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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+        "compact_leaf": "297620240363722999999999702129759636277/250000000000000000000000000000000000000000000000000000000000000000",
+        "compact_reservations": "1/4000000000000000000000000",
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+        "gamma_sublinear": "1/2000000000000000000000000",
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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..217224417e
--- /dev/null
+++ b/research/coordinated-crossover-pr200/BANK-CHART-PROOF.md
@@ -0,0 +1,47 @@
+# Final302 frame changes, 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, with302 exact operation-frame changes. 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 all24602 used-frame 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-three 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 Lean formalization or publication occurred.
+
+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..1289eb2003
--- /dev/null
+++ b/research/coordinated-crossover-pr200/BOOTSTRAP-REVIEW.md
@@ -0,0 +1,38 @@
+# 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 three levels, use a3 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 three exact recurrence steps, their strict ordering, the two paid moments and all 47 arithmetic constraints. The expression for a3 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/CROSSOVERS.md b/research/coordinated-crossover-pr200/CROSSOVERS.md
new file mode 100644
index 0000000000..abb23e7dab
--- /dev/null
+++ b/research/coordinated-crossover-pr200/CROSSOVERS.md
@@ -0,0 +1,31 @@
+# Accepted crossover construction and rejected alternatives
+
+The deliverable's conditional assembly certifies kappa = 1366380910073/2000000000000000 = 0.0006831904550365. The selected construction combines actual operation-frame optimization with completed dirty-bank packing on PR200, then composes the full supplier with PR202's complex supplier. Its bit coarse saving is 0.000683657523331095; the complex saving is 0.000700918443859411. The bit side binds. All 47 strict assembly inequalities pass.
+
+## Contributions actually used
+
+1. Coordinated frame changes: 302 operation-frame changes alter actual internal children while preserving the fixed source/target endpoints. The complete histogram, rather than a sum of local gains, is priced.
+2. Dirty-bank completion: for a coordinate projector P, the literal partial swap is S_P(x,y) = (x-Px+Py, y-Py+Px). Disjoint blocks partitioning all 72 coordinates compose to the full swap on arbitrary correlated dirty contents. Every bank uses the same weighted chart, making intermediate conjugations cancel. Genuine entrance residuals have sizes 4 and 24; nine replicas give 1,100 four-coordinate banks and 44,742 twenty-four-coordinate banks per stage, filling every bank exactly.
+3. Finite ordinary-leaf composition: three acyclic stopped-leaf levels improve the admitted ordinary saving while preserving fixed-depth execution and strict routing/row tolls. The final root is reassembled from the bit and complex suppliers with unchanged constraints.
+
+The resulting bit normalization is W=56,402, rank=4,055,136, deficit=5,808, m=72, max child=22. The factor-three moment normalization and nine physical replicas are different operations and are not counted twice.
+
+## Exact tested exclusions
+
+| Route | Proven finite fact | Actual failed or neutral check | Included gain |
+|---|---|---|---|
+| Shared totals and rank-two face compensation | All three compensated 8-by-8 matrices are exact; 22,569 additions become 22,339 | 260 target read roots fall outside actual paid caps; source interaction rank becomes 2 and total span 19 instead of 17 | None |
+| Integral 165-to-55 harmonic residual | All 27,225 integer factor entries and all 385 literal source/target/dirty columns pass | 9 feature spans have rank 19 and 46 rank 20; all outputs rank 20 exceed rank-18 side caps | None |
+| Ballot/octahedral residual transport | Coarse four-dimensional trade identity remains exact | Each of 110 trades expands to 64 real paired source ports of rank 10 | None |
+| Full-cube source-chart shortcut | Rank-two radical shear fixes the full twisted source Lagrangian | Real path [3,19,1] replaces [2,18,1], adds 7,920 rank mass and changes deficit 1,320 to -6,600 | None |
+| Full source-assisted port to PR200 | High-interface equality passes; changed resources are honestly recounted | Vertex (0,141004) needs three old donors but has one new retired donor; low recount adds 440 rows | None |
+| Selective A0 source rewrite | Exact high-interface checks pass | Adds 440 rows; entropy increases by 11,050.526501520155 | None |
+| Selective G12 source rewrite | Exact high-interface checks pass | Histogram, stock and entropy are unchanged | None |
+
+The generalized dirty-safe quotient completion theorem supplies a concrete sufficient method when new source availability and donor stock are actually instantiated. Its 300 finite cases and 1,605 formal columns support that constructor; they do not certify the failed global port.
