diff --git a/.gitattributes b/.gitattributes
index 41aef290fd..76e425cdaf 100644
--- a/.gitattributes
+++ b/.gitattributes
@@ -8,3 +8,8 @@ upstream/** -whitespace
 references/copied-centers/pr29/** -whitespace
 references/copied-centers/pr33/** -whitespace
 references/semantic-bulk/** -whitespace
+
+# Preserve the exact bytes of pinned upstream proof and witness snapshots.
+references/partial-gauge/pr97/** whitespace=-blank-at-eof
+# Preserve the exact pinned PR117 source bytes and legal notices.
+references/three-stage-cover/pr117/** -whitespace
diff --git a/Makefile b/Makefile
index 84fe8290f3..3c154f295d 100644
--- a/Makefile
+++ b/Makefile
@@ -15,6 +15,8 @@ community-followup-check:
 verify:
 	$(MAKE) verify-community
 	$(MAKE) verify-producers
+	$(MAKE) verify-partial-gauge
+	$(MAKE) verify-three-stage-cover
 	$(MAKE) verify-certificates
 	$(MAKE) verify-ternary
 	$(MAKE) verify-research
@@ -30,6 +32,7 @@ verify-community: community-audit-check community-followup-check copied-reversed
 	$(MAKE) climbed-48-verify
 
 verify-producers:
+	$(MAKE) stopped-product-verify
 	$(MAKE) copied-centers-verify
 	$(MAKE) structured-bulk-verify
 	$(MAKE) endpoint-gauge-producer endpoint-gauge-certificate
@@ -428,3 +431,38 @@ pair-assembly-verify: pair-assembly-producer pair-assembly-check
 .PHONY: verify-pair
 verify-pair: pair-assembly-verify
 	python3 scripts/audit_pair_candidate.py --check docs/research/community-pair-arithmetic.json
+
+.PHONY: stopped-product-producer stopped-product-certificate stopped-product-verify
+stopped-product-producer:
+	python3 scripts/stopped_product_producer.py
+
+stopped-product-certificate:
+	python3 scripts/stopped_product_network.py
+
+stopped-product-verify: stopped-product-producer stopped-product-certificate
+
+.PHONY: partial-gauge-bit partial-gauge-producer partial-gauge-certificate partial-gauge-verify
+partial-gauge-bit:
+	python3 scripts/partial_gauge_bit.py
+
+partial-gauge-producer:
+	python3 scripts/partial_gauge_producer.py
+
+partial-gauge-certificate:
+	python3 scripts/partial_gauge_network.py
+
+partial-gauge-verify: partial-gauge-bit partial-gauge-producer partial-gauge-certificate
+
+.PHONY: verify-partial-gauge
+verify-partial-gauge: partial-gauge-verify
+
+.PHONY: three-stage-cover-producer three-stage-cover-certificate three-stage-cover-verify verify-three-stage-cover
+three-stage-cover-producer:
+	python3 scripts/three_stage_cover_producer.py
+
+three-stage-cover-certificate:
+	python3 scripts/three_stage_cover_network.py
+
+three-stage-cover-verify: three-stage-cover-producer three-stage-cover-certificate
+
+verify-three-stage-cover: three-stage-cover-verify
diff --git a/NOTICE b/NOTICE
index b565637758..c719bbed04 100644
--- a/NOTICE
+++ b/NOTICE
@@ -302,3 +302,60 @@ 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/.
diff --git a/PATCHING.md b/PATCHING.md
index b60f3f101b..a02e230256 100644
--- a/PATCHING.md
+++ b/PATCHING.md
@@ -1,6 +1,6 @@
 # Applying the inherited manuscript patch
 
-This patch is the PR #10 baseline. The current copied-center extension is
+This patch is the PR #10 baseline. The PR #36 copied-center extension is
 presented in `notes/copied-centers-note.tex`; it is not included
 in this historical combined manuscript patch.
 
@@ -30,3 +30,6 @@ alternatives for the same base. PR7 is pinned at
 `6725c6a17b17871a35353fd29157f4ed851bc114` for the retained producer sources.
 The old uniform recurrences keep their original exponents. The active consumers
 use `lem:batched-chunk-swap` and `prop:batched-simultaneous-layer`.
+
+The stopped product-ring extension is supplied separately in
+`notes/stopped-product-note.tex`; it does not regenerate this historical patch.
diff --git a/README.md b/README.md
index 035c259b68..9aff3ee61b 100644
--- a/README.md
+++ b/README.md
@@ -3,6 +3,87 @@
 **Community research maintained by Douglas Colkitt — conditional on the original
 OpenAI #109 framework.**
 
+## Three-stage cover extension
+
+The construction contributed by **icekylinx**, extending
+[PR #115](https://github.com/CrocSwap/integer-mult-bounds/pull/115), gives
+
+$$
+T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad
+\kappa=\frac{3146011}{10000000000}=3.146011\times10^{-4}.
+$$
+
+Three signed shears on a regular Cayley cover align all interstage data
+frames. The complex branch combines eumemic's PR #117 local DAG with
+arbitrary-subspace Clifford frames; the bit branch retains the PR #97 /
+Swapnil physical word and uses a batched weighted q-adic cover. Local
+transitions, copied centers, rare-class fallback, finite routers and
+internal row borrowing are charged. The retained analytic and fixed-tape
+hypotheses still apply.
+
+[Proof source](notes/three-stage-cover-note.tex) ·
+[Exact certificate](certificates/three-stage-cover-network.json) ·
+[Incremental reproduction](docs/three-stage-cover.md)
+
+```sh
+make three-stage-cover-verify
+```
+
+## Partial-gauge extension
+
+The construction contributed by **icekylinx**, extending
+[PR #104](https://github.com/CrocSwap/integer-mult-bounds/pull/104), gives
+
+$$
+T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad
+\kappa=\frac{7237}{78125000}=9.26336\times10^{-5}.
+$$
+
+It applies the stopped whole-projector bit interface to the physical deferred
+word of Zhihao Chen's PR #97, based on Swapnil Jain's witness. The new complex
+producer combines cyclic interval contractions, pair-first cube assembly,
+compatible frame enlargement and partial source gauges. Every residual,
+endpoint correction and target-data transition is charged. The bound retains
+the analytic and fixed-tape hypotheses of the preceding construction.
+
+[Proof source](notes/partial-gauge-note.tex) ·
+[Exact certificate](certificates/partial-gauge-network.json) ·
+[Incremental reproduction](docs/partial-gauge.md)
+
+```sh
+make partial-gauge-verify
+```
+
+## Stopped product-ring extension
+
+The new construction contributed by **icekylinx**, extending merged
+[PR #36](https://github.com/CrocSwap/integer-mult-bounds/pull/36), gives
+
+$$
+T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad
+\kappa=\frac{194869}{2500000000}=7.79476\times10^{-5}.
+$$
+
+It combines a stopped product-ring bit interchange on `(23,23)` with an
+all-disjoint rational-center complex network on `(24,24)`. The new generic
+opposite-bank factorization pays one reversed child per projector rank;
+atom adapters, ordinary leaves, endpoint copies and the exact denominator-21
+grid are included in the proof. The bound retains the original analytic
+and fixed-tape hypotheses.
+
+[Proof source](notes/stopped-product-note.tex) ·
+[Exact certificate](certificates/stopped-product-network.json) ·
+[Incremental reproduction](docs/stopped-product.md)
+
+```sh
+make stopped-product-verify
+```
+
+The maintainer-reviewed community checkpoint below remains its own result
+and validation record.
+
+## Reviewed community checkpoint
+
 The reviewed community witness gives
 
 $$
diff --git a/SOURCES.json b/SOURCES.json
index 6096d5aff9..ab7ec8e74c 100644
--- a/SOURCES.json
+++ b/SOURCES.json
@@ -234,5 +234,190 @@
       }
     ],
     "continuing_analytic_dependencies": "See structured_bulk_extension.adopted and references/semantic-bulk/MANIFEST.json."
+  },
+  "stopped_product_extension": {
+    "contributor": "icekylinx",
+    "base_commit": "56b66d58297deca1d7dd130247d720e960f77a37",
+    "proof_predecessor_PR": "https://github.com/CrocSwap/integer-mult-bounds/pull/36",
+    "proof_predecessor_commit": "11817ccacb564bb7f98789c20dc11d3fece207e3",
+    "predecessor_merged": true,
+    "archive": "integer_mult_round5_kappa_7_79476e-5_complete.zip",
+    "archive_sha256": "714cbffa0cccba3c18f046a63e62d6981683cb386b5d5cd09761be7b70f0b0be",
+    "kappa": "194869/2500000000",
+    "proof": "notes/stopped-product-note.tex",
+    "certificate": "certificates/stopped-product-network.json",
+    "producer": "scripts/stopped_product_producer.py",
+    "assistance": "Substantial OpenAI GPT-6 Astra and Codex assistance.",
+    "scope": "Conditional on the retained analytic and fixed-tape framework; new product-ring and exact rational-grid arguments supplied as written proofs.",
+    "continuing_dependencies": [
+      "copied_centers_extension",
+      "structured_bulk_extension"
+    ],
+    "icekylinx_lineage": [
+      {
+        "pr": 10,
+        "commit": "62691e395a0458ce089a1c7b5d89e74291e95e29"
+      },
+      {
+        "pr": 18,
+        "commit": "f2ab41aebad47861caf6316282c1793e5513845e"
+      },
+      {
+        "pr": 24,
+        "commit": "ed90fd940279c336ebc45968631bdebfb087b505"
+      },
+      {
+        "pr": 32,
+        "commit": "0ef3aeb61f55cc0b321ce6a0ef00acee25cefe52"
+      },
+      {
+        "pr": 36,
+        "commit": "11817ccacb564bb7f98789c20dc11d3fece207e3"
+      }
+    ],
+    "curation": "Selected h23/h24 inputs, generator, exact certificate and proofs only; excludes raw graphs, research alternatives and recursive predecessor copies."
+  },
+  "partial_gauge_extension": {
+    "contributor": "icekylinx",
+    "predecessor_pr": 104,
+    "predecessor_commit": "948ce1510df750f4c18b96bdaef436a86f8bf834",
+    "archive": "integer_mult_round6_kappa_9_26336e-5_complete.zip",
+    "archive_sha256": "e701a98b10864dc4b1414d9fcb1fc7c67b28e2363cb2a2292f9fed577a081d2d",
+    "kappa": "7237/78125000",
+    "proof": "notes/partial-gauge-note.tex",
+    "certificate": "certificates/partial-gauge-network.json",
+    "producer": "scripts/partial_gauge_producer.py",
+    "assistance": "Substantial OpenAI GPT-6 Astra and Codex assistance.",
+    "adopted": [
+      {
+        "pr": 97,
+        "contributor": "Zhihao Chen (jacklightChen)",
+        "commit": "e15350b66e4bf796153a68c240a6cc1b788abe3e",
+        "role": "Actual deferred bit word, reflected physical ledger, fan-order clarification, gauged exits and signed endpoint correction"
+      },
+      {
+        "contributor": "Swapnil Jain",
+        "repository": "Swapnil-jain/integer-mult-kappa",
+        "commit": "741e7aa078392553815df7926ee17ac5e25a8c38",
+        "role": "Underlying deferred readouts, V leaves, lifted frames and frozen witness"
+      },
+      {
+        "pr": 62,
+        "contributor": "Avi Eisenberg (ikeboy)",
+        "commit": "ad0f25ff7b23cff7f08ad237c2254e6ecf74257e",
+        "role": "Cyclic interval and core-aware pair primitives"
+      }
+    ],
+    "icekylinx_lineage": [
+      {
+        "pr": 10,
+        "commit": "62691e395a0458ce089a1c7b5d89e74291e95e29"
+      },
+      {
+        "pr": 18,
+        "commit": "f2ab41aebad47861caf6316282c1793e5513845e"
+      },
+      {
+        "pr": 24,
+        "commit": "ed90fd940279c336ebc45968631bdebfb087b505"
+      },
+      {
+        "pr": 32,
+        "commit": "0ef3aeb61f55cc0b321ce6a0ef00acee25cefe52"
+      },
+      {
+        "pr": 36,
+        "commit": "11817ccacb564bb7f98789c20dc11d3fece207e3"
+      },
+      {
+        "pr": 104,
+        "commit": "948ce1510df750f4c18b96bdaef436a86f8bf834"
+      }
+    ],
+    "continuing_dependencies": [
+      "stopped_product_extension",
+      "copied_centers_extension",
+      "structured_bulk_extension"
+    ],
+    "curation": "Selected generators, exact counts, proof sources and pinned dependencies; excludes alternatives, large generated graphs and recursive handoff copies.",
+    "scope": "Conditional on retained analytic and fixed-tape framework; finite verification and written proofs have separate scopes."
+  },
+  "three_stage_cover_extension": {
+    "contributor": "icekylinx",
+    "predecessor_pr": 115,
+    "predecessor_commit": "cf60442983c153a3a0f5272ea4713f99eba13c86",
+    "archives": [
+      {
+        "filename": "integer_mult_round7_kappa_1_080522e-4_incremental.zip",
+        "sha256": "ba7a7d10be57f614fdb4df4074ec5ffd2feddeac583f1fdf2da12ae7c02f6058"
+      },
+      {
+        "filename": "integer_mult_round8_kappa_3_146011e-4_incremental.zip",
+        "sha256": "8e730754c2076bf3ee7ff1bd320486172ef6ab1c9897ddd633aef02a9ce5257a"
+      }
+    ],
+    "kappa": "3146011/10000000000",
+    "proof": "notes/three-stage-cover-note.tex",
+    "general_subspace_proof": "notes/general-clifford-frames.tex",
+    "certificate": "certificates/three-stage-cover-network.json",
+    "producer": "scripts/three_stage_cover_producer.py",
+    "assistance": "Construction supplied with substantial OpenAI GPT-6 Astra assistance; repository integration with Codex assistance.",
+    "adopted": [
+      {
+        "pr": 117,
+        "contributor": "eumemic",
+        "commit": "cbb05ce504d571546d9b7794c186a613c659c3bf",
+        "role": "Pinned positive h24 local DAG and its exact support/label replay; original Claude disclosure retained",
+        "dag_sha256": "3c034d0aae388ef567a454826f4f48b26fd8a94c71e8ffed4835271b349a783b"
+      },
+      {
+        "pr": 97,
+        "contributor": "Zhihao Chen (jacklightChen)",
+        "commit": "e15350b66e4bf796153a68c240a6cc1b788abe3e",
+        "role": "Retained physical deferred bit word and reflected ledger"
+      },
+      {
+        "contributor": "Swapnil Jain",
+        "commit": "741e7aa078392553815df7926ee17ac5e25a8c38",
+        "role": "Underlying deferred word, lifted frames and frozen witness"
+      }
+    ],
+    "icekylinx_lineage": [
+      {
+        "pr": 10,
+        "commit": "62691e395a0458ce089a1c7b5d89e74291e95e29"
+      },
+      {
+        "pr": 18,
+        "commit": "f2ab41aebad47861caf6316282c1793e5513845e"
+      },
+      {
+        "pr": 24,
+        "commit": "ed90fd940279c336ebc45968631bdebfb087b505"
+      },
+      {
+        "pr": 32,
+        "commit": "0ef3aeb61f55cc0b321ce6a0ef00acee25cefe52"
+      },
+      {
+        "pr": 36,
+        "commit": "11817ccacb564bb7f98789c20dc11d3fece207e3"
+      },
+      {
+        "pr": 104,
+        "commit": "948ce1510df750f4c18b96bdaef436a86f8bf834"
+      },
+      {
+        "pr": 115,
+        "commit": "cf60442983c153a3a0f5272ea4713f99eba13c86"
+      }
+    ],
+    "continuing_dependencies": [
+      "partial_gauge_extension",
+      "stopped_product_extension",
+      "structured_bulk_extension"
+    ],
+    "scope": "Conditional on retained analytic, precision, ordinary-leaf and fixed-tape contracts. New general Clifford frames, weighted routing, local-ring fallback and internal row borrowing supplied as written proofs.",
+    "curation": "Selected round7 PR117 input and frame theorem plus round8 three-stage cover only. Excludes PR112 reclaim, two-stage alternatives, half-sign motif, research screens, raw generated graphs and PDFs."
   }
 }
diff --git a/certificates/joint-dual-kappa.json b/certificates/joint-dual-kappa.json
index d92f653fe4..d721fd7b84 100644
--- a/certificates/joint-dual-kappa.json
+++ b/certificates/joint-dual-kappa.json
@@ -842,7 +842,7 @@
     "scripts/experiments/pin_joint_dual_sources.py": "4d31cc0e38b7902709ca5f8f714cba109760d2592346dd7f8dc0b900a2d9fe00",
     "scripts/experiments/verify_joint_dual.py": "7a31bba1e9c0ded32ae285df2ca9bc91ef20c330f6789fdba361876f3c7a1985"
   },
-  "source_manifest_sha256": "fd1aab7d75753b4b55b2377f26119dae74f367b0b4fab66cba42d944b3042cc9",
+  "source_manifest_sha256": "f9a5b92f86b1eee61cb5f867b7a4097c5d4b91c33e618804754cd259f187625a",
   "sources": [
     {
       "files": {
diff --git a/certificates/joint-dual-validation.json b/certificates/joint-dual-validation.json
index 99adb4cf74..50fbc39e9a 100644
--- a/certificates/joint-dual-validation.json
+++ b/certificates/joint-dual-validation.json
@@ -226,7 +226,7 @@
     }
   },
   "bit_saving": "2382370177/50000000000000",
-  "certificate_sha256": "2ed658f05365b28825e24a6e7bac9fc9cc015d5a5d63d9579f4ac6e0640ae559",
+  "certificate_sha256": "ed18aa80841734755a141d59853b6fc5d8a0e164f0407e368537527480f7f3fe",
   "kappa": "4764513337/100000000000000",
   "margins": 7,
   "producer_words_regenerated_and_matched": true,
diff --git a/certificates/partial-gauge-bit-input.json b/certificates/partial-gauge-bit-input.json
new file mode 100644
index 0000000000..f01ac32417
--- /dev/null
+++ b/certificates/partial-gauge-bit-input.json
@@ -0,0 +1,142 @@
+{
+  "N": 3136441,
+  "R": 28866,
+  "W": 108516254,
+  "deficit": 1344189,
+  "dimensions": [
+    23,
+    23
+  ],
+  "exit_nullity_histogram": {
+    "1": 1760,
+    "10": 96,
+    "11": 335,
+    "12": 28,
+    "13": 116,
+    "14": 36,
+    "15": 319,
+    "16": 5,
+    "17": 91,
+    "18": 4,
+    "19": 164,
+    "2": 1596,
+    "20": 83,
+    "21": 49,
+    "22": 154,
+    "23": 17301,
+    "3": 1307,
+    "4": 3871,
+    "5": 806,
+    "6": 177,
+    "7": 447,
+    "8": 30,
+    "9": 91
+  },
+  "local_rank_histograms": {
+    "aux": {
+      "1": 55349,
+      "10": 162,
+      "11": 299,
+      "12": 208,
+      "13": 202,
+      "14": 38,
+      "15": 649,
+      "16": 208,
+      "17": 1309,
+      "18": 599,
+      "19": 533,
+      "2": 64873,
+      "20": 2310,
+      "21": 413,
+      "3": 13926,
+      "4": 11248,
+      "5": 6151,
+      "6": 1617,
+      "7": 883,
+      "8": 1243,
+      "9": 458
+    },
+    "center": {
+      "22": 23
+    },
+    "data": {
+      "1": 11386,
+      "10": 86,
+      "11": 93,
+      "12": 97,
+      "13": 65,
+      "14": 26,
+      "15": 43,
+      "16": 218,
+      "17": 38,
+      "18": 8,
+      "19": 1547,
+      "2": 2402,
+      "21": 1,
+      "22": 2,
+      "3": 1144,
+      "4": 2013,
+      "5": 521,
+      "6": 475,
+      "7": 276,
+      "8": 382,
+      "9": 130
+    }
+  },
+  "m": 529,
+  "max_child": 528,
+  "rank_mass": 57403754177,
+  "schema": 1,
+  "source": "PR97 e15350b66e4bf796153a68c240a6cc1b788abe3e; opposite-bank whole-projector compiler applied to its actual physical idempotent increments.",
+  "source_manifest": "references/partial-gauge/pr97/SOURCE.json",
+  "source_manifest_sha256": "68fb539abcd6df21d142e4e204b6e9482eb7f9cd3907650f54bd058070266596",
+  "v": 1771,
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+}
diff --git a/certificates/partial-gauge-complex-input.json b/certificates/partial-gauge-complex-input.json
new file mode 100644
index 0000000000..b404923928
--- /dev/null
+++ b/certificates/partial-gauge-complex-input.json
@@ -0,0 +1,101 @@
+{
+  "h": 24,
+  "v": 2024,
+  "R": 34307,
+  "q": 8120,
+  "c": 146284,
+  "matched": 120097,
+  "loss": 552,
+  "selected_roles": 6341,
+  "selected_rank_histogram": {
+    "22": 850,
+    "21": 70,
+    "20": 770,
+    "19": 81,
+    "18": 1503,
+    "16": 761,
+    "14": 487,
+    "12": 406,
+    "10": 374,
+    "8": 257,
+    "6": 294,
+    "4": 252,
+    "2": 236
+  },
+  "remaining_internal_histogram": [
+    27033,
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+    83148,
+    13608,
+    17316,
+    6534,
+    5740,
+    2708,
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+    223,
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+    352,
+    1944,
+    305,
+    1566,
+    852,
+    188,
+    22,
+    24,
+    0
+  ],
+  "source_data_histogram": {
+    "23": 2024
+  },
+  "target_data_histogram": {
+    "23": 18,
+    "12": 188,
+    "11": 4,
+    "16": 82,
+    "2": 5576,
+    "3": 802,
+    "10": 239,
+    "6": 641,
+    "4": 3354,
+    "8": 634,
+    "1": 908,
+    "5": 141,
+    "7": 82,
+    "14": 89,
+    "9": 63,
+    "13": 2,
+    "17": 1,
+    "19": 3,
+    "18": 8
+  },
+  "copied_centers_already": true,
+  "phase1_operations": 32446,
+  "total_M_operations": 178567,
+  "untouched_non_source_roles": 8079,
+  "selection": "Descending first-frame rank; intersect all incident latest target frames; retain maximal nondegenerate summand",
+  "partial_first_transition_histogram": {
+    "0": 2241,
+    "1": 371,
+    "2": 1793,
+    "3": 221,
+    "4": 636,
+    "5": 150,
+    "7": 25,
+    "6": 534,
+    "8": 259,
+    "10": 73,
+    "12": 5,
+    "9": 29,
+    "13": 2,
+    "11": 2
+  },
+  "distinct_caps": 2278,
+  "repaired_caps": 369,
+  "scalar_readout": "Exact rational C=J M; Boolean support is a safe superset only",
+  "chronological_contract": "Nondeferred reads at0; source V; center-closed M phase1; copied centers at0; selected partial-gauge reads in reverse selection order; remaining M; side scatter; inverse M/V atfull F"
+}
diff --git a/certificates/partial-gauge-input.json b/certificates/partial-gauge-input.json
new file mode 100644
index 0000000000..158600c7e9
--- /dev/null
+++ b/certificates/partial-gauge-input.json
@@ -0,0 +1,170 @@
+{
+  "assembly": {
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+    "margins": {
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+      "bulk_exposure": "92650807349/1000000000000000",
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+      "coordinate_movement": "92650807349/1000000000000000",
+      "dimension": "68493150000000000000000000000/68505842576158799709657523187",
+      "original_prefix": "634663057688300803047390957523187/6850584257615879970965752318700000000",
+      "scalar_work": "10153992421197191567218839942476813/54804674060927039767726018549600000000"
+    },
+    "minimum_margin": "6345945518457246442476813/68505842576158799709657523187",
+    "parameters": {
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+      "eta": "1/100000000",
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+      "coordinate_movement_above_kappa": "17207349/1000000000000000",
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+      "guard_width": "12692576158799709657523187/68505842576158799709657523187",
+      "lambda_above_internal": "92650807349/100000000000000000000000",
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+      "lambda_prime_above_lambda": "92650807349/100000000000000000000000",
+      "lambda_prime_below_one": "4632540274799192651/50000000000000000000000",
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+      "row_product_gap": "196189/25",
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+      "small_field_exposure": "6346630640342463215046374/68505842576158799709657523187"
+    }
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+  "finite_bridge": {
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+      "halving_degree": 367,
+      "m": 529,
+      "maxchild": 528,
+      "wire_bits": 27
+    },
+    "complex": {
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+      "coefficient_bound": 28,
+      "coefficient_denominator_divides": 42,
+      "halving_degree": 200,
+      "local_group_upper": 99202668726680,
+      "m": 576,
+      "maxchild": 574,
+      "s": 84709241408,
+      "scalar_group_upper": 401572403009697216,
+      "wire_bits": 28
+    },
+    "ordinary_leaf_row_degree": 252,
+    "rows": {
+      "coefficient": 15761,
+      "contract": "W_complex^D_complex * W_coarse^D_coarse * W_old^D_old; one preceding prefix and one padding; sequential reuse",
+      "degree": 40000,
+      "degree_gap": "196189/25",
+      "suffix_slope": 160000
+    },
+    "semantic": {
+      "B": 4144494358077648199899551022812574000922568626809480832,
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+      "C1": 1,
+      "E": 4144494358077648199899551022812574000922568542100239424,
+      "exact_grid": "One common dyadic grid times the fixed odd divisor to K=G*(D_complex+1); no child rounding",
+      "fixed_odd_divisor": 21,
+      "induction_gap": 12433483074232944599698653068437722002767705880428442496,
+      "literal_charge": 17371189839801017404792510229752260,
+      "strict_literal_gap": 4144494358077648199882179832972772983517776031870487164
+    }
+  }
+}
diff --git a/certificates/partial-gauge-network.json b/certificates/partial-gauge-network.json
new file mode 100644
index 0000000000..fae9fef678
--- /dev/null
+++ b/certificates/partial-gauge-network.json
@@ -0,0 +1,557 @@
+{
+  "assembly": {
+    "absorption_gap": "210905811144819709520320681/5352018951262406227316993998984375000",
+    "margins": {
+      "Gaussian_arithmetic": "7615340165060206281200290342476813/54804674060927039767726018549600000000",
+      "bulk_exposure": "92650807349/1000000000000000",
+      "compact_phase_layer": "6345945518457246442476813/68505842576158799709657523187",
+      "coordinate_movement": "92650807349/1000000000000000",
+      "dimension": "68493150000000000000000000000/68505842576158799709657523187",
+      "original_prefix": "634663057688300803047390957523187/6850584257615879970965752318700000000",
+      "scalar_work": "10153992421197191567218839942476813/54804674060927039767726018549600000000"
+    },
+    "minimum_margin": "6345945518457246442476813/68505842576158799709657523187",
+    "parameters": {
+      "C0": 316603394779621611419893923923079016464961761204405165715149703864163276249149334956086075455540666229511598112768,
+      "C1": 1,
+      "a_bit": "92650807349/1000000000000000",
+      "a_complex": "92651/1000000000",
+      "actual_bit_saving": "1240183559/10000000000000",
+      "alpha_squared_power": "9519260838628478050000000/68505842576158799709657523187",
+      "beta": "1/1000000",
+      "c": "463254032112459539899192651/5000000000000000000000000000000",
+      "delta": "1/800000000",
+      "epsilon": "68493150000000000000000000000/68505842576158799709657523187",
+      "eta": "1/100000000",
+      "kappa": "7237/78125000",
+      "lambda_": "99990734919357750807349/100000000000000000000000",
+      "lambda_prime": "49995367459725200807349/50000000000000000000000",
+      "q": "4632540274799192651/50000000000000000000000",
+      "sigma": "999907349/1000000000",
+      "tau": "999907349192651/1000000000000000"
+    },
+    "recurrence": {
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+      "leaf": "999907349092651/1000000000000000",
+      "reservations": "4999536745967887540460100807349/5000000000000000000000000000000"
+    },
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+      "K_dominates_log": "634594558191670162704927742476813/6850584257615879970965752318700000000",
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+      "alpha_below_one": "68496323315320171231607523187/68505842576158799709657523187",
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+      "lambda_above_internal": "92650807349/100000000000000000000000",
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+      "lambda_above_tau": "92650807349/100000000000000000000000",
+      "lambda_prime_above_lambda": "92650807349/100000000000000000000000",
+      "lambda_prime_below_one": "4632540274799192651/50000000000000000000000",
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+      "record_suffix": "12692576158799709657523187/68505842576158799709657523187",
+      "row_product_gap": "196189/25",
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+      "short_record_fallback": "68486802878377195389187669443684432762498737/68505842576158799709657523187000000000000000",
+      "small_field_exposure": "6346630640342463215046374/68505842576158799709657523187"
+    }
+  },
+  "bit": {
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+}
diff --git a/certificates/stopped-product-bit-axis.json b/certificates/stopped-product-bit-axis.json
new file mode 100644
index 0000000000..5824891de9
--- /dev/null
+++ b/certificates/stopped-product-bit-axis.json
@@ -0,0 +1,38 @@
+{
+  "h": 23,
+  "v": 1771,
+  "c": 34764,
+  "q": 5336,
+  "baseline_R": 40100,
+  "matching": 1324,
+  "R": 38776,
+  "loss": 506,
+  "histogram": [
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+    3096,
+    2344,
+    3628,
+    9435,
+    2110,
+    23,
+    23
+  ],
+  "label_source": "positive",
+  "description": "Retained-total aligned triple producer, base-two recursion, original maximum carrier matching, and compatible positive labels."
