075d517eaa263c5dd3d4fbd4c69bfba34ee1cb55e837f",
+    "joint-plateau-result-prdf958.json": "cd5731260af0daa62952c3b82debdb26cfb79e62a5c615688d8ccac6f7a95fcc",
+    "joint-plateau-search-prdf958.py": "80f3f9e8ba1972b911b701a4de4f811d6d25c9b4c0bc6282d640c2f44caafd0d",
+    "opframe-bases.json": "387ae1b5e69004426d094f4947b29c532d60fdf31f727f5e8830468278fe0f92"
+  },
+  "final_frame_sha256": "9a2bba5c72aebc9ee4f21e59b10ce810b2da98a1fa1748fbc44c50a080e584ff",
+  "raw_child_delta": {
+    "1": -27,
+    "2": -36,
+    "3": 51,
+    "4": -30,
+    "7": 30,
+    "8": -15,
+    "9": -15,
+    "10": 24,
+    "11": -24,
+    "12": 30,
+    "15": -15
+  }
+}
diff --git a/research/five-stage-source527-banks/source527/pr230/active-opframe-bases.json b/research/five-stage-source527-banks/source527/pr230/active-opframe-bases.json
new file mode 100644
index 0000000000..1c2c64e3ef
--- /dev/null
+++ b/research/five-stage-source527-banks/source527/pr230/active-opframe-bases.json
@@ -0,0 +1 @@
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new file mode 100644
index 0000000000..395f00fb92
--- /dev/null
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+          1,
+          1
+        ],
+        [
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+          1,
+          0,
+          0,
+          1,
+          1
+        ],
+        [
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+          0,
+          0,
+          0,
+          1,
+          1,
+          1
+        ]
+      ],
+      "origin": "extra21"
+    },
+    {
+      "targets": [
+        56,
+        57,
+        58,
+        59
+      ],
+      "pivot": 56,
+      "roles": [
+        14073
+      ],
+      "frame_basis": [
+        [
+          1,
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+          0,
+          0,
+          0,
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+          0,
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+          1,
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+          0,
+          0,
+          0,
+          0,
+          0,
+          -1
+        ],
+        [
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+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          1,
+          0,
+          0,
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+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          -1
+        ],
+        [
+          0,
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+          0,
+          0,
+          0,
+          0,
+          0,
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+          1,
+          0,
+          0,
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+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          -1
+        ],
+        [
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          1,
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+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
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+        ],
+        [
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+        ],
+        [
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+          0,
+          0,
+          0,
+          -1
+        ],
+        [
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+          0,
+          0,
+          0,
+          0,
+          0,
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+          0,
+          0,
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+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          -1
+        ],
+        [
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+          0,
+          0,
+          0,
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+          0,
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+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
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+        ],
+        [
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+        [
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+          0,
+          0,
+          0,
+          -1
+        ],
+        [
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+          0,
+          0,
+          0,
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+          0,
+          1,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          -1
+        ],
+        [
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+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
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+          0,
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+          0,
+          0,
+          1,
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+          0,
+          0,
+          0,
+          0,
+          -1
+        ],
+        [
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+          0,
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+          0,
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+          0,
+          0,
+          0,
+          0,
+          1,
+          1,
+          0,
+          0,
+          0,
+          1
+        ],
+        [
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          1,
+          0,
+          0,
+          -1
+        ],
+        [
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          1,
+          0,
+          -1
+        ],
+        [
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
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+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          0,
+          1,
+          -1
+        ]
+      ],
+      "origin": "extra21"
+    }
+  ]
+}
diff --git a/research/five-stage-source527-banks/source527/targetagg/upstream_setup.py b/research/five-stage-source527-banks/source527/targetagg/upstream_setup.py
new file mode 100644
index 0000000000..6c48fbd555
--- /dev/null
+++ b/research/five-stage-source527-banks/source527/targetagg/upstream_setup.py
@@ -0,0 +1,53 @@
+"""Exact setup for493 source heads,82 equal groups and7 integer rank groups.
+Prepared with substantial OpenAI Codex assistance; Apache-2.0.