+
+## Scope
+
+This is a sufficient finite supplier construction and conditional root assembly. It does not prove the inherited all-size compiler, leaf, routing, prime, precision or analytic interfaces. No isolated rank factor, canceled coefficient, scalar addition count, configured process or local control is treated as a new paid supplier. No new Lean certificate is claimed.
+
+Pinned sources: PR200 a1175449f34d39ff933d9d8ab23ced1f32b290ec and PR202 8d8d67bcf69c5ea67d3a29dbc64ba588156d6e8d. The newest public comparator is PR205, with displayed conditional kappa 0.000683061299399923; this package exceeds it by approximately 0.0189%. PR205 independently presents completed-bank packing on PR200, so packing is not claimed as exclusively original here. PR204's earlier claim 0.0006773148 remains a historical comparison, not an input certificate. PR205's proof and certificate were compared at pinned head 95c58706ad231d532bd7038308946a0344d08186: its coarse saving exactly matches the earlier local bank-only baseline. The new paid gain is the coordinated internal-frame refinement. Its different bank scheduling changes a fixed routing constant, not an independent exponent gain.
diff --git a/research/coordinated-crossover-pr200/FILES.json b/research/coordinated-crossover-pr200/FILES.json
new file mode 100644
index 0000000000..529b6a62bb
--- /dev/null
+++ b/research/coordinated-crossover-pr200/FILES.json
@@ -0,0 +1,33 @@
+{
+  "files": {
+    "BANK-CHART-PROOF.md": "3026341b38b322f8514f22497abb54bc2eb73324cb75af9d6c9c4cd9378d07cc",
+    "BASELINE.json": "34d948f1f3724370d50605e1f8784407c6ee8f95f34e3135ad6afae07d7fcd0a",
+    "BOOTSTRAP-REVIEW.md": "fa555f609d1a9d5bdc37f868f0828e084193c829406a4bf3d67b20f62233ff66",
+    "CROSSOVERS.md": "423c86f2406ea0540f25b336a467235099ff847d2e9f185de9126b479a8c61a9",
+    "LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+    "NOTICE": "c9850418622aea8d1382a230306c940947b5da818f4854b4b272ed60cf239f6c",
+    "PR-STATEMENT.md": "7396fa4635dd4a09eefddfdd2d4cc66fcda021d31c17e7fb131a69f2fd7033e9",
+    "PR205-COMPARISON.md": "6439f550614aae7549f505467aa3f6798b299a6411a340d8d5416e97f0e98cd9",
+    "PROOF.md": "0fa8d2d5a0affc673311e624701b13883fe5d09de890da5a11d7efddeb3d1a49",
+    "README.md": "4ad7dfac61595bd30f6ce405cbad3660d62fb822494fafa2de5cae8a5b313979",
+    "arithmetic.py": "3a138f67954162087618a3cf38fb0e4c29564c1e039b1a25439f5c4f1e884868",
+    "baseline-pr202.part01": "f5beb67709bf1235a2b6483883aa997a150ed4be47db978ab8be6040114cf924",
+    "baseline-pr202.part02": "21e1ff2a0ddc3e8febad3ca19d210d2e83d575f0968bb5125ab62bf6d00794fe",
+    "baseline-pr202.part03": "218c880af743a75e272cd51b3bd1ce5f36583f332823c2b5606d21d5d8b72e98",
+    "baseline-pr202.part04": "e0f95fcdd581a7617eabd53f4d0ea9e7cf0551a31fcd081e3d2154750740f114",
+    "baseline-pr202.part05": "9b64c7b8b5b7893eae071733b7ca14e280fe08b3db942c1c6d00e88d6e615286",
+    "certificate.json": "23416f66749bf92dc7b47c69d5a94125effc26a043dd9ffbe83479e4d42e5e3a",
+    "frames/opframe-bases.json": "65b6b30748fe8db9b472c6c4bd34ba6b6ec757f4440ad0d78f8cad139cbd906b",
+    "frames/verify-bit-frames.py": "e480ceee488b96e95fefb6ceee6280f3650c6d408f864f36615e7046873a8cc7",
+    "geometry/BANK-PROOF-PR200.md": "a1019da230d9d9c452c23a68e29c6647ddf73ec1a4be17f77cc9e9bcbe16431f",
+    "geometry/audit-gauge-charts.py": "111710eaa3cbef146def51419e66077e103366198790e06a434d75c99a17b508",
+    "geometry/base_two_moment.py": "aee70c242a2f1d225528fabe2002e1564d1dc884797c07d28a831c7878ee9098",