+}
diff --git a/certificates/stopped-product-complex-input.json b/certificates/stopped-product-complex-input.json
new file mode 100644
index 0000000000..7b1d99c518
--- /dev/null
+++ b/certificates/stopped-product-complex-input.json
@@ -0,0 +1,41 @@
+{
+  "h": 24,
+  "v": 2024,
+  "c": 45764,
+  "q": 8120,
+  "baseline_R": 53884,
+  "matched": 8966,
+  "R": 44918,
+  "orientation_changes": 7338,
+  "rank_sum": 1079136,
+  "loss": 552,
+  "histogram": [
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+  ],
+  "central_disjoint": 24,
+  "center_denominator": 21
+}
diff --git a/certificates/stopped-product-input.json b/certificates/stopped-product-input.json
new file mode 100644
index 0000000000..f0a910a495
--- /dev/null
+++ b/certificates/stopped-product-input.json
@@ -0,0 +1,253 @@
+{
+  "bit": {
+    "dimensions": [
+      23,
+      23
+    ],
+    "m": 529,
+    "N": 3136441,
+    "W": 143617474,
+    "L": 1792252,
+    "total_rank": 75972299557,
+    "deficit": 1344189,
+    "maxchild": 506,
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+      "20": 33418770,
+      "21": 7473620,
+      "22": 12627230,
+      "484": 6272882,
+      "506": 137344592
+    }
+  },
+  "complex": {
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+    ],
+    "m": 576,
+    "N": 4096576,
+    "B1": 90914032,
+    "B2": 90914032,
+    "W": 190021216,
+    "L": 2234496,
+    "total_rank": 109450358336,
+    "deficit": 1862080,
+    "child_multiplicities": {
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+      "20": 16418688,
+      "21": 1068672,
+      "22": 2283072,
+      "23": 16483456
+    },
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+  },
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+      "wire_bits": 28
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+    "ordinary_leaf_row_degree": 252,
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+          "invocations": 2024,
+          "local_group_upper": 408200,
+          "copied_center_extra_groups": 48
+        },
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+        }
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+      "scalar_group_upper": 1656684480
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+      "induction_gap": 18943983431358232713000942240128,
+      "fixed_odd_divisor": 21,
+      "exact_grid": "One common dyadic grid times the fixed odd divisor to K=G*(D_complex+1); no child rounding"
+    },
+    "rows": {
+      "coefficient": 1176,
+      "degree": 4000,
+      "suffix_slope": 16000,
+      "degree_gap": "40024/25",
+      "contract": "W_complex^D_complex * W_coarse^D_coarse * W_old^D_old; one preceding prefix and one padding; sequential reuse"
+    }
+  },
+  "assembly": {
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+      "kappa": "194869/2500000000"
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+      "complex_below_one_over32": "779301/25000000",
+      "beta_positive": "1/1000000",
+      "beta_below_one": "999999/1000000",
+      "leaf_saving_above_bit": "1/10000000000",
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+      "compact_leaf": "126948995551/1250000000000000000000",
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+      "epsilon_positive": "124999998750000000000000000000/125019489955217650665401004449",
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+      "guard_width": "19491205217650665401004449/125019489955217650665401004449",
+      "K_geometry": "974622765755022147550220501004449/12501948995521765066540100444900000000",
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+      "small_field_exposure": "9746227754999996102008898/125019489955217650665401004449",
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+      "row_product_gap": "40024/25",
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+      "dimension_above_kappa": "312475634452011916691632483971664027819/312548724888044126663502511122500000000"
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+      "scalar_work": "15592839154630577103152893798995551/100015591964174120532320803559200000000",
+      "dimension": "124999998750000000000000000000/125019489955217650665401004449"
+    },
+    "minimum_margin": "9744977462650669298995551/125019489955217650665401004449",
+    "absorption_gap": "20668543364879972849164027819/312548724888044126663502511122500000000",
+    "recurrence": {
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+      "leaf": "24998051001949/25000000000000",
+      "reservations": "124990255022342449779498995551/125000000000000000000000000000"
+    }
+  }
+}
diff --git a/certificates/stopped-product-network.json b/certificates/stopped-product-network.json
new file mode 100644
index 0000000000..998a8a7ea2
--- /dev/null
+++ b/certificates/stopped-product-network.json
@@ -0,0 +1,482 @@
+{
+  "assembly": {
+    "absorption_gap": "20668543364879972849164027819/312548724888044126663502511122500000000",
+    "margins": {
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+      "bulk_exposure": "1948995551/25000000000000",
+      "compact_phase_layer": "9744977462650669298995551/125019489955217650665401004449",
+      "coordinate_movement": "1948995551/25000000000000",
+      "dimension": "124999998750000000000000000000/125019489955217650665401004449",
+      "original_prefix": "974622765755022147550220501004449/12501948995521765066540100444900000000",
+      "scalar_work": "15592839154630577103152893798995551/100015591964174120532320803559200000000"
+    },
+    "minimum_margin": "9744977462650669298995551/125019489955217650665401004449",
+    "parameters": {
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+      "C1": 1,
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+      "actual_bit_saving": "803380799/10000000000000",
+      "alpha_squared_power": "14618091340150667350000000/125019489955217650665401004449",
+      "beta": "1/1000000",
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+      "delta": "1/800000000",
+      "epsilon": "124999998750000000000000000000/125019489955217650665401004449",
+      "eta": "1/100000000",
+      "kappa": "194869/2500000000",
+      "lambda_": "2499805100446848995551/2500000000000000000000",
+      "lambda_prime": "1249902550224398995551/1250000000000000000000",
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+      "sigma": "24998051/25000000",
+      "tau": "24998051004449/25000000000000"
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+      "leaf": "24998051001949/25000000000000",
+      "reservations": "124990255022342449779498995551/125000000000000000000000000000"
+    },
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+      "K_dominates_log": "974497756010044392550224398995551/12501948995521765066540100444900000000",
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+      "alpha_below_one": "125004871863877499998051004449/125019489955217650665401004449",
+      "alpha_below_one_fourth": "124961017589857047996001004449/500077959820870602661604017796",
+      "alpha_positive": "14618091340150667350000000/125019489955217650665401004449",
+      "artificial_boundary": "700128635079475474301789581367398995551/100015591964174120532320803559200000000",
+      "beta_below_one": "999999/1000000",
+      "beta_positive": "1/1000000",
+      "bit_positive": "1948995551/25000000000000",
+      "bulk_exposure_above_kappa": "305551/25000000000000",
+      "c_below_one": "124990255022342449779498995551/125000000000000000000000000000",
+      "c_positive": "9744977657550220501004449/125000000000000000000000000000",
+      "cell_above_band": "124995125636122500001948995551/250038979910435301330802008898",
+      "compact_leaf": "126948995551/1250000000000000000000",
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+      "degree": 4000,
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+  },
+  "kappa": "194869/2500000000",
+  "proof_predecessor_commit": "11817ccacb564bb7f98789c20dc11d3fece207e3",
+  "scope": "Exact finite moments, stopped savings and assembly; the new complex producer is checked separately. Opposite-bank factorization, atom streaming, ordinary conversion, common exact grid and retained analytic/tape interfaces are written proof dependencies.",
+  "source_sha256": {
+    "certificates/copied-centers-bit-axes.json": "ed307cef595091b11ba8f8e6ff11302ad8db190227a2fbf34c232fd680206254",
+    "certificates/copied-centers-network.json": "7dbd675e42e22ac076280360f8750031851564bb7a56079e66236fee5d0a6cdf",
+    "certificates/stopped-product-bit-axis.json": "52f493a0019b7e4ce69a4c26bd3d35af5d3397461126abc4be44d1430ae8930d",
+    "certificates/stopped-product-complex-input.json": "5bcf00402bbfb768bd2fc0935f32ad329ee0780a3dfe5827bb010831d7909c4d",
+    "certificates/stopped-product-input.json": "8185c7bf8c307728f32484c5972af9a1b948ac5d0838e02f85c86cb438060417",
+    "notes/stopped-product-assembly.tex": "1c3c7147851f30f1ff97679cfcda157b1cbe2655b8766e34f1345f20c630eaec",
+    "notes/stopped-product-complex.tex": "4738089103a4b263b3b0ef458f5ef35519c696f1ea871ae79e508d3720f5870a",
+    "notes/stopped-product-factorization.tex": "70d5a4405e1be9c7b6cfb326bb4ffdf88e5585d1fb3d16c7bb5ad62ac421fae3",
+    "notes/stopped-product-interface.tex": "454a8f3924afc26e575b2a8cec6d0860397cb7f2dc431f3ba63fb8a7142cc96b",
+    "notes/stopped-product-note.tex": "486d14c599f4c2551c019f3358898f72e6ecafeb04ad590726ea69ed987fe025",
+    "scripts/copied_centers/physical.py": "837345d1f8753eab0e70f105b1d1bb0db965b993da037b1825ddef3672609d7d",
+    "scripts/copied_centers_network.py": "409ebfbd8a41c4c4d089aa0949fcf5a949415dc26f57440ffd48b7bd8881e6a9",
+    "scripts/partial_swap_network.py": "acdd81b4dea2815f8838a0c3ea8ac4968f2bc13fd7c99173473d865779096782",
+    "scripts/stopped_product/README.md": "6c1805dda45b01ed0ef420c674ec2cd0048fb9d7a1b015ab6cd837067fe66176",
+    "scripts/stopped_product/__init__.py": "7b794c22bccc00bb6ae8ba4d3379f469570565bbcb1a86090699133b8cf1b7d3",
+    "scripts/stopped_product/complex.py": "407650d56dceb5cb09d74928658cebed1bb42f4d7e28ca9e3fd36eea075f1ad0",
+    "scripts/stopped_product_network.py": "a8baaa87dd7679db4f2dd1a18a720397ee6f443262ad04fa26cf51b39909b74c",
+    "scripts/stopped_product_producer.py": "2c5c71399136ebcfebb8e5277317d5f334eb8df5716ad287391d0996fcc1492c",
+    "scripts/structured_bulk_assembly.py": "e99467e14211fba699878ed036e1e2362eb8e395dc22f9ecdcecc7fbf2cc736d"
+  },
+  "status": "Conditional stopped product-ring multiplication witness"
+}
diff --git a/certificates/three-stage-cover-bit-input.json b/certificates/three-stage-cover-bit-input.json
new file mode 100644
index 0000000000..1956ae0e06
--- /dev/null
+++ b/certificates/three-stage-cover-bit-input.json
@@ -0,0 +1,84 @@
+{
+  "R": 28866,
+  "center_loss": 506,
+  "description": "Unchanged exact physical local ranks from the previously adopted PR97 bit producer; the new construction changes ambient geometry, finite cover, and recursive weighted routing only.",
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+  "h": 23,
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+    "center": {
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+    }
+  },
+  "source": "PR97 e15350b66e4bf796153a68c240a6cc1b788abe3e; matched to source-bound physical paths by partial_gauge_bit.reconstruct.",
+  "v": 1771
+}
diff --git a/certificates/three-stage-cover-complex-input.json b/certificates/three-stage-cover-complex-input.json
new file mode 100644
index 0000000000..d1b89bcb71
--- /dev/null
+++ b/certificates/three-stage-cover-complex-input.json
@@ -0,0 +1,115 @@
+{
+  "h": 24,
+  "v": 2024,
+  "R": 28705,
+  "q": 8120,
+  "c": 91770,
+  "matched": 71185,
+  "loss": 552,
+  "selected_roles": 4599,
+  "selected_rank_histogram": {
+    "22": 120,
+    "21": 359,
+    "20": 794,
+    "19": 174,
+    "18": 381,
+    "17": 147,
+    "16": 409,
+    "15": 57,
+    "14": 157,
+    "12": 218,
+    "13": 57,
+    "11": 131,
+    "10": 190,
+    "7": 100,
+    "8": 187,
+    "9": 113,
+    "5": 126,
+    "6": 137,
+    "4": 126,
+    "3": 181,
+    "2": 184,
+    "1": 251
+  },
+  "remaining_internal_histogram": [
+    26680,
+    69592,
+    67235,
+    29094,
+    30564,
+    11421,
+    4563,
+    2083,
+    2592,
+    1678,
+    937,
+    710,
+    794,
+    435,
+    479,
+    172,
+    281,
+    197,
+    755,
+    540,
+    85,
+    0,
+    183,
+    24,
+    0
+  ],
+  "source_data_histogram": {
+    "23": 2024
+  },
+  "target_data_histogram": {
+    "1": 1907,
+    "3": 1772,
+    "18": 23,
+    "4": 1500,
+    "15": 36,
+    "13": 70,
+    "2": 3107,
+    "11": 148,
+    "6": 695,
+    "5": 876,
+    "12": 168,
+    "19": 20,
+    "14": 81,
+    "9": 235,
+    "7": 426,
+    "10": 202,
+    "23": 14,
+    "8": 269,
+    "17": 14,
+    "16": 79,
+    "22": 4,
+    "21": 2,
+    "20": 16
+  },
+  "copied_centers_already": true,
+  "phase1_operations": 30251,
+  "total_M_operations": 118451,
+  "untouched_non_source_roles": 6401,
+  "selection": "Full unrestricted intersections and source gauges; descending first-frame rank, then reached-target count, then role index; general_clifford_subspace_frames.txt proves all required ranks",
+  "partial_first_transition_histogram": {
+    "0": 1425,
+    "1": 572,
+    "2": 536,
+    "3": 176,
+    "4": 274,
+    "6": 214,
+    "5": 205,
+    "7": 150,
+    "9": 244,
+    "8": 170,
+    "11": 185,
+    "10": 178,
+    "12": 157,
+    "13": 109,
+    "14": 4
+  },
+  "distinct_caps": 1792,
+  "repaired_caps": 0,
+  "scalar_readout": "Exact rational C=J M; Boolean support is a safe superset only",
+  "chronological_contract": "Nondeferred reads at0; source V; center-closed M phase1; copied centers at0; selected partial-gauge reads in reverse selection order; remaining M; side scatter; inverse M/V atfull F"
+}
\ No newline at end of file
diff --git a/certificates/three-stage-cover-input.json b/certificates/three-stage-cover-input.json
new file mode 100644
index 0000000000..4277c8ee2c
--- /dev/null
+++ b/certificates/three-stage-cover-input.json
@@ -0,0 +1,105 @@
+{
+  "absorption_gap": "427438661618132345110228739011/125078699795119603047628520369990000000000",
+  "assembly_minimum": "3934989696955306336847963001/12507869979511960304762852036999",
+  "bit_child_multiplicities": {
+    "1": 200205,
+    "10": 744,
+    "11": 1176,
+    "12": 915,
+    "13": 801,
+    "14": 192,
+    "15": 2076,
+    "16": 1278,
+    "17": 4041,
+    "18": 1821,
+    "19": 6240,
+    "2": 201825,
+    "20": 6930,
+    "21": 1242,
+    "22": 75,
+    "3": 45210,
+    "4": 39783,
+    "44": 51903,
+    "45": 462,
+    "46": 147,
+    "47": 249,
+    "48": 492,
+    "49": 12,
+    "5": 20016,
+    "50": 273,
+    "51": 15,
+    "52": 957,
+    "53": 108,
+    "54": 348,
+    "55": 84,
+    "56": 1005,
+    "57": 288,
+    "58": 273,
+    "59": 90,
+    "6": 6276,
+    "60": 1341,
+    "61": 531,
+    "62": 2418,
+    "63": 11613,
+    "64": 3921,
+    "65": 4788,
+    "66": 5280,
+    "7": 3477,
+    "8": 4875,
+    "9": 1764
+  },
+  "bit_moment_gap": "9940153528978648563021016615239768808198774866649/15679126900749033641597254460382631690240000000000000000",
+  "complex_child_multiplicities": {
+    "1": 214497,
+    "10": 3417,
+    "11": 2574,
+    "12": 2886,
+    "13": 1515,
+    "14": 1680,
+    "15": 624,
+    "16": 1080,
+    "17": 633,
+    "18": 2334,
+    "19": 1680,
+    "2": 211026,
+    "20": 303,
+    "21": 6,
+    "22": 561,
+    "23": 6186,
+    "3": 92598,
+    "4": 96192,
+    "46": 72318,
+    "47": 753,
+    "48": 552,
+    "49": 543,
+    "5": 36891,
+    "50": 378,
+    "51": 378,
+    "52": 411,
+    "53": 300,
+    "54": 561,
+    "55": 339,
+    "56": 570,
+    "57": 393,
+    "58": 654,
+    "59": 171,
+    "6": 15774,
+    "60": 471,
+    "61": 171,
+    "62": 1227,
+    "63": 441,
+    "64": 1143,
+    "65": 522,
+    "66": 2382,
+    "67": 1077,
+    "68": 360,
+    "7": 7527,
+    "8": 8583,
+    "9": 5739
+  },
+  "complex_group_order": 6689091708829555045758824829998182442380949567092062603242444969909078551334138157882363702009196947404919488369449744746991926866237084072342349052621959272314109341860982597079245951051060915136612492730495464103716652777049004952774932308881916158490684984763171132527532852346752575317647723807415492585356459791712931903073469266911277499226776278127225601421860692008492054366318924455033501842970829052238301059912292523870500029776308231873343329850749563566739947724729692531852159045336304497880188189155544256335559916592125266647930688623519281982850918244666405377532274750402883540845161453986633761588402168669904233211290018920284019914861652664495645151328501875434571709289695239534437990400000000000000000000,
+  "complex_moment_gap": "59660505965587856912406691094331138186546269153644104866458597552665571068577116382634727949704110939016528739306556665498819711235700737563102396203975205221339133504724694432491851145335328474764447111340057237370263560869716809394705046078151888885896465104252464291177722544058434227599509153650009184180706030887062493946461013321850105545236094560014790613346572797863224674797577418235580531379991780630697040554957397988009682182045601295362730851922874373402414598770202370029517008428953007230108852814349067642665708147118783306275789878773811811445256495511373788061555564732214081272243700795208443535449470081033281653673439322265496376162021685578206559390236098486450856162033470234056213985410270945906907471137066433711096177567495215981265605045461916455727272303712611652753551670063284645382934158696277492599602169286405620938102814288075749650451212554459344538961569338671639261965819931295434033557288521941097120781270873/258864312882296500031055289607308876918948194191739080393665208773437327950361705506204269262727624369607191381066070736950721387722343455305319397218060826215806055566676458467985854943815840460405474061110154037491708121904679791617548836252416330123113259519318454367469789088759143697885963048633822970601575393057295335974413285977207878402333467407939617985467006958820688402850530703846606121734476356557987835722081138227469842662438908972992927393171772790668096312894309081073575846448722623251554492048017857602287936132027784462360677641358992824901206117953947391105954258456029157280133485124255285284951748169636791141814335580522461373398332705279009190357387940809475590086743023351702184232232736981858998991300824787343216349598086601789879924718839538480440090437137911672142043373939595769029469714536428078367503524111596722215987131151839686520104495974436734700813521784031923667417309936799035132679343460868275000000000000000000000",
+  "kappa": "3146011/10000000000",
+  "row_coefficient": 68505,
+  "row_degree": 160000
+}
diff --git a/certificates/three-stage-cover-network.json b/certificates/three-stage-cover-network.json
new file mode 100644
index 0000000000..5fcdaf7420
--- /dev/null
+++ b/certificates/three-stage-cover-network.json
@@ -0,0 +1,592 @@
+{
+  "assembly": {
+    "absorption_gap": "427438661618132345110228739011/125078699795119603047628520369990000000000",
+    "margins": {
+      "Gaussian_arithmetic": "4722025175696264927919695237147963001/10006295983609568243810281629599200000000",
+      "bulk_exposure": "786997963001/2500000000000000",
+      "compact_phase_layer": "3934989696955306336847963001/12507869979511960304762852036999",
+      "coordinate_movement": "786997963001/2500000000000000",
+      "dimension": "12499999875000000000000000000000/12507869979511960304762852036999",
+      "original_prefix": "393511477565510145645101062952036999/1250786997951196030476285203699900000000",
+      "scalar_work": "6296071101698264298321324836347963001/10006295983609568243810281629599200000000"
+    },
+    "minimum_margin": "3934989696955306336847963001/12507869979511960304762852036999",
+    "parameters": {
+      "C0": 30696165898473906616464229210844217420767910202529753490360540450228768688163253600168840067667831942752384026854675608973799795380956593399141534112289239093330959827972896749165296184292677835464515042706184824146384636242947281779666181300241070877513766533217348531891170932662020307626188895470709615022057145970927098960139074253683920944753842216343194789843869048667245739946268044502169449739240047151298139657615969333261157765099433401579431741292725398235380507016225359633946900127581928474596898499722343371238746933858246860365082965753170791881602435981187257576524581883923220228449525753564243954428480950015663870403438565692691899173956579577361970732572037793733768891886401674483018689425094811761512045899886454943670633875852161010090639506253307030648261144912781663778645598021750465250333034689058743324535580864008727198856438637208832861170670419216209536983422886557419704292735318095822309918687019441302913514700304362634673588909066954341683971827907137382210012518361438056275277057064232991212789748905422006367435262834520504766612238813884074429782434316694786048082334035090608690501223482525285086525145461908703567251407765400438424335197761011186633090084188216941628069753763950559961548629403475265038319959658909620955172461605059173858444185012893723129298952614330442388531596408731782960746915881084072080603129410307130756509497968197206236172746714331895144445306055703171094812858528748519947672355467464128161052256129172799973643988142608069782604426724336348775294337633428178396224161239290663179656843957209639958446124597724968528581361144955015530528759188904934377208174674019599277955634609851614438383703272677733192621976769701286010377503716067478820818649918994787686527537194606647407039743222442541788507026896213132795524049193565701496785320578474168708305848642077472629345077346366218508632492830924321018579299074045972357348213174517250439981737800501020223163331916298028476359702006548313059616793218842383239493235457007274959132148479678539408282659646982187565056313462378252737774344983914114735063811593255146137319287480976858827473123868891197560843786009039915710780050550420460943164924231991989265412042858447481018196941754397663918947609234828077558511570244262179574004855616253459989663860102596373456031167093421946704741585786919199557045158439097213323953265846259553979220422046668529371017045876005336436479780234457930853838058795062002900011332052685428656525216784710243056129526713814497116290605431924667772478168683313660127033196982460371452963204231617886166630916523188238575759041967947979643981673386101779048771374925593700054204227929165478258872448575069729031897223414543223724259924992665669280726345289899457744940274678838535789203591966880452913448419201468164512694692355085936457303202300411660069424774391870920342021411572490632052161486961063479441431802719136822510030209619757474602682641529902941548456029770010345465738687308400303061060830666158934115070701723064182053653475143883887978182801499074168859056945574903272433563512135170640483603680837439264441595964867966549719628372663760876967360366308213970577592654353990010342197845485343012932469684817188969407289165866432168136962769959068185160314472824869765084912962897046125994447014189628438994121412862411165243833306895011648126457335989626426973787547689408286670235593322870322971810657403364020863426313792495249182801111830025188105697456742708832183524873981641173125187913100582226355310906961666595892837069563291972552191243444641098106071819143340241823732627414969135657845994558312942717869715326560449568658089273632302303332211170482991384787227702625642934010483232956902905766650492835947745649529830083281994505534654387479899352704510171536563654325312867482128978655653891744796926263680448177948352148536123483694563541642088601802880699196021313662811984922825257581647851754983351962386608433561854054647401336779281658633497726631247223605482115751711431087053212427473902081904556540321107521160756853582886163401164221981970038166439381638958549108016125060070284677401583644946753917796741174529439722340672459507806727576418354017624578135910854664726107600816947963640384551908501288584527261793442907319029799761559801783790863087943637491543750862440543535999534299571916325334049661352934275006506806261536051826429744056546054814346338065165529044305154718958072650379376680236770874295849492870591826116705604017007000175593740743193727398961093309618647336965924686189812963596962945033033582874236791813195084478338267034792908409419966724251751555218509766268153392566999396990244951342912001278944978288677854232438043616411418906149079698403993711785777631754898882908535194064092844260914921258456944304147862615175426906678308035529355051278798414906405945164735775359269076276985611302847179482677217251426707809544982432278670716514552848640103139161902168229660364050471078410543795331939638213953000309376353130390959610186193720452546227933796976891909253999601128929969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+      "C1": 1,
+      "a_bit": "786997963001/2500000000000000",
+      "a_complex": "786999/2500000000",
+      "actual_bit_saving": "3731649599/10000000000000",
+      "alpha_squared_power": "5902547104457805549850000000/12507869979511960304762852036999",
+      "beta": "1/1000000",
+      "c": "3934989775655101062952036999/12500000000000000000000000000000",
+      "delta": "1/800000000",
+      "epsilon": "12499999875000000000000000000000/12507869979511960304762852036999",
+      "eta": "1/100000000",
+      "kappa": "3146011/10000000000",
+      "lambda_": "249921300204486897963001/250000000000000000000000",
+      "lambda_prime": "124960650102636947963001/125000000000000000000000",
+      "q": "39349897363052036999/125000000000000000000000",
+      "sigma": "2499213001/2500000000",
+      "tau": "2499213002036999/2500000000000000"
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+      "exact_grid": "One common dyadic grid times 21^(-K), K=G*(D_complex+1); no child rounding",
+      "fixed_odd_divisor": 21,
+      "induction_gap": 111055305777576981079333088091528984853344234564695309861346615796681376984862595584385566814825723225065607620096765225351194906472983970593633490176627056143721106422216376755528119362946940294138718438389502442272575254888926946359465958092642454461528315668094171494868458070625567251310275709029976281859969802035024973630345605898977698541411492686452560470896362667488229038144227261802720013756008122230380971218084541261262657695655823668658950184225427955063214585074277389610648356907705994058405417887296296081460681795384769917569659912628146292790816155218539674416938232085789850310171387459249808896608889201657263141215880658854662744055015571588799834330147393335377433664022354602401083559392946127860513044648674705406919568666223816015468903965667025340144119898972040872907203271174785033232387665847763108009852161530142716332160067066751837213707179574252591067714594053544058250767700675229431707994132250951767159823880930497637110625485449912246835158803804381905960868264331668097823945790431917913581064268403484452805162153450641620990563468628336466310668991185880644314032649403430172062823279155783609645549955960073962729119570348715614972556047638785947887408140560829970697541152469893627224267809283418449452726720870731356561951538279893448059167527629826444039497465186815595777650550569683085978973215957641756509217487490299866164550074777393205341280756036138035699078147210796210410311267654583249641781697396157048712105568341990194442414384343978539177853722698463098527122077792030797445101272568641850205564721500979758996350089259471060798861398700259666592458073683378558355029991761888715422417068499336795061334343576346495754377262800468424884892315062440860206370624567956984890331860058081042470833447653633321049837921692854046139223298543861639675597354995850508859932311157956484561482373856907142627114211849356794423726471326447622124353190163367013621165941341204612004415252324756057407706546587887432096571449396406537871078499358394022000185376309958673357085215678239275339440428250059544485801768844432268234401142242422179995068758409331624787482333237154753129298212691959202974878244579561902394653944301077795056291145847155572379461058376014013433509638477668262814026109997068688688862925756416891490834110770803131842448717697999783176970400776755881870625561288673883845002434404789633133347284764143606345327352007357839147496208044247051216690022543968404644094168790736461730906139472825594749348196894044957708319856882799600381475919920189999653851383248069523540433884047333160862017656352433059763269699113355978412167829255798419719711790386309356030363196360494713947343462279669772189322873117640824491855045603035073599002406708970846851003486075195901727988383981549822258882130360972649597625325336052190859970270351883947109922148789771137350078582695336364508752558559909075047492013359764961469000046539345139154116867915672819070627671087115745227284484716067794747424679828786349978406104553411497003360168223115846366565852062171682797115302555511986768271289889538656176987073711609923999680746087859958155377720458219333144356918498989871309558571961281827195754735164897898439765065047966008970757788262057115263477029525579731066098218853296797050262370730526982102535883656985888665854536244133823838418009487156630573081773171311664311617988135048929117216731203042265451969903411852272221970444073770129656135682305250089367989058263878161642679859949467697153039825899201373308762169431448975537834085922032304481156204554227141039994009857088944360278469876431808752629397462236412000219233660046055973838129880578708956491953260821318624578345569184889813070192457876695541858132041130777307984129489628334586859639493684942574253992097303592844714909015896296482674756504031418599158553221773188034926625396857260903261878307905372874612848347540683995335159545128317882603325896188545222235978735007016860893357709283788163455082912370982400426341362702008872400551280825422269097566453712528758994504945650540249217498070923831844598267419752342110289825722952355243810308353333079195669922481418558143835271314909198434758233868828986616418060218640849839011109292650177754767791950900767208612899787574741181682059837568829258764729800119342223741623142490714266858722191898513982763470972130009550787149092732897948358791151902066155445576903588311987739858760918732667250155209815280844800000000000068718912,
+      "literal_charge": 6061317467480298914166412413635509035200825324641462576528500918260505197048529008335411593478234130210827057096014978142654834128399860733918726118791773922544231627186584932607180837437985697148325446613654772127558622768215964281375124396090186525633003482521737660857117057240380408058143301159395745008292344044487337285943700394911933590972408446368912334264475733554299876270081563954488872295078335957930985305442753589850599233939824365300285962223023588247381049686123738701973261608696653853640271295980430937712764175125304472239135234455542038111815182015673494498712783967998518258154295470432336395361110853163893932642029796415156619769016056023201425897551251393123598093723935966354950958705012418775872953351008819532305000414367477670368808844583692409434398245075205495223168715503573269586583420103111590096600671365292561495345243553257247298264986356518377914831220801501017951433236345152310406984299832525720293608660467685802945483512669025302907128271609978978462701143150604820858999336153620386383894468943290542669059358179441300249076561668526454073442875717677916326520380838342739697951112608267914738492672336157550028283025113961492707126404199254138271260685098789681348025487639773424867242919160106456506225534345420260373319252745281061156015856899471184592970267481712361162351271741573628066610930691160032539534168785670646828732270689456404423559701174212880588017796628230376017675290791793374993860591404265466456680648175654814058715126532658208137745454919267048288763951796324118745993820549985758936031091665141162389848923851841799059583105683402276350936999910323347488009519439806868167328926974701226839113335255880808750186544931465301588153899893702599669867866186021256769597944650784226039304068822923262564211182177780591366866646362124875884880995517669656219522666002703053336351359891487652867447175318019352977089273823249142319226929813637473276418387127197612183733738386142721845626159231038325296874200627272574306187965929458860133905538784309005014926007818655156818488946449124798464099925756731475826598449776290085204577274569078978042280310875204761493427427197988866936787709324177115415689567097667988137061963313037882715845129855200144933656979765875835424049864470446024907245877422963999708688816979203564558062413402109110136866845080992848753919979905561317141584585229824048025921909547308796975243130649305866894488417110756862463558580639319802212486767915649173681178903044454862485635704362237681574148245978149890284574577126487264579106426185408914448845493504091361788063449572056526660614284169584892130984109554414961055346550364120714926497961868347353913614754951268350956156592460248884512851335524123604981906157784294005113951272816555231962286602696050339075796628150139399920930876099280442740618192335848877634784032742506145082992546749815280528246791731174104728858887400270864819582995045694383013425221340653158400000000000000002244,
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+    }
+  },
+  "kappa": "3146011/10000000000",
+  "predecessor_commit": "cf60442983c153a3a0f5272ea4713f99eba13c86",
+  "scope": "Exact local inventories, strict moments, contaminated fallback charge and semantic assembly. Local finite reconstruction is checked separately; group geometry, uniform weighted routing, borrowed rows and inherited analytic/tape contracts are written proof dependencies.",
+  "source_sha256": {
+    "certificates/copied-centers-network.json": "7dbd675e42e22ac076280360f8750031851564bb7a56079e66236fee5d0a6cdf",
+    "certificates/partial-gauge-bit-input.json": "367b9844db23f8fa64002637903bcc5ce28ffc8879d274b5c54c0390fb0234e4",
+    "certificates/partial-gauge-network.json": "e2e6c48bc283b1417c13adaa35c36d782191e21a33f7279a4ed1558ca60518be",
+    "certificates/stopped-product-network.json": "43ddb7ee033d316375102cf2d97bb2cf47f879f36a005d3726fa6647a2cbecc5",
+    "certificates/three-stage-cover-bit-input.json": "d0d496b64d45408d6e0773ebbcd4d5dcca918a596073a51e5fc186b82c3b8520",
+    "certificates/three-stage-cover-complex-input.json": "afecf8ce2d8eb259074e61cd0c386b464ca2f57c61b00979eac808fa26e0b770",
+    "certificates/three-stage-cover-input.json": "633e108352d3a880c3dc05e265f40270698ea7ebbe655706f5f8675070c86bd2",
+    "notes/general-clifford-frames.tex": "c9bb9655b594583c6b4bab8cb485233f1d1e588f83858cf1e4c3530cee8fed54",
+    "notes/three-stage-cover-assembly.tex": "ccf34a6d37c4af14bebeab15be50ab204eb1caebe546385520050ec30089441d",
+    "notes/three-stage-cover-bit.tex": "47123a756f823c83a0d48cca2bf5add6472cd88410b13aed345e70c8834fb511",
+    "notes/three-stage-cover-complex.tex": "1c5b638aa9b75365b8addec9c8eeee6bc5e499722901079b94c65fce033fde8a",
+    "notes/three-stage-cover-note.tex": "db8078687825bcd43c6cc2fb69712ebb12840057b77cd5af8f67f53e4ffffa92",
+    "notes/three-stage-cover-rows.tex": "6ad97a5cd7d6aecf978a81226e1c2763d37840caca130c61f3bf5c4ac111c5c8",
+    "references/partial-gauge/pr97/SOURCE.json": "68fb539abcd6df21d142e4e204b6e9482eb7f9cd3907650f54bd058070266596",
+    "references/three-stage-cover/pr117/LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+    "references/three-stage-cover/pr117/NOTICE": "a0030510cd3b5334fb113eae759aadd2c25951bd353dfb69ad9cb780ae5adf30",
+    "references/three-stage-cover/pr117/SOURCE.json": "74e9f4b4ae0529d21d19c159b2b50ee0923e93e0d7eb22cf208ad41dd5992275",
+    "references/three-stage-cover/pr117/dag.json.gz": "3c034d0aae388ef567a454826f4f48b26fd8a94c71e8ffed4835271b349a783b",
+    "references/three-stage-cover/pr117/replayed.py": "99c1e3694ae06a7ce7a370b04ecb45ea3fcfe69a6e6e7f4c4606e0a6c74c3184",
+    "scripts/partial_gauge/binary.py": "b29e2f37d7f32a959a378190d7f1da3c2da534fbdaf878044800301957bffc03",
+    "scripts/partial_gauge/lift.py": "895874399113faa090d168d6168f0664136df6e21205671e7f7f9239cad93d1d",
+    "scripts/partial_gauge/match.cpp": "12b8b7c93bb78324e7328e624b166be2067ae8861540531c024040220d86f368",
+    "scripts/partial_gauge/word.py": "96015b7b08a261673ab8be1ea47f89302f8a1d62a822d657a09f6a1fcfb51e58",
+    "scripts/partial_gauge_bit.py": "7d58e86061da2d73991531e9bc431589019359391adea866c61c7dd1f99ac855",
+    "scripts/partial_swap_network.py": "acdd81b4dea2815f8838a0c3ea8ac4968f2bc13fd7c99173473d865779096782",
+    "scripts/structured_bulk_assembly.py": "e99467e14211fba699878ed036e1e2362eb8e395dc22f9ecdcecc7fbf2cc736d",
+    "scripts/three_stage_cover/README.md": "b39a28fa8da5d7d3188dfed4c0e0af1e2397f0f5c8f69cfb69671ed27ddca4bb",
+    "scripts/three_stage_cover/__init__.py": "b964b59af038e70cc15af655a22bcf6aaafc9c842b213765781bc8bea7166430",
+    "scripts/three_stage_cover/lift.py": "ae5f4c56cd32b5232d6dcdcaed1e24efb81fe20467a2d915ee51479bced06cfc",
+    "scripts/three_stage_cover/select.py": "457e636ced9bbd368198f78e015660a929ba1aaccbe34122d18da09ebb488f33",
+    "scripts/three_stage_cover/verify.py": "4bd2347e87e4edb699968bc31dfb94f2ab89bf1227576e59f43c779ba74e4e06",
+    "scripts/three_stage_cover_network.py": "7e0ad6fd1921b6905a53ce1eeaa03d3bb3a8a8bbaa0382516ea98b636f46335e",
+    "scripts/three_stage_cover_producer.py": "ea6dd4f06173304bbcb74b8e5bcc02357b7bb162a6d26b4a6f7a9e886158123f"
+  },
+  "status": "Conditional three-stage cover multiplication witness"
+}
diff --git a/docs/partial-gauge.md b/docs/partial-gauge.md
new file mode 100644
index 0000000000..dd5c670a51
--- /dev/null
+++ b/docs/partial-gauge.md
@@ -0,0 +1,65 @@
+# Partial-gauge reproduction
+
+This extension inherits PR #104 at
+`948ce1510df750f4c18b96bdaef436a86f8bf834` and gives the conditional saving
+`7237/78125000 = 9.26336e-5`. Proofs are supplied as LaTeX source.