+"""
+from pathlib import Path
+from collections import defaultdict
+import importlib.util,json,sys
+sys.dont_write_bytecode=True
+D=Path(__file__).resolve().parent;P=D.parent
+sp=importlib.util.spec_from_file_location('ordered493_setup',P/'extra/setup.py');source=importlib.util.module_from_spec(sp);sp.loader.exec_module(source)
+m=source.m;extra=source.extra;W=m.W;C=m.C;groups=json.loads((D/'selection.json').read_text())['groups'];agg=[];role_groups=defaultdict(list);owner={}
+for k,g in enumerate(groups):
+ F=W.register(g['frame_basis']);ts=g['targets'];assert C.nondeg(F)and g['pivot']in ts
+ for t in ts:assert t not in owner;owner[t]=k
+ for s in g['roles']:
+  assert set(ts)<=set(m.adj[s])and len({m.adj[s][t]for t in ts})==1 and C.sub(W.gauge[s]['frame'],F);role_groups[s].append(k)
+ agg.append(dict(targets=ts,pivot=g['pivot'],roles=g['roles'],frame=F))
+assert len(agg)==82 and len(owner)==316
+byrole={r['role']:r for r in extra};rf={}
+for k,g in enumerate(agg):
+ for role in g['roles']:
+  r=byrole.get(role);f=W.gauge[role]['frame']
+  if r and'split_gauge_basis'in r and r['early_role']not in g['roles']:f=r['mix_frame']
+  assert C.sub(f,g['frame']);rf[role,k]=f
+for role,ks in role_groups.items():ks.sort(key=lambda k:C.dimf[rf[role,k]])
+m.agg=agg;m.role_groups=role_groups;m.target_group=owner;m.read_frames=rf;m.extra_byrole=byrole
+rankselection=json.loads((P/'rank/selection.json').read_text());rankgroups=[];rankowner={}
+for k,g in enumerate(rankselection['groups']):
+ F=W.register(g['frame_basis']);dep={int(t):{int(p):c for p,c in cs.items()}for t,cs in g['dependent'].items()};assert not(set(dep)&{p for cs in dep.values()for p in cs})
+ for t in g['targets']:assert t not in owner and t not in rankowner;rankowner[t]=k;assert all(W.module.dot(C.cov[t],b)==0 for b in C.B[F])
+ for s in g['prefix_roles']:
+  assert C.sub(W.gauge[s]['frame'],F)
+  for t,cs in dep.items():assert m.adj[s].get(t,0)==sum(c*m.adj[s].get(p,0)for p,c in cs.items())
+ rankgroups.append(dict(targets=g['targets'],dependent=dep,roles=set(g['prefix_roles']),frame=F))
+assert len(rankgroups)==7 and len(rankowner)==56 and sum(len(g['dependent'])for g in rankgroups)==27
+m.rankgroups=rankgroups;m.rankowner=rankowner
+
+echelonselection=json.loads((P/'echelon/selection.json').read_text());echelon=[];eowner={}
+def addrow(a,c,b):
+ z=a.copy()
+ for s,v in b.items():
+  z[s]=z.get(s,0)+c*v
+  if not z[s]:del z[s]
+ return z
+for k,r in enumerate(echelonselection['groups']):
+ F=W.register(r['frame_basis']);R={int(t):{int(s):c for s,c in a.items()}for t,a in r['transformed_response'].items()};moves=[tuple(v)for v in r['moves']]
+ A={t:{s:m.adj[s].get(t,0)for s in r['prefix_roles']if m.adj[s].get(t,0)}for t in r['targets']}
+ for t,p,c in moves:assert t!=p;A[t]=addrow(A[t],c,A[p])
+ assert A==R and C.nondeg(F)
+ for t in r['targets']:assert t not in owner and t not in rankowner and t not in eowner;eowner[t]=k;assert all(W.module.dot(C.cov[t],b)==0 for b in C.B[F])
+ for s in r['prefix_roles']:assert C.sub(W.gauge[s]['frame'],F)
+ echelon.append(dict(targets=r['targets'],roles=set(r['prefix_roles']),response=R,moves=moves,frame=F))
+assert len(echelon)==4 and len(eowner)==20
+m.echelon=echelon;m.eowner=eowner
diff --git a/research/five-stage-source527-banks/source527/word.py b/research/five-stage-source527-banks/source527/word.py
new file mode 100644
index 0000000000..95b9affd44
--- /dev/null
+++ b/research/five-stage-source527-banks/source527/word.py
@@ -0,0 +1,263 @@
+def replay(mode='F2',direction=1,bits=24,mutation=None):
+ major=mode=='bound';binary=mode=='F2';v=W.v
+ unit=(lambda i:1)if major else(lambda i:1<<i)if binary else(lambda i:1<<(bits*i))
+ def add(x,c,y):return x+abs(c)*y if major else x^y if binary and c&1 else x if binary else x+c*y
+ target_frames=[None]*v;target_paths=[[]for _ in range(v)];center_reads=[];target_hist=Counter()
+ aggregation_active=[True]*len(agg);aggregation_read=[set()for _ in agg];aggregation_restores=[];aggregation_pivot_reads=0