+    "geometry/check-normalizer-routing.py": "c9cec5576df65336cae37753ef37311be1c1a28cb6075c4489182e2278c3d0ee",
+    "geometry/pack-admitted-frames.py": "b130f84bd79fcecc1189a6bd294e53c7de99882a54eb5c44ef05fe6b496aa435",
+    "geometry/pack-pr200.py": "ebd1765d5edc7804bc55e36ebac6c40500d8d393c40aa76cab6668954b6d4e70",
+    "geometry/retained-packing.py": "56822a01bfc646b0f3b28d3ebc60aeb6acf380d53bbed96780992ea66ad1bec7",
+    "requirements.txt": "fdf15f8e56a3349c4cf3a05e4d073f03512a65deff2383767a4eaed1c445310f",
+    "verify.py": "bd0a759be666d68d1f1ce588c8d7e3c51c92b86d8d4a2951b3803c64e5042a76",
+    "verify_inner.py": "d03b7b8b76a1244df91fa9aac17739bc61004751e937fcf0bb5bbf38d0dcc8e8"
+  }
+}
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 @@
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diff --git a/research/coordinated-crossover-pr200/NOTICE b/research/coordinated-crossover-pr200/NOTICE
new file mode 100644
index 0000000000..ba5ea80dd7
--- /dev/null
+++ b/research/coordinated-crossover-pr200/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/PR-STATEMENT.md b/research/coordinated-crossover-pr200/PR-STATEMENT.md
new file mode 100644
index 0000000000..562fe75be0
--- /dev/null
+++ b/research/coordinated-crossover-pr200/PR-STATEMENT.md
@@ -0,0 +1,44 @@
+Title:
+
+Conditional kappa 6.831904550365e-4: coordinated frame optimization and completed entrance banks
+
+Body:
+
+## Change
+
+Add `research/coordinated-crossover-pr200`, a finite supplier certificate and conditional assembly claiming
+
+```
+kappa = 1366380910073 / 2000000000000000
+      = 0.0006831904550365.
+```
+
+This is approximately 0.0189% above the newly displayed [PR205](https://github.com/CrocSwap/integer-mult-bounds/pull/205) claim of 0.000683061299399923. PR205 also supplies a completed-bank construction on PR200; no exclusive originality is asserted for bank packing. The earlier comparison with PR204 was approximately +0.8675% and is now historical. It combines a new, explicitly paid bit supplier with the retained source-assisted complex supplier; component improvements are not added.
+
+The bit construction starts from [PR200](https://github.com/CrocSwap/integer-mult-bounds/pull/200), pinned at `a1175449f34d39ff933d9d8ab23ced1f32b290ec`. Coordinated changes to 302 operation frames reduce the actual internal child histogram while keeping the source and target interfaces fixed. Completed entrance banks then replace the real rank-4 and rank-24 dirty residual completions with explicit coordinate partitions of width 72. Nine physical replicas make every bank full. Internal alias recipients, source births and deleted terminals are excluded from fresh bank allocation. Arbitrary incoming dirty contents and spectator fields are restored.
+
+The resulting integer-normalized bit profile has width 72, stock 56,402, rank mass 4,055,136, deficit 5,808 and largest child 22. Its certified paid coarse saving is `136731504666219/200000000000000000`. The complex side uses [PR202](https://github.com/CrocSwap/integer-mult-bounds/pull/202), pinned at `8d8d67bcf69c5ea67d3a29dbc64ba588156d6e8d`, with saving `700918443859411/10^18`.
+
+Exactly three stopped-leaf bootstrap levels are used, following [PR185](https://github.com/CrocSwap/integer-mult-bounds/pull/185) and [PR197](https://github.com/CrocSwap/integer-mult-bounds/pull/197). Each level calls only its predecessor for ordinary leaves and routing. The unattained limiting saving is not substituted. The final paired assembly uses eta = beta = 10^-24 and preserves the original strict scalar, router, row and recovery constraints.