+
+From the repository root, with Python 3.11+ and a C++17 compiler:
+
+```sh
+make partial-gauge-verify
+```
+
+CI runs this extension in its own `partial-gauge` matrix group. The
+incremental target performs three checks:
+
+- `partial_gauge_bit.py` checks pinned PR #97 source hashes, reconstructs
+  auxiliary and data ranks from its actual frozen paths, matches the physical
+  ledger receipt, and rebuilds the whole-projector child list. The unchanged
+  upstream complete-basis scalar and rational frame audits are inherited.
+- `partial_gauge_producer.py` rebuilds the selected h24 interval/cube DAG,
+  compatible matching, backward enlarged frames, physical word and partial
+  source gauges. Independent checks cover exact supports and signed decoder,
+  frame nondegeneracy and nesting, center closure, untouched source roles,
+  reverse readout order and every positive-rank transition. The deterministic
+  construction also reproduces the complete histogram, including zero-rank
+  no-ops; see [the producer guide](../scripts/partial_gauge/README.md).
+- `partial_gauge_network.py` certifies both strict moments, the stopped
+  ordinary-bit saving, denominator-21 semantic charge, product row stock,
+  all 47 strict constraints and seven assembly margins. It writes the
+  deterministic, source-bound `certificates/partial-gauge-network.json`.
+
+For arithmetic and the pinned bit profile alone:
+
+```sh
+make partial-gauge-certificate
+```
+
+Generated complex graphs are temporary by default. To retain them and a
+compact report outside the repository:
+
+```sh
+python3 scripts/partial_gauge_producer.py --work-dir /tmp/partial-gauge-work --output /tmp/partial-gauge-verification.json
+```
+
+The selected proof is [partial-gauge-note.tex](../notes/partial-gauge-note.tex).
+The source notices and compact PR #97 witness remain under
+`references/partial-gauge/`. Unselected experiments, large generated graphs,
+recursive handoff copies and PDFs are excluded.
+
+## Shared source bindings
+
+The joint-dual manifest also binds `Makefile`, `README.md` and `NOTICE`.
+This round refreshes those three digests and the two derived certificate
+digests. Its producer, physical words, frame profiles and mathematical
+certificate fields are unchanged from PR #104's successful local replay.
+Incremental validation checks that equality and reruns the joint arithmetic;
+it does not repeat the unchanged graph and complete-basis suites.
+
+After reviewing changes to those shared files, the explicit manifest refresh
+is `python3 scripts/experiments/pin_joint_dual_sources.py`. Its derived
+`joint-dual-kappa.json` and `joint-dual-validation.json` must also be updated.
+The full affected replay command, when needed, is `make verify-joint`.
+CI checks the committed source bindings and requires deterministic output;
+it never silently repins sources. Stage the final artifacts before the
+reproducibility check and require `git diff --exit-code` to remain clean.
diff --git a/docs/reproducibility.md b/docs/reproducibility.md
index e48f4190cd..8f70a8f12b 100644
--- a/docs/reproducibility.md
+++ b/docs/reproducibility.md
@@ -1,5 +1,18 @@
 # Reproducing the result
 
+For the three-stage cover extension, run `make three-stage-cover-verify`.
+The [incremental guide](three-stage-cover.md) covers the new PR #117 local
+word, cover moments and exact assembly.
+
+For the partial-gauge extension, run `make partial-gauge-verify`. Its
+[incremental guide](partial-gauge.md) separates the pinned PR #97 bit input,
+new complex producer and exact assembly.
+
+For the new stopped product-ring extension, run `make stopped-product-verify`.
+Its [incremental reproduction guide](stopped-product.md) covers the h24
+rational-center producer, new moments, stopping parameters and assembly.
+The community checkpoint and historical targets below retain their own scope.
+
 The selected release is documented in the [joint-frame community review](research/community-round2-review.md)
 and [current status](research/current-status.md), with the
 [selected parameter certificate](../research/matrix-exponent-synthesis/candidate/arithmetic.json).
diff --git a/docs/stopped-product.md b/docs/stopped-product.md
new file mode 100644
index 0000000000..5bc1cbc88e
--- /dev/null
+++ b/docs/stopped-product.md
@@ -0,0 +1,67 @@
+# Stopped product-ring reproduction
+
+The extension gives the conditional saving `194869/2500000000 = 7.79476e-5`.
+It starts from merged PR #36 and is submitted on upstream main at
+`56b66d58297deca1d7dd130247d720e960f77a37`.
+
+From the repository root, with Python 3.11+, Make and a C++17 compiler:
+
+```sh
+make stopped-product-verify
+```
+
+The incremental checks:
+
+- Match the unchanged positive-label h23 bit producer to its verified
+  copied-center input, preserving the same matching and histogram.
+- Rebuild only the new h24 all-disjoint complex producer with aligned pair
+  stars, exact rational center decoding, support and binary-frame checks,
+  then reproduce every matching count and histogram entry.
+- Reconstruct both child lists and strict rational moments; derive the
+  stopped ordinary-bit saving and check that the atom-adapter toll is lower.
+- Check the denominator-21 semantic charge, product row stock `p^4000`,
+  seven assembly margins and all 47 strict constraints.
+
+Intermediate DAGs and the matcher executable are temporary. To retain them:
+
+```sh
+python3 scripts/stopped_product_producer.py --work-dir /tmp/stopped-product-producers
+```
+
+For arithmetic alone, use `make stopped-product-certificate`. The inherited
+`make verify-producers` also includes this target; the full `make verify`
+retains the community and historical checks. Those unchanged suites are
+not part of this round's local incremental validation.
+
+The [proof](../notes/stopped-product-note.tex) is supplied as LaTeX source;
+no PDF is generated. Its common-basis factorization, atom streaming and
+stopping argument, outer ordinary conversion and shared exact rational
+grid are written proofs. Finite checks verify the selected scalar data and
+arithmetic, not the complete all-size theorem.
+
+[SOURCES.json](../SOURCES.json) and [NOTICE](../NOTICE) preserve contribution
+credits and source pins. Existing two-stage, ordinary-leaf and semantic
+proof dependencies remain in `references/copied-centers/` and
+`references/semantic-bulk/`. Research alternatives, raw binary DAGs and
+recursive predecessor copies are excluded from this submission.
+
+## Shared source bindings before submission
+
+The inherited `research/joint-dual/SOURCE.json` also binds the repository's
+`Makefile`, `README.md` and `NOTICE`. Adding a verification target or editing
+those documents therefore affects the joint-frame check even when its
+mathematical construction is unchanged.
+
+After reviewing such source changes, refresh the manifest and regenerate
+its dependent certificate and validation receipt:
+
+```sh
+python3 scripts/experiments/pin_joint_dual_sources.py
+make verify-joint
+```
+
+Include `research/joint-dual/SOURCE.json`, `certificates/joint-dual-kappa.json`
+and `certificates/joint-dual-validation.json` in the same submission. Check
+that their changes are the expected source bindings, then rerun the affected
+check against the staged artifacts and require a clean diff. CI verifies
+the committed bindings; it does not refresh them automatically.
diff --git a/docs/three-stage-cover.md b/docs/three-stage-cover.md
new file mode 100644
index 0000000000..38554e1fb3
--- /dev/null
+++ b/docs/three-stage-cover.md
@@ -0,0 +1,74 @@
+# Three-stage cover reproduction
+
+The extension inherits PR #115 at
+`cf60442983c153a3a0f5272ea4713f99eba13c86` and gives the conditional saving
+`3146011/10000000000 = 3.146011e-4`.
+
+From the repository root, using Python 3.11+ and a C++17 compiler:
+
+```sh
+make three-stage-cover-verify
+```
+
+CI runs this target in its own `three-stage-cover` matrix group. Its
+incremental scope is:
+
+- Rebuild the new pinned PR #117 local DAG, compatible matching, full backward
+  subspace intersections, actual scalar word and partial source gauges.
+  Independently check scalar supports, signed coefficients, divisor 21,
+  carrier containment, center closure, untouched source roles, actual frame
+  transitions and reverse target chains. Intermediate frames can be
+  degenerate; their validity follows from the new general Clifford proof.
+- Reconstruct the unchanged PR #97 local physical ranks from its pinned
+  paths, then compose the new three-stage bit inventory. The unchanged
+  upstream complete-basis and rational-frame suites are inherited checks.
+- Certify the bit moment with the full rare-class local-ring fallback,
+  stopped ordinary saving, complex moment, full finite group order and
+  router charge, exact semantic guard, row reserve, 47 strict inequalities
+  and seven assembly margins.
+
+For the moments and arithmetic alone:
+
+```sh
+make three-stage-cover-certificate
+```
+
+The group is specified by a written construction; verification does not
+materialize its vertices. The complex group size and finite routing allowance
+remain in the certificate. The bit group depends on atom width; its uniform
+batching, weighted children and internally borrowed rows are proved in
+`notes/three-stage-cover-bit.tex` and `notes/three-stage-cover-rows.tex`.
+Finite arithmetic does not by itself verify those all-size interfaces.
+
+To retain regenerated local files outside the repository:
+
+```sh
+python3 scripts/three_stage_cover_producer.py --work-dir /tmp/three-stage-cover-work --output /tmp/three-stage-cover-verification.json
+```
+
+The [main proof](../notes/three-stage-cover-note.tex) includes
+[general Clifford frames](../notes/general-clifford-frames.tex), exact data
+chronology and phases, complete dirty cleanup, weighted factorization,
+fallback, row restoration and final composition. Sources are included
+without a PDF. `SOURCES.json`, `NOTICE` and `references/three-stage-cover/`
+retain the adopted author's source pins and original legal notices.
+
+The previous submitted production code and certificates remain unchanged.
+Unselected round7/round8 research and large generated graphs are excluded.
+Only the new selected local construction and changed cover/assembly are
+verified this round.
+
+## Shared source digests
+
+Changes to `Makefile`, `README.md` and `NOTICE` also affect the inherited
+joint-dual source manifest. After reviewing their final content, explicitly
+run `scripts/experiments/pin_joint_dual_sources.py`, regenerate the joint
+arithmetic certificate, and update its validation receipt's certificate
+digest. The unchanged mathematical fields and prior replay payload must
+match the preceding submission. A full producer replay is required when
+those producer or verifier inputs change; the full command is
+`make verify-joint`.
+
+The final reproducibility check runs against staged artifacts and requires
+`git diff --exit-code` to remain clean. CI verifies committed hashes and
+does not silently refresh source pins.
diff --git a/notes/general-clifford-frames.tex b/notes/general-clifford-frames.tex
new file mode 100644
index 0000000000..0367eeb005
--- /dev/null
+++ b/notes/general-clifford-frames.tex
@@ -0,0 +1,102 @@
+\subsection{Preimage frames for arbitrary binary subspaces}
+This frame construction is due to icekylinx with OpenAI GPT-6 Astra assistance;
+the present exposition and integration use Codex.
+Let $H=\mathbb F_2^h$ with the standard dot product. In the binary
+symplectic space $H\oplus H$ put
+\[
+ L_0=\{(0,z)\},\qquad
+ F=\begin{pmatrix}I&I\\0&I\end{pmatrix},\qquad
+ L_F=FL_0=\{(z,z)\}.
+\]
+We also write $F$ for the exact Gaussian-dyadic operator $C^{\otimes h}$.
+Its exact square is a Pauli translation; only its binary symplectic square
+is the identity. For an exact Clifford operator $T$ the frame label is
+$L(T)=T^{-1}L_0$. The physical transition from $T$ to $S$ is $ST^{-1}$,
+and its Fourier rank is
+$d(L(S),L(T))$, where $d(L,L')=h-\dim(L\cap L')$.
+This follows by transporting the intersection by $S$: the intersection
+with $L_0$ computes the kernel dimension of the $B$ block of $ST^{-1}$.
+
+\begin{lemma}
+For every subspace $U\subseteq H$, including degenerate subspaces,
+\[
+ L_U=\{(u,u+v):u\in U,\ v\in U^\perp\}
+\]
+is Lagrangian, and
+\[
+ d(L_U,L_V)=\dim U+\dim V-2\dim(U\cap V),\qquad
+ FL_U=L_{U^\perp}.
+\]
+\end{lemma}
+\begin{proof}
+The displayed parametrization is injective and has dimension $h$.
+Its symplectic pairing is $u\cdot v'+v\cdot u'=0$.
+In an intersection, $u$ belongs to $U\cap V$ and $v$ to
+$(U+V)^\perp$, giving the asserted dimension and distance.
+Finally $F(u,u+v)=(v,u+v)$ proves the complement identity.
+\end{proof}
+
+\subsection{Exact representatives and nested children}
+Choose a deterministic invertible binary matrix $G$ carrying the first
+$r=\dim U$ coordinate space $E_r$ to $U$, and set
+\[
+ K_G=\begin{pmatrix}G&0\\G+G^{-T}&G^{-T}\end{pmatrix},\qquad
+ T_U=K_G T_{E_r}K_G^{-1}.
+\]
+Here $T_{E_r}$ is the exact coordinate $C$ operator, and the $K_G$ are
+exact Gaussian-dyadic Clifford lifts. The matrix $K_G$ is symplectic
+because $G^TG+I$ is symmetric. It fixes both $L_0,L_F$ and maps $L_V$
+to $L_{GV}$. Thus both $K_G$ and $FK_GF^{-1}$ have Fourier rank zero;
+their lifts are fixed binary address permutations with unit quadratic
+phases. In particular,
+\[
+ L(T_U)=L_U,\quad \operatorname{rank}B(T_U)=r,\quad
+ \operatorname{rank}B(FT_UF^{-1})=r,\quad T_U^2\text{ is Pauli}.
+\]
+Set $V_U=GE_r^\perp$, a direct complement of $U$. The binary action gives
+$T_U^{-1}L_F=L_{V_U}$ and hence
+$\operatorname{rank}B(FT_U^{-1})=h-r$.
+No assertion $V_U=U^\perp$ is needed.
+For the norm-one source lines and their orthogonal target hyperplanes,
+retain the inherited exact representatives; set $T_0=I,T_H=F$.
+At every common scalar gate use one identical exact representative on all
+incident registers. If separately transported representatives have the
+same preimage, their physical ratio has rank zero and is implemented as
+an actual left adapter before the gate.
+
+For $U\subseteq V$, the exact transition $T_VT_U^{-1}$ has rank
+$\dim V-\dim U$. Clifford elimination expresses it as two rank-zero
+adapters surrounding that many coordinate Fourier factors. These factors
+are a single $C^{\otimes r}$ child: the scaled Hadamard satisfies
+\[
+ \frac{1+i}{2}\begin{pmatrix}1&1\\1&-1\end{pmatrix}
+   =SCS,\qquad S=\operatorname{diag}(1,i).
+\]
+Thus no new irrational field is introduced. All adapters are charged
+finite movement and scalar operations. Exact transitions telescope;
+commutativity between distinct $T_U$ is unnecessary.
+
+\subsection{Dirty tails and reverse/complement words}
+A dirty role entering with gauge $T_\sigma$ and whose last active frame
+is $T_U$, with $\sigma\subseteq U$, is assigned terminal gauge
+$FT_\sigma$. Its logical value is restored by the complete transparent
+scalar word, so its physical action is exactly $F$. The tail is
+$FT_\sigma T_U^{-1}$. Its rank is
+\[
+ d(L_U,L_{V_\sigma})=h-\dim U+\dim\sigma.
+\]
+Indeed $H=\sigma\oplus V_\sigma$ and $U\supseteq\sigma$ imply
+$\dim(U\cap V_\sigma)=\dim U-\dim\sigma$.
+An inactive exterior adds $m-h$, giving $m-\dim U+\dim\sigma$.
+
+For the complete time-reversed word use
+\[
+ D_U=T_UF^{-1},\qquad L(D_U)=L_{U^\perp},\qquad
+ D_UD_V^{-1}=T_UT_V^{-1}.
+\]
+The last equality is exact, so reversal preserves every inverse transition
+rank. Reverse the scalar signs and the whole transparent word, including
+old-value corrections and final cleanup. The endpoints $D_0=F^{-1}$
+and $F$ differ by a Pauli, and the inherited norm-one endpoints are
+reconciled by actual rank-zero adapters. Neither equality of binary
+matrices nor equality of frame cosets substitutes for these operators.
diff --git a/notes/partial-gauge-assembly.tex b/notes/partial-gauge-assembly.tex
new file mode 100644
index 0000000000..4a09f324a3
--- /dev/null
+++ b/notes/partial-gauge-assembly.tex
@@ -0,0 +1,101 @@
+\subsection{Literal two-stage inventory and semantic precision}
+Let $B_a$ count selected gauges of dimension $a$, and $H_r$ the remaining
+internal histogram, including each residual first transition. Let $D_r^x$
+and $D_r^y$ be the source and target data-chain histograms. Put
+\[
+ m=h^2=576,\quad N=v^2=4096576,\quad
+ W=2N+2vR=147067888.
+\]
+The child inventory consists of $2v(R-\sum_aB_a)$ exteriors of width
+$m-h$, $2vB_a$ exteriors of width $m-h+a$, $2vH_r$ internal children,
+$2v(D_r^x+D_r^y)$ data-chain children, $2N$ connectors of width
+$(h-1)^2$, and $N$ paid correction children of width one. Zero-width
+transitions require no recursive call. Per-role telescoping gives
+\[
+ s=\sum_r rn_r=Wm-N+2v\ell=84709241408,
+ \qquad r_{\max}=574.
+\]
+Writing $F$ for the full transform and $T$ for the source-line transform,
+the retained signed endpoint is
+$X'=-Fy$, $Y'=FT^{-2}x+FT^{-1}y$.
+Pretranslate the initial $x$ bank by the fixed Pauli translation $T^2$,
+then add $T^{-1}X'$ to $Y'$ using the paid rank-one copy. Sign and bank
+corrections yield both $F$ outputs. The translation $T^2$ is retained.
+
+The rational logarithm and exponential enclosures in
+\path{scripts/partial_gauge_network.py} prove, for $b=92651/10^9$,
+\[
+ \frac1W\sum_r n_r(r/m)^{1-b}
+ <1-4.1393037848\times10^{-11}.
+\]
+The finite receipt supplies a scalar-operation bound
+$G=401572403009697216$, including direct rational dirty reads and the
+signed correction. Set
+\[
+ E=64(W+m+G+1)^3,\quad B=s+E,\quad C_0=32mB^2.
+\]
+It verifies $2GW^2+8s+4W+4+32m<E$ and
+$2B(m-574)>s+E$. Thus the inherited completed-child induction applies
+with semantic $C_1=1$. On one exact grid $2^{-P}21^{-K}$,
+$K=G(D_c+1)$, unfinished calls can accumulate at most $K$ odd-denominator
+factors. Completed children are Gaussian-dyadic $C^{\otimes u}$ and add
+none. The root dyadic-input divisibility invariant makes each local
+21-division exact; only the outer boundary scales and unscales, at cost
+$O(V_{\rm bits}\log d)$. No recursive boundary rounds or re-encodes values.
+
+\subsection{Joint rows and conditional transfer}
+The coarse bit, ordinary leaf and complex families have respective
+halving-degree and wire-bit pairs $(367,27),(9,28),(200,28)$.
+Their product stock has coefficient
+\[
+ 367\cdot27+9\cdot28+200\cdot28=15761,
+ \qquad40000-\frac{51}{25}\,15761=\frac{196189}{25}>0.
+\]
+One $p^{40000}$ reserve, one preceding prefix and one final padding suffice
+under the retained eventual row contract. Sequential calls reuse restored
+rows and preserve complete spectators.
+
+Choose
+\[
+ \eta=10^{-8},\quad\beta=10^{-6},\quad
+ a=(1-\beta)b-10^{-10}=\frac{92650807349}{10^{15}},
+ \quad\tau=1-a,\quad\sigma=1-b,
+\]
+\[
+ q_0=a(1-2\eta),\quad c=q_0(1+\eta),\quad
+ \epsilon=\frac{1-\eta}{1+c+q_0},\quad
+ \lambda'=1-q_0,\quad\lambda=(\tau+\lambda')/2,
+\]
+\[
+ g=\epsilon q_0,\qquad r=(g+1-\epsilon)/2,\qquad\delta=\eta/8.
+\]
+Here $a<a_b$ and $a<(1-\beta)b$. The exact assembly receipt checks all
+47 inherited strict conditions, including the seven final savings
+\[
+ 1-\epsilon(1+c),\quad a,\quad g,\quad a,\quad
+ \min(1-\epsilon-\delta,r-\delta),\quad
+ 1-\epsilon-\delta,\quad\epsilon.
+\]
+Their minimum is $g$, and
+\[
+ g-\frac{7237}{78125000}
+ =\frac{210905811144819709520320681}
+ {5352018951262406227316993998984375000}>0.
+\]
+\begin{theorem}
+Under the retained analytic, semantic-precision, exact-recovery and
+fixed-tape interfaces, the selected construction gives
+\[
+ T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad
+ \boxed{\kappa=\frac{7237}{78125000}=9.26336\times10^{-5}}.
+\]
+\end{theorem}
+\begin{proof}
+The stopped word supplies the required ordinary interchange contract;
+the signed framed program supplies the exact completed complex tensor
+contract. The common grid and product reserve meet the retained precision
+and row requirements. Substitution of the displayed parameters into the
+unchanged semantic/bulk transfer yields its seven savings. Their strict
+excess over $\kappa$ absorbs the fixed logarithmic overheads, including
+outer grid conversions, and proves the stated conditional bound.
+\end{proof}
diff --git a/notes/partial-gauge-construction.tex b/notes/partial-gauge-construction.tex
new file mode 100644
index 0000000000..147fc5ab91
--- /dev/null
+++ b/notes/partial-gauge-construction.tex
@@ -0,0 +1,60 @@
+\subsection{Stopped opposite-bank bit word}
+Apply the factorization in \path{notes/stopped-product-factorization.tex}
+to each rank increment of PR97's literal deferred word: a rank-$r$
+projector increment is one recursive rank-$r$ reversed child, with the
+fixed atom adapters charged as in \path{notes/stopped-product-interface.tex}.
+The selected finite profile has
+\[
+ m_b=529,\quad W_b=108516254,\quad s_b=57403754177,
+ \qquad r_{\max,b}=528.
+\]
+Its rational moment certificate proves coarse saving
+$15513/125000000$, with normalized gap exceeding $1.1417\times10^{-10}$.
+The atom exponent $10^{-3}$ and retained ordinary leaf saving
+$384599/10^{10}$ give the stopped ordinary saving
+\[
+ a_b=(1-10^{-3})\frac{15513}{125000000}
+       +10^{-3}\frac{384599}{10^{10}}
+     =\frac{1240183559}{10^{13}}.
+\]
+The exact inputs and calculation are
+\path{certificates/partial-gauge-bit-input.json}, checked by
+\path{scripts/partial_gauge_bit.py}.
+
+\subsection{Interval and pair-first cube producer}
+For disjoint strip values $v_0,\ldots,v_{k-1}$ construct cyclic intervals
+$I(a,1)=v_a$ and $I(a,l)=I(a,l-1)+v_{a+l-1\bmod k}$ for $2\le l<k$.
+Then $I(j+1,k-1)$ is the all-but-$j$ sum; add the omitted term once for
+a total. Singleton and empty cases use the corresponding direct values.
+Every addition has disjoint formal source supports.
+
+For two binary choices $u,v\in\{0,1\}$, write the desired four outputs as
+$F+A_u+B_v+D_{uv}$. Form $P_u=F+A_u$ and $Q_v=F+B_v$, then
+\[
+ Y_1=Q_0+A_0,\quad Y_2=P_0+B_1,\quad
+ Y_3=P_1+D_{10},\quad Y_4=Q_1+D_{11},
+\]
+followed by $Y_1+D_{00},Y_2+D_{01},Y_3+B_0,Y_4+A_1$.
+These are exactly the four outputs. This uses twelve additions and four
+compatible retained-carrier links, with compatibility witnessed by the
+actual source and target labels.
+
+For three binary choices there are 27 disjoint pieces: one $F$, six
+$A_u$, twelve $B_{uv}$ and eight $D_{uvw}$. Apply the pair primitive once
+to the plane with no third coordinate and once to each of its two choices.
+Add corresponding plane outputs. The eight results are
+$F+A_u+A_v+A_w+B_{uv}+B_{uw}+B_{vw}+D_{uvw}$.
+There are 44 additions, twelve local links and eight designated uses,
+hence forty local roles. The surrounding producer hash-conses equal sums
+and retains only ancestors of designated outputs. Its selected counts are
+\[
+ h=24,\quad v=\binom{24}{3}=2024,\quad c=146284,\quad q=8120,
+ \qquad |\mathcal M|=120097,\quad R=c+q-|\mathcal M|=34307.
+\]
+The retained centers are $A_i=\sum_{T\not\ni i}X_T$.
+Since $\sum_i A_i=21\sum_T X_T$, their rational decoder has fixed odd
+divisor 21. The identity decoder and copied-center proof in
+\path{notes/stopped-product-complex.tex} apply unchanged, with loss
+$\ell=24\cdot23=552$. Matching edges are actual physical dependencies.
+The proof below uses their chosen frames and physical word, not only these
+scalar counts.
diff --git a/notes/partial-gauge-frames.tex b/notes/partial-gauge-frames.tex
new file mode 100644
index 0000000000..86615eca33
--- /dev/null
+++ b/notes/partial-gauge-frames.tex
@@ -0,0 +1,95 @@
+\subsection{Backward enlargement with nondegenerate repair}
+Work in $H=\mathbb F_2^{24}$ with its standard bilinear form.
+Include scalar dependencies and retained-carrier edges in the acyclic
+forward dependency graph. Impose the side output hyperplanes $t_T^\perp$
+and the retained-center coordinate hyperplanes as output constraints.
+For a vertex with old nondegenerate frame $E$, intersect these constraints
+with its already chosen successor frames to obtain $L$. The original
+containments and backward induction give $E\subseteq L$.
+
+\begin{lemma}
+There is a maximal nondegenerate subspace $\widehat E\subseteq L$
+containing $E$. Choosing such spaces in reverse topological order preserves
+all dependency containments.
+\end{lemma}
+\begin{proof}
+Nondegeneracy of $E$ gives
+$L=E\mathbin{\perp}(L\cap E^\perp)$ by orthogonal projection.
+On the second summand repeatedly split off a vector of norm one, or, if
+all norms vanish, a pair $u,v$ with $\langle u,v\rangle=1$.
+The corresponding Gram blocks are $(1)$ and
+$\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)$.
+After orthogonal elimination the remainder is radical. Discarding it
+leaves a nondegenerate space of the maximum possible dimension, containing
+$E$. Each chosen frame lies in the already chosen successor frames by its
+definition. This proves nesting, including the carrier edges.
+\end{proof}
+The supplied old bases are orthonormal, so projection is explicitly
+$x\mapsto x+\sum_{u\in\mathcal B_E}\langle x,u\rangle u$.
+New alternating blocks are permitted: the retained general quadratic
+Gauss compiler implements every nondegenerate residual of rank $r$ by one
+$C^{\otimes r}$ child, fixed input/output binary maps and unit phases.
+No orthonormal-basis claim is made for the enlarged frames.
+
+\subsection{Dirty readouts and partial first-frame gauges}
+Let $M$ be the literal invertible scalar addition word on the $R$ roles,
+$V$ the source injection, $J$ the rational designated decoder, and $C=JM$.
+For arbitrary dirty contents $z$, the sequence
+\[
+ y\leftarrow y-Cz,\quad z\leftarrow z+Vx,\quad z\leftarrow Mz,
+ \quad y\leftarrow y+Jz,\quad z\leftarrow M^{-1}z,
+ \quad z\leftarrow z-Vx
+\]
+adds $JMVx$ to $y$ and restores $z$. These are signed operations over
+$\mathbb Q(i)$. Take the predecessor closure, under per-slot operation
+precedence, of the final operations producing retained centers. Execute
+this phase first, then their positive copied readouts. A role outside both
+the injected sources and the phase's touched slots still holds its
+original dirty coordinate, so its negative column readout can be delayed.
+
+Propagating unions of designated target sets backwards through $M$ gives
+a superset of each column's support in $JM$. An eligible untouched role
+cannot influence a retained-center output: such an influence would place
+its first operation in the center predecessor closure. Thus every target
+in its propagated support is a side output. Dependency nesting implies
+that its first frame $A_s$ is contained in every such target hyperplane.