+ rank_active=set(range(len(rankgroups)));rank_seen=[set()for _ in rankgroups];rank_restores=[];rank_skipped=0
+ echelon_active=set(range(len(echelon)));echelon_seen=[set()for _ in echelon];echelon_restores=[];echelon_reads=0
+ def target_move(t,f):
+  previous=target_frames[t];assert previous is None or C.sub(previous,f),('actual target chronology',t,previous,f)
+  assert all(W.module.dot(C.cov[t],b)==0 for b in C.B[f]),('actual target cap',t,f)
+  d=C.dimf[f]-(0 if previous is None else C.dimf[previous]);assert d>=0
+  if d:target_hist[d]+=1
+  target_frames[t]=f;target_paths[t].append(f)
+ X0=[unit(i)for i in range(v)];Y0=[unit(v+i)for i in range(v)];Z0={r:unit(idx[r])for r in regs};x=X0[:];y=Y0[:];z=Z0.copy();top=1;done=[]
+ def track(value):
+  nonlocal top
+  if major:top=max(top,value)
+  return value
+ def value(s):
+  s=W.phys[s];return x[borrow[s]]if s in borrow else z[s]
+ def assign(s,val):
+  s=W.phys[s]
+  if s in borrow:x[borrow[s]]=track(val)
+  else:z[s]=track(val)
+ source_paths={role:[W.w['source_frame'][r['source']],r['mix_frame']]for role,r in extra_byrole.items()}
+ pair_dirty_paths={r:[zero]for r in pair_bydirty}
+ def source_move(s,f):
+  role=W.phys[s]
+  if role in pair_dirty_paths:
+   assert C.sub(pair_dirty_paths[role][-1],f),('paired dirty chronology',role,pair_dirty_paths[role][-1],f)
+   pair_dirty_paths[role].append(f)
+  if role not in source_paths:return
+  assert C.sub(source_paths[role][-1],f),('orderedsource chronology',s,role,source_paths[role][-1],f)
+  source_paths[role].append(f)
+ responses=W.adjoint() if mutation=='old_adjoint' else adj
+ def aggregate_restore(k,trigger):
+  if not aggregation_active[k]:return
+  group=agg[k];pivot=group['pivot'];assert aggregation_read[k]==set(group['roles']),('early aggregation inverse',k,trigger,aggregation_read[k])
+  for t in group['targets']:target_move(t,group['frame'])
+  if mutation!='omit_aggregation_inverse' or k!=39:
+   for t in group['targets']:
+    if t!=pivot:y[t]=track(add(y[t],1,y[pivot]))
+  aggregation_active[k]=False;aggregation_restores.append(dict(group=k,before_role=trigger,frame=group['frame']))
+ def rank_restore(k,trigger):
+  group=rankgroups[k];assert k in rank_active and rank_seen[k]==group['roles']
+  for t in group['targets']:target_move(t,group['frame'])
+  if mutation!='omit_rank_inverse'or k!=0:
+   for t,cs in group['dependent'].items():
+    for p,c in cs.items():y[t]=track(add(y[t],c,y[p]))
+  rank_active.remove(k);rank_restores.append(dict(group=k,before_role=trigger,frame=group['frame']))
+ def echelon_restore(k,trigger):
+  group=echelon[k];assert echelon_seen[k]==group['roles']
+  for t in group['targets']:target_move(t,group['frame'])
+  if mutation!='omit_echelon_inverse'or k!=0:
+   for t,p,c in reversed(group['moves']):y[t]=track(add(y[t],-c,y[p]))
+  echelon_active.remove(k);echelon_restores.append(dict(group=k,before_role=trigger,frame=group['frame']))
+ def read(s):
+  nonlocal aggregation_pivot_reads,rank_skipped,echelon_reads
+  if s in pair_byalias:
+   if mutation=='subtract_paired_alias'and s==next(iter(pair_byalias)):
+    for t,c in adj[s].items():y[t]=track(add(y[t],-direction*c,value(s)))
+   return
+  if s in fresh:
+   if mutation=='omit_transformed_source_read'and s==next(iter(fresh)):return
+   for t,c in fresh[s]['transformed_response'].items():
+    t=int(t);source_move(s,W.gauge[s]['frame']);target_move(t,W.gauge[s]['frame']);y[t]=track(add(y[t],-direction*c,value(s)))
+   return
+  echelon_skip=set()
+  if s in W.gauge:
+   for k,group in enumerate(echelon):
+    if k not in echelon_active:continue
+    if s not in group['roles']:
+     if set(responses[s])&set(group['targets']):echelon_restore(k,s)
+     continue
+    echelon_seen[k].add(s);echelon_skip.update(group['targets'])
+    for t,a in group['response'].items():
+     c=a.get(s,0)
+     if not c:continue
+     if mutation=='flip_echelon_sign'and k==0 and c<0:c=-c
+     source_move(s,W.gauge[s]['frame']);target_move(t,W.gauge[s]['frame']);y[t]=track(add(y[t],-direction*c,value(s)));echelon_reads+=1