+
+## Validation and scope
+
+The included verifier checks the actual finite frame/source/gauge chronology, the original and terminal-modified binary and defining integer columns, explicit bank assignments and inverses, frame prime witnesses, adverse controls, rational supplier moments, the finite ordinary-leaf recurrence and all 47 strict assembly inequalities. The next final 10^-18 grid point fails for these fixed suppliers and parameters; this does not assert global optimality.
+
+The claimed consequence is conditional `T(n) = O(n (log n)^(1-kappa))` under the retained all-size interfaces. General Clifford/tensor realization, uniform weighted local-ring compilation, completed restored-row and fixed-tape leaf guarantees, paid routing/layout, prime supply, precision/recovery and analytic transfer remain disclosed conditions. A finite column replay and positive moment do not independently establish these conditions. This submission makes no new Lean-certification or practical-speedup claim.
+
+Offline replay from the repository root uses Python 3.11+ with the pinned NumPy/SciPy requirements:
+
+```sh
+python -B research/coordinated-crossover-pr200/verify.py
+```
+
+An optional `--temp-root` chooses scratch storage. The replay never fetches sources. The bank chart/router bound `K = 21,900,544,398` remains paid through the ordinary routing toll and its strict exponent gap.
+
+## Other routes tested
+
+Shared-total/rank-two compensation saves 230 scalar additions but fails 260 actual target-cap membership checks. The integral 165-to-55 residual factor is exact, but its real paired-address features require ranks 19/20 and its outputs rank 20 against side caps of rank 18. Source-assisted rewrites also fail to improve this bit word: the full port lacks required donor stock at an actual vertex, the A0-only rewrite adds 440 rows, and the G12-only rewrite leaves the paid profile unchanged. None of these gains is credited.
+
+## Sources
+
+The package retains the pinned upstream sources and their license/attribution notices. It builds on the public PR200 physical bit construction, PR202 source-assisted complex construction, completed-bank work and finite-bootstrap argument. No files or claims were published by the local research process; this is a manually uploaded research addition.
diff --git a/research/coordinated-crossover-pr200/PR205-COMPARISON.md b/research/coordinated-crossover-pr200/PR205-COMPARISON.md
new file mode 100644
index 0000000000..ed62ebbc2e
--- /dev/null
+++ b/research/coordinated-crossover-pr200/PR205-COMPARISON.md
@@ -0,0 +1,13 @@
+# 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 `129155636577/10^18`, 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..7b2fb0db68
--- /dev/null
+++ b/research/coordinated-crossover-pr200/PROOF.md
@@ -0,0 +1,13 @@
+# Sufficient conditional composition
+
+Assume the all-width completed suppliers, restored weighted selectors, router, analytic fixed-tape hypotheses and finite bridge used by the public PR185/197/202 construction. The conclusion is the same conditional multiplication bound, with kappa `1366380910073/2000000000000000`.
+
+1. **Fixed finite bit word.** Start with PR200's actual word; replace exactly 302 operation frames by the included rational bases. Full replay checks every original source, target and dirty-register formal column, the defining integer decoder and F2 decoder, chronology, all 1760 compensated aliases, and 34 terminal actions. The scalar operations and all bank endpoints are unchanged. Exact Gram/factor witnesses are recomputed for all 24,602 used frames; every inherited prime q>2^80 remains allowed.
+2. **Complete bank endpoint.** Apply the concrete ring-general partial-swap construction in BANK-CHART-PROOF.md. Exactly 2200 genuine rank20 entrance gauges give rank4 residuals; the 14914 other retained physical chains give rank24 residuals. Nine replicas fill 18-by-4 and 3-by-24 banks, private to each of three stages. All 462078 role-copy assignments occur once. Arbitrary correlated dirty contents obey the full-swap identity and inverse restoration. Sources, spliced recipient gauges and deleted terminals are excluded. Only the 2200 rank60 entrance exterior children are removed.
+3. **No free routing.** All 220 gauge charts have literal inverse and elementary-factor witnesses. Explicit normalizers H_b=(b+1)Pi_b diag(B^-1,I48) are distinct modulo every permitted prime, witnessed by an outside-stage unit column. The nine-replica routing bill K=21900544398<2^40 includes charts, permutations and scalar factors. Its fixed ordinary-selector cost is absorbed by the inherited positive stopped-leaf atom/row toll, changing the finite stopping constant rather than the exponent. This remains an explicit inherited all-width assumption, not a consequence of the finite checks alone.