+This justifies initializing a target's latest constraint to all of $H$;
+intersecting with $A_s$ already enforces its output constraint.
+
+Select roles in descending $\dim A_s$, breaking ties by support size and
+role index. Intersect $A_s$ with the latest selected frame of each incident
+target, and retain a maximal nondegenerate summand $\sigma_s$ by the lemma's
+block elimination with $E=0$. If its dimension is positive, select it and
+replace every incident target's latest frame by $\sigma_s$.
+For each target, successive selections therefore form a descending chain
+of actual subspaces. Execute the selected reads in \emph{reverse selection
+order}. Each target then has an ascending chain from zero, ending in its
+side hyperplane; equal-dimensional members coincide. Complete the remaining
+$M$, the positive side reads, and the inverse $M,V$ at the full frame.
+The phase closure allows this reordering because omitted predecessors
+never touch the slots of the phase operations.
+
+For a selected role set $r=\dim A_s$, $a=\dim\sigma_s$.
+The first transition becomes the nondegenerate orthogonal difference
+$A_s\cap\sigma_s^\perp$, of rank $r-a$. Choose input gauge $C_{\sigma_s}$
+and output gauge $C_HC_{\sigma_s}$, retaining the full-frame inverse
+cleanup. The auxiliary exterior width changes from $m-h$ to $m-h+a$;
+the first transition is replaced, not removed unless $a=r$.
+The 6,341 selected roles and their order are regenerated by
+\path{scripts/partial_gauge_producer.py}; with \texttt{--work-dir DIR},
+\texttt{selection.json} retains the selected list and \texttt{word.json}
+retains \texttt{ops}, \texttt{sources}, \texttt{rootroles}, and
+\texttt{phase1}. The construction and independent checker are in
+\path{scripts/partial_gauge/}. The remaining internal and both data-chain
+histograms are recorded in
+\path{certificates/partial-gauge-complex-input.json}.
+
+Boolean support is used only for safe containment tests; readout
+coefficients are those of $JM$ over $\mathbb Q$. At the first copy into a
+fresh role choose any source present in the copied nonempty value. Its
+coefficient dominates that role's original-dirty coefficient thereafter,
+since subsequent additions are positive. Each raw designated output is
+a disjoint source sum, so the dirty coefficient is at most one.
+Injected source roles use their own source as witness. Consequently
+$|(JM)_{t,s}|\le\sum_j|J_{t,j}|\le h+4$, with denominator dividing
+$2(h-3)$. Partial direct-readout sums are polynomially bounded in $R,h$.
diff --git a/notes/partial-gauge-note.tex b/notes/partial-gauge-note.tex
new file mode 100644
index 0000000000..7592d4970b
--- /dev/null
+++ b/notes/partial-gauge-note.tex
@@ -0,0 +1,60 @@
+\documentclass[11pt]{article}
+\usepackage[margin=1in]{geometry}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern,amsmath,amssymb,amsthm,mathtools,microtype}
+\usepackage[hidelinks]{hyperref}
+\setlength{\emergencystretch}{3em}
+\allowdisplaybreaks
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\title{A conditional integer-multiplication saving of $7237/78125000$\\
+\large Constructive frames and partial source gauges}
+\author{icekylinx\\\small With substantial OpenAI GPT-6 Astra and Codex assistance}
+\date{9 October 2026}
+\begin{document}\maketitle
+\begin{abstract}
+A cancellation-free interval and cube producer, backward nondegenerate
+frame enlargement, and chronologically ordered partial source gauges give
+a strict complex saving $92651/10^9$. Applying the stopped opposite-bank
+compiler to a literal deferred bit word supplies the bit interface.
+Under the retained analytic, semantic-precision and fixed-tape interfaces,
+the exact assembly yields $T(n)=O(n(\log n)^{1-\kappa})$ with
+$\kappa=7237/78125000=9.26336\times10^{-5}$.
+\end{abstract}
+
+\section{Scope and attribution}
+This note supplements the stopped product-ring proof
+\path{notes/stopped-product-note.tex}, the public counterpart of the
+preceding complete handoff being icekylinx's
+\href{https://github.com/CrocSwap/integer-mult-bounds/pull/104}{PR104}.
+Its opposite-bank factorization, stopped ordinary interface, rational
+retained-center decoder, general binary Gauss normal form, and signed
+two-stage endpoint are retained. The analytic and machine proof map is
+\path{notes/structured-bulk-assembly.tex}; all its analytic, ordered-affine,
+exact-recovery, prime-existence and eventual setup hypotheses remain.
+Finite exact certificates establish the displayed arithmetic margins;
+they do not replace those hypotheses or a proof of the physical schedule.
+
+The bit word is Zhihao Chen's
+\href{https://github.com/CrocSwap/integer-mult-bounds/pull/97}{PR97},
+pinned at \texttt{e15350b66e4bf796153a68c240a6cc1b788abe3e},
+using Swapnil Jain's underlying witness pinned at
+\texttt{741e7aa078392553815df7926ee17ac5e25a8c38}.
+The interval and core-aware pair primitive is taken from
+\href{https://github.com/CrocSwap/integer-mult-bounds/pull/62}{PR62},
+\texttt{ikeboy/integer-mult-bounds}, commit
+\texttt{ad0f25ff7b23cff7f08ad237c2254e6ecf74257e}.
+The present cube integration, frame and gauge construction, and composition
+are by icekylinx with substantial OpenAI GPT-6 Astra assistance and Codex
+integration. The predecessor's attribution of Aurel Prosz, Swapnil Jain,
+Zhihao Chen, Rohan Arun, Dominik Scholz, RaD/hipotures and the underlying
+framework remains in force; source notices are retained in
+\texttt{NOTICE} and \texttt{SOURCES.json}.
+
+\section{Finite producer and bit interface}
+\input{notes/partial-gauge-construction.tex}
+\section{Constructive frames and chronological gauges}
+\input{notes/partial-gauge-frames.tex}
+\section{Precision, rows and exact assembly}
+\input{notes/partial-gauge-assembly.tex}
+\end{document}
diff --git a/notes/stopped-product-assembly.tex b/notes/stopped-product-assembly.tex
new file mode 100644
index 0000000000..3932d3ade5
--- /dev/null
+++ b/notes/stopped-product-assembly.tex
@@ -0,0 +1,108 @@
+\subsection{Simultaneous row reserves}
+Use a conservative global width bound $e$ for every adapter and child.
+For each recursive family $j$, the least integer $t_j$ with
+$m_j^{t_j}>2r_j^{t_j}$ gives depth at most
+$t_j\lceil\log_2\max(2,e)\rceil$ plus a fixed constant.
+The new coarse bit family has $(t_j,\operatorname{bitlength}W_j)=(16,28)$,
+the ordinary leaf family $(9,28)$, and the complex family $(17,28)$.
+The required reserve is the product of all three stocks:
+\[
+ Q=W_c^{D_c}W_b^{D_b}W_{\rm old}^{D_{\rm old}}.
+\]
+After the complex and coarse divisions, the old stock remains available
+to every peeled-atom or stopped-leaf call. Sequential copies and outer
+ordinary wrappers reuse restored rows; they introduce no independent
+row factor. The combined coefficient is
+\[
+ 16\cdot28+9\cdot28+17\cdot28=1176,
+ \qquad4000-\frac{51}{25}\,1176=\frac{40024}{25}>0.
+\]
+Thus the retained logarithmic comparison, for eventual $e\le Cp$,
+supplies a sufficient $p^{4000}$ stock including the fixed constants.
+Use one physically preceding transformed prefix and one complete-row
+padding, as in \path{notes/structured-bulk-assembly.tex}. Complete
+spectators and parked fields are preserved by every descendant.
+The suffix condition
+$b_{\rm in}^{1-\epsilon}>16000(\log_2b_{\rm in}+8)$ is eventual
+for every fixed $\epsilon<1$.
+
+\subsection{Analytic transfer and selected parameters}
+The retained analytic proof map in
+\path{notes/structured-bulk-assembly.tex} supplies the active-masked
+arbitrary-source router with paid repair scans, the rounded phase-cell
+inverse, persistent bulk halos with original period cuts, sequential
+overlap tapes and factor pages, prime packing and exact recovery.
+The new ordinary bit primitive has precisely the arbitrary-width,
+complete-spectator and restored-row contract consumed by that router.
+The new complex primitive supplies the exact completed tensor contract
+on one common grid. Its outer grid conversions add only logarithmic
+work. The changed finite implementations therefore meet the same
+analytic interfaces, including the full bulk bound
+\[
+ O\!\left(Tp\left[p^\tau\operatorname{polylog}p+
+ d\operatorname{polylog}p+dp^\delta+w_G^2p^\delta+
+ dw_G^2p^\delta/L\right]\right)+n^{o(1)},
+\]
+where $L=2^{\lceil8\log_2p\rceil}$, Gaussian band width
+$w_G^2=O(p^{1-r})$, and normalization retains $\epsilon+r<1$.
+Here $w_G$ is unrelated to the address atom width.
+
+Use the proved ordinary bit saving $a_b=803380799/10^{13}$ and complex
+saving $b=1949/25000000$. To instantiate the retained assembly, weaken
+the bit parameter explicitly to
+\[
+ \beta=10^{-6},\qquad
+ a=(1-\beta)b-10^{-10}=\frac{1948995551}{25000000000000}<a_b.
+\]
+Thus the bit time bound also holds at $\tau=1-a$, and the complex
+leaf saving exceeds $a$ by exactly $10^{-10}$. Put $\sigma=1-b$ and
+\[
+\begin{gathered}
+ \eta=10^{-8},\quad q_0=a(1-2\eta),\quad c=q_0(1+\eta),\quad
+ \epsilon=\frac{1-\eta}{1+c+q_0},\\
+ \lambda'=1-q_0,\quad\lambda=(\tau+\lambda')/2,\quad
+ g=\epsilon q_0,\quad r=(g+1-\epsilon)/2,\quad\delta=\eta/8.
+\end{gathered}
+\]
+The notation $q_0$ denotes the assembly saving parameter, not the
+fixed address prime. Since $\sigma<\tau$, the internal paid-adapter
+exponent is $\tau$; the stopped complex leaf exponent is
+$\sigma+\beta(1-\sigma)$. Both are strictly below $\lambda'$.
+
+\begin{theorem}[Conditional fixed-tape multiplication bound]
+Under the retained analytic, exact-recovery and fixed-tape interfaces,
+the selected construction gives
+\[
+ T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad
+ \boxed{\kappa=\frac{194869}{2500000000}=0.0000779476}.
+\]
+\end{theorem}
+\begin{proof}
+The exact certificate checks 47 strictly positive conditions of the
+retained semantic/bulk assembly with the new finite constants. They
+include the primitive and leaf orderings, compact reservations and
+geometry, linear precision, row product, short-record fallback,
+normalization, phase-cell/band separation, prime supply and all
+artificial-boundary charges. The seven final savings are
+\[
+ 1-\epsilon(1+c),\quad a,\quad g,\quad a,\quad
+ \min(1-\epsilon-\delta,r-\delta),\quad
+ 1-\epsilon-\delta,\quad\epsilon.
+\]
+Their exact minimum is $g$, and the prefix margin is $g+\eta$.
+Exact rational subtraction gives
+\[
+ g-\kappa=
+ \frac{20668543364879972849164027819}
+      {312548724888044126663502511122500000000}
+ >6.6129027953\times10^{-11}>0.
+\]
+The strict gaps absorb fixed logarithms, including the common-grid
+conversions. The stopped atom recurrence supplies the ordinary
+interchange interface, the rational-center construction supplies the
+exact complex interface, and the product stock supports all nested
+recursions. The unchanged analytic transfer then proves the stated
+bound, with its constructive setup, descriptor and catalogue domination,
+prime-existence and exact-recovery thresholds retained as eventual
+conditions.
+\end{proof}
diff --git a/notes/stopped-product-complex.tex b/notes/stopped-product-complex.tex
new file mode 100644
index 0000000000..4a9037137c
--- /dev/null
+++ b/notes/stopped-product-complex.tex
@@ -0,0 +1,137 @@
+\subsection{All-disjoint retained centers}
+Use two factors $h=24$. For triple inputs write $G=\sum_TX_T$,
+$T_i=\sum_{T\ni i}X_T$, and retain all $A_i=G-T_i$.
+Each input triple avoids exactly 21 points, so
+\[
+ \sum_{i=1}^{24}A_i=21G,\qquad G=\frac1{21}\sum_i A_i.
+\]
+For a target triple $S$ the central correction is
+\[
+ G-\frac12\sum_{i\in S}A_i
+       =\frac{\sum_{i\in S}T_i-G}{2}.
+\]
+An input triple meeting $S$ in $j$ points has central coefficient
+$(j-1)/2$. Adding half the disjoint sum and subtracting half the
+intersection-two sum leaves coefficient one exactly when $j=3$.
+The scalar source-to-scatter transfer is therefore the identity.
+The transparent old-value subtraction and new-value addition use this
+identity only on fresh contributions and restore arbitrary old values.
+
+The base-two paired exclusion and aligned pair-star producer retains
+nested binary labels: common-pair supports have their orthonormal triple
+span, other ordinary sums have their coordinate cover. Each retained
+$A_i$ has the nondegenerate coordinate hyperplane excluding $i$,
+of dimension 23. Hence all copied-center arguments and general
+nondegenerate binary residuals of the preceding proof apply. The
+selected local counts are
+\[
+ v=2024,\quad c_h=45764,\quad q_h=8120,\quad
+ |\mathcal M|=8966,\quad R=44918,\quad\ell=552.
+\]
+The old internal histogram, starting at rank zero, is
+\begin{small}\begin{align*}
+[&8930,49042,30844,15972,11883,8272,6399,4356,6682,2263,\\
+ &5109,1808,5317,1733,5391,1757,4908,3290,6233,4363,\\
+ &4056,264,564,24,24].
+\end{align*}\end{small}
+It has rank mass $24R+2\ell$. Replace 24 rank-24 original transitions
+by 24 rank-one transitions; all rank-23 copied transforms remain.
+The new mass is $24R+\ell$.
+
+\subsection{Physical endpoints and strict complex moment}
+The copied-center two-stage complex contract in
+\path{notes/copied-centers-complex.tex} is unchanged by the scalar
+center decoding. For the tensor triple line $U$, let
+$T=C_U$, $F=C_I$, $E=C_{U^\perp}=FT^{-1}$.
+The two data outputs are $A=-Fy$ and $B=FT^{-2}x+Ey$.
+Add $T^{-1}$ applied to a copy of $A$ into $B$, paying one rank-one
+child per data pair. The inherited input/output $Z$ phases and
+$i^{9f}$ scalar normalize $FT^{-2}$ to $F$; sign and bank corrections
+complete the exact tensor $C$. Auxiliary source/sink operators remain
+$C_{D_0}$ and $FC_{D_0}$ in the second stage. Every original role
+therefore returns the exact selected-axis tensor operation on arbitrary
+inputs. This algebraic statement holds over $\mathbb Q(i)$ as well as
+over Gaussian dyadics.
+
+With $m_c=576$, $N_c=v^2=4096576$, $B_c=vR=90914032$, put
+\[
+ W_c=2N_c+2B_c=190021216,\quad L_c=2v\ell=2234496,
+\]
+\[
+ s_c=W_cm_c-N_c+L_c=109450358336,\qquad N_c-L_c=1862080.
+\]
+The complete list is $2B_c$ width-552 auxiliary exteriors,
+$2N_c$ width-529 data macros, $2vH'_r$ width-$r$ internal children,
+$4N_c$ width-23 data fronts, and $N_c$ width-one endpoint copies.
+These disjoint classes include both kinds of paid copies and sum to
+$s_c$. All binary residuals, including alternating forms, use the retained
+one-child Gauss normal form with paid bit adapters and unit phases.
+At $a_c=1949/25000000$ the exact rational enclosure gives
+\[
+ \sum_r\frac{n_r}{W_c}(r/m_c)^{1-a_c}
+ <1-1.3338373969\times10^{-10}.
+\]
+
+\subsection{One exact odd-denominator grid}
+The only new odd divisor is 21. Let $d$ bound the root axis count,
+$D$ bound its maximum complex recursion depth, and $G_0$ bound the
+number of grouped scalar operations per node. Choose
+\[
+ K=G_0(D+1),\qquad \text{common grid }2^{-P}21^{-K}.
+\]
+Root inputs and arbitrary dirty auxiliaries start Gaussian dyadic.
+Multiply their integer numerators by $21^K$ once at the outer boundary.
+Every local coefficient contains at most one factor 21 in its denominator,
+so a local finite program adds at most $G_0$ to the odd-denominator
+exponent. A completed $u$-axis child has Gaussian-dyadic coefficients
+and therefore preserves its input's existing odd-denominator exponent;
+it adds no new odd factor. Only the active chain of unfinished calls
+can accumulate new factors. Its length is at most $D+1$, proving that
+all intermediates lie on the common grid and every local division by
+21 is an exact integer division of its encoded numerator.
+
+This divisibility claim uses the root dyadic-input invariant, not an
+assumption about arbitrary numerators on the finest grid. Internal values
+may have odd denominators. They are neither rounded nor re-encoded when
+a child returns. At the completed outer return the operator is the exact
+Gaussian-dyadic tensor on root dyadic inputs, so the final numerators
+are divisible by $21^K$; exact unscaling restores dyadic encoding.
+
+Since the largest child is $552<576$, $D=O(\log d)$ and
+$K=O(\log d)$ with fixed constants. Constant-divisor long division
+is linear in numerator length. Initial and final scaling by $K$
+successive constant multiplications/divisions cost $O(V\log d)$ and
+occur only at the outer boundary. The extra numerator length is
+$O(\log d)$, preserving the retained $O(p)$ word bound. This overhead
+is subordinate to the positive-power complex layer bound.
+
+For explicit semantic constants use
+\[
+ g_h=4(c_h+v)+10v+4hv+4h^2+8h+8=408200,
+\]
+\[
+ G_0=N_c+2v(g_h+2h)=1656684480,
+\]
+where endpoint additions and copied-center scalar work are included.
+The complete child contributes at most $u$ binary denominator bits
+and row-norm factor $2^u$, with no additional odd exponent. A local
+21-division contributes fewer than five denominator-height bits,
+covered by the fixed scalar charge. Set
+\[
+\begin{aligned}
+ E_0&=64(W_c+m_c+G_0+1)^3=403063477262941121443761178688,\\
+ \mathcal E&=2G_0W_c^2+8s_c+4W_c+4+32m_c\\
+           &=119639333593035096270287748<E_0,\\
+ B_0&=s_c+E_0,\qquad C_0=32m_cB_0^2.
+\end{aligned}
+\]
+With $f=\lfloor e/m_c\rfloor$, the completed-child induction is
+\[
+ A(e)\le A(552f)+s_cf+E_0\le2B_0e,
+\]
+since $2B_0(m_c-552)\ge s_c+E_0$. It counts total denominator height
+and magnitude, including the local odd factor. The retained prefix and
+normalization allowance $18e$ fits inside $C_0e$. Together with the
+$O(\log d)$ common-grid encoding overhead this gives $C_1=1$ and
+$O(p)$ records for $d=\lfloor b_{\rm in}^{\epsilon}\rfloor$,
+$p=6b_{\rm in}$, fixed $\epsilon<1$.
diff --git a/notes/stopped-product-factorization.tex b/notes/stopped-product-factorization.tex
new file mode 100644
index 0000000000..94f7b81397
--- /dev/null
+++ b/notes/stopped-product-factorization.tex
@@ -0,0 +1,112 @@
+\subsection{Atom algebra and the global frame convention}
+Fix an odd prime $q$ and an atom width $w\ge1$ throughout one invocation.
+An atom is a consecutive $w$-digit radix-$q$ field. A field of $f$ atoms
+is the product ring $R_f=(\mathbb Z/q^w\mathbb Z)^f$, with rational
+unit coefficients acting componentwise. Let $C_f$ reverse the atoms,
+preserving their internal digits. The target is
+\[
+ F_n(H,D)=(C_nD,C_nH).
+\]
+It acts on two contiguous chunks with arbitrary complete spectators.
+The product-ring interpretation preserves the original complete address
+set and its lexicographic order. No standalone reversal is assumed free.
+
+At a node with $n=mf$ atoms, conjugate every frame, source gauge and
+endpoint of the finite network by the same $\operatorname{diag}(I,C_n)$.
+For a projector $P$ the new partial swap is
+\[
+ D'_P=\begin{pmatrix}
+ (I-P)\otimes I_f&(PC_m)\otimes C_f\\
+ (C_mP)\otimes C_f&C_m(I-P)C_m\otimes I_f
+ \end{pmatrix}.
+\]
+Global conjugation preserves common-frame scalar identities, nested
+residuals, copied-center schedules and arbitrary auxiliary restoration.
+The full endpoint is now $F_n$. These frame choices define the logical
+interpretation of physical inputs; they are not preprocessing passes.
+
+\subsection{Simultaneous generic basis and full involution factorization}
+For every proper nonzero residual projector $P_j$ of rank $r_j$, require
+all leading principal minors of sizes $1,\ldots,r_j$ to be nonzero in
+one common rational basis. For each individual minor, similarity to
+$\operatorname{diag}(I_{r_j},0)$ witnesses nonvanishing. Clearing
+denominators and taking the finite product on irreducible
+$\operatorname{GL}_m$ proves a common rational choice exists; exact
+rational enumeration terminates. Include exteriors, data residuals,
+internal edges, copied centers and endpoint corrections in this finite
+list. The basis is fixed label preparation, not an executed dense map.
+
+For $M=PC_m$, every northeast $i$-by-$j$ rectangle has rank
+$\min(i,j,r)$: after reversing its columns it is the leading rectangle
+of $P$ and contains the corresponding nonsingular leading square.
+Hence physical lower-triangular matrices $L,R$ give
+\[
+ LMR=\Pi=EF^t,\quad E=(e_1,\ldots,e_r),\quad
+ F=(e_m,e_{m-1},\ldots,e_{m-r+1}).
+\]
+The full involution, not just this corner, must be normalized.
+At $f=1$, put $Q=\operatorname{diag}(L,R^{-1})$ and
+$g=QD'_PQ^{-1}$. Since $\operatorname{rank}(g-I)=r$ and its upper-right
+block is $EF^t$, there are $A\in\mathbb Q^{r\times m}$ and
+$B\in\mathbb Q^{m\times r}$ with
+\[
+ g-I=\binom EB(A,F^t),\qquad AE+F^tB=-2I_r.
+\]
+The second identity follows from $g^2=I$ and the full column and row
+ranks of the factorization. Define
+\[
+ K=-(B+F)E^t+FA(I-EE^t),\qquad
+ S=\begin{pmatrix}I&0\\K&I\end{pmatrix}.
+\]
+Then $KE=-B-F$, $F^tK=A+E^t$, and direct multiplication yields
+\[
+ SgS^{-1}=I+\binom E{-F}(-E^t,F^t)
+ =\begin{pmatrix}I-EE^t&EF^t\\FE^t&I-FF^t\end{pmatrix}.
+\]
+This swaps precisely the first $r$ head coordinates and last $r$ tail
+coordinates in opposite orders, fixing all others.
+
+For $f$ atoms use $Q_f=\operatorname{diag}(L\otimes I_f,R^{-1}\otimes I_f)$
+and $S_f=\left(\begin{smallmatrix}I&0\\K\otimes C_f&I\end{smallmatrix}\right)$.
+The rank factorization lifts with $\bar E=E\otimes I_f$,
+$\bar F=F\otimes C_f$, $\bar A=A\otimes I_f$ and
+$\bar B=B\otimes C_f$. Its middle cross block is $\Pi\otimes C_f$.
+Since $C_r\otimes C_f=C_{rf}$ on the selected physical intervals,
+it is exactly one child $F_{rf}$ of the same family. There is no
+recursive orientation conversion or extra gathering.
+
+Choose $q$ after the finite basis and factorizations, excluding all
+needed nonzero numerators, denominators and pivots. The identities hold
+on every atom ring. They use only componentwise scalar arithmetic;
+reversal commutes with those scalars, without inter-atom carries.
+
+\subsection{Selected finite network}
+Use the retained compatible positive-label producer on both axes $h=23$:
+\[
+ v=1771,\quad c=34764,\quad q_h=5336,\quad
+ |\mathcal M|=1324,\quad R=38776,\quad\ell=506.
+\]
+The copied-center lemma changes $H_{23}$ by $-23$ and $H_1$ by $+23$.
+Thus $\sum_r rH'_r=23R+506$; the rank-22 copies are still paid.
+The new common basis preserves the producer's scalar map, label
+inclusions, matching and endpoint identities. With
+\[
+ m_b=529,\quad N_b=3136441,\quad B_b=68672296,\quad
+ W_b=143617474,\quad L_b=1792252,
+\]
+the complete list is $2B_b$ width-506 auxiliary exteriors,
+$2N_b$ width-484 data residuals, $4N_b$ width-22 data fronts,
+$2vH'_r$ width-$r$ local residuals, and $N_b$ width-one endpoint copies.
+Every call has volume fraction $1/W_b$, and
+\[
+ s_b=W_bm_b-N_b+L_b=75972299557,\qquad W_bm_b-s_b=1344189.
+\]
+Every width is at most $506<m_b$. For $a_*=4019/50000000$,
+$\tau_*=1-a_*$, the exact rational moment certificate gives
+\[
+ \mu_{\tau_*}:=\sum_r\frac{n_r}{W_b}(r/m_b)^{\tau_*}<1,
+ \qquad 1-\mu_{\tau_*}>4.2909639537\times10^{-10}.
+\]
+It uses the retained 24-term positive atanh-series logarithm enclosure,
+its geometric tail, range reduction and upward rational exponential
+bound. Also $\mu_1=s_b/(W_bm_b)<1$. These are the two gaps used below.
diff --git a/notes/stopped-product-interface.tex b/notes/stopped-product-interface.tex
new file mode 100644
index 0000000000..20c44ce486
--- /dev/null
+++ b/notes/stopped-product-interface.tex
@@ -0,0 +1,95 @@
+\subsection{Paid atom adapters and remainders}
+Expand each fixed lower-triangular formal factor into unit scalings and
+additions from earlier to later formal fields. Execute intrabank targets
+in descending formal-field order, so earlier controls still have their
+required original values. Every atom update reads only physically earlier
+controls. In a cross-bank shear the update is
+\[
+ D_{i,t}\leftarrow D_{i,t}+k_{ij}H_{j,f-1-t}.
+\]
+Even with the reversed source index, the source lies in the earlier head
+bank. The retained ordered-affine streaming primitive applies to each
+ordinary width-$w$ atom update, costing $O(V)$ per pass.
+There are $O(f)$ passes per fixed matrix and thus $O(Vn)$ per finite
+node. The reversed control is an indexed earlier atom, not a reversed
+field operation. Atom loops use binary descriptors on a fixed tape set;
+all inactive digits remain complete spectators. The inherited parking
+schedule never scans through parked ancestors, and polynomial descriptor
+work is absorbed by the complete stream volume.
+
+For $n=m_bf+\delta$, $0\le\delta<m_b$, pair the first $\delta$
+head atoms with the last $\delta$ tail atoms in reverse order. Each pair
+uses the retained ordinary width-$w$ interchange. The remaining two
+contiguous $m_bf$-atom intervals receive the finite construction.
+When $n<m_b$, at most $m_b$ such ordinary calls complete $F_n$.
+Both cases cost $O(Vw^{1-a_{\rm old}})$ in addition to the node adapters,
+where $a_{\rm old}=384599/10^{10}$. A radix-$q$ atom can be padded to
+the next binary power and treated by that ordinary interface, with
+volume factor below four and binary width $O(w)$, so the old leaf
+algorithm need not use the same address prime.
+
+\subsection{Stopped recurrence}
+Let $T(n,w)$ denote time normalized by logical volume. The complete
+recursive list, including copied streams, yields
+\[
+\begin{aligned}
+ T(n,w)&\le\sum_r\frac{n_r}{W_b}
+       T(r\lfloor n/m_b\rfloor,w)+C n+Cw^{1-a_{\rm old}}
+                         &&(n\ge m_b),\\
+ T(n,w)&\le Cw^{1-a_{\rm old}}&&(1\le n<m_b).
+\end{aligned}
+\]
+The constant includes finite scalar gates, split/merge, parking, all
+adapter passes and remainders at every node. Choose constants $A,B$
+using the positive gaps $1-\mu_1$ and $1-\mu_{\tau_*}$.
+Strong induction, with child widths at most $(r/m_b)n$, gives
+\[
+ T(n,w)\le An+Bn^{\tau_*}w^{1-a_{\rm old}}.
+\]
+Indeed the linear child terms use at most $\mu_1 An$, leaving the
+linear toll, and the other child terms use at most
+$\mu_{\tau_*}Bn^{\tau_*}w^{1-a_{\rm old}}$, leaving the peeled-atom
+toll since $n^{\tau_*}\ge1$. Enlarge $B$ for the bounded base cases.
+All constants are independent of both $n$ and $w$.
+
+\subsection{Outer ordinary wrapper and arbitrary widths}
+Write $C=C_n$. The map $F_n(H,D)=(CD,CH)$ obeys
+$F_nLF_n(H,D)=(H+D,D)$ for $L(H,D)=(H,D+H)$.
+Accordingly the chronological sequence
+\[
+ D\leftarrow D-H;\quad F_n;\quad D\leftarrow D+H;\quad
+ F_n;\quad D\leftarrow H-D
+\]
+is exactly $(H,D)\mapsto(D,H)$. The three lower updates cost $O(Vn)$
+in the atom model. Both calls reuse their restored row reserve.
+This is a constant outside the entire recursion, so it does not multiply
+any recursive child count or characteristic.
+
+For a radix-$q$ width $e$ choose $\theta=1/1000$,
+$w=\lceil e^\theta\rceil$ and $n=\lfloor e/w\rfloor$.
+Apply the ordinary wrapper to the atom body and the old ordinary
+algorithm to the fewer than $w$ residual digits. For large $e$ the cost is
+\[
+ O\!\left(V\left[e^{1-\theta}+
+ e^{(1-\theta)(1-a_*)+\theta(1-a_{\rm old})}\right]\right).
+\]
+The remainder is bounded by the second term. Thus the ordinary saving is
+\[
+ a_b=(1-\theta)a_*+\theta a_{\rm old}
+     =\frac{803380799}{10^{13}}=0.0000803380799.
+\]
+Since $\theta>a_b$, the paid adapter term is strictly lower order.
+All bounded widths use the old interface. Numerical padding of a binary
+chunk range to the least suitable radix-$q$ range has volume factor
+below $q^2$; it preserves numerical order and is removed after the
+completed ordinary wrapper, whose endpoint is ordinary interchange.
+
+If external rows are unavailable, use the inherited ordinary high-digit
+borrowing construction from \path{notes/batched-bit-rows.tex}.
+Borrow $O(\log e)$ high digits, exchange and join them into a preceding
+row field, pad that field once, act on the body, and separate the high
+parts. It applies literally to the completed ordinary wrapper; no claim
+about borrowed high positions is made for an unwrapped reversed call.