+  if s in W.gauge:
+   for k in sorted({rankowner[t]for t in responses[s]if t in rankowner}):
+    if k not in rank_active:continue
+    if s not in rankgroups[k]['roles']:rank_restore(k,s)
+    else:rank_seen[k].add(s)
+  if mutation=='omit_gauge_compensation'and s==new_gauge_selection[-1]['role']:return
+  if mutation=='omit_extra_compensation'and s==extra_selection[0]['role']:return
+  grouped=role_groups.get(s,[])
+  for k in sorted({target_group[t]for t in responses[s]if t in target_group}) if s in W.gauge else []:
+   if k not in grouped:aggregate_restore(k,s)
+  skipped=set()
+  for k in grouped:
+   group=agg[k];pivot=group['pivot'];assert aggregation_active[k]
+   assert set(group['targets'])<=set(responses[s])and len({responses[s][t]for t in group['targets']})==1
+   c=responses[s][pivot];source_move(s,read_frames[s,k]);target_move(pivot,read_frames[s,k]);y[pivot]=track(add(y[pivot],-direction*c,value(s)))
+   if mutation=='repeat_aggregation_read'and k==39:
+    t=next(t for t in group['targets']if t!=pivot);target_move(t,read_frames[s,k]);y[t]=track(add(y[t],-direction*c,value(s)))
+   skipped.update(group['targets']);aggregation_read[k].add(s);aggregation_pivot_reads+=1
+  for t,c in responses[s].items():
+   if t in skipped or t in echelon_skip:continue
+   rk=rankowner.get(t)
+   if s in W.gauge and rk in rank_active and t in rankgroups[rk]['dependent']:
+    rank_skipped+=1
+    if mutation!='repeat_rank_read'or rk!=0:continue
+   if s in W.gauge:source_move(s,W.gauge[s]['frame']);target_move(t,W.gauge[s]['frame'])
+   else:assert target_frames[t]is None,'initial correction leaves target atD0'
+   y[t]=track(add(y[t],-direction*c,value(s)))
+ def gate(i,sign):
+  a,b,_=W.ops[i]
+  if sign==1:
+   source_move(a,W.opframe[i]);source_move(b,W.opframe[i])
+  assign(a,add(value(a),sign,value(b)))
+ for s in range(W.R):
+  if s not in W.gauge and s not in borrow and s not in removed:read(s)
+ for n,s in W.source.items():assign(s,add(value(s),1,x[n]))
+ for r in gauge_selection:
+  if mutation=='omit_transformed_source_mix'and r is next(iter(fresh.values())):continue
+  if mutation=='omit_extra_mix'and r is extra_selection[0]:continue
+  if mutation!='omit_gauge_mix' or r is not new_gauge_selection[-1]:x[r['partner']]=track(add(x[r['partner']],1,x[r['source']]))
+ def forward(i):
+  if i not in omitted:gate(i,1);done.append(i)
+ for k in schedule['center_prefix_events']:
+  event=schedule['events'][k];i=event['op']
+  if event['kind']=='early_mix':
+   r=early[i];a,b=r['partner'],r['source']
+   if mutation!='omit_early_mix' or r is not selection[0]:x[a]=track(add(x[a],1,x[b]))
+  else:forward(i)
+ for r,s in zip(W.g['roots'],W.w['rootroles']):
+  if r['kind']=='center':
+   for t in r['targets']:
+    assert all(f is None for f in target_frames),'copied centers scatter before any nonzero target movement'
+    center_reads.append((s,t));y[t]=track(add(y[t],direction,value(s)))
+ for k,group in enumerate(echelon):
+  assert all(target_frames[t]is None for t in group['targets'])
+  if mutation!='omit_echelon_setup'or k!=0:
+   for t,p,c in group['moves']:y[t]=track(add(y[t],c,y[p]))
+ for k,group in enumerate(rankgroups):
+  assert all(target_frames[t]is None for t in group['targets'])
+  if mutation!='omit_rank_setup'or k!=0:
+   for t,cs in group['dependent'].items():
+    for p,c in cs.items():y[t]=track(add(y[t],-c,y[p]))
+ for k,group in enumerate(agg):
+  assert all(target_frames[t]is None for t in group['targets']), 'aggregation setup after all D0 center scatters'
+  if mutation!='omit_aggregation_setup'or k!=39:
+   for t in group['targets']:
+    if t!=group['pivot']:y[t]=track(add(y[t],-1,y[group['pivot']]))
+ for e in terminal:
+  assert all(target_frames[t]is None for t in e['targets']),'terminal setup afterD0center scatter'
+  for t in e['targets']:
+   if t!=e['pivot']:y[t]=track(add(y[t],-1,y[e['pivot']]))
+ # Original dirty compensation happened at D0 before these source injections.