+4. **Paid profile.** The normalized bit profile is m=72, W=56402/3, rank=1351712, deficit=1936, largest child22. Integer moment normalization scales W, histogram and deficit by three; actual persistent stock is 169206 on nine replicas. Include the full 10^-16 bad-class envelope with 32*72^2 fallback children per paid edge. Two independent exact rational log/exp enclosures prove coarse c=136731504666219/200000000000000000 and reject c+10^-18. The ordinary leaf starts at the fully paid a0=546573370825043616599485038841961823/(8*10^38), not at the coarse root.
+5. **Acyclic finite bootstrap.** Construct A0 -> A1 -> A2 -> A3. At each fixed level, internally borrowing/restoring spectators gives a_j=(1-c)c+c*a_(j-1). This is the PR185/197 completed-leaf composition under their all-width interfaces. Each a_(j-1)<a_j<c<1-a_j is checked exactly. There is no recursive self-assumption or added exterior reserve. Depth three is independent of input size.
+6. **Complex supplier.** Replay PR202's frozen source-assisted flow witness with the unchanged exact local lift checker and fresh-column contract. Each local map is invertible; the transport DAG, source controls, read chronology, target chains and paid histogram are checked. The exact scalar-program hash equals the public certificate. Regenerated byte/provenance hashes are bound locally rather than assumed equal across Windows and POSIX. Recompute b=700918443859411/10^18 from its actual profile; b+10^-18 is excluded. A global scalar transcript/full Clifford replay remains outside the retained contract.
+7. **Balanced assembly.** With eta=beta=10^-24, weakening 10^-30, choose a=min(a3,(1-beta)b-10^-30), q=(1-2eta)a, and k strictly below (1-eta)q/(1+q). Execute the unchanged public assembly and preserve all 47 positive strict slacks, seven margins and finite bridge. The resulting k=1366380910073/2000000000000000 passes; k+10^-18 fails on this fixed parameter choice. This is a grid-optimal price for the stated profile/parameters, not a global impossibility bound.
+
+The finite proof joins explicit admitted witnesses under the stated sufficient all-size hypotheses. Exact witness checks do not establish those analytic or machine foundations unconditionally.
diff --git a/research/coordinated-crossover-pr200/README.md b/research/coordinated-crossover-pr200/README.md
new file mode 100644
index 0000000000..e963b15dee
--- /dev/null
+++ b/research/coordinated-crossover-pr200/README.md
@@ -0,0 +1,25 @@
+# Coordinated frame cuts and entrance banks on PR200
+
+Conditional kappa: **0.0006831904550365**, exactly `1366380910073/2000000000000000`.
+
+Latest pinned comparator: PR205 at `95c58706ad231d532bd7038308946a0344d08186`, kappa 0.000683061299399923. This package is approximately **0.01890835% higher**. Public PR205 independently uses completed PR200 banks; the additional gain here comes from the admitted 302-frame refinement.
+
+This composes 302 admitted rational operation-frame changes and completed stage-private entrance banks on PR200 with PR202's source-assisted complex supplier, then applies three fixed ordinary-leaf bootstrap levels and the unchanged balanced assembly. Gains are composed through actual profiles; they are not added.
+
+## Reproduce
+
+Python 3.11 or later; assertions enabled. Install `numpy==2.3.5` and `scipy==1.17.0`, then run:
+
+```sh
+python -B research/coordinated-crossover-pr200/verify.py
+```
+
+The pinned source and witnesses are included in five archive parts. No git, network, account, or download is used by verification. Allow about 250 MB scratch space; on Windows with a nearly full system drive, use `--temp-root F:/your-existing-scratch-directory`. Verification uses a temporary extraction and leaves the submitted files unchanged.
+
+The verifier executes the complete bit formal-column replay, modified rational frames, terminal actions, actual prime witnesses, explicit bank assignment and chart/inverse/routing checks. It replays the exact complex local lifts and fresh-column/target contract against pinned witnesses, recomputes paid moments with two independent rational enclosures, and checks all 47 strict assembly inequalities. Rank corruption and adjacent-grid controls must fail. The numerical complex search is not rerun; its frozen witness is rechecked exactly.
+
+## Scope
+
+This is a **conditional finite supplier certificate**, with the same all-size compiler, completed weighted/restored selector, routing, precision, fixed analytic tape and finite bridge conditions as the cited public work. It does not independently prove those conditions or provide a new Lean certificate. The complex branch retains the source-assisted local-flow contract: no newly flattened global scalar transcript or full Clifford/router replay is claimed.