+Single-digit moves cost $O(V\log e)$, and small widths use the old
+algorithm. Atom indices and rational coefficients are finitely specified;
+the algorithm uses the same fixed tape set and complete spectator ranges.
diff --git a/notes/stopped-product-note.tex b/notes/stopped-product-note.tex
new file mode 100644
index 0000000000..a8b23254c8
--- /dev/null
+++ b/notes/stopped-product-note.tex
@@ -0,0 +1,76 @@
+\documentclass[11pt]{article}
+\usepackage[margin=1in]{geometry}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern,amsmath,amssymb,amsthm,mathtools,booktabs,longtable,microtype}
+\usepackage[hidelinks]{hyperref}
+\setlength{\emergencystretch}{3em}
+\allowdisplaybreaks
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\title{A conditional integer-multiplication saving of $194869/2500000000$\\
+\large Stopped product-ring interchange and rational retained centers}
+\author{icekylinx\\\small With substantial OpenAI GPT-6 Astra and Codex assistance}
+\date{9 October 2026}
+\begin{document}\maketitle
+\begin{abstract}
+A common rational basis and opposite bank orders reduce each projector
+residual to one contiguous reversed child over a fixed atom product ring.
+Ordered atom adapters are paid explicitly; stopping at a growing atom
+width and using the retained ordinary primitive supplies an ordinary
+bit interface. An independent 24-dimensional complex producer uses
+rational retained-center decoding on one exact common grid. The selected
+interfaces and retained semantic/bulk assembly give the conditional bound
+$T(n)=O(n(\log n)^{1-\kappa})$, where
+$\kappa=194869/2500000000=0.0000779476$.
+\end{abstract}
+
+\section{Scope and retained dependencies}
+The model is the retained fixed finite-alphabet multitape Turing machine.
+Finite graphs, rational bases, address primes, coefficient tables and tape
+counts are fixed independently of input length. Bit payload arithmetic is
+binary, while its address geometry is rational and then evaluated on atom
+rings. Complex payloads use exact Gaussian rational arithmetic with the
+specified common grid; their address labels remain binary.
+
+The proof predecessor is icekylinx's merged copied-center PR36,
+\texttt{11817ccacb564bb7f98789c20dc11d3fece207e3}, whose proof is
+\path{notes/copied-centers-note.tex}. Its ordinary interchange saving
+$a_{\rm old}=384599/10^{10}$, copied-center schedule in both orientations,
+two-stage data and auxiliary endpoint identities, general binary residual
+normal form, and fixed-tape complete-row contracts are retained.
+The present opposite-bank factorization, stopped product-ring interface,
+rational-center specialization and exact composition are by icekylinx,
+with substantial OpenAI GPT-6 Astra assistance and Codex integration.
+Earlier own contributions include PR10 batching, PR18 partial swaps,
+PR24 endpoint gauges and PR32 structured blocks and semantic composition.
+
+Aurel Prosz (Paureel) supplies the two-stage topology and paid endpoint
+copy, with Swapnil Jain's linked development credited by Zhihao Chen's
+PR29. The retained leaf proof also credits Rohan Arun's PR31 corner method
+and Dominik Scholz's PR33 parameterization. Zhihao Chen's PR21/23 and the
+RaD/hipotures PR20 source provide the phase endpoints, semantic guard,
+arbitrary-source routing, phase-cell inverse and bulk locality/tape
+interfaces. Their pinned proof map remains in
+\path{notes/structured-bulk-assembly.tex}; adopted sources and notices are
+in \path{references/semantic-bulk/} and \path{references/copied-centers/}.
+The underlying Douglas Colkitt framework, OpenAI manuscript, eumemic,
+Bortlesboat, dleen, earlier finite contributions and Harvey--van der Hoeven
+analytic work retain their credits and recorded AI assistance in
+\texttt{NOTICE} and \texttt{SOURCES.json}.
+
+The conclusion remains conditional on those analytic, ordered-affine,
+exact-recovery and machine interfaces and their eventual setup and
+prime-existence thresholds. The new algebraic and tape constructions
+below establish the changed finite interfaces; exact arithmetic certifies
+their selected counts and margins.
+
+\section{One reversed child for each projector}
+\input{notes/stopped-product-factorization.tex}
+\section{Stopped atom recursion and ordinary interchange}
+\input{notes/stopped-product-interface.tex}
+\section{Rational centers and exact semantic precision}
+\input{notes/stopped-product-complex.tex}
+\section{Product rows and exact assembly}
+\input{notes/stopped-product-assembly.tex}
+\end{document}
diff --git a/notes/three-stage-cover-assembly.tex b/notes/three-stage-cover-assembly.tex
new file mode 100644
index 0000000000..983ffb6ac8
--- /dev/null
+++ b/notes/three-stage-cover-assembly.tex
@@ -0,0 +1,117 @@
+\subsection{Local inventory and strict supplier moments}
+The selected complex local word has
+\[
+ h=24,\quad v=2024,\quad c=91770,\quad q_{\rm roots}=8120,
+ \quad |\mathcal M|=71185,\quad R=28705,\quad\ell=552.
+\]
+Its complete scalar word, full backward subspace intersections and 4,599
+partial source gauges are regenerated from the pinned compressed DAG by
+\path{scripts/three_stage_cover_producer.py}. The source and target chains,
+remaining first transitions, inverse cleanup and copied centers remain in
+\path{certificates/three-stage-cover-complex-input.json}.
+At each cover vertex its child inventory is exactly the three-stage table
+proved above. Thus $m_c=70$, $w_c=90163$, $s_c=6309018$ and
+$w_cm_c-s_c=2392$; the largest child is 68.
+
+For $a_c=786999/2500000000$, the rational enclosure proves
+\[
+ \frac1{w_c}\sum_r n_r(r/70)^{1-a_c}
+ <1-2.3047018455\times10^{-10}.
+\]
+The bit local ledger has $h=23,v=1771,R=28866,\ell=506$.
+Three copies of its physical local increments and exteriors give
+$m_b=67,w_b=90140,s_b=6037356$, with deficit 2024.
+The contaminated moment includes the entire $32\cdot67^2$-child fallback
+on a fraction $10^{-16}$ of every ideal edge. At $a_0=747/2000000$ its
+strict gap exceeds $6.3397366396\times10^{-7}$, and its rank moment also
+contracts. The stopped ordinary saving is
+\[
+ a_b=(1-10^{-3})a_0+10^{-3}\frac{384599}{10^{10}}
+     =\frac{3731649599}{10^{13}}.
+\]
+These are exact rational certificates; the lower tolerances displayed
+here do not substitute for their saved fractions.
+
+\subsection{Full finite router and completed-child guard}
+Let $V_0=|O(70,2)|$, $N=V_0v$, $W=V_0(2v+3R)$ and $s=V_0s_c$.
+All these finite counts appear in the certificate. Define
+\[
+ L=4(c+v)+10v+4hv+4h^2+8h+8+2h
+       +8Rv(M+16),
+\]
+where $M$ is the number of actual local mixer operations.
+The source-witness bound for the positive DAG gives dirty-readout
+coefficients of magnitude at most $h+4$, with denominators dividing
+$2(h-3)$. The last term conservatively pays their binary-addition
+implementation. Include the finite group and fixed Clifford routers by
+\[
+ G=N+3V_0L+64(m_c+1)^3W^2.
+\]
+A fixed role permutation has at most $W^2$ elementary exchanges; the
+polynomial factor also covers the binary matrix and quadratic-phase
+adapters. These finite costs retain their ordinary movement charge.
+Set
+\[
+ E=64(W+m_c+G+1)^3,\quad B=s+E,\quad C_0=32m_cB^2.
+\]
+The exact calculation checks
+\[
+ 2GW^2+8s+4W+4+32m_c<E,\qquad
+ 2B(m_c-68)\ge s+E,\qquad 2B+18<C_0.
+\]
+Every completed child is the exact Gaussian-dyadic $C^{\otimes u}$
+operator on all roles, giving the inherited semantic constant $C_1=1$.
+One grid $2^{-P}21^{-K}$, with $K=G(D_c+1)$, contains every unfinished
+call's rational-center intermediates. Completed children preserve the
+incoming odd-denominator exponent. Only the completed outer operation on
+dyadic input justifies unscaling; recursive returns are not rounded.
+The existing outer exact-integer conversion costs $O(V_{\rm bits}\log d)$.
+
+\subsection{External rows and exact composition}
+The complex group has 2,415 bits and its role stock 2,431 bits.
+The least halving degree for $70\to68$ is 24, contributing 58,344.
+Including conservative previous bit and ordinary-leaf reserves gives
+\[
+ 58344+9909+252=68505,\qquad
+ 160000-\frac{51}{25}\,68505=20249.8>0.
+\]
+Thus the fixed external reserve $p^{160000}$ meets the inherited product
+stock contract. The width-dependent q-adic selectors are supplied by
+the separate internal borrowing proof, rather than multiplied into this
+external polynomial stock. One preceding prefix and one final padding
+suffice, and sequential completed calls reuse restored rows.
+
+Put $b=a_c$, $\eta=10^{-8}$, $\beta=10^{-6}$ and
+\[
+ a=(1-\beta)b-10^{-10}<a_b,\quad \tau=1-a,\quad\sigma=1-b,
+ \quad q_0=a(1-2\eta),\quad c_0=q_0(1+\eta),
+\]
+\[
+ \epsilon=\frac{1-\eta}{1+c_0+q_0},\quad
+ \lambda'=1-q_0,\quad\lambda=(\tau+\lambda')/2,
+ \quad g=\epsilon q_0,\quad r=(g+1-\epsilon)/2,\quad\delta=\eta/8.
+\]
+All 47 retained strict inequalities hold. The seven assembly savings are
+\[
+ 1-\epsilon(1+c_0),\ a,\ g,\ a,\
+ \min(1-\epsilon-\delta,r-\delta),\ 1-\epsilon-\delta,\ \epsilon.
+\]
+Their minimum $g$ exceeds $3146011/10^{10}$ by more than
+$3.4173577301\times10^{-12}$.
+\begin{theorem}
+Under the retained analytic, semantic-precision, exact-recovery and
+fixed-tape interfaces, the selected construction gives
+\[
+ T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad
+ \boxed{\kappa=\frac{3146011}{10000000000}=3.146011\times10^{-4}}.
+\]
+\end{theorem}
+\begin{proof}
+The weighted bit construction supplies completed ordinary interchange,
+including its internally restored rows. The signed complex cover supplies
+the exact completed tensor on every role. The literal semantic charge and
+external reserve satisfy the unchanged precision and row contracts.
+Substituting the displayed parameters in the retained semantic/bulk
+transfer gives the seven savings above. Their positive excess absorbs
+the fixed logarithmic overheads and yields the stated conditional bound.
+\end{proof}
diff --git a/notes/three-stage-cover-bit.tex b/notes/three-stage-cover-bit.tex
new file mode 100644
index 0000000000..f05225d21f
--- /dev/null
+++ b/notes/three-stage-cover-bit.tex
@@ -0,0 +1,264 @@
+\subsection{Atom contract and common endpoint}
+Fix an odd prime $q$, chosen below, and put $R_w=\mathbb Z/q^w\mathbb Z$.
+An atom is a complete consecutive $w$-digit address field. Write $C_n$ for
+reversal of the $n$ atom positions, without reversing digits inside an atom.
+The recursive contract is the involution
+\[
+ F_A(H,D)=(AC_nD,C_nA^{-1}H),
+\]
+where $A$ is a diagonal array of units on the head atoms. Its entries may
+depend on exposed ancestor row fields and the atom index, but not on the
+fresh role fields of this invocation or on the head/data values. Only the
+polynomial-time descriptors generated by the construction are required;
+no constant-cost oracle for arbitrary functions of a prefix is assumed.
+Every call preserves all complete spectator fields and restores its row
+coordinate. The original unweighted call has $A=I$.
+
+For a split idempotent $P$ on the head coordinates set
+\[
+ D_P=\begin{pmatrix}I-P&PC_n\\C_nP&I-C_nPC_n\end{pmatrix},
+ \qquad F=D_I.
+\]
+These are involutions, and $D_P-I$ has split rank $\operatorname{rank}P$.
+Nested commuting-projector identities are polynomial identities and require
+no symmetry of $P$. For $n=mf$ and $G=g\otimes I_f$ define
+\[
+ Q_g=\operatorname{diag}(G,C_nGC_n).
+\]
+Then $Q_gF=FQ_g$ and $Q_gD_PQ_g^{-1}=D_{GPG^{-1}}$.
+Consequently conjugating every base frame by any $g\in\operatorname{GL}_m(R_w)$
+leaves the same full physical endpoint $F$.
+
+Put $Q_A=\operatorname{diag}(A,I)$. Conjugate the entire completed scalar
+word by $Q_A$. Scalar gates mix roles having the same address map, so they
+commute with $Q_A$. A current role routing also commutes with $Q_A$, since
+it changes only fresh fields. Thus only address residuals change, and the
+completed endpoint is $Q_AFQ_A^{-1}=F_A$. More explicitly, when a port
+changes charts from $g$ to $gR_s$, its local label changes from $M$ to
+$R_s^{-1}MR_s$, and
+\[
+ (gR_s)(R_s^{-1}MR_s)(gR_s)^{-1}=gMg^{-1}.
+\]
+This routing is a change of row/chart, not an uncharged dense address map.
+The adapted chart $Q_AQ_gQ_A^{-1}$ commutes with $F_A$. None of these
+conjugations is executed as an extra normalization pass.
+
+\subsection{Explicit rational three-stage geometry}
+Use the retained PR97 local producer with
+\[
+ h=23,\qquad v=1771,\qquad R_{\rm loc}=28866,\qquad \ell=506.
+\]
+Its physical projector ranks and scalar word are unchanged. On
+$\mathcal A=\mathbb Q^{23}$ take the form $H_0=(I-J/9)/2$.
+For a triple $S$ let $q_S=\chi_S$; then $q_S^tH_0q_S=1$.
+Fix $q_0=\chi_{\{1,2,3\}}$ and $K=q_0^\perp$. A rational basis is
+\[
+ k_i=e_i-(e_i^tH_0q_0)q_0,\qquad 2\le i\le23.
+\]
+Projection along $q_0$ is injective on the span of $e_2,\ldots,e_{23}$,
+since $q_0$ has nonzero first coordinate. Thus this is indeed a basis.
+Let $\mathcal B,\mathcal C$ be orthogonal copies of $K$ and put
+$\mathcal E=\mathcal A\perp\mathcal B\perp\mathcal C$, of dimension $m=67$.
+
+Choose a coordinate permutation $\pi_S$ taking $\{1,2,3\}$ to $S$.
+It preserves $H_0$. The rational isometry $R_{12,S}$ fixes $q_S$ and
+$\mathcal C$, and exchanges $\pi_Sk_i\in q_S^\perp\subset\mathcal A$
+with $k_i\in\mathcal B$. Define $R_{23,S}$ by exchanging the same vectors
+with their copies in $\mathcal C$, fixing $q_S$ and $\mathcal B$.
+Both maps are involutions. A hyperedge $(g,S)$ uses the invocations
+$gR_{12,S},g,gR_{23,S}$, with active spaces
+\[
+ g(\mathcal B+\mathbb Qq_S),\qquad g\mathcal A,
+ \qquad g(\mathcal C+\mathbb Qq_S).
+\]
+Their projectors commute; their pairwise and common intersection is the
+projector $P_{gq_S}$, and their sum is $I+2P_{gq_S}$.
+This follows from the orthogonal decomposition
+$\mathbb Qq_S\perp q_S^\perp\perp\mathcal B\perp\mathcal C$ before
+conjugation. These are the cumulative-projector three-stage transfer
+identities: the three scalar shears complete the interchange, with zero
+interstage connectors and no rank-one endpoint correction. Each port
+family visits the full cover bijectively, by right multiplication with
+its displayed involution. The physical ledger is therefore three copies
+of the local auxiliary, data and center histograms, and three copies of
+the exterior histogram $m-\text{nullity}$.
+
+\subsection{One weighted child on a generic residue class}
+Work first on one atom column. Let $g$ be its weighted residual involution,
+of split rank $r$ relative to $I$. If the cross block has generic northeast
+rank profile, unit lower row and column elimination, without normalizing
+pivots, gives
+\[
+ Lg_{XY}R=ED_pF_0^t,
+ \quad E=(e_1,\ldots,e_r),\quad
+ F_0=(e_m,\ldots,e_{m-r+1}),
+\]
+with invertible diagonal $D_p$. Conjugate by $\operatorname{diag}(L,R^{-1})$.
+Split rank and the unit pivot minor give a factorization
+\[
+ g-I=\binom EB(A_0,D_pF_0^t),\qquad
+ A_0E+D_pF_0^tB=-2I_r.
+\]
+For clarity, if $g-I=UV$, then $g^2=I$ gives
+$U(VU+2I)V=0$. The unit minor supplies a left inverse of $U$ and a right
+inverse of $V$, proving the identity over the local ring itself.
+Set
+\[
+ K=-(B+F_0D_p^{-1})E^t+F_0D_p^{-1}A_0(I-EE^t),
+ \qquad S=\begin{pmatrix}I&0\\K&I\end{pmatrix}.
+\]
+The identities $KE=-B-F_0D_p^{-1}$ and
+$D_pF_0^tK=A_0+E^t$ give
+\[
+ SgS^{-1}=\begin{pmatrix}
+ I-EE^t&ED_pF_0^t\\F_0D_p^{-1}E^t&I-F_0F_0^t
+ \end{pmatrix}.
+\]
+This is one weighted reversed interchange on the first $r$ head and last
+$r$ tail coordinates. All other factors are unit lower intrabank maps or
+lower cross-bank additive maps. Thus variable diagonal units remain in
+the child's weight array rather than becoming unsupported routing passes.
+
+For $n=mf$, group atoms by coarse coordinate $i$ and column $j$.
+If $L_P,R_P,D_P$ are unweighted normalizers for the current projector and
+$A_j=\operatorname{diag}(A[if+j])$, then
+$L_j=A_jL_PA_j^{-1}$ and $R_j=R_P$ are unit lower normalizers for
+$A_jPC_m$. The child's exact contiguous index rule is
+\[
+ A_{\rm child}[kf+j]=A[i_kf+j]D_P[k].
+\]
+The cross-bank atom pairing is reversed, and $C_r\otimes C_f=C_{rf}$.
+Hence the middle operation is exactly one $rf$-atom call of the same
+contract, with no interleaving or gathering. Its coefficients depend only
+on parent prefix fields, which become ancestor spectators at the child;
+the child's fresh roles come from disjoint unused row stock.
+
+All unit lower adapters require $O(n)$ atom passes. The retained ordered
+affine lemma implements each later-target shear in $O(V)$, including
+prefix-controlled offsets. The generated descriptors have polynomial cost
+in $w$: good-class evaluation follows one ancestral index per level;
+fallback evaluation branches by a constant per level. Since depth is
+$O(\log e)$ and $w=\lceil e^{1/1000}\rceil$, this costs $e^{O(1)}=w^{O(1)}$.
+Stored ancestor data use $O(w\log e)$ bits. The proof of the ordered affine
+lemma still applies with these controls: polynomial preparation per prefix
+is absorbed by the complete $q^w$ target fiber. Unrelated spectator counters
+are incremented by ripple counting rather than read in their entirety.
+
+\subsection{Weighted leaves and a genuine local-ring fallback}
+At one atom write $J_a=\left(\begin{smallmatrix}0&a\\a^{-1}&0\end{smallmatrix}\right)$,
+$U(a)=\left(\begin{smallmatrix}1&a\\0&1\end{smallmatrix}\right)$ and
+$L(b)=\left(\begin{smallmatrix}1&0\\b&1\end{smallmatrix}\right)$.
+Then
+\[
+ J_a=\operatorname{diag}(1,-1)U(a)L(-a^{-1})U(a),
+ \qquad U(a)=J_1L(a)J_1.
+\]
+Thus a weighted leaf uses four old ordinary atom interchanges, three
+prefix-controlled shifts and one fixed sign. With
+$a_{\rm old}=384599/10^{10}$ it costs $O(Vw^{1-a_{\rm old}})$.
+This constant occurs only at stopped leaves.
+
+A bad residue class cannot be treated by field Bruhat elimination:
+an entry zero modulo $q$ need not vanish modulo $q^w$. Instead Gaussian
+eliminate the full invertible residual of size $N_0=2m$ with unit pivots.
+Each invertible Schur complement has a unit in its next pivot column;
+row swaps and unnormalized additions clear that column. There are at most
+$N_0(N_0-1)$ additions and $N_0$ swaps. Pivot choices are determined by
+the unweighted low residue class: diagonal conjugation scales every
+Schur-complement row and column by units and preserves its zero pattern.
+The final diagonal has product a fixed sign, since the residual has
+determinant $(-1)^r$ and row swaps change only the sign.
+
+A lower addition is a paid additive adapter. An upper addition is
+conjugated into a lower addition by two ordinary coarse-field swaps.
+For a field of $f$ atoms, let $R_{\rm tail}$ reverse the later field.
+Then the ordinary weighted swap is
+\[
+ J_A=R_{\rm tail}F_AR_{\rm tail},\qquad
+ J_A(H,D)=(AD,A^{-1}H).
+\]
+Each $R_{\rm tail}$ is one unweighted reversed call on the first and last
+$\lfloor f/2\rfloor$ atoms, fixing a possible middle atom. Thus $J_A$
+costs at most three children of widths $f,\lfloor f/2\rfloor,
+\lfloor f/2\rfloor$. A paired diagonal scaling is $J_aJ_1$, at most six
+children. The full diagonal is a product of $N_0-1$ such pairs and a fixed
+sign. Across the two banks the reversed pairing already makes the swaps
+and diagonal pairs directly reversed calls; the preceding ordinary bounds
+remain conservative. Reindexing a tail weight by $j\mapsto f-1-j$ changes
+only its descriptor. Charging every addition as upper therefore gives
+\[
+ 6N_0(N_0-1)+3N_0+6(N_0-1)<32m^2.
+\]
+Every child has width at most $f$, hence ratio at most $1/m$. Coefficients
+are computed from constant-sized local-ring matrices and parent weight
+descriptors; only units are inverted. This proves the fallback cost,
+including its actual reversals, rather than assuming nonunit pivots away.
+
+\subsection{Finite covers, batching, and the strict moment}
+Take $q>2^{80}$ avoiding all fixed rational denominators and rank witnesses.
+At atom width $w$ use the finite cover $\operatorname{GL}_m(R_w)$ and its
+right-multiplication edges $g\mapsto gR_s$. Every vertex has the unique chart
+\[
+ g=g_0+qG_1,\quad g_0\in\operatorname{GL}_m(\mathbb F_q),
+ \quad G_1\in(\mathbb Z/q^{w-1}\mathbb Z)^{m^2},
+\]
+using fixed representatives for $g_0$. Right multiplication permutes $g_0$
+and acts affinely on $G_1$, with a carry determined by $g_0$. Its fixed
+dimensional rational linear part factors into fixed scalings, ordered
+shifts and a constant number of old width-$O(w)$ interchanges. Thus each
+fixed port routing costs $O(Vw^{1-a_{\rm old}})$.
+
+Let $W_0=|\operatorname{GL}_m(\mathbb F_q)|(2v+3R_{\rm loc})$.
+The full role count is $W(w)=W_0q^{m^2(w-1)}$. Split preceding rows into
+low role type, high matrix and unused stock. Batch \emph{all} high matrices
+for each low type in one stream. A frame edge is one child on this whole
+batch, of volume $V/W_0$, and routing is executed once per fixed operation
+type. Consequently neither the number of tapes nor the number of child
+programs grows with $w$. High cardinality cancels in the normalized moment,
+but its $m^2(w-1)$ digits are charged in the row reserve.
+
+For each fixed projector require the leading minors of sizes $1$ through
+$r$ to be units. Each cleared minor polynomial in $g_0$ is nonzero:
+similarity to $\operatorname{diag}(I_r,0)$ supplies a witness after the
+fixed rank is preserved modulo $q$. Their total degree is at most $m^3$.
+The polynomial zero bound and conditioning on invertibility (a factor at
+most two) give bad fraction at most $2m^3/q<10^{-16}$ for each edge type.
+No simultaneous union bound over edge types is needed, since their moments
+are averaged separately. Good classes use the whole-rank child; bad
+classes use the full $32m^2$-child fallback just proved.
+
+Let $H_r$ be the three-stage histogram described above and
+$W=2v+3R_{\rm loc}=90140$. The retained ledger gives
+\[
+ \sum_r rH_r=6037356,\qquad Wm-\sum_r rH_r=2024=2v-3\ell,
+ \qquad \max\{r:H_r>0\}=66.
+\]
+For $a=747/2000000$, $\tau=1-a$, and $b=10^{-16}$ the conservative moment is
+\[
+ \mu_\tau\le
+ \sum_r\frac{H_r}{W}(r/m)^\tau+
+ b\frac{32m^2\sum_rH_r}{W}(1/m)^\tau<1.
+\]
+The entire fallback is added; no displaced good contribution is subtracted.
+The exact script \path{scripts/three_stage_cover_network.py} and certificate
+\path{certificates/three-stage-cover-network.json}, from the retained axis
+\path{certificates/three-stage-cover-bit-input.json}, give
+$1-\mu_\tau>6.3397366396\cdot10^{-7}$.
+The certificate uses the 32-term positive atanh logarithm series with its
+geometric tail, $\exp(u)\le1+u+u^2/(2(1-u/3))$, and upward rounding to
+the grid $2^{-120}$. All inputs have $0\le u<3$; these are rational upper
+bounds, not floating-point root estimates. Since every child ratio is at
+most one and $\tau<1$, this same strict bound implies $\mu_1<1$.
+
+With $T(n,w)$ denoting time divided by logical volume, the fixed low-type
+profile $\{c_r\}$, including fallback classes, satisfies
+\[
+ T(n,w)\le\sum_rc_rT(r\lfloor n/m\rfloor,w)+Cn+Cw^{1-a_{\rm old}}
+ \quad(n\ge m),\qquad
+ T(n,w)\le Cw^{1-a_{\rm old}}\quad(n<m).
+\]
+Peeling fewer than $m$ leftover atoms uses the weighted leaf algorithm.
+The constants are independent of $w,n$ and ancestor values. Induction,
+using $\mu_1<1$ for the linear term and $\mu_\tau<1$ for the stopped term,
+gives $T(n,w)=O(n+n^\tau w^{1-a_{\rm old}})$. The maximum proper ratio
+$66/67$ has halving degree $47$, so recursion depth is $O(\log n)$.
diff --git a/notes/three-stage-cover-complex.tex b/notes/three-stage-cover-complex.tex
new file mode 100644
index 0000000000..c54cb3d5ff
--- /dev/null
+++ b/notes/three-stage-cover-complex.tex
@@ -0,0 +1,152 @@
+\subsection{Three invocation banks and a finite orthogonal cover}
+Use the completed $h=24$ triple word contributed by eumemic with Claude
+(PR117), with its retained licenses and notices. Its local profile is
+\[
+ v=2024,\quad R=28705,\quad \ell=552.
+\]
+The global cover below retains that word, its dirty readouts and copied
+centers. Let $E=A\perp B\perp C$ be the standard binary space with
+$\dim A=h$, $\dim B=\dim C=h-1$, and $m=3h-2$.
+For a triple $S\subseteq[h]$ put $q=q_S=\sum_{i\in S}e_i$.
+Choose $d\notin S$. The orthogonal involution
+$\tau_{q+e_d}(x)=x+(x\cdot(q+e_d))(q+e_d)$ sends $e_d$ to $q$;
+its images of the other coordinate vectors give an explicit orthonormal
+basis of $A\cap q^\perp$.
+Define $R_{12,S}$ to exchange this basis with the fixed basis of $B$,
+fixing $q,C$; define $R_{23,S}$ similarly using $C$ and fixing $q,B$.
+Both are orthogonal involutions, with
+\[
+ R_{12,S}A=B\oplus\langle q\rangle,\qquad
+ R_{23,S}A=C\oplus\langle q\rangle.
+\]
+Take $G=O(E)$ and three distinct invocation sets equal to $G$.
+The data position $(g,S)$ meets vertices $gR_{12,S},g,gR_{23,S}$,
+respectively. At a fixed vertex of any stage, the map from port $S$ to its
+position is bijective. All three local source vectors equal $gq_S$.
+Thus $N=|G|v$ and the persistent-role count is
+$W=|G|(2v+3R)$, using independent auxiliary banks.
+
+\subsection{Exact chronology and phase endpoints}
+Write $A-q=A\cap q^\perp$, and transport the following table by $g$.
+The middle stage is the complete reverse/complement word with source and
+target interchanged and the scalar sign reversed.
+\[
+\begin{array}{c|c|c|c|c}
+\text{stage}&\text{logical shear}&\text{active space}&X&Y\\\hline
+1&Y\gets Y+X&B+q&q\longrightarrow B+q&0\longrightarrow B\\
+2&X\gets X-Y&A&B+q\longrightarrow A+B&B\longrightarrow(A-q)+B\\
+3&Y\gets Y+X&C+q&A+B\longrightarrow E&(A-q)+B\longrightarrow E-q
+\end{array}
+\]
+The offsets are respectively $0$, $B$, and $(A-q)+B$, each orthogonal
+to its active space. They are fixed at each invocation: if
+$k=gR_{23,S}$, the third offset is $k(B+C)$, independently of its port.
+Every exit label is precisely the next entrance label. Any difference
+between exact representatives is reconciled by an actual rank-zero
+adapter; consequently there are no positive-rank data connectors.
+The forward stages retain their two data-chain histograms; reversing,
+complementing and exchanging the middle chains preserves their combined
+rank histogram.
+
+Let $T$ be the inherited norm-one phase on the actual transported source
+$gq$, and now let $F=C^{\otimes m}$.
+The exact entrance gauges are $(T,I)$ and the exact terminal gauges are
+$(F,FT^{-1})$, after the endpoint adapters just described.
+The scalar composition is
+\[
+ (X,Y)\mapsto(X,Y+X)\mapsto(-Y,X+Y)\mapsto(-Y,X).
+\]
+Thus physical input $(x,y)$ produces
+\[
+ (-Fy,FT^{-2}x).
+\]
+There is no remaining mixed term. Since $T^2=X_{gq}$ is an exact fixed
+Pauli translation, applying $T^2$ to the input $x$ bank, negating the
+first output and exchanging the banks yields $(Fx,Fy)$ exactly.
+Equivalently one may use the inherited fixed-sign output normalization.
+The translation uses the actual support of $gq$; it assumes no fixed
+integer weight. These fixed permutations and signs incur scalar/router
+charges, without recursive Fourier children.
+
+\subsection{Arbitrary dirty roles and exterior ranks}
+The retained local word is transparent for arbitrary logical auxiliaries:
+with source injection $V$, scalar addition circuit $M$, decoder $J$ and
+$D=JM$, the signed sequence
+\[
+ y\gets y-Dz,\quad z\gets z+Vx,\quad z\gets Mz,\quad
+ y\gets y+Jz,\quad z\gets M^{-1}z,\quad z\gets z-Vx
+\]
+restores $z$ and adds $JMVx$ to $y$. Its chronological deferred readouts
+and copied-center branches are part of the completed word. The inverse
+word also restores arbitrary auxiliaries. Adjoining a fixed orthogonal
+offset applies the same exact gauge to every register at its scalar gates.
+
+A forward dirty role with selected local entrance gauge $T_\sigma$
+has exterior rank $m-h+\dim\sigma$ by the preceding tail lemma.
+In the middle reversed word its actual first local frame is $D_H=I$
+and its actual last local frame is $D_\sigma=T_\sigma F_A^{-1}$.
+Start the role with the common-offset gauge $F_B$. Its logical value is
+restored, and assign the desired terminal gauge $F F_B$, making its total
+physical action $F$. The preimage of the actual last frame is
+$B\oplus(A\cap\sigma^\perp)$, whereas the desired terminal preimage is
+$A\oplus C$. Their intersection is $A\cap\sigma^\perp$, of dimension
+$h-\dim\sigma$. The preimage distance is therefore
+\[
+ m-h+\dim\sigma.