+ for r in pair_selection:
+  b,f=r['aliased_role'],r['mix_frame']
+  for a,c in r['retained_dirty_coefficients'].items():
+   source_move(a,f);source_move(b,f)
+   if mutation=='omit_sparse_anchor'and r is pair_selection[14]and a==next(iter(r['retained_dirty_coefficients'])):continue
+   if mutation=='wrong_sparse_sign'and r is pair_selection[14]and a==next(iter(r['retained_dirty_coefficients'])):c=-c
+   if mutation!='omit_pair_initialization'or r is not pair_selection[0]:assign(a,add(value(a),c,value(b)))
+ for j,i in enumerate(W.rest):
+  for s in at[j]:
+   if mutation!='late_phase1_gauge' or s not in new_phase_roles:read(s)
+  if mutation=='late_phase1_gauge'and j==168:
+   for s in sorted(new_phase_roles):read(s)
+  if i in bywrite:
+   e=bywrite[i];a,b,n=W.ops[i];source_move(b,W.opframe[i]);target_move(e['pivot'],W.opframe[i]);y[e['pivot']]=track(add(y[e['pivot']],direction,value(b)))
+  else:forward(i)
+  if i in afterwrite:
+   e=afterwrite[i]
+   for t in e['targets']:
+    target_move(t,e['root_frame'])
+    if t!=e['pivot']:y[t]=track(add(y[t],1,y[e['pivot']]))
+ for s in at[len(W.rest)]:read(s)
+ assert not any(aggregation_active),'all quotient inverses precede original side/K reads'
+ assert not rank_active,'rank quotient inverses precede side/K reads'
+ assert not echelon_active,'echelon inverses precede side/K reads'
+ for j,(r,s)in enumerate(zip(W.g['roots'],W.w['rootroles'])):
+  if r['kind']=='side' and j not in deletedroots:
+   for k in sorted({target_group[t]for t in r['targets']if t in target_group}):aggregate_restore(k,('root',j))
+   source_move(s,W.w['root_frame'][j])
+   for t in r['targets']:target_move(t,W.w['root_frame'][j]);y[t]=track(add(y[t],direction,value(s)))
+  for e in deliveries[j]:
+   a,b=e['carrier'],e['passive']
+   if b not in by_source:x[a]=track(add(x[a],1,x[b]))
+   for k in sorted({target_group[t]for t in e['receivers']if t in target_group}):aggregate_restore(k,('K',j))
+   for t in e['receivers']:target_move(t,e['deliver_frame']);y[t]=track(add(y[t],direction,x[a]))
+ assert not any(aggregation_active),'all target inverses completed beforecleanup'
+ for e in W.k['entries']:
+  if e['passive']not in by_source:x[e['carrier']]=track(add(x[e['carrier']],-1,x[e['passive']]))
+ if mutation=='undo_before_restore':
+  for e in W.k['entries']:
+   if e['passive']in by_source:x[e['carrier']]=track(add(x[e['carrier']],-1,x[e['passive']]))
+ for role in source_paths:source_move(role,W.w['full_frame'])
+ for role in pair_dirty_paths:source_move(role,W.w['full_frame'])
+ for i in reversed(done):gate(i,-1)
+ for r in reversed(pair_selection):
+  b=r['aliased_role']
+  for a,c in reversed(list(r['retained_dirty_coefficients'].items())):
+   c=-c
+   if mutation=='wrong_pair_restore_sign'and r is pair_selection[0]:c=-c
+   if mutation!='omit_pair_restoration'or r is not pair_selection[0]:assign(a,add(value(a),c,value(b)))
+ if mutation!='undo_before_restore':
+  for e in W.k['entries']:
+   if e['passive']in by_source:x[e['carrier']]=track(add(x[e['carrier']],-1,x[e['passive']]))
+ for n,s in W.source.items():assign(s,add(value(s),-1,x[n]))
+ if binary:want=[Y0[t]^X0[t]for t in range(v)]
+ else:
+  want=Y0[:];cache={}
+  for root in W.g['roots']:
+   n=root['node']
+   if n not in cache:
+    mask=C.sup[n];val=0
+    while mask:low=mask&-mask;mask-=low;val=add(val,1,X0[low.bit_length()-1])
+    cache[n]=val
+   for t in root['targets']:want[t]=add(want[t],direction,cache[n])
+  for e in W.k['entries']:
+   val=add(X0[e['carrier']],1,X0[e['passive']])
+   for t in e['receivers']:want[t]=add(want[t],direction,val)
+ # Finish every local target at its codimension-one cap, then reverse the exact
+ # annihilator stream. All copied-center scatters lie before these movements,
+ # hence atD0 forward and atD1 after the complementary movements reverse.