+
+Base bit: PR200, `a1175449f34d39ff933d9d8ab23ced1f32b290ec`. Base complex/assembly: PR202, `8d8d67bcf69c5ea67d3a29dbc64ba588156d6e8d`. The included tracked source is the LF checkout of PR202; its PR200 bit program agrees with the separate PR200 pin (only SOURCE.json differs). Original attribution and licensing are retained. Prepared with substantial OpenAI Codex assistance; no new personal byline.
diff --git a/research/coordinated-crossover-pr200/arithmetic.py b/research/coordinated-crossover-pr200/arithmetic.py
new file mode 100644
index 0000000000..f7eee99ae4
--- /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(3):
+ 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/baseline-pr202.part01 b/research/coordinated-crossover-pr200/baseline-pr202.part01
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diff --git a/research/coordinated-crossover-pr200/baseline-pr202.part05 b/research/coordinated-crossover-pr200/baseline-pr202.part05
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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..5e882a3fbb
--- /dev/null
+++ b/research/coordinated-crossover-pr200/certificate.json
@@ -0,0 +1,199 @@
+{
+  "a_bit": "4375408149318106538866216854671930632866706725794304565219529058788314814256861959639757/6400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+  "a_complex": "700918443859411/1000000000000000000",
+  "adjacent_kappa_rejected": true,
+  "assembly": {
+    "absorption_gap": "134281643351448469855595122809435201472739986862783885293176478497632435149777090241037553043786318538250810999419972289/246322131082666081020725623388609831999871293422821561354667625512514417816371066596945447670129629672549080027711000000000000000000000000",
+    "margins": {
+      "Gaussian_arithmetic": "269255886111883479314991099644648533713048750891109557636200948729645684632351887487203757789591105011931098889929870370327450919972289/394115409732265729633160997421775731199794069476514498167468200820023068506193706555112716272207407476078528044337600000000000000000000000",
+      "balanced_prefix": "168284928819927174572023388609831999871293422821561354667625512514417816371066596945447670129629672549080027711/246322131082666081020725623388609831999871293422821561354667625512514417816371066596945447670129629672549080027711",
+      "bulk_exposure": "4375408149318106538866216854671930632866706725794304565219529058788314814256861959639757/6400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "compact_phase_layer": "168284928819927174571777066478749333790272697198172744835625641220994994809711929319935155711813301482483082263329870370327450919972289/246322131082666081020725623388609831999871293422821561354667625512514417816371066596945447670129629672549080027711000000000000000000000000",
+      "coordinate_movement": "4375408149318106538866216854671930632866706725794304565219529058788314814256861959639757/6400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "dimension": "246153846153846153846153600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000/246322131082666081020725623388609831999871293422821561354667625512514417816371066596945447670129629672549080027711",
+      "scalar_work": "1346279430559417396575940786747573332889326656949102227509004228821919709407178107938068846619221009326043276240329870370327450919972289/1970577048661328648165804987108878655998970347382572490837341004100115342530968532775563581361037037380392640221688000000000000000000000000"
+    },
+    "minimum_margin": "168284928819927174571777066478749333790272697198172744835625641220994994809711929319935155711813301482483082263329870370327450919972289/246322131082666081020725623388609831999871293422821561354667625512514417816371066596945447670129629672549080027711000000000000000000000000",
+    "parameters": {
+      "C0": {
+        "base": 2,
+        "exponent": 210000
+      },
+      "C1": 1,
+      "a_bit": "4375408149318106538866216854671930632866706725794304565219529058788314814256861959639757/6400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "a_complex": "700918443859411/1000000000000000000",
+      "alpha_squared_power": "336569857639854349143800455088581333661566120019734099503251153735412811180778526265382825841442974031563109974329870370327450919972289/492644262165332162041451246777219663999742586845643122709335251025028835632742133193890895340259259345098160055422000000000000000000000000",
+      "beta": "1/1000000000000000000000000",
+      "c": "2187704074659053269433904051927815998326814496680297610679131662687431612823865760290819711685185743138040360243/3200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "delta": "1/8000000000000000000000000",
+      "epsilon": "246153846153846153846153600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000/246322131082666081020725623388609831999871293422821561354667625512514417816371066596945447670129629672549080027711",
+      "eta": "1/1000000000000000000000000",
+      "kappa": "1366380910073/2000000000000000",