+\]
+This proves the same exterior rank in the reversed stage without assuming
+commuting intermediate representatives. It includes the inverse of the
+complete cleanup, not just the positive producer DAG. The three auxiliary
+banks are independent; all copied centers remain and contribute
+$3|G|\ell$ to the rank loss.
+
+\subsection{Finite rank certificate and group order}
+Let $s_a$ count selected gauges of dimension $a$, let $H_r$ be the
+remaining internal histogram, and $D_r,T_r$ the literal data histograms.
+Per group vertex the complete positive-width multiset has
+\[
+ 3\Bigl(R-\sum_a s_a\Bigr)\text{ children of width }m-h,\qquad
+ 3s_a\text{ of width }m-h+a,\qquad
+ 3(H_r+D_r+T_r)\text{ of width }r.
+\]
+Its weighted sum is $(2v+3R)m-2v+3\ell$.
+At $h=24$, $m=70$, this gives
+\[
+ W/|G|=90163,\quad s/|G|=6309018,\quad
+ (Wm-s)/|G|=2392,\quad r_{\max}=68.
+\]
+The normalized rational moment certificate uses
+$a_c=786999/2500000000$. Rational upper bounds for
+$\log(m/r)$ and
+$e^x\le 1+x+x^2/(2(1-x/3))$ prove
+\[
+ \sum_r\frac{n_r r}{90163\cdot70}
+        (70/r)^{a_c}<1,
+\]
+with certified gap exceeding $2.3047\times10^{-10}$.
+The finite group multiplicity cancels only from this normalized moment.
+
+For a standard nonalternating binary space of even dimension $2n$,
+its characteristic vector $\Omega$ is isotropic and fixed by every
+orthogonal map. Choose $e$ with $e\cdot\Omega=1$ and let
+$U=\langle\Omega,e\rangle^\perp$; $U$ is a nondegenerate alternating
+space of dimension $2n-2$. The induced quotient action is symplectic.
+For each $S\in\operatorname{Sp}(U)$, all lifts are parametrized by
+$u\in U,c\in\mathbb F_2$ through
+\[
+ \Omega\mapsto\Omega,\quad e\mapsto e+u+c\Omega,\quad
+ v\mapsto Sv+(u\cdot Sv)\Omega\quad(v\in U).
+\]
+Checking the pairings verifies these are exactly the orthogonal lifts.
+Counting symplectic bases then gives
+\[
+ |O(2n,2)|=2^{2n-1}|\operatorname{Sp}(2n-2,2)|,\qquad
+ |G|=2^{69+1156}\prod_{i=1}^{34}(2^{2i}-1).
+\]
+This group order has 2415 bits and $W$ has 2431 bits. The least integer
+$d$ satisfying $70^d>2\,68^d$ is 24, so the complex row coefficient is
+$24\cdot2431=58344$. These full finite role counts are retained in the
+scalar guard and row stock. Group elements and involutions are specified
+by their binary matrices; the definition needs no numerical enumeration.
+All fixed address adapters, endpoint Paulis and invocation permutations
+are implemented inside the finite scalar/router guard of the assembly.
diff --git a/notes/three-stage-cover-note.tex b/notes/three-stage-cover-note.tex
new file mode 100644
index 0000000000..f5ad09a450
--- /dev/null
+++ b/notes/three-stage-cover-note.tex
@@ -0,0 +1,63 @@
+\documentclass[11pt]{article}
+\usepackage[margin=1in]{geometry}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern,amsmath,amssymb,amsthm,mathtools,microtype}
+\usepackage[hidelinks]{hyperref}
+\setlength{\emergencystretch}{3em}
+\allowdisplaybreaks
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\title{Three-stage covers for a conditional integer-multiplication saving\\
+\large $\kappa=3146011/10000000000$}
+\author{icekylinx\\\small Construction with OpenAI GPT-6 Astra assistance;
+integration with Codex assistance}
+\date{9 October 2026}
+\begin{document}\maketitle
+\begin{abstract}
+Three signed shears on a regular Cayley cover align all interstage data
+frames. Arbitrary-subspace Clifford frames support the selected complex
+word; weighted reversed interchanges on a batched finite local-ring cover
+support the bit word. Exact rational moments and the retained semantic/bulk
+assembly give the conditional saving $\kappa=3.146011\times10^{-4}$.
+\end{abstract}
+
+\section{Scope and source continuity}
+This extends icekylinx's
+\href{https://github.com/CrocSwap/integer-mult-bounds/pull/115}{PR115},
+\texttt{cf60442983c153a3a0f5272ea4713f99eba13c86}, retaining
+\path{notes/stopped-product-note.tex}, \path{notes/partial-gauge-note.tex}
+and the semantic/analytic proof map in
+\path{notes/structured-bulk-assembly.tex}.
+The analytic reduction, exact recovery, ordered-affine streaming,
+prime-packing, ordinary leaves and eventual fixed-tape setup remain
+inherited hypotheses. The new group geometry, weighted compiler and
+row-borrowing arguments are supplied below, rather than counted as free
+operations. Finite arithmetic does not formalize the all-size theorem.
+
+The complex local DAG is eumemic's
+\href{https://github.com/CrocSwap/integer-mult-bounds/pull/117}{PR117},
+\texttt{cbb05ce504d571546d9b7794c186a613c659c3bf}, with its original
+Apache-2.0 notice and Claude assistance disclosure preserved.
+The bit physical word is Zhihao Chen's
+\href{https://github.com/CrocSwap/integer-mult-bounds/pull/97}{PR97},
+\texttt{e15350b66e4bf796153a68c240a6cc1b788abe3e}, based on Swapnil Jain's
+\texttt{741e7aa078392553815df7926ee17ac5e25a8c38} witness.
+The generalized Clifford theorem, cover construction, weighted interface
+and composition are contributed by icekylinx with OpenAI GPT-6 Astra assistance;
+this repository integration uses Codex assistance.
+Prior Aurel Prosz/Paureel, Zhihao Chen, RaD/hipotures, Rohan Arun,
+Dominik Scholz and framework credits remain recorded in
+\texttt{NOTICE} and \texttt{SOURCES.json}.
+
+\section{Frames for every binary subspace}
+\input{notes/general-clifford-frames.tex}
+\section{Three-stage complex cover}
+\input{notes/three-stage-cover-complex.tex}
+\section{Weighted local-ring bit interface}
+\input{notes/three-stage-cover-bit.tex}
+\section{Uniform stock and ordinary output}
+\input{notes/three-stage-cover-rows.tex}
+\section{Finite precision and strict assembly}
+\input{notes/three-stage-cover-assembly.tex}
+\end{document}
diff --git a/notes/three-stage-cover-rows.tex b/notes/three-stage-cover-rows.tex
new file mode 100644
index 0000000000..590da4b892
--- /dev/null
+++ b/notes/three-stage-cover-rows.tex
@@ -0,0 +1,95 @@
+\subsection{Internal row reserve and ordinary conversion}
+The finite cover requires a row divisor $W(w)$ at each coarse level.
+All children have ratio at most $66/67$ and depth
+$D(n)\le47\lceil\log_2\max(2,n)\rceil+O(1)$.
+A sufficient common divisor is $W(w)^{D(n)}$ times the inherited old-leaf
+reserve. The old supplier has its retained complete-row contract and
+logarithmic reserve. Therefore a sufficient integer $S(e,w)$ satisfies
+\[
+ \log_qS(e,w)=O(w\log e).
+\]
+Unused stock is divided only into fresh descendant role fields; exposed
+parent low/high fields are spectator prefixes. Their weights cannot depend
+on those fresh roles. Every completed call restores its row coordinate,
+so successive residuals and the outer wrapper reuse one reserve instead
+of multiplying reserves for sequential calls.
+
+The unweighted reversed endpoint $F_n(H,D)=(C_nD,C_nH)$ converts to an
+ordinary interchange by the chronological sequence
+\[
+ D\leftarrow D-H;\quad F_n;\quad D\leftarrow D+H;\quad
+ F_n;\quad D\leftarrow H-D.
+\]
+Indeed the successive values are $(H,D-H)$, $(C_n(D-H),C_nH)$,
+$(C_n(D-H),C_nD)$, $(D,D-H)$, and $(D,H)$.
+The three additive maps cost $O(Vn)$ in the atom model. The two calls
+reuse restored rows; their constant multiplicity is outside the recursion
+and does not multiply its characteristic moment.
+
+\subsection{Borrowing and restoring the stock}
+Set $\beta=1/1000$, $w=\lceil e^\beta\rceil$, and
+$k=\lceil\log_qS(e,w)\rceil=O(w\log e)$, using the depth bound for the
+original width $e$. For sufficiently large $e$, $k<e/2$. Write the two
+original complete radix-$q$ chunks as $H=(H_{\rm hi},H_{\rm body})$ and
+$D=(D_{\rm hi},D_{\rm body})$, both high parts having $k$ digits. The
+physical order, including arbitrary original complete spectators $Z$, is
+\[
+ H_{\rm hi},\ H_{\rm body},\ Z,\ D_{\rm hi},\ D_{\rm body}.
+\]
+The complete $q^k$ range of $H_{\rm hi}$ already precedes both bodies.
+Reinterpret it as a preceding row field; $D_{\rm hi}$ remains an intervening
+spectator. Pad only this row range to $\lceil q^k/S\rceil S$. As $q^k\ge S$,
+the padding factor is less than two.
+
+Execute the completed \emph{ordinary} body wrapper with this row reserve,
+then remove the padding. Each row is restored, so the genuine row range
+is invariant at the completed endpoint. The order of values is now
+\[
+ H_{\rm hi},\ D_{\rm body},\ Z,\ D_{\rm hi},\ H_{\rm body}.
+\]
+One inherited ordinary width-$k$ interchange of the two high parts,
+with the bodies and $Z$ as spectators, gives
+\[
+ D_{\rm hi},\ D_{\rm body},\ Z,\ H_{\rm hi},\ H_{\rm body}.
+\]
+This is precisely the original full-chunk interchange. Its extra cost is
+$O(Vk^{1-a_{\rm old}})\le O(Vw\log e)$, conservatively. No full body is
+moved once per borrowed digit. Padding occurs only in this outer wrapper;
+it contributes one volume constant, not a recursive branching factor.
+The wrapper does not rely on a raw reversed call preserving borrowed high
+positions. Small widths use the retained supplier directly.
+
+\subsection{Uniform saving and return to binary addresses}
+Select atom width from the original $e$. Since $k=o(e)$, the body width
+$e-k$ has the same asymptotic powers as $e$. Apply the ordinary wrapper to
+its complete atom body and the old ordinary supplier to fewer than $w$
+remaining digits. The stopped recurrence and the borrowing cost give
+\[
+ O\!\left(V\left[e^{1-\beta}+
+ e^{(1-\beta)(1-a)+\beta(1-a_{\rm old})}+e^\beta\log e\right]\right).
+\]
+Here $a=747/2000000$ and $a_{\rm old}=384599/10^{10}$. Set
+\[
+ a_{\rm bit}=(1-\beta)a+\beta a_{\rm old}
+ =\frac{3731649599}{10^{13}}=0.0003731649599.
+\]
+The inequalities $\beta>a_{\rm bit}$ and $\beta<1-a_{\rm bit}$ make both
+the atom-adapter and borrowing terms lower order. The remainder cost
+$O(Vw^{1-a_{\rm old}})$ is also covered. Thus the ordinary radix-$q$
+interchange has cost $O(Ve^{1-a_{\rm bit}})$, without an externally supplied
+superpolynomial stock of rows.
+
+For an original binary chunk range $[2^e]$, let
+$t=\lceil e/\log_2q\rceil$ and embed it numerically into $[q^t]$.
+Pad both exchanged ranges while retaining every spectator. The total
+logical-volume increase is below $q^2$. Execute the completed ordinary
+radix-$q$ interchange, then remove the padding. Its endpoint preserves the
+rectangular genuine set $[2^e]\times[2^e]$; intermediate maps may mix genuine
+and padded addresses, which does not affect this endpoint justification.
+As $q$ is fixed and $t=\Theta(e)$, the binary interchange has the same
+saving exponent. All schedules use a fixed number of finite-alphabet
+tapes; low residue tables and port types are fixed independently of $w$.
+
+The conclusion retains the original local-producer, ordered-affine,
+ordinary-leaf and multiplication-framework hypotheses. It adds no free
+unit-scaling, dense chart transformation, or external row-stock primitive.
diff --git a/references/partial-gauge/pr62/LICENSE b/references/partial-gauge/pr62/LICENSE
new file mode 100644
index 0000000000..261eeb9e9f
--- /dev/null
+++ b/references/partial-gauge/pr62/LICENSE
@@ -0,0 +1,201 @@
+                                 Apache License
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diff --git a/references/partial-gauge/pr62/NOTICE b/references/partial-gauge/pr62/NOTICE
new file mode 100644
index 0000000000..fd30e610b0
--- /dev/null
+++ b/references/partial-gauge/pr62/NOTICE
@@ -0,0 +1,185 @@
+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 new research note, scripts, certificates, patches, and documentation
+were prepared by Douglas Colkitt with assistance from OpenAI Codex.
+This project is not an official OpenAI release or an OpenAI endorsement.
+
+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.
+
+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.
+
+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.
+
+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.
+
+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.
diff --git a/references/partial-gauge/pr62/PROOF.md b/references/partial-gauge/pr62/PROOF.md
new file mode 100644
index 0000000000..26e1166562
--- /dev/null
+++ b/references/partial-gauge/pr62/PROOF.md
@@ -0,0 +1,221 @@
+# Interval strips and core-aware pair assembly: conditional saving 4.986133 × 10^-5
+
+Under OpenAI's base theorem and the retained analytic, fixed finite-alphabet
+and fixed multitape interfaces, the construction below gives
+
+\[
+T(n)=O\bigl(n(\log n)^{1-\kappa}\bigr),\qquad
+\boxed{\kappa=\frac{4986133}{10^{11}}=4.986133\times10^{-5}>2^{-15}.}
+\]
+
+The exact bit saving is 4986382/10^11. The next grid value of the bit saving
+fails the exact moment test, and the next κ grid value fails the assembly.
+
+This is **6.782% above PR #57** (4669442391/10^14), **10.092% above PR #54**
+(141532521/3125000000000) and **10.849% above PR #53** (4498144/10^11). It
+does not use PR #57's compiler or PR #54's clones. These compare asymptotic
+exponent savings, not running times. They are conditional research claims
+without independent mathematical review or formal verification.
+
+Section "Stacking with PR #57's frame compiler" below gives a second, larger
+witness, κ = 5101691/10^11, that also uses PR #57's compiler.
+
+## What changes
+
+Only the scalar producer DAG changes, in `pair_graph.py`, together with its
+pinned carrier matching, `links-{23,25}.uses`. Everything else is PR #53's
+`research/skip-strips` pipeline, which is PR #48's `research/copied-fixed`
+pipeline, with constants recomputed for the new DAG.
+
+A role is a physical scratch wire. With c additions, q designated outputs and
+μ retained carrier links, the role count is R = c + q − μ. A link lets a
+donor addition x = u + w continue its carrier into a later consumer t of u,
+when the original envelope of x lies inside the envelope of t: the common core
+of t lies inside the core of x, and the cover of x lies inside the cover of t.
+So R counts the additions that cannot continue a carrier, plus q.
+
+### 1. Interval strips
+
+The paired-exclusion recursion (icekylinx PR #18/#36) needs, for each vertex,
+every leave-one-out sum of its k strip values v_1, …, v_k. PR #53 used the
+skip-prefix layout s_j = A_(j−2) + (v_(j−1) + B_(j+1)). We build every cyclic
+interval of the values instead, each from an interval one item shorter:
+
+    I(a, m) = I(a, m−1) + v_(a+m−1),   2 ≤ m ≤ k−1,
+
+and output s_j = I(j+1, k−1), with the total I(0, k−1) + v_(k−1). The interval
+I(a, m) with m < k−1 has the later consumer I(a−1, m+1) of its last item
+v_(a+m−1), and its cover lies inside that consumer's cover, so it can continue
+a carrier. Only the k outputs and the total stay unmatched.
+
+This costs k(k−2)+1 additions instead of about 4k. The role count still falls,
+because R counts unmatched additions. In a model of one strip, an exact integer
+program over every leave-one-out circuit shows that k−1 unmatched additions is
+the minimum for k ≤ 8. Skip-prefix leaves 2k−3 unmatched; intervals leave k−1
+or k. Intervals are used for every strip with k ≥ 3 nonzero values; smaller
+strips keep PR #53's layout. Value orders are PR #53's: reversed at the top
+level, sorted by (support size, ±support) below it.
+
+### 2. Core-aware pair assembly
+
+For groups i < j with members (a, a2) and (b, b2), let F = far[i,j] and let
+Sa, Sa2, Sb, Sb2 be the strip outputs that omit the other group. Each of the
+four outputs is a sum of four pieces, for example
+out[a,b] = F + Sa + Sb + e(a2,b2). PR #53 computes left(a,j) = F + Sa,
+then out[a,b] = left(a,j) + (Sb + e(a2,b2)). None of those ten additions can
+continue a carrier. We compute
+
+    F+Sa, F+Sa2, F+Sb, F+Sb2,
+    Y1 = (F+Sb) + Sa,      Y2 = (F+Sa) + Sb2,
+    Y3 = (F+Sa2) + e(a,b2), Y4 = (F+Sb2) + e(a,b),
+    out[a,b]  = e(a2,b2) + Y1,   out[a,b2] = e(a2,b) + Y2,
+    out[a2,b] = Y3 + Sb,         out[a2,b2] = Y4 + Sa2.
+
+Each F+S node then has a later consumer of its strip operand whose envelope
+contains its own: the same common core and a larger cover. An exact integer
+program over every assembly of these nine pieces, with their actual cores and
+covers, shows that 8 unmatched additions per group pair is optimal. The cores
+matter: a node such as Sa + e(a2,b2) has the two-point core {c, a2}, so a
+one-point-core donor cannot continue into it. Each output needs its own
+addition and at least one more addition that serves only that output, and
+neither can continue a carrier, so 8 is also a lower bound for this
+decomposition.
+
+### Counts
+
+| h | v | additions c | outputs q | matched μ | roles R | loss |
+|---|---|---|---|---|---|---|
+| 23 | 1771 | 62294 | 5336 | 38911 | 28719 | 506 |
+| 25 | 2300 | 86595 | 6925 | 55844 | 37676 | 600 |
+
+PR #53 had R23 = 32946 and R25 = 43409. The physical constants are
+m = 575, N = 4,073,300, W = 140,924,496 (PR #53: 160,799,739) and
+L = 2,226,400. The complete rank is Wm − N + L = 81,029,738,300. The deficit
+N − L is unchanged.
+
+The weighted matching is pinned in `links-{h}.uses`. `optimize_matching.py`
+regenerates it from Rohan Arun's PR #44 optimizer with only the graph import
+changed. Its output is not trusted: `profiles.cpp` replays every pinned link,
+and `producer.py` recounts every link independently.
+
+## Why the inherited proofs apply
+
+The new DAG enters the inherited proofs only through properties and
+quantities that `producer.py` recomputes, as for PR #41, #43, #47, #48 and #53:
+
+- **Scalar outputs.** Every addition joins disjoint supports, every node has a
+  common point, and every output is the exact leave-two-out sum or retained
+  total, checked on dense global supports.
+- **Carrier links.** Every pinned link lies in the admissible adjacency
+  (causal order and envelope inclusion), with distinct donors and uses.
+- **Physical timeline.** The full physical timeline and every dirty basis
+  vector are checked in both orientations at h = 23 and 25.
+- **Fixed-basis profiles.** These are recomputed from the new DAGs and pinned
+  matchings: 37,728 and 51,927 distinct transition matrices; 6,026 and 7,981
+  of them CRT-checked; zero modular disagreements. Exactly h full center
+  cleanups appear, so the copied-center replacement applies unchanged.
+- **Data corners.** These do not depend on the producer. All 4,073,300 pairs
+  are replayed, and the ten primary-prime failures are recovered by exact
+  rational elimination.
+
+The integer programs explain why the links exist; correctness does not depend
+on them.
+
+## Exact arithmetic
+
+`verify.py` uses PR #43/#48's exact logarithm and exponential enclosures:
+
+- The moment at a_b = 4986382/10^11 is below 1; at 4986383/10^11 its exact
+  lower bound exceeds 1.
+- Each of the following is excluded at the new bit saving by an exact lower
+  bound: the complete child lists of PR #57, #54, #53, #48, #47, #46, #44,
+  #43, #42, #41 and #40, and the PR #36 baseline.
+- The inherited balanced assembly passes all 47 strict conditions and seven
+  margins at κ = 4986133/10^11 and fails at the next grid value. Its
+  parameters are β = 1/20, h = 10^-12, a_c = 717/10^7, C1 = 1 and row stock
+  p^2000.
+
+## Stacking with PR #57's frame compiler
+
+PR #57 (eumemic, with OpenAI Codex assistance) compiles each set of nodes that
+share one original envelope as a single invertible binary map, matches
+retained signals to later uses, and reclaims retired roles with paid XOR
+clearing. Its compiler, `scripts/experiments/binary_frame_compiler.py`, is
+included unchanged through PR #57's own commit on this branch.
+`frame/frame_compile.py` feeds it the hash-pinned `pair_graph.py` instead of
+PR #53's graph, exactly as PR #57's driver feeds it PR #53's graph.
+
+The compiled words give the following role counts:
+
+| h | roles R, this graph | roles R, compiled | shared-frame regions | retained signals | reclaimed roles |
+|---|---|---|---|---|---|
+| 23 | 28719 | 27918 | 52473 | 37025 | 801 |
+| 25 | 37676 | 36586 | 73695 | 53194 | 1090 |
+
+With W = 137,151,806 the stacked witness gives
+
+\[
+\kappa=\frac{5101691}{10^{11}}=5.101691\times10^{-5},
+\]
+
+with exact bit saving 5101952/10^11. This is **9.257% above PR #57** and
+2.318% above the plain witness above. PR #57's own result on PR #53's graph
+is 4669442391/10^14.
+
+`frame/frame_verify.py` checks, as PR #57's verifier does:
+
+- both serialized XOR words, replayed independently on every input and dirty
+  basis vector in both orientations;
+- every frame transition, reconstructed from the actual XOR word;
+- all fixed-basis profiles, with bounded-minor and CRT checks;
+- the exact moment, the rejection of the next bit-saving grid value, the
+  exclusion of PR #53's and PR #57's networks at the new saving, all 47
+  strict assembly inequalities, seven margins, and the rejection of the next
+  κ grid value.
+
+This stacked witness also depends on PR #57's compiler argument, which has
+the same review status as PR #57.
+
+## Reproduce
+
+```sh
+python3 research/pair-assembly/producer.py --work-dir /tmp/pair-assembly
+python3 research/pair-assembly/verify.py
+python3 research/pair-assembly/frame/frame_verify.py
+python3 research/pair-assembly/optimize_matching.py --work-dir /tmp/pair-opt   # optional, needs scipy
+python3 research/pair-assembly/frame/frame_compile.py                          # optional, rewrites the words
+```
+
+`producer.py --record` rewrites the JSON records. Without `--record` it checks
+them against fresh regeneration. `make pair-assembly-verify` runs the first
+three commands and the tests.
+
+## Scope and attribution
+
+This is a conditional finite certificate. It covers the same written proof
+obligations as PR #43–#53: the base theorem, the residual compiler, fixed-tape
+recursion, the physical meaning of carrier continuations and copied centers,
+the analytic/semantic/routing/recovery interfaces and eventual thresholds.
+None of these are re-proved here. There is no formal verification or claim of
+global optimality. The stacked witness additionally depends on PR #57's
+compiler mechanism and its written argument.
+
+The interval strips and core-aware pair assembly are new here, by Avi
+Eisenberg with assistance from Claude (Anthropic). Everything else is
+inherited, with all original notices:
+
+- Avi Eisenberg PR #53: skip-prefix strips and this pipeline's adaptation.
+- eumemic PR #57: the joint frame compiler, used unchanged for the stacked
+  witness.
+- Chafik Boukhalfa PR #43/#46/#48: pipeline, recovery, the sorting idea and
+  verification code.
+- Rohan Arun PR #44: weighted matching; also PR #37/#39/#40/#42/#47.
+- RaD / hipotures PR #41: alternating order and changed DAGs.
+- Dominik Scholz PR #35/#38.
+- James Chang PR #34.
+- icekylinx PR #18/#24/#32/#36.
+- Zhihao Chen PR #21/#23/#29.
+- Paureel / Aurel Prosz, Swapnil Jain, Douglas Colkitt, OpenAI, and
+  Harvey–van der Hoeven.
+
+Apache-2.0.
diff --git a/references/partial-gauge/pr62/SOURCE.json b/references/partial-gauge/pr62/SOURCE.json
new file mode 100644
index 0000000000..c4a7d407e0
--- /dev/null
+++ b/references/partial-gauge/pr62/SOURCE.json
@@ -0,0 +1,11 @@
+{
+  "repository": "ikeboy/integer-mult-bounds",
+  "commit": "ad0f25ff7b23cff7f08ad237c2254e6ecf74257e",
+  "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/62",
+  "scope": "Primitive proof and unchanged attribution; implementation adapted in scripts/partial_gauge/interval.py.",
+  "files": {
+    "LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+    "NOTICE": "95d7dabb54a327216e007f1e0cf37dd2f138ab31527f2cc5ee0de55e87f274ca",
+    "PROOF.md": "30685624bf922bb9d9f7ed4b0d367a660fe373cc9ee28fec33585bd8c7023e75"
+  }
+}
diff --git a/references/partial-gauge/pr97/LICENSE b/references/partial-gauge/pr97/LICENSE
new file mode 100644
index 0000000000..261eeb9e9f
--- /dev/null
+++ b/references/partial-gauge/pr97/LICENSE
@@ -0,0 +1,201 @@
+                                 Apache License
+                           Version 2.0, January 2004
+                        http://www.apache.org/licenses/
+
+   TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION
+
+   1. Definitions.
+
+      "License" shall mean the terms and conditions for use, reproduction,
+      and distribution as defined by Sections 1 through 9 of this document.
+
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diff --git a/references/partial-gauge/pr97/NOTICE b/references/partial-gauge/pr97/NOTICE
new file mode 100644
index 0000000000..b565637758
--- /dev/null
+++ b/references/partial-gauge/pr97/NOTICE
@@ -0,0 +1,304 @@
+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.
diff --git a/references/partial-gauge/pr97/PROOF.md b/references/partial-gauge/pr97/PROOF.md
new file mode 100644
index 0000000000..58824db895
--- /dev/null
+++ b/references/partial-gauge/pr97/PROOF.md
@@ -0,0 +1,246 @@
+# Deferred bit endpoints and a signed two-stage complex interface
+
+Under the analytic and fixed finite-alphabet multitape interfaces already
+retained by main, the networks in this package give the conditional bound
+
+\[
+T(n)=O\bigl(n(\log n)^{1-\kappa}\bigr),\qquad
+\kappa=63965813/10^{12}>2^{-14}.
+\]
+
+The finite checks do not formalize these inherited interfaces or establish an
+unconditional integer-multiplication theorem. Repository integration validation
+is recorded separately; mathematical certificates and validation receipts have
+different purposes. This is not a current-record or global-priority claim.
+
+The scalar/frame inputs are Swapnil Jain's round7 bit network and round6 complex
+network, pinned at `741e7aa078392553815df7926ee17ac5e25a8c38`. The original source,
+notes, licenses and assistance disclosures are preserved in `swapnil-round7/`.
+That source reports a slightly larger final number under its own analytic
+stack. We instead use main's retained semantic guard and balanced assembly.
+Our additions are explicit reflected scheduling, the commuting fan-order
+clarification, gauged exits, a signed complex endpoint correction and a bridge
+to the existing main interfaces. No separately researched local role saving is
+added to these networks without recompilation.
+
+## Bit word, geometry and common basis
+
+The pinned producer at h=23 has v=1771 and R=28866. Its frozen supports, links,
+deferred readouts, V leaves and rational frames are checked by the original
+`check_word.py`, `check_frames.py` and `check_lifted.py`. In particular, the
+deferred scratch slots are untouched before their deferred readout; their
+entrance frames lie in every required target hyperplane. These are conditions
+on the complete physical word, not just its ordinary-input support.
+
+`round7_literal_frame_ledger.py` expands that word into 1080616 events and
+checks the forward shear on all 32408 ordinary/dirty F2 basis vectors. Gates
+with several fan recipients are expanded in stable order of their starting
+frame dimension. The recipients are distinct targets of the same unchanged
+source, so these XORs commute. The separately checked chains prove nesting;
+rank order alone is insufficient. This changes the internal order of160 fan
+groups and avoids otherwise uncharged frame retreats, preserving their scalar
+map and all costs. The imported source is left intact.
+
+Reverse time, complement the factor frames and exchange the data banks for
+stage two. All ordinary gates retain their XOR direction after the exchange.
+The new ledger checks continuity and the complete opposite shear; a logical
+transpose alone would not prove this statement. A copied-center scatter is a
+macro: copy the unchanged center, transport the copy, scatter and discard the
+copy. Its reflection creates a new copy at the complemented source and moves
+it to the complemented target. No unknown dirty register is erased.
+
+For an idempotent A write D_A=[[I-A,A],[A,I-A]]. Commuting idempotents satisfy
+D_A D_B=D_(A+B-2AB). Let P,Q be the two rank-one triple projectors, sigma the
+local auxiliary entrance projector of rank f, m=h², and r=h-f. Stage one's
+incoming gauge is A=sigma tensor Q, its last live frame is T=I tensor Q,
+and its terminal gauge is I-A. Its exterior is
+
+\[
+E_1=I-T+A=I-(I-\sigma)\otimes Q,\qquad\operatorname{rank}E_1=m-r.
+\]
+
+Direct multiplication gives D_(I-A)D_T=D_E1 and D_(I-A)D_A=D_I. For stage two
+put B=(I-P) tensor I and lift a factor frame K to B+P tensor K. Its entrance
+is B, last live frame is I-P tensor sigma, terminal gauge is I-B, and
+
+\[
+E_2=B+P\otimes\sigma,\qquad\operatorname{rank}E_2=m-r.
+\]
+
+Again D_(I-B)D_(I-P tensor sigma)=D_E2 and D_(I-B)D_B=D_I. These are genuine
+commuting-idempotent exits, not an assumption that all endpoint movements are
+nested. Scalar dirty restoration and complementary gauges give the full
+address swap. Both exteriors are charged as block m-2r and the inner(r,h)
+profile of their complementary small residual.
+
+With U=P tensor Q, stage-one data ends at (I tensor Q,(I-P) tensor Q), while
+stage two begins at (B+U,B). Each connector is (I-P) tensor (I-Q), of rank
+(h-1)², and both are charged. The final F2 scalar shear pair gives (y,x+y);
+the temporary rank-one correction uses D_U D_I=D_(I-U) to remove the extra
+term, retaining one paid rank-one child per pair.
+
+The common rational tensor basis and inner residual profiles use the preserved
+`notes/flag-basis.tex`, `notes/partial-swap-batching.tex` and the round7 lifted
+checks. The staircase corner adds runs h-2,h-5,1^6, alongside the outer block
+m-4h+2. `check_stair.py` checks its finite witnesses and symbolic cut bounds.
+The GL_h×GL_h family acts transitively on line pairs; nonzero polynomial
+witnesses, together with identically imposed triangular zero conditions, give
+a common rational point for the finite family. This is an existence argument,
+not a claim that sampled pairs enumerate the family or that one modular basis
+is already a common integer basis. Rational enumeration avoiding the finitely
+many nonzero determinant polynomials is an effective fixed finite choice.