+ reflected=Counter()
+ for t,chain in enumerate(target_paths):
+  assert chain
+  d=W.h-1-C.dimf[chain[-1]];assert d>=0
+  if d:target_hist[d]+=1
+  previous=[C.cov[t]];dimension=1
+  for f in reversed(chain):
+   assert all(W.module.dot(a,b)==0 for a in previous for b in C.B[f])
+   basis=C.A[f];step=len(basis)-dimension;assert step>=0
+   if step:reflected[step]+=1
+   previous=basis;dimension=len(basis)
+  if W.h>dimension:reflected[W.h-dimension]+=1
+ assert reflected==target_hist
+ assert len(center_reads)==sum(len(r['targets'])for r in W.g['roots']if r['kind']=='center')
+ assert sum(k*n for k,n in target_hist.items())==W.v*(W.h-1)
+ source_histogram=Counter();source_reflected=Counter()
+ for role,path in source_paths.items():
+  assert C.dimf[path[0]]==1 and C.dimf[path[-1]]==W.h
+  for a,b in zip(path,path[1:]):
+   assert C.sub(a,b)and C.nondeg(a)and C.nondeg(b)
+   step=C.dimf[b]-C.dimf[a]
+   if step:source_histogram[step]+=1
+  for a,b in zip(reversed(path),list(reversed(path))[1:]):
+   assert all(W.module.dot(x,y)==0 for x in C.A[a]for y in C.B[b]);step=len(C.A[b])-len(C.A[a]);assert step>=0
+   if step:source_reflected[step]+=1
+ assert source_histogram==source_reflected and sum(k*n for k,n in source_histogram.items())==len(source_paths)*(W.h-1)
+ paired_hist=Counter();paired_reflected=Counter()
+ for role,path in pair_dirty_paths.items():
+  assert C.dimf[path[0]]==0 and C.dimf[path[-1]]==W.h
+  for a,b in zip(path,path[1:]):
+   assert C.sub(a,b)and C.nondeg(a)and C.nondeg(b)
+   d=C.dimf[b]-C.dimf[a]
+   if d:paired_hist[d]+=1
+  for a,b in zip(reversed(path),list(reversed(path))[1:]):
+   assert all(W.module.dot(x,y)==0 for x in C.A[a]for y in C.B[b]);d=len(C.A[b])-len(C.A[a])
+   if d:paired_reflected[d]+=1
+ assert paired_hist==paired_reflected and sum(k*n for k,n in paired_hist.items())==len(pair_dirty_paths)*W.h
+ if major:return max(top,max(a+b for a,b in zip(y,want)),max(z[r]+Z0[r]for r in regs),max(a+b for a,b in zip(x,X0)))
+ assert y==want,('wrong targets',mode,[t for t in range(v)if y[t]!=want[t]][:10])
+ assert x==X0,('source restoration',mode,[s for s in range(v)if x[s]!=X0[s]][:10])
+ assert z==Z0,('dirty restoration',mode)
+ return dict(mode=mode,direction=direction,formal_columns=2*v+len(regs),sources=v,targets=v,dirty=len(regs),all_targets=True,all_source_and_dirty_restored=True,copied_center_target_reads=len(center_reads),all_forward_centers_at_D0=True,all_reflected_centers_at_D1=True,actual_target_histogram={str(k):n for k,n in sorted(target_hist.items())},actual_reflected_target_histogram={str(k):n for k,n in sorted(reflected.items())},aggregation_groups=len(agg),aggregation_pivot_reads=aggregation_pivot_reads,aggregation_restores=aggregation_restores,actual_extra_source_paths=source_paths,rank_groups=len(rankgroups),rank_targets=len(rankowner),rank_restores=rank_restores,rank_removed_reads=rank_skipped,actual_extra_source_histogram=dict(source_histogram),actual_extra_reflected_source_histogram=dict(source_reflected),echelon_restores=echelon_restores,echelon_reads=echelon_reads,paired_dirty_paths=pair_dirty_paths,paired_dirty_histogram=dict(paired_hist),paired_dirty_reflected_histogram=dict(paired_reflected))
diff --git a/research/five-stage-source527-banks/verify.py b/research/five-stage-source527-banks/verify.py
new file mode 100644
index 0000000000..dbea044263
--- /dev/null
+++ b/research/five-stage-source527-banks/verify.py
@@ -0,0 +1,82 @@
+#!/usr/bin/env python3
+"""Immutable source527 five-stage banks: regenerate every mandatory finite stage.