+      "lambda_": "6395624591850681893461133787520736218685239832140422550106711103807918410980047703259889301788314814256861959639757/6400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "lambda_prime": "3197812295925340946730566895948072184001673185503319702389320868337312568387176134239709180288314814256861959639757/3200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "q": "2187704074659053269433104051927815998326814496680297610679131662687431612823865760290819711685185743138040360243/3200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "sigma": "999299081556140589/1000000000000000000",
+      "tau": "6395624591850681893461133783145328069367133293274205695434780470941211685185743138040360243/6400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"
+    },
+    "recurrence": {
+      "internal": "6395624591850681893461133783145328069367133293274205695434780470941211685185743138040360243/6400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "leaf": "999299081556140589000000000700918443859411/1000000000000000000000000000000000000000000",
+      "reservations": "3197812295925340946730566095948072184001673185503319702389320868337312568387176134239709180288314814256861959639757/3200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"
+    },
+    "strict_constraints": {
+      "Gaussian_arithmetic_above_kappa": "214850777155596201368600808930470355522283178903230270162019178396787203748294034208300043037326711738979101128519972289/394115409732265729633160997421775731199794069476514498167468200820023068506193706555112716272207407476078528044337600000000000000000000000",
+      "K_dominates_log": "168284928819927174571838604940287795328734235598172744835625641220994994809711929319935155711813301482483082263329870370327450919972289/246322131082666081020725623388609831999871293422821561354667625512514417816371066596945447670129629672549080027711000000000000000000000000",
+      "K_geometry": "184783669544204542559187223388609831999871293422821561354667625512514417816371066596945447670129629672549080027711/246322131082666081020725623388609831999871293422821561354667625512514417816371066596945447670129629672549080027711000000000000000000000000",
+      "alpha_below_one": "492307692307692307692307446322131082666081020725623388609831999871293422821561354667625512514417816371066596945447670129629672549080027711/492644262165332162041451246777219663999742586845643122709335251025028835632742133193890895340259259345098160055422000000000000000000000000",
+      "alpha_below_one_fourth": "122824495683693186161219011239216334666274080591391046577830561602521796097004754772207341009223371862242976903881170129629672549080027711/492644262165332162041451246777219663999742586845643122709335251025028835632742133193890895340259259345098160055422000000000000000000000000",
+      "alpha_positive": "336569857639854349143800455088581333661566120019734099503251153735412811180778526265382825841442974031563109974329870370327450919972289/492644262165332162041451246777219663999742586845643122709335251025028835632742133193890895340259259345098160055422000000000000000000000000",
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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/frames/opframe-bases.json b/research/coordinated-crossover-pr200/frames/opframe-bases.json
new file mode 100644
index 0000000000..dfbfb9f407
--- /dev/null
+++ b/research/coordinated-crossover-pr200/frames/opframe-bases.json
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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/geometry/BANK-PROOF-PR200.md b/research/coordinated-crossover-pr200/geometry/BANK-PROOF-PR200.md
new file mode 100644
index 0000000000..5f4075b8a2
--- /dev/null
+++ b/research/coordinated-crossover-pr200/geometry/BANK-PROOF-PR200.md
@@ -0,0 +1,57 @@
+# 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/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/verify.py b/research/coordinated-crossover-pr200/verify.py
new file mode 100644
index 0000000000..56885a8401
--- /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 manual-upload package; no network, git, or publication 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..ce3a8630e1
--- /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']}
+# Every source, target and dirty-register formal column, exact frames and all primes.
+run(HERE/'frames/verify-bit-frames.py','--package',ROOT/'research/paired-cube-diagonal-bit-168','--frames',HERE/'frames/opframe-bases.json','--out',HERE/'frames/checked','--full')
+run(HERE/'geometry/pack-admitted-frames.py','--candidate',HERE/'frames')
+run(HERE/'geometry/audit-gauge-charts.py');run(HERE/'geometry/check-normalizer-routing.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, three finite leaf levels, and unchanged 47 constraints.
+run(HERE/'arithmetic.py');assert read(HERE/'assembly.json')==read(HERE/'certificate.json'),'Derived final certificate differs'
+cert=read(HERE/'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/'geometry/PR200-GAUGE-CHART-AUDIT.json');assert chart['conservative_extra_selector_calls']==21900544398<2**40
+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