+Its denominators and required minors exclude finitely many odd primes, as in
+the retained transfer. All choices are independent of n. The two orientations
+and newly gauged residuals occur among the checked inner families.
+
+The actual event histogram, including exits, connectors and corrections,
+matches the pinned one with m=529, W=108516254 and s=57403754177. An independent
+rational moment enclosure proves saving a_b=31987/500000000.
+
+## Complex scalar word and phases
+
+For the h24 paired triple-exclusion producer, let M be its invertible role
+mixer, V the source injection and A the complete signed readout. The retained
+totals are T and E_i=sum of triples avoiding i, for i<h-1. Twice the center
+scatter at S is 2T-sum_(i in S)E_i if h-1 is absent, and
+(5-h)T+sum_(i<h-1,i notin S)E_i otherwise. Adding the signed side pieces gives
+2x_S. The exact dense support audit checks all4096576 coefficients of AMV=I.
+
+On arbitrary scratch z, the literal word first subtracts AMz from y, restores
+z, injects Vx, adds AM(z+Vx), and then restores z by M^-1 and -V. Thus the
+complete shear is y+=x with arbitrary dirty restoration. Reverse order AND
+negate every coefficient for the inverse; exchanging the data banks gives
+x-=y. This proof applies to the actual fixed h24 word. The independent small
+h8/h10 checks replay all formal ordinary/dirty inputs using exact signed
+integer encodings with coefficient bounds, and reject missing cancellation
+and wrong inverse signs. They are controls, not substitutes for AMV=I.
+
+`round6_complex_literal_ledger.py` explicitly traces all535744 events, the
+24 temporary center copies and124369 actual geometric transitions at h24.
+Each ordinary scalar gate has equal frames. For the reflected word every
+frame and copied-center endpoint is checked. Binary frame inclusion and
+nondegenerate nonalternating residuals are verified exactly. Tensor lifting
+by an odd triple line preserves the Gram form and oddness. The stage-one
+auxiliary exit and stage-two auxiliary entrance each cost m-h=552. Both
+data connectors have rank529. Each inner move and center-copy move is a
+whole-residual child with its sign wrappers, as in the retained complex
+normal form. The rebuilt histogram equals the pinned histogram exactly:
+W=207387136, m=576, s=119453132304 and largest child552. Its independent
+rational moment certifies a_c=36926111/500000000000.
+
+Complex phases require an additional endpoint argument. Write
+C_U=H diag(i^q_U)H, q_U(z)=wt(P_U z) modulo4. For nested nondegenerate
+orthogonal sums these phases add. At a tensor data pair let u=q_U(z) and
+w=wt(z) modulo4. The two signed shears, with source frames (C_U,I) and
+terminal frames (C_F,C_(Uperp)), give
+
+\[
+X'=-i^w y,\qquad Y'=i^{w-2u}x+i^{w-u}y.
+\]
+
+Pre-translate x by P_U*1, multiplying its Fourier coordinate by (-1)^u.
+Copy X', apply C_U^-1, add to Y', and negate X'. The outputs are C_F y and
+C_F x, so after the role exchange every data role has the same completed
+C_F contract as every restored auxiliary role. The correction is one inverse
+rank-one child. Translation and the minus sign have no recursive child.
+This works because q_U(z) modulo2 equals z dot P_U*1. For a tensor of two
+triple lines, its generator has weight9. No repeated phase2q_U is discarded.
+The retained identity C^-1=-i Z C Z executes inverse children with unit
+wrappers. Both stage connectors have the same orthogonal increments as on
+the bit side, so their phase difference is q_C. Exhaustive controls over all
+65536 Fourier addresses at h4 reject omission of the translation or use of
+C_U instead of its inverse, each on32768 addresses.
+
+## Integer recursion, precision and fixed tapes
+
+Use the existing whole-residual compiler, compact source/target adapters and
+integer-width recursion. At width e, first process e mod m axes individually;
+all children then have t floor(e/m) axes for the paid widths t<m. A residual
+basis orders its selected axes consecutively. The bit common-basis profiles
+above use the inherited aligned and shifted runs. The complex wrappers are
+binary address maps and unit phases. All finite descriptors and coefficients
+are fixed constants. Unchanged analytic routing/error/recovery assumptions
+remain exactly those in main's `notes/structured-bulk-assembly.tex`.
+
+Every completed complex child on u axes is C^tensor u (or its wrapped inverse)
+on EVERY returned role, including arbitrary scratch. Its coefficients lie in
+2^-u Z[i] and its absolute row sum is at most2^u. This is now applicable to
+the changed two-stage word by the signed endpoint proof above. Originals of
+temporary copies and all other operands remain parked on the fixed stack
+tapes. A completed copy has the same bound as any other completed child.
+There is no return rounding, encoding change or promotion of an unfinished
+child intermediate to a parent operand.
+
+The actual h24 word has374340 literal scalar updates per invocation. With
+two stages, center-copy overhead and four endpoint groups per pair, use
+G=1531908928. Coefficients have magnitude at most19/2 and denominator at
+most2. A weighted update can be expanded into at most19 unit additions and
+one half; its magnitude/denominator excess is at most a fixed20 units.
+The conservative charge2GW² dominates these updates and temporary copies,
+since W=207387136. Unit phases and address translations add no denominator
+loss. With all other retained overhead paid by8s+4W+4+32m, set
+
+\[
+E=64(W+m+G+1)^3,\quad B=s+E,\quad C_0=32mB^2.
+\]
+
+The exact bridge checks literal charge<E. For f=floor(e/576), the largest
+active child has552f axes, while completed children contribute at most sf
+to the semantic exponent. Consequently
+
+\[
+A(e)\le A(552f)+sf+E\le2Be,
+\]
+
+because2B(576-552)>=s+E and direct leaves obey A(e)<=8e. Prefix kernels and
+outer normalization add at most18e, below the C0e allowance. Thus the same
+fixed fine-grid proof yields C1=1; it does not concatenate temporary internal
+excess of already completed children. E and C0 are large but fixed.
+
+The actual maximum child and width determine each least halving degree d_j
+and bitlength w_j. Reserve the product W_c^D_c W_b^D_b, with
+D_j=d_j ceil(log2(e)), rather than the maximum of the two stocks. The existing
+physically preceding transformed prefix and one complete-row padding supply
+the recomputed stock; nested bit stock is restored before another complex
+child. Sequential temporary copies use this existing row field and do not
+introduce a separately live recursive row reservation. The certificate gives
+the exact coefficient and sufficient stock degree. These operations preserve
+the fixed alphabet, fixed tape count and polynomial descriptor bounds of the
+inherited interfaces; the number of fixed-role streams is not the number of
+physical tapes.
+
+## Final exact combination
+
+Use beta=1/10 and eta=10^-8. The stopped complex saving (1-beta)a_c exceeds
+a_b. Put q=a_b(1-2eta), c=q(1+eta), epsilon=(1-eta)/(1+c+q), lambda'=1-q,
+lambda=((1-a_b)+lambda')/2, g=epsilon*q,
+r=(g+1-epsilon)/2 and delta=eta/8. The inherited balanced combination now
+has all47 strict constraints and all7 margins above63965813/10^12; the
+next10^-12 grid point is rejected. This includes the recomputed scalar
+charge, product-row gap, compact reservations, phase-cell separation, prime
+packing and exact recovery allowances. A second moment implementation
+checks both complete recursive histograms with exact rational logarithm and
+exponential enclosures. Passing arithmetic does not by itself establish any
+of the physical or analytic lemmas above.
+
+## Attribution and scope
+
+Zhihao Chen (jacklightChen): explicit reflected event ledgers, commuting
+fan-order clarification, nonzero auxiliary endpoint proof, signed complex
+phase controls and corrected rank-one endpoint, and integration with the
+retained balanced semantic assembly, with Codex assistance. Subsequent work
+using these specific contributions should explicitly acknowledge Zhihao Chen.
+Historical GPT-6 Astra assistance attribution on PR7/PR23/PR29 remains;
+no claim about the current runtime model is made.
+
+Swapnil Jain supplies deferred readouts, V leaves, lifted frames, the h24
+producer, common flag/staircase family and frozen witnesses, with the original
+Claude assistance disclosure. Avi Eisenberg/ikeboy PR62 supplies interval
+strips and core-aware pair assembly; RaD/hipotures PR41 supplies alternating
+order/links, and Rohan Arun PR44 the weighted-link precedent. Aurel Prosz/
+Paureel supplies two-stage/complement scheduling; icekylinx PR36 supplies
+copied-center and bridge precedents; James Chang PR34 supplies balanced
+combination and reversed geometry; Zhihao Chen PR23/PR29 and RaD supply
+semantic precision, bulk routing and transfer. All earlier author, license
+and AI disclosures in the preserved source and main remain in force.
+
diff --git a/references/partial-gauge/pr97/README.md b/references/partial-gauge/pr97/README.md
new file mode 100644
index 0000000000..ce8b472195
--- /dev/null
+++ b/references/partial-gauge/pr97/README.md
@@ -0,0 +1,35 @@
+This is a compact, unmodified snapshot of the PR97 physical bit ledger and
+the underlying frozen round-seven input. `SOURCE.json` pins the commits and
+every imported file's SHA-256. Each imported byte sequence also matches the
+round-six handoff's original manifest. Original LICENSE and NOTICE are retained.
+
+Zhihao Chen / jacklightChen supplies the PR97 literal forward/reflected physical
+ledger and proof, with the Codex assistance disclosed in those files. Swapnil
+Jain supplies the underlying deferred word, frame witnesses and frame/lifted
+checking sources. Earlier authors and assistance disclosures remain credited
+in NOTICE, PROOF.md, and the original source headers.
+
+The snapshot preserves `witness_23.json.gz` and `deferred_23.json.gz` (about
+2.3 MB together), not the large expanded event/path dumps. The archived
+`bit-ledger-result.json` is a validation receipt, not a new test result. Its
+historical elapsed time is preserved verbatim. The original scripts retain
+their upstream directory assumptions; they are source/proof witnesses here,
+not a relocated full-audit harness.
+
+`scripts/partial_gauge_bit.py` verifies every byte pin, loads the compressed
+inputs explicitly, reconstructs auxiliary and data-chain rank increments,
+checks them against the actual physical ledger, reconstructs exit nullities
+from each physical slot's deferred frame, and composes the new whole-projector
+histogram. It compares every coefficient to the round-six input certificate.
+It does not repeat the unchanged complete-basis replay or rational geometry
+audits; rank differences alone would not establish those geometric hypotheses.
+
+The whole-projector reconstruction counts two stages of v physical invocations,
+with each local rank increment kept whole, each exterior assigned rank
+`m-(h-dim sigma)`, two bank-pair entrance projectors of rank `(h-1)^2`, and
+one rank-one copy correction per source pair. This replaces the old corner
+splitting while preserving the exact physical total rank mass.
+
+New reconstruction and integration: copyright 2026 icekylinx, Apache-2.0,
+prepared with OpenAI Codex assistance. No independent review or endorsement
+by the predecessor authors is implied.
diff --git a/references/partial-gauge/pr97/SOURCE.json b/references/partial-gauge/pr97/SOURCE.json
new file mode 100644
index 0000000000..970000a79f
--- /dev/null
+++ b/references/partial-gauge/pr97/SOURCE.json
@@ -0,0 +1,31 @@
+{
+  "archive": "integer-mult-research/pro/round6-20261009",
+  "archive_manifest_sha256": "b2edf27c7eea66982a75cb224887da8bce25e18ce9b3857ff45c4666ee4b4ea5",
+  "archive_source_inputs_sha256": "ed58db13eafeb47401c6c962291e1fe760cbd25a563ca2796f5389b03a9714f3",
+  "pr97": {
+    "commit": "e15350b66e4bf796153a68c240a6cc1b788abe3e",
+    "repository": "jacklightChen/integer-mult-bounds",
+    "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/97"
+  },
+  "scope": "Unmodified hash-bound physical-ledger receipt, source and compressed witness inputs. Full upstream frame/lifted/scalar audits are inherited, not repeated by the incremental checker.",
+  "sha256": {
+    "LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+    "NOTICE": "1c53970fb4face631a1b331ec66d33111eccd280a129f9b64c2da6df96861c3f",
+    "PROOF.md": "f2c95747d9a5858da43932d857be999e6166d4740f25e2c57f99ce05a8902352",
+    "bit-ledger-result.json": "c3dd79e5e6d72e49d8d1be2d4d9ad808cd7d24b7a725190d24100cbd3644394a",
+    "check_frames.py": "1cb2b8cde2a8b3f12008da0d335dcdd38005cc692367423d9589aa1f4d6306e8",
+    "check_lifted.py": "c3e87ad6b31f377df549421f53d22b99131d2228dcd08a1956913211383cbd6c",
+    "deferred-readout.tex": "4e848f7f06c975045629da7cca9f79e10f255d08dc17a72629c9bed522565d52",
+    "deferred.py": "9b2acaeb6f1beb2a53c15e00c027f9ba22397a12071ad0b71131e12012a98036",
+    "deferred_23.json.gz": "8c38e947ff9e021e308000dd82bb5e2194eb265d8b75194e22ce783d3e75317c",
+    "linalg.py": "56cc779911a3c5c901bc5194ccaf97f720157a2c4ba602b07ee8642af2676b8b",
+    "round7_literal_frame_ledger.py": "02617ba608f971e3bdd2b45d4cbfc3d0d629873a0b9f7c22b12f347d099fdb59",
+    "witness_23.json.gz": "b2486aa2bb222bacea6e52162a3920780e45dd8edc5d7cb03eabea336a1c6588"
+  },
+  "snapshot_prefix": "round6_work/references/round6_pr97/",
+  "underlying": {
+    "commit": "741e7aa078392553815df7926ee17ac5e25a8c38",
+    "repository": "Swapnil-jain/integer-mult-kappa",
+    "url": "https://github.com/Swapnil-jain/integer-mult-kappa/tree/741e7aa078392553815df7926ee17ac5e25a8c38"
+  }
+}
diff --git a/references/partial-gauge/pr97/bit-ledger-result.json b/references/partial-gauge/pr97/bit-ledger-result.json
new file mode 100644
index 0000000000..708e83ebd4
--- /dev/null
+++ b/references/partial-gauge/pr97/bit-ledger-result.json
@@ -0,0 +1,272 @@
+{
+  "h": 23,
+  "roles": 28866,
+  "formal_basis": 32408,
+  "events": 1080616,
+  "frame_keys": 90173,
+  "rank_histograms": {
+    "aux": {
+      "1": 55349,
+      "5": 6151,
+      "10": 162,
+      "4": 11248,
+      "2": 64873,
+      "3": 13926,
+      "6": 1617,
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+      "12": 208,
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+      "18": 599,
+      "21": 413,
+      "7": 883,
+      "11": 299,
+      "14": 38,
+      "15": 649,
+      "13": 202,
+      "17": 1309,
+      "19": 533,
+      "20": 2310
+    },
+    "data": {
+      "4": 2013,
+      "5": 521,
+      "3": 1144,
+      "1": 11386,
+      "2": 2402,
+      "6": 475,
+      "7": 276,
+      "8": 382,
+      "9": 130,
+      "10": 86,
+      "11": 93,
+      "12": 97,
+      "13": 65,
+      "14": 26,
+      "15": 43,
+      "16": 218,
+      "17": 38,
+      "18": 8,
+      "19": 1547,
+      "21": 1,
+      "22": 2
+    },
+    "center": {
+      "22": 23
+    }
+  },
+  "fan_groups_reordered_internally": [
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+    66700,
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+    67070,
+    67072,
+    67074
+  ],
+  "complete_forward_F2": true,
+  "complete_reflected_F2": true,
+  "reflected_frame_continuity": true,
+  "all_scalar_gates_equal_frame_keys": true,
+  "exterior_formula": "E1=I-I_tensor_Q+sigma_tensor_Q; E2=(I-P)_tensor_I+P_tensor_sigma; rank=m-h+dim(sigma)",
+  "histogram_matches_external": true,
+  "histogram": {
+    "1": 1690297301,
+    "3": 1356586,
+    "19": 1639946,
+    "17": 8476006,
+    "15": 7948248,
+    "13": 2164162,
+    "11": 5093396,
+    "7": 3580962,
+    "21": 6906900,
+    "9": 1526602,
+    "5": 354200,
+    "483": 61280142,
+    "23": 61280142,
+    "525": 5653032,
+    "527": 6233920,
+    "489": 293986,
+    "521": 13711082,
+    "523": 4629394,
+    "491": 580888,
+    "485": 545468,
+    "487": 173558,
+    "493": 14168,
+    "499": 1129898,
+    "495": 322322,
+    "505": 99176,
+    "503": 410872,
+    "501": 127512,
+    "507": 1186570,
+    "509": 340032,
+    "497": 17710,
+    "513": 106260,
+    "517": 626934,
+    "515": 1583274,
+    "511": 322322,
+    "519": 2854852,
+    "439": 6272882,
+    "18": 6272882
+  },
+  "recursive_rank": 57403754177,
+  "elapsed": 9.80610633303877,
+  "limit_seconds": 180,
+  "exact_chain_geometry": "Relies on separately replayed pinned check_frames and check_lifted. Rank labels alone do not prove geometry.",
+  "scope": "Explicit one-axis forward/reflected events, scalar complete basis and exact accounting. Tensor endpoint identities proved separately; full complex/tape/precision and final assembly integration not established.",
+  "new_multiplication_bound": false
+}
+
diff --git a/references/partial-gauge/pr97/check_frames.py b/references/partial-gauge/pr97/check_frames.py
new file mode 100644
index 0000000000..f22b6998f9
--- /dev/null
+++ b/references/partial-gauge/pr97/check_frames.py
@@ -0,0 +1,262 @@
+"""Round-seven check, part 2 (stdlib only): every frame of the reordered word, exactly over Q.
+
+The base frames (lifted node frames M_n, late-copy intersections, root frames t_T^perp and U_c) are rebuilt from the
+witness with check_lifted.py's exact machinery. The design-B frames (deferred readout frames sigma_u, V-leaf starts
+s_i) are the primitive integer bases of the data file. Checked:
+ X. the frame dimensions recorded in the data file (node frames, late-copy intersections) equal the exact ones; every
+    slot's node list moves along slot successors of the witness (gate uses and links, for which check_lifted.py proves
+    nesting), its first node is where its start says, its last node reaches its root frame directly;
+ Q. sigma_u <= F0(u) (the slot's start frame); sigma_u <= t_T^perp for every target of u (Z supports, a superset of
+    the F2 supports); on every y_T the deferred levels are nested in time order (readout order); t_S <= s_i <= N_i
+    (the first frame of the use); every X_S chain <t_S>, early V frames, deferred V frames, F is nested in time order;
+    G = I - J/9 nondegenerate on every new frame (Gram determinant nonzero mod a prime, so nonzero over Z);
+ S. side lemma at check_lifted's integer point (U, V of point(h, 1), the point of check_stair.py too): every distinct
+    high-rank step (2r > h) of the reordered word that check_lifted.py does not already cover (slot starts, gauged
+    corners sigma_u -> F, y_T and X_S chain steps, base steps that only the new schedule uses) has the generic
+    north-east pivots under both conjugations, mod 2^31-1;
+ F. the child-width histogram rebuilt from the exact dimensions (Z and F2 y supports), rank sum s = W m - N + L,
+    moment certificate and kappa, with the round-six and the staircase entrance corners;
+ N. negative controls: reversed readout order, readouts at F0(u) instead of sigma_u, perturbed sigma frames, X_S
+    chains with the deferred V gates first, side lemma with U = V = I.
+Usage: python3 check_frames.py [h]"""
+import sys, os, time, random
+from fractions import Fraction as Fr
+from math import gcd
+from collections import Counter, defaultdict
+HERE = os.path.dirname(os.path.abspath(__file__))
+sys.path[:0] = [HERE, os.path.join(HERE, '..', '..', 'scripts'), os.path.join(HERE, '..', 'two-stage-bit')]
+import check_lifted as cl
+import deferred as dr
+from linalg import null_exact, rank_mod, inv_mod_matrix, rightmost_pivots, predicted, Q31, Echelon
+
+T0 = time.time()
+def log(*a): print('[%5.0fs]' % (time.time() - T0), *a, flush=True)
+OK = {}
+def report(name, ok):
+    OK[name] = bool(ok); log('%-100s %s' % (name, 'PASS' if ok else 'FAIL'))
+
+
+def prim(r):
+    r = [Fr(x) for x in r]; den = 1
+    for x in r: den = den * x.denominator // gcd(den, x.denominator)
+    z = [int(x * den) for x in r]; g = 0
+    for x in z: g = gcd(g, x)
+    return [x // g for x in z] if g else z
+
+
+class ModQ7(cl.ModQ):
+    """check_lifted's mod-q frames, extended by the design-B frames ('sigma', s) and ('v', s) (integer bases)."""
+    def __init__(self, fr, ld, extra, control=False):
+        super().__init__(fr, ld, control); self.extra = extra
+
+    def basis(self, key):
+        if key[0] in ('sigma', 'v') and key not in self.bcache:
+            E = Echelon(self.q)
+            for r in self.extra[key]: E.insert([x % self.q for x in r])
+            if len(E.rows) != len(self.extra[key]): self.dimfail += 1
+            self.bcache[key] = E.rows
+        return super().basis(key)
+
+
+def main(h=23):
+    W, D = dr.load(h); S = dr.Schedule(W, D); trip = S.trip; tv = lambda T: [int(q in T) for q in range(h)]
+    FULL = [[int(i == j) for j in range(h)] for i in range(h)]
+    pr = cl.Prog(W); bas, dn = cl.spans(pr); users = cl.users_of(pr, dn)
+    fr = cl.Frames(pr, dn, bas, users); fr.cobases(); d, _ = fr.dims(); ld, _ = cl.late_frames(fr, pr, users)
+    log('exact base frames: %d node frames (%d lifted), %d late-copy intersections' % (len(d), len(pr.j), len(ld)))
+
+    # ------------------------------------------------------------------ X. recorded dims, slot paths
+    okd = all(S.node_dims[n] == d[n] for n in pr.act) and len(S.node_dims) == len(pr.act)
+    okd &= all(S.late_dims[n, lt] == ld[('c', n, lt)] for (n, lt) in S.late_dims) and len(S.late_dims) == len(ld)
+    report('X. recorded frame dims == exact dims (%d node frames, %d late-copy intersections)' % (len(d), len(ld)), okd)
+    def ukey(n, k):
+        u = users[n][k]; return ('n', u[1]) if u[0] == 'gate' else cl.root_key(u)
+    badp = 0
+    for s in range(S.R):
+        hl = S.hold[s]
+        for a, b in zip(hl, hl[1:]): badp += b not in fr.succ[a]
+        root = [k for k in S.chain_keys(s) if k[0] in ('out', 'ret')]
+        if root: badp += fr.rid[root[0]] not in fr.droots[hl[-1]]
+        st = S.start[s]
+        if st[0] == 'fresh':                                          # a copy starts at the use it serves
+            nxt = ('n', hl[1]) if len(hl) > 1 else root[0]
+            badp += ukey(hl[0], st[2]) != nxt
+    report('X. every slot moves along slot successors of the witness, starts at its use, ends at its root (%d slots)' % S.R,
+           badp == 0)
+
+    # ------------------------------------------------------------------ exact frames
+    _eb = {}
+    def exact(key):
+        if key not in _eb:
+            k = key[0]
+            if k == 'sigma': B = S.sigma[key[1]]
+            elif k == 'v': B = S.vstart[key[1]]
+            elif k == '0': B = []
+            elif k == 'F': B = FULL
+            elif k == 'c':
+                n, lt = key[1], key[2]
+                B = exact(ukey(n, lt[-1]))
+                for kk in reversed(lt[:-1]): B = fr.intersect_exact(exact(ukey(n, kk)), B)
+                B = [prim(r) for r in B]; assert len(B) == ld[key]
+            else: B = [prim(r) for r in fr.exact_basis(key)]
+            _eb[key] = B
+        return _eb[key]
+    _nz = {}
+    def nullZ(B):
+        kb = tuple(map(tuple, B))
+        if kb not in _nz: _nz[kb] = [prim(z) for z in null_exact(B, h)] if B else FULL
+        return _nz[kb]
+    def inside(A, B):
+        if not A or len(B) == h: return True
+        return all(sum(a * z for a, z in zip(ra, rz)) == 0 for ra in A for rz in nullZ(B))
+    def nondeg(B, p=Q31):                 # det(9 B G B^T) != 0 mod p  =>  != 0 over Z
+        if not B or len(B) == h: return True
+        sm = [sum(b) for b in B]
+        Gr = [[9 * sum(x * y for x, y in zip(a, b)) - sa * sb for b, sb in zip(B, sm)] for a, sa in zip(B, sm)]
+        return rank_mod(Gr, p) == len(B)
+    gout = {T: [9 * x - 3 for x in tv(T)] for T in trip}
+    tperp = lambda T: ('out', T[0], T)                    # t_T^perp is the root frame of every output (c, T)
+
+    # ------------------------------------------------------------------ Q. exact facts
+    indep = all(rank_mod(B, Q31) == len(B) for B in list(S.sigma.values()) + list(S.vstart.values()))
+    report('Q. sigma_u and s_i bases independent (rank mod 2^31-1 == row count; %d + %d frames, max |entry| %d)' % (
+        len(S.sigma), len(S.vstart), max(abs(x) for B in list(S.sigma.values()) + list(S.vstart.values()) for r in B for x in r)), indep)
+    sel = S.readout; F0 = {s: S.start_key(s) for s in range(S.R)}
+    report('Q. sigma_u <= F0(u) exactly (%d deferred slots)' % len(sel), all(inside(S.sigma[s], exact(F0[s])) for s in sel))
+    CZ = S.adjoint(); CF = [{t: 1 for t, x in dd.items() if x % 2} for dd in CZ]
+    capbad = sum(1 for s in sel for t in CZ[s] if any(sum(a * g for a, g in zip(r, gout[trip[t]])) for r in S.sigma[s]))
+    report('Q. sigma_u <= t_T^perp exactly for every Z-support target (a superset of the F2 targets)', capbad == 0)
+    def ychains(co, order, frame=lambda s: S.sigma[s], limit=None):
+        byT = defaultdict(list)
+        for s in order:
+            for t in co[s]: byT[t].append(s)
+        bad = 0; levels = {}
+        for t, ss in list(byT.items())[:limit]:
+            seq = [frame(s) for s in ss]
+            for A, B in zip(seq, seq[1:]):
+                if A is B: continue
+                if len(A) > len(B) or not inside(A, B): bad += 1
+            lv = []
+            for s in ss:
+                if not lv or S.f[lv[-1]] != S.f[s]: lv.append(s)
+            levels[t] = lv
+        return bad, levels
+    bZ, levZ = ychains(CZ, sel); bF, levF = ychains(CF, sel)
+    report('Q. deferred levels on every y_T nested in time order, exactly (Z targets %d bad, F2 targets %d bad)' % (bZ, bF),
+           bZ == 0 and bF == 0)
+    nd = sum(1 for B in list(S.sigma.values()) + list(S.vstart.values()) if not nondeg(B))
+    report('Q. G nondegenerate on every sigma_u and s_i (Gram det != 0 mod 2^31-1 => over Z)', nd == 0)
+    bad_t = bad_n = 0
+    for s, n in S.srcop.items():
+        bad_t += not inside([tv(trip[n - 1])], S.vstart[s])
+        N1 = S.chain_keys(s)[2]                                       # first frame of the use after its start
+        bad_n += not inside(S.vstart[s], exact(N1))
+    report('Q. V leaves exactly: t_S <= s_i (%d bad), s_i <= N_i, the first frame of the use (%d bad), %d slots' % (
+        bad_t, bad_n, len(S.srcop)), bad_t == bad_n == 0)
+    def xchains(order_fn):
+        bad = 0; steps = []
+        for n, us in S.xs:
+            seq = [[tv(trip[n - 1])]] + [S.vstart[s] for s in order_fn(us)] + [FULL]
+            keys = [('n', n)] + [('v', s) for s in order_fn(us)] + [('F',)]
+            for (A, ka), (B, kb) in zip(zip(seq, keys), zip(seq[1:], keys[1:])):
+                if len(A) > len(B) or not inside(A, B): bad += 1
+            steps.append(keys)
+        return bad, steps
+    bx, xkeys = xchains(lambda us: us)
+    report('Q. X_S chains nested exactly in time order (<t_S>, early V gates, deferred V gates, F; %d leaves)' % len(S.xs), bx == 0)
+
+    # ------------------------------------------------------------------ S. side lemma on the new high-rank steps
+    ops0, chains0, _, R0 = cl.compile_(pr, dn, users)
+    def dimof0(k):
+        if k == ('0',): return 0
+        if k == ('F',): return h
+        if k[0] in ('out', 'ret'): return h - 1
+        if k[0] == 'c': return ld[k]
+        return d[k[1]]
+    _, base_steps = cl.chain_steps(chains0, dimof0, h)
+    base_frames = {k for e in base_steps for k in e}
+    def dimx(k): return len(exact(k)) if k[0] in ('sigma', 'v') else dimof0(k)
+    new = {}
+    def add_chain(keys, kind):
+        prev = keys[0]; pd = dimx(prev)
+        for k in keys[1:]:
+            dk = dimx(k)
+            if dk > pd:
+                if (prev, k) not in base_steps and 2 * (dk - pd) > h: new.setdefault((prev, k), (dk - pd, kind))
+                prev, pd = k, dk
+            else: prev = k
+    for s in range(S.R):
+        add_chain(S.chain_keys(s), 'slot')
+        if S.f[s]: add_chain([('sigma', s), ('F',)], 'corner')
+    for t, lv in levZ.items():
+        add_chain([('0',)] + [('sigma', s) for s in lv] + [tperp(trip[t])], 'ydata')
+    for t, lv in levF.items():
+        add_chain([('0',)] + [('sigma', s) for s in lv] + [tperp(trip[t])], 'ydata')
+    for keys in xkeys: add_chain(keys, 'xdata')
+    extra = {('sigma', s): B for s, B in S.sigma.items()}; extra.update({('v', s): B for s, B in S.vstart.items()})
+    mq = ModQ7(fr, ld, extra)
+    used = {k for e in new for k in e} - base_frames
+    ndq = sum(1 for k in used if not mq.nondeg(k))
+    report('S. frames used only by the new steps: %d, G-degenerate mod 2^31-1 %d, reduction dimension mismatches %d' % (
+        len(used), ndq, mq.dimfail), ndq == 0 and mq.dimfail == 0)
+    pr_ = predicted(h); q = mq.q
+    def lemma(items, m):
+        bad = 0
+        for (A, B), (r, _) in items:
+            QA, QB = m.conjugated(A), m.conjugated(B)
+            for t in range(2):
+                a, b = QA[t], QB[t]
+                M = [[(b[i * h + c] - a[i * h + c]) % q for c in range(h)] for i in range(h)]
+                if rightmost_pivots(M, q) != pr_[r]: bad += 1
+        return bad
+    items = sorted(new.items(), key=lambda e: (str(e[0][1]), str(e[0][0])))
+    badL = lemma(items, mq)
+    report('S. side lemma (generic NE pivots, both conjugations, mod 2^31-1) on all %d new high-rank steps %s' % (
+        len(items), dict(Counter(k for _, k in new.values()))), badL == 0)
+
+    # ------------------------------------------------------------------ F. histogram from exact dims
+    import moment
+    from certificate_round3 import evaluate
+    def dim_exact(k): return dimx(k)
+    rk = dr.chain_ranks(S, dim_exact)
+    xd = S.xdata(lambda s: len(exact(('v', s))))
+    okx = all(sum(x) == h - 1 for x in xd)
+    ylZ = S.ylevels(CZ); ylF = {t: sorted({S.f[s] for s in sel if t in CF[s]}) for t in range(S.v)}
+    res = {}
+    for corner_name, corner in (('round-six corner', dr.ROUND6_CORNER(h)), ('staircase corner', dr.STAIRCASE_CORNER(h))):
+        for yname, yl in (('Z', ylZ), ('F2', ylF)):
+            c = dr.histogram(S, rk, yl, xd, corner)
+            rs = sum(w * n for w, n in c['hist'].items())
+            a, _ = moment.certify(c)
+            k = evaluate(a, Fr(36926111, 5 * 10**11), Fr(1, 1000), 'crude', m_c=576, s_c=119453132304)
+            res[corner_name, yname] = (a, k['kappa'], k['ok'], rs == c['s'] and max(c['hist']) < c['m'])
+    for corner_name in ('round-six corner', 'staircase corner'):
+        (aZ, kZ, oZ, sZ), (aF, kF, oF, sF) = res[corner_name, 'Z'], res[corner_name, 'F2']
+        report('F. %s: rank sum == s (Z and F2 y supports), a_b = %s (%.6e), kappa = %s (%.7e) ok=%s, F2 a_b = %s' % (
+            corner_name, aZ, float(aZ), kZ, float(kZ), oZ, aF), sZ and sF and oZ and oF and okx)
+    report('F. claims: a_b 12599/200000000, kappa 6299103187973/10^17; staircase a_b 31987/500000000, kappa 1599247689723/(2.5*10^16)',
+           res['round-six corner', 'Z'][:2] == (Fr(12599, 200000000), Fr(6299103187973, 10**17)) and
+           res['staircase corner', 'Z'][:2] == (Fr(31987, 500000000), Fr(1599247689723, 25 * 10**15)))
+
+    # ------------------------------------------------------------------ N. negative controls
+    bZr, _ = ychains(CZ, list(reversed(sel)))
+    report('N. NEG reversed readout order breaks the y_T chains (%d bad)' % bZr, bZr > 0)
+    bF0, _ = ychains(CF, sorted(sel, key=lambda s: (dimx(F0[s]), s)), frame=lambda s: exact(F0[s]), limit=300)
+    report('N. NEG readouts at F0(u) instead of sigma_u break the y_T chains (%d breaks on 300 targets)' % bF0, bF0 > 0)
+    rng = random.Random(5); pert = tot = 0
+    for s in [s for s in sel if dimx(F0[s]) < h][:200]:
+        B = [r[:] for r in S.sigma[s]]; B[0] = [x + rng.randint(1, 3) for x in B[0]]; tot += 1
+        pert += not inside(B, exact(F0[s]))
+    report('N. NEG perturbed sigma frames fail sigma <= F0 (%d / %d)' % (pert, tot), tot and pert >= 0.9 * tot)
+    early = set(S.srcop) - S.sel
+    bxr, _ = xchains(lambda us: [s for s in us if s not in early] + [s for s in us if s in early])
+    report('N. NEG deferred V gates before the early ones break the X_S chains (%d bad)' % bxr, bxr > 0)
+    mq0 = ModQ7(fr, ld, extra, control=True)
+    badc = lemma(items[:300], mq0)
+    report('N. NEG side lemma with U = V = I fails (%d / %d)' % (badc, 2 * min(300, len(items))), badc > 0)
+    log('ALL %s' % ('PASS' if all(OK.values()) else 'FAIL: ' + str([k for k, x in OK.items() if not x])))
+    return all(OK.values())
+
+
+if __name__ == '__main__':
+    sys.exit(0 if main(int(sys.argv[1]) if len(sys.argv) > 1 else 23) else 1)
diff --git a/references/partial-gauge/pr97/check_lifted.py b/references/partial-gauge/pr97/check_lifted.py
new file mode 100644
index 0000000000..4a19240e2f
--- /dev/null
+++ b/references/partial-gauge/pr97/check_lifted.py
@@ -0,0 +1,677 @@
+"""Release check (stdlib only) of the lifted-frame + late-copy bit side program, stacked with the staircase corner.