+Prepared with substantial OpenAI Codex assistance; Apache-2.0.
+"""
+import sys
+if not __debug__:raise SystemExit('Assertions required; refusing-O')
+sys.dont_write_bytecode=True
+from pathlib import Path
+from datetime import datetime,timezone
+from fractions import Fraction
+import argparse,hashlib,importlib.util,json,time,traceback
+ROOT=Path(__file__).resolve().parent
+REQUIRED={'raw','bit','scalar','primes','banks','complex','math','finite'}
+
+def progress(message):print(datetime.now(timezone.utc).strftime('%H:%M:%S UTC')+'  '+message,flush=True)
+def sha(p):return hashlib.sha256(p.read_bytes()).hexdigest()
+def serial(x):
+ if isinstance(x,Fraction):return str(x)
+ if isinstance(x,dict):return {str(k):serial(v)for k,v in x.items()}
+ if isinstance(x,(list,tuple)):return list(map(serial,x))
+ return x
+
+def load(name):
+ s=importlib.util.spec_from_file_location('source527_'+name,ROOT/(name+'.py'));m=importlib.util.module_from_spec(s);sys.modules[s.name]=m;s.loader.exec_module(m);return m
+
+def integrity():
+ p=ROOT/'MANIFEST.json';m=json.loads(p.read_text());actual={}
+ for q in ROOT.rglob('*'):
+  assert not q.is_symlink(),'Symlink in package'
+  if q.is_file()and q.name!='MANIFEST.json':actual[q.relative_to(ROOT).as_posix()]=sha(q)
+ # Nested source manifests are also ordinary pinned inputs.
+ for q in ROOT.rglob('MANIFEST.json'):
+  if q!=p:actual[q.relative_to(ROOT).as_posix()]=sha(q)
+ assert actual==m['files'],'Missing, changed or unpinned package/source file'
+ return sha(p),len(actual)
+
+def validate_required(results):
+ assert set(results)==REQUIRED,'Missing mandatory proof stage'
+ assert all(isinstance(results[k],dict)and results[k]for k in REQUIRED)
+
+def controls():
+ out=[];good={k:{'present':True}for k in REQUIRED}
+ for k in sorted(REQUIRED):
+  bad=good.copy();bad.pop(k)
+  try:validate_required(bad)
+  except AssertionError:out.append('omitted '+k+' stage')
+  else:raise AssertionError('Missingstageaccepted')
+ return out
+
+def main():
+ ap=argparse.ArgumentParser(description=__doc__);ap.add_argument('--output',type=Path,required=True);args=ap.parse_args()
+ import sympy
+ assert sys.version_info>=(3,11)and sympy.__version__=='1.14.0'
+ digest,count=integrity();output=args.output.resolve();assert not output.is_relative_to(ROOT)and not output.exists(),'Output must be new and outside package';output.mkdir(parents=True)
+ results={};timings={};start=time.monotonic()
+ def save(name,value):(output/name).write_text(json.dumps(serial(value),sort_keys=True,indent=2)+'\n')
+ def stage(name,call):
+  t=time.monotonic();value=call();results[name]=value;timings[name]=time.monotonic()-t;save(name+'.json',value);progress(name+' PASS '+format(timings[name],'.1f')+'s');return value
+ try:
+  progress('Regenerating source527 from pinned source recipes')
+  context=load('prepare').prepare()
+  raw=stage('raw',lambda:load('raw_ledger').run(context))
+  portable=load('portable_bit');bit=portable.run(prepared_context=context,raw=raw,output_dir=output,run_geometry=True,package_root=ROOT);raw=bit['raw'];results['raw']=raw;save('raw.json',raw);results['bit']=portable.summary(bit);timings['bit']=bit['seconds'];save('bit.json',results['bit']);progress('bit physical/global/geometry PASS')
+  scalar=stage('scalar',lambda:load('scalar_check').run(context,bit['records'],progress))
+  primes=stage('primes',lambda:load('prime_check').run(context,bit['physical'],progress))
+  bankresult=stage('banks',lambda:load('bank_check').run(context,bit['global_result'],bit['lower']))
+  banks=bankresult['banks'];banked=bankresult['banked_result']
+  complex_result=stage('complex',lambda:load('portable_complex').run(package_root=ROOT,progress=progress))
+  assert complex_result['precision_guard']['retained_row_coefficient']==20161
+  math=stage('math',lambda:load('math_check').run(raw,complex_result['five_stage_histogram'],banks,bit['global_result']))
+  finite=stage('finite',lambda:load('finite_check').run(raw,bit['physical'],scalar,primes,math,banks,banked,bit['global_result']))
+  assert math['mathematics']['finite_bridge']['rows']['coefficient']==complex_result['precision_guard']['retained_row_coefficient']==finite['row_reserve']['external_complex_coefficient']
+  validate_required(results);assert len(scalar['controls'])==25 and len(primes['controls'])==6
+  assert math['mathematics']['kappa']=='710069340338651/1000000000000000000'
+  after,_=integrity();assert after==digest,'Package changed during replay'
+  certificate=dict(schema='source527-parity-fused-five-stage-banks/1',**math['mathematics'],scope='Conditionalfiniteconstruction; inheritedallsizecompiler,weightedselector,commonancestorchart,restoredrow,routing,prime,recovery,complexsymbolicandanalyticinterfacesremainassumptions.')
+  save('certificate.json',certificate)
+  report=dict(status='PASS_IMMUTABLE_PARITY_FUSED_SOURCE527_FIVE_STAGE_BANKED_CONSTRUCTION',manifest_sha256=digest,files=count,kappa=math['mathematics']['kappa'],kappa_scientific=math['mathematics']['kappa_scientific'],fresh_stages=sorted(REQUIRED),inputs_unchanged=True,timings=timings,seconds=time.monotonic()-start,negative_controls=controls())
+  save('verification.json',report);print(json.dumps(report),flush=True)
+ except BaseException as exc:
+  save('failure.json',dict(status='FAIL',error=str(exc),completed_stages=sorted(results),timings=timings,traceback=traceback.format_exc()));raise
+if __name__=='__main__':main()
diff --git a/research/five-stage527-frame-semantics.md b/research/five-stage527-frame-semantics.md
new file mode 100644
index 0000000000..d1991829c7
--- /dev/null
+++ b/research/five-stage527-frame-semantics.md
@@ -0,0 +1,17 @@
+# Bank routes and the retained frame coordinate system
+
+The full-address bank adapter changes stream ownership. It does not change the cover variable used to interpret an emitted MOVE or ADD frame.
+
+An original stage invocation at class d uses the exact checked global projector d*A*d^-1. Its helper formerly lived at class d*tau_i. The banked helper lives at h=d*tau_i*N^-1. The frame still means d*A*d^-1. A consumer must not substitute h for d or conjugate that frame by N a second time.
+
+This convention gives the required endpoint directly. Write R0=diag(I24-sigma,0,...,0). Since d=h*N*tau_i, the helper residual is
+
+    d*R0*d^-1 = h*N*(tau_i*R0*tau_i)*N^-1*h^-1.
+
+The checked normalizer turns tau_i*R0*tau_i into the assigned coordinate bank projector. Thus the endpoint is the intended block at bank class h. During the actual scalar word, the inherited original d-frame remains the one used for equal-frame additions and paid rank increments. Chart routing and its charged selector factors move complete streams to that invocation; they are not a second change of projector coordinates.
+
+`BankLowerer.iter_stage` delegates to `BankPlan.map_stage_row`, which preserves every original mathematical field and changes only stream operand slots. The same original local cover d is present in every resulting address word. Boundary projectors already contain the original data-route conjugacies and are interpreted at their physical class d. Every idle and bridge remains a separate all-cover phase.
+
+The conditional common ancestor chart is unchanged throughout each bank's completed block schedule, as required by the inherited bank compiler theorem.
+
+This note accompanies `research/five-stage-source527-banks/`. Prepared by eumemic with substantial OpenAI Codex assistance. The five-stage source composition is Henry Grant / hcg890’s PR234; the bank compiler lineage is retained in that package’s notices.