+
+Input: certificates/round7/witness_<h>.json.gz: the side DAG (leaves = triples of [h], additions), outputs (c, T),
+retained totals, links (donor x -> use (y, k)), the flag matrix F (integer h x h), the lift index j(n) of every
+lifted addition and the late-copy decisions (early users, ordered late users) per node.
+
+What is checked, from the witness alone:
+ A. DAG semantics: disjoint supports; output (c,T) support = {S : c in S, S cap T = {c}}; retained = star(c).
+ B. Compile: one slot per role, schedule order (dim span, id), links hand the donor's non-pivot slot to the linked
+    use; late copies: the pivot of n climbs C_1 <= ... <= C_k (C_i = frames of users late[i:] intersected) and the
+    copy for late user i is a gate (pivot, fresh slot) at C_i, placed in time as LATE as possible (just before the
+    first of: its user's gate, the pivot's consumer gate, the next copy's slot). R = adds + outputs + retained - links.
+ C. Dirty-scratch replay of the compiled program: word L J L^-1 V L J L^-1 V^-1 and its mirror, arbitrary scratch,
+    over F2 and over Z, reads at the slot's last touch and at the end; targets must receive J L V x exactly
+    (DAG values) and every slot is restored; over F2 every target also receives exactly x_T.
+ D. Frames (exact over Q). Frame of addition n: M_n = span(n) + (U_n cap F_j(n)), U_n = common kernel of the root
+    covectors reachable from n ((9I-J) t_T for an output to Y_T, 1 - 3 e_c for a retained use of c), F_j = first j
+    rows of F. Nesting is a THEOREM given the combinatorial facts checked here:
+      j(n) <= j(t) on every slot successor pair n -> t;  span(x) <= span(t) on every link pair (exact);
+      span(n) <= every root frame n reaches directly (exact integer dot products);  F invertible.
+    Then U_n <= U_t, so M_n <= M_t, M_n <= U_n <= every reachable root frame, and every late-copy frame C_i
+    (an intersection of frames that all contain M_n) is nested. Dimensions (exact, see linalg.py):
+      dim M_n = rank[span(n); F_j] - rank(K_n F_j^T)   (modular law; span(n) <= U_n; K_n spans the root covectors),
+    with ranks by mod-p + Hadamard certificates, Bareiss when the mod-p rank is not full; late-copy intersections by
+    exact Fraction bases. Rank of every chain step = difference of dimensions (nested), histogram vs claim.
+ E. Frame reductions mod q = 2^31-1 (dimension-matched, so they are the reductions of the rational frames),
+    G = I - J/9 nondegenerate on every frame of every step (Gram matrix invertible mod q => over Q), and the side
+    lemma at ONE integer point (U, V of point_UV(h, 1), the random stream of independent/two-stage-bit/flag_existence.py): for every distinct step A -> B of
+    rank r with 2r > h (retained copy edges 0 -> U_c included), U^T (P_B - P_A) U^-T and V^T (..) V^-T have the
+    generic north-east pivots of f_r. (NE ranks mod q <= over Q <= f_r, so equality mod q proves it over Q.)
+ F. Certificate: rt_histogram with the round-6 entrance corner and with the staircase corner (m-4h+2, h-2, h-5, 1^6),
+    rank sum == s, moment.certify (exact), kappa = certificate_round3.evaluate(a_b, round-6 complex values).
+Negative controls (CONTROL=1): see controls().
+Usage: python3 check_lifted.py h [--no-generic]"""
+import sys, os, json, random, time
+from itertools import combinations
+from collections import Counter, OrderedDict
+from fractions import Fraction as Fr
+from array import array
+HERE = os.path.dirname(os.path.abspath(__file__))
+sys.path.insert(0, HERE)
+from linalg import *
+
+T0 = time.time()
+def load_witness(h):
+    import gzip
+    with gzip.open(os.path.join(HERE, '..', '..', 'certificates', 'round7', 'witness_%d.json.gz' % h), 'rt') as fh:
+        return json.load(fh)
+
+
+def log(*a):
+    print('[%5.0fs]' % (time.time() - T0), *a, flush=True)
+
+
+# ----------------------------------------------------------------------------------------------- A. DAG
+class Prog:
+    def __init__(self, W):
+        self.W = W; h = self.h = W['h']
+        self.trip = list(combinations(range(h), 3)); v = self.v = len(self.trip)
+        self.args = {int(n): (tuple(a) if a else None) for n, a in W['args'].items()}
+        self.act = sorted(self.args)
+        for n in self.act:
+            if self.args[n] is None: assert 1 <= n <= v and tuple(W['leaf'][str(n)]) == self.trip[n - 1]
+            else: assert all(a < n and a in self.args for a in self.args[n]), n
+        self.sup = {}
+        for n in self.act:
+            if self.args[n] is None: self.sup[n] = 1 << (n - 1)
+            else:
+                a, b = self.args[n]; assert not self.sup[a] & self.sup[b], 'overlapping supports'
+                self.sup[n] = self.sup[a] | self.sup[b]
+        self.outputs = {(c, tuple(T)): n for c, T, n in W['outputs']}
+        self.retained = {c: n for c, n in W['retained']}
+        star = [sum(1 << i for i, t in enumerate(self.trip) if q in t) for q in range(h)]
+        assert len(self.outputs) == 3 * v and len(self.retained) == h
+        for (c, T), n in self.outputs.items():
+            A, B = [q for q in T if q != c]; assert c in T and self.sup[n] == star[c] & ~star[A] & ~star[B]
+        for c, n in self.retained.items(): assert self.sup[n] == star[c]
+        roots = set(self.outputs.values()) | set(self.retained.values()); reach = set(); st = list(roots)
+        while st:
+            n = st.pop()
+            if n in reach: continue
+            reach.add(n)
+            if self.args[n]: st.extend(self.args[n])
+        assert reach == set(self.act), 'inactive nodes in witness'
+        self.mL = {x: (y, k) for x, y, k in W['links']}
+        self.j = {int(n): j for n, j in W['j'].items()}
+        self.late = {int(n): (list(e), list(l)) for n, (e, l) in W['late'].items()}
+        self.F = W['flag']
+
+
+def trow(h, S): return [int(q in S) for q in range(h)]
+
+
+def spans(pr):
+    """independent triple subsets (exact: 0/1 rows with three ones, every minor < 3^(h/2) < Q31) and dims."""
+    h = pr.h; assert 3 ** h < Q31 ** 2
+    bas = {}; dn = {}
+    for n in pr.act:
+        if pr.args[n] is None: bas[n] = (n - 1,); dn[n] = 1; continue
+        a, b = pr.args[n]; E = Echelon(Q31); keep = []
+        for i in bas[a] + bas[b]:
+            if len(E) == h: break
+            if E.insert(trow(h, pr.trip[i])): keep.append(i)
+        bas[n] = tuple(keep); dn[n] = len(keep)
+    return bas, dn
+
+
+def users_of(pr, dn):
+    key = lambda n: (dn[n], n); users = {n: [] for n in pr.act}
+    for n in sorted(pr.act, key=key):
+        if pr.args[n]:
+            for pos, x in enumerate(pr.args[n]): users[x].append(('gate', n, pos))
+    for (c, T), n in sorted(pr.outputs.items()): users[n].append(('out', (c, T)))
+    for c, n in sorted(pr.retained.items()): users[n].append(('ret', c))
+    return users
+
+
+# ----------------------------------------------------------------------------------------------- B. compile
+def root_key(u): return ('out',) + u[1] if u[0] == 'out' else ('ret', u[1])
+
+
+def compile_(pr, dn, users, late=True, defer=True, tamper=None):
+    """returns ops (scalar program), chains (frame keys per slot), reads, R. tamper: negative-control hooks."""
+    key = lambda n: (dn[n], n); order = sorted(pr.act, key=key); gpos = {n: i for i, n in enumerate(order)}
+    linked = {r: x for x, r in pr.mL.items()}
+    chains = []; edge = {}; reads = []; ops = []; deferred = []
+    def new(first): chains.append(list(first)); return len(chains) - 1
+    def assign(s, n, k):
+        u = users[n][k]
+        if u[0] == 'gate': edge[u[1], u[2]] = s
+        else:
+            chains[s] += [root_key(u), ('F',)]
+            reads.append((s, [u[1][1]] if u[0] == 'out' else [T for T in pr.trip if u[1] in T]))
+    def ufr(n, k):
+        u = users[n][k]; return ('n', u[1]) if u[0] == 'gate' else root_key(u)
+    for n in order:
+        if pr.args[n] is None:
+            piv = new([('0',), ('n', n)]); ops.append(('src', piv, n, n))
+        else:
+            a, b = pr.args[n]; sa, sb = edge.pop((n, 0)), edge.pop((n, 1))
+            if n in pr.mL and pr.mL[n][0] == a: sa, sb = sb, sa          # the linked operand stays on the non-pivot
+            piv, oth = sa, sb
+            chains[piv].append(('n', n)); chains[oth].append(('n', n)); ops.append(('add', piv, oth, n))
+            if n in pr.mL: y, k = pr.mL[n]; assign(oth, y, k)
+            else: chains[oth].append(('F',))
+        ks = [k for k in range(len(users[n])) if (n, k) not in linked]
+        if late and n in pr.late:
+            early, lt = pr.late[n]; assert sorted(early + lt) == ks, ('bad late decision', n)
+        else: early, lt = ks[1:], [ks[0]]
+        for k in early:
+            f = new([('0',), ('n', n)]); ops.append(('copy', piv, f, n)); assign(f, n, k)
+        for i, k in enumerate(lt):
+            Ck = ufr(n, k) if i == len(lt) - 1 else ('c', n, tuple(lt[i:]))
+            if tamper == 'nointersect' and i < len(lt) - 1: Ck = ufr(n, k)
+            chains[piv].append(Ck)
+            if i < len(lt) - 1:
+                f = new([('0',), Ck]); deferred.append((n, i, f, k, lt[-1])); assign(f, n, k)
+        k = lt[-1]
+        if users[n][k][0] == 'gate': edge[users[n][k][1], users[n][k][2]] = piv
+        else: assign(piv, n, k)
+    assert not edge
+    # place deferred copies: as late as possible, in order, before the user's gate and the pivot consumer's gate
+    INF = len(order); byn = {}
+    for (n, i, f, k, kl) in deferred: byn.setdefault(n, []).append((i, f, k, kl))
+    ins = {}
+    for n, lst in byn.items():
+        lst.sort(); D = []
+        for i, f, k, kl in lst:
+            u, ul = users[n][k], users[n][kl]
+            d = min(gpos[u[1]] if u[0] == 'gate' else INF, gpos[ul[1]] if ul[0] == 'gate' else INF)
+            D.append(d)
+        for t in range(len(D) - 2, -1, -1): D[t] = min(D[t], D[t + 1])
+        for (i, f, k, kl), d in zip(lst, D):
+            if not defer: d = gpos[n] + 1                                   # immediate copy (still a valid program)
+            if tamper == 'aftercons' and n == tamper_node[0] and i == 0:
+                ul = users[n][kl]; d = gpos[ul[1]] + 1                       # copy AFTER the pivot was consumed
+            ins.setdefault(d, []).append((n, i, f))
+    pivslot = {}
+    for o in ops:
+        if o[0] in ('src', 'add'): pivslot[o[3]] = o[1]
+    out = []
+    for o in ops:
+        if o[0] in ('src', 'add'):
+            for (n, i, f) in sorted(ins.get(gpos[o[3]], []), key=lambda z: (gpos[z[0]], z[1])):
+                out.append(('copy', pivslot[n], f, n))
+        out.append(o)
+    for (n, i, f) in sorted(ins.get(INF, []) + [z for d, zs in ins.items() if d > INF for z in zs], key=lambda z: (gpos[z[0]], z[1])):
+        out.append(('copy', pivslot[n], f, n))
+    if tamper == 'dropcopy':
+        idx = next(i for i, o in enumerate(out) if o[0] == 'copy' and o[3] in pr.late); out.pop(idx)
+    R = len(chains); adds = sum(1 for n in pr.act if pr.args[n])
+    assert R == adds + len(pr.outputs) + len(pr.retained) - len(pr.mL), 'role count'
+    return out, chains, reads, R
+
+tamper_node = [None]
+
+
+# ----------------------------------------------------------------------------------------------- C. replay
+def replay(pr, ops, reads, R, ring, rng, mirror=False, midread=True):
+    trip, v = pr.trip, pr.v; tid = {T: i for i, T in enumerate(trip)}
+    mod = (lambda z: z & 1) if ring == 2 else (lambda z: z)
+    last = {}
+    for i, o in enumerate(ops):
+        last[o[1]] = i
+        if o[0] != 'src': last[o[2]] = i
+    readers = {}
+    for s, Ts in reads: readers.setdefault(last[s] if midread else len(ops) - 1, []).append((s, Ts))
+    x = [rng.randrange(-10**6, 10**6) for _ in range(v)]
+    a0 = [rng.randrange(-10**6, 10**6) for _ in range(R)]; y0 = [rng.randrange(-10**6, 10**6) for _ in range(v)]
+    a = list(a0); y = list(y0); src = [(o[1], o[2]) for o in ops if o[0] == 'src']
+    def L(sg=None):
+        for i, o in enumerate(ops):
+            if o[0] == 'add': a[o[1]] += a[o[2]]
+            elif o[0] == 'copy': a[o[2]] += a[o[1]]
+            if sg is not None:
+                for s, Ts in readers.get(i, ()):
+                    for T in Ts: y[tid[T]] += sg * a[s]
+    def Li():
+        for o in reversed(ops):
+            if o[0] == 'add': a[o[1]] -= a[o[2]]
+            elif o[0] == 'copy': a[o[2]] -= a[o[1]]
+    def V(sg):
+        for s, n in src: a[s] += sg * x[n - 1]
+    if not mirror: L(-1); Li(); V(+1); L(+1); Li(); V(-1)
+    else: V(+1); L(+1); Li(); V(-1); L(-1); Li()
+    if any(mod(p - q) for p, q in zip(a, a0)): return False
+    val = {}
+    for n in pr.act: val[n] = x[n - 1] if pr.args[n] is None else val[pr.args[n][0]] + val[pr.args[n][1]]
+    want = list(y0)
+    for (c, T), n in pr.outputs.items(): want[tid[T]] += val[n]
+    for c, n in pr.retained.items():
+        for T in trip:
+            if c in T: want[tid[T]] += val[n]
+    if any(mod(p - q) for p, q in zip(y, want)): return False
+    if ring == 2 and any(mod(y[i] - y0[i] - x[i]) for i in range(v)): return False
+    return True
+
+
+# ----------------------------------------------------------------------------------------------- D. frames
+class Frames:
+    def __init__(self, pr, dn, bas, users):
+        self.pr = pr; self.dn = dn; self.bas = bas; self.users = users; h = self.h = pr.h
+        self.rootcov = {}
+        for (c, T) in pr.outputs: self.rootcov[('out', c, T)] = [9 * int(q in T) - 3 for q in range(h)]
+        for c in pr.retained: self.rootcov[('ret', c)] = [1 - 3 * int(q == c) for q in range(h)]
+        rid = {k: i for i, k in enumerate(sorted(self.rootcov))}; self.rkeys = sorted(self.rootcov)
+        self.rid = rid
+        # slot successors (same for base and late compile: every free use of n is reached by one slot from n,
+        # a donor's non-pivot slot goes on to its linked use) and direct roots
+        linked = {r: x for x, r in pr.mL.items()}
+        self.succ = {n: set() for n in pr.act}; self.droots = {n: set() for n in pr.act}
+        def to(n, use):
+            if use[0] == 'gate': self.succ[n].add(use[1])
+            else: self.droots[n].add(rid[root_key(use)])
+        for n in pr.act:
+            for k, u in enumerate(users[n]):
+                if (n, k) not in linked: to(n, u)
+            if n in pr.mL: y, k = pr.mL[n]; to(n, users[y][k])
+
+    def check_structure(self):
+        """combinatorial / exact facts behind the nesting theorem."""
+        pr, h = self.pr, self.h; bad_j = bad_link = bad_root = 0
+        J = lambda n: self.pr.j.get(n, 0)
+        for n in pr.act:
+            if pr.args[n] is None: assert J(n) == 0
+            for t in self.succ[n]:
+                if J(n) > J(t): bad_j += 1
+        for x, (y, k) in pr.mL.items():                                   # span(x) <= span(linked use target)
+            u = self.users[y][k]
+            if u[0] == 'gate':
+                t = u[1]; E = Echelon(Q31)
+                for i in self.bas[t]: E.insert(trow(h, pr.trip[i]))
+                if not all(E.contains(trow(h, pr.trip[i])) for i in self.bas[x]): bad_link += 1
+        for n in pr.act:                                                  # span(n) <= direct root frames
+            for r in self.droots[n]:
+                g = self.rootcov[self.rkeys[r]]
+                if any(sum(gq * tq for gq, tq in zip(g, trow(h, pr.trip[i]))) for i in self.bas[n]): bad_root += 1
+        # F invertible over Q (a nonzero determinant mod a prime)
+        F_ok = rank_mod(self.pr.F, Q31) == h
+        return bad_j, bad_link, bad_root, F_ok
+
+    def cobases(self):
+        """K_n: independent subset of the root covectors reachable from n, spanning all of them over Q
+        (selection mod M127, valid by the Hadamard certificate on root covectors)."""
+        pr, h = self.pr, self.h
+        rows = [self.rootcov[k] for k in self.rkeys]
+        mx = max(sum(x * x for x in r) for r in rows); assert mx ** h < M127 ** 2, 'Hadamard bound for K'
+        key = lambda n: (self.dn[n], n); K = {}
+        for n in sorted(pr.act, key=key, reverse=True):
+            cand = sorted(self.droots[n]); srcs = sorted((K[t] for t in self.succ[n]), key=len, reverse=True)
+            cap = h - self.dn[n]                                          # span(n) <= U_n
+            if not cand and len(srcs) == 1: K[n] = srcs[0]; continue
+            E = Echelon(M127); keep = []
+            seq = list(srcs[0]) if srcs else []
+            for r in seq: E.insert(rows[r]); keep.append(r)                # srcs[0] is already independent
+            for lst in [cand] + srcs[1:]:
+                for r in lst:
+                    if len(keep) >= cap: break
+                    if E.insert(rows[r]): keep.append(r)
+            K[n] = tuple(keep)
+        self.K = K
+        return K
+
+    def dims(self):
+        """exact dim of every addition frame M_n."""
+        pr, h = self.pr, self.h; F = pr.F; d = dict(self.dn); stats = Counter()
+        rows = [self.rootcov[k] for k in self.rkeys]
+        self.KF = {}
+        for n, j in pr.j.items():
+            Fj = F[:j]; K = [rows[r] for r in self.K[n]]
+            KF = [[sum(a * b for a, b in zip(kr, fr)) for fr in Fj] for kr in K]
+            SF = [trow(h, pr.trip[i]) for i in self.bas[n]] + Fj
+            r1 = self._rank(SF, min(len(SF), h), stats, 'SF'); r2 = self._rank(KF, min(len(K), j), stats, 'KF')
+            d[n] = r1 - r2; self.KF[n] = (r2, KF)
+            assert self.dn[n] <= d[n] <= h - len(K)
+        self.d = d
+        return d, stats
+
+    @staticmethod
+    def _rank(M, full, stats, tag):
+        if not M or full == 0: return 0
+        r = rank_mod(M, Q31)
+        if r == full: stats[tag + ' full mod q'] += 1; return r          # rank mod q <= rank over Q <= full
+        if hadamard_ok(M, M521): stats[tag + ' M521'] += 1; return rank_mod(M, M521)
+        stats[tag + ' Bareiss'] += 1; return rank_exact(M)
+
+    # exact rational bases, used for the late-copy intersections
+    def exact_basis(self, key):
+        pr, h = self.pr, self.h
+        if key[0] in ('out', 'ret'):
+            return null_exact([self.rootcov[key]], h)
+        n = key[1]
+        S = [trow(h, pr.trip[i]) for i in self.bas[n]]
+        j = pr.j.get(n, 0)
+        if not j: return S
+        _, KF = self.KF[n]; F = pr.F[:j]
+        X = null_exact(KF, j) if KF else [[int(i == t) for t in range(j)] for i in range(j)]
+        vecs = S + [[sum(c * F[i][q] for i, c in enumerate(x)) for q in range(h)] for x in X]
+        # independent subset: rows independent mod q are independent over Q; d of them span M_n (dim M_n = d exactly)
+        E = Echelon(Q31); out = []
+        for vv in vecs:
+            if E.insert(vv): out.append(vv)
+        assert len(out) == self.d[n], ('exact basis dim', n, len(out), self.d[n])
+        return out
+
+    def intersect_exact(self, A, B):
+        h = self.h
+        if not A or not B: return []
+        # z = (x, y): x A = y B  <=>  [A; -B]^T z = 0
+        M = [[A[i][q] for i in range(len(A))] + [-B[i][q] for i in range(len(B))] for q in range(h)]
+        Z = null_exact(M, len(A) + len(B))
+        # A and B are bases, so z -> x A is injective on the null space: the images are a basis of A cap B
+        return [[sum(z[i] * A[i][q] for i in range(len(A))) for q in range(h)] for z in Z]
+
+
+def late_frames(fr, pr, users):
+    """exact dims of every late-copy intersection frame ('c', n, late users)."""
+    cache = {}; dims = {}
+    def eb(k):
+        if k not in cache: cache[k] = fr.exact_basis(k)
+        return cache[k]
+    def ufr(n, k):
+        u = users[n][k]; return ('n', u[1]) if u[0] == 'gate' else root_key(u)
+    for n, (early, lt) in pr.late.items():
+        for i in range(len(lt) - 1):
+            B = eb(ufr(n, lt[-1]))
+            for k in reversed(lt[i:-1]): B = fr.intersect_exact(eb(ufr(n, k)), B)
+            dims[('c', n, tuple(lt[i:]))] = len(B)
+    return dims, cache
+
+
+def chain_steps(chains, dimof, h):
+    rk = Counter(); steps = {}
+    for ch in chains:
+        prev = ch[0]; pd = dimof(prev)
+        for k in ch[1:]:
+            dk = dimof(k)
+            assert dk >= pd, ('decreasing chain', prev, k, pd, dk)
+            if dk > pd: rk[dk - pd] += 1; steps[prev, k] = dk - pd; prev, pd = k, dk
+            else: prev = k                                                 # equal dims, nested => equal frames
+        assert pd == h
+    return rk, steps
+
+
+# ----------------------------------------------------------------------------------------------- E. mod-q frames, side lemma
+def point_UV(h, seed):
+    """the U, V drawn by independent/two-stage-bit/flag_existence.py with this seed (same random stream)."""
+    CR = 10**6; rng = random.Random(seed)
+    nz = lambda: rng.choice([-1, 1]) * rng.randint(1, CR)
+    for b in range(h):
+        for i in range(h):
+            for jj in range(h):
+                if i <= b and jj <= b: nz()
+    for b in range(h):
+        for i in range(h):
+            for jj in range(h):
+                if i >= b and jj >= b: nz()
+    U = [[rng.randint(-CR, CR) for _ in range(h)] for _ in range(h)]
+    V = [[rng.randint(-CR, CR) for _ in range(h)] for _ in range(h)]
+    return U, V
+
+
+class ModQ:
+    def __init__(self, fr, latedims, control=False):
+        self.fr = fr; self.h = h = fr.h; self.q = q = Q31; self.latedims = latedims
+        U, V = point_UV(h, 1)
+        if control: U = [[int(i == k) for k in range(h)] for i in range(h)]; V = [r[:] for r in U]
+        self.conj = []
+        for A in (U, V):
+            Ai = inv_mod_matrix(A, q); assert Ai is not None
+            self.conj.append(([list(c) for c in zip(*A)], Ai))          # columns of A, rows of A^-1
+        self.inv9 = pow(9, q - 2, q); self.bcache = {}; self.qcache = OrderedDict(); self.cap = 6000
+        self.degenerate = 0; self.dimfail = 0
+
+    def basis(self, key):
+        if key in self.bcache: return self.bcache[key]
+        fr, h, q = self.fr, self.h, self.q; pr = fr.pr
+        if key == ('0',): B = []
+        elif key == ('F',): B = [[int(i == t) for t in range(h)] for i in range(h)]
+        elif key[0] in ('out', 'ret'): B = null_mod([fr.rootcov[key]], h, q)
+        elif key[0] == 'c':
+            n, lt = key[1], key[2]
+            B = None
+            for k in lt:
+                u = fr.users[n][k]; kk = ('n', u[1]) if u[0] == 'gate' else root_key(u)
+                Bk = self.basis(kk)
+                B = Bk if B is None else self._cap(B, Bk)
+            if len(B) != self.latedims[key]: self.dimfail += 1
+        else:
+            n = key[1]; S = [trow(h, pr.trip[i]) for i in fr.bas[n]]; j = pr.j.get(n, 0)
+            E = Echelon(q)
+            for r in S: E.insert(r)
+            if j:
+                r2, KF = fr.KF[n]
+                if KF and rank_mod(KF, q) != r2: self.dimfail += 1
+                X = null_mod(KF, j, q) if KF else [[int(i == t) for t in range(j)] for i in range(j)]
+                Fj = pr.F[:j]
+                for x in X: E.insert([sum(c * Fj[i][t] for i, c in enumerate(x)) % q for t in range(h)])
+            B = E.rows
+            if len(B) != fr.d[n]: self.dimfail += 1
+        self.bcache[key] = B
+        return B
+
+    def _cap(self, A, B):
+        q, h = self.q, self.h
+        if not A or not B: return []
+        M = [[A[i][t] for i in range(len(A))] + [(-B[i][t]) % q for i in range(len(B))] for t in range(h)]
+        Z = null_mod(M, len(A) + len(B), q); E = Echelon(q)
+        for z in Z: E.insert([sum(z[i] * A[i][t] for i in range(len(A))) % q for t in range(h)])
+        return E.rows
+
+    def gram_inv(self, B):
+        q = self.q; BG = [[(x - sum(b) * self.inv9) % q for x in b] for b in B]
+        Gr = [[sum(x * y for x, y in zip(bg, c)) % q for c in B] for bg in BG]
+        return BG, inv_mod_matrix(Gr, q)
+
+    def nondeg(self, key):
+        B = self.basis(key)
+        if not B or len(B) == self.h: return True
+        return self.gram_inv(B)[1] is not None
+
+    def conjugated(self, key):
+        """(U^T P U^-T, V^T P V^-T) for the G-orthogonal projector P onto the frame, flattened."""
+        if key in self.qcache: self.qcache.move_to_end(key); return self.qcache[key]
+        h, q = self.h, self.q; B = self.basis(key)
+        if not B: out = (array('q', [0] * h * h),) * 2
+        elif len(B) == h: I = array('q', [int(i == t) for i in range(h) for t in range(h)]); out = (I, I)
+        else:
+            BG, Gi = self.gram_inv(B)
+            if Gi is None: raise ZeroDivisionError(key)
+            res = []
+            for Acols, Ai in self.conj:
+                LU = [[sum(x * y for x, y in zip(ac, b)) % q for b in B] for ac in Acols]       # A^T B^T  (h x d)
+                RU = [[sum(x * y for x, y in zip(bg, ai)) % q for ai in Ai] for bg in BG]      # B G A^-T (d x h)
+                Mi = matmul(Gi, RU, q)
+                Mt = list(zip(*Mi))
+                res.append(array('q', [sum(x * y for x, y in zip(l, c)) % q for l in LU for c in Mt]))
+            out = tuple(res)
+        self.qcache[key] = out
+        if len(self.qcache) > self.cap: self.qcache.popitem(last=False)
+        return out
+
+
+def side_lemma(mq, steps, h, retained):
+    q = mq.q; pr_ = predicted(h); E = dict((k, r) for k, r in steps.items() if 2 * r > h)
+    for c in retained: E[('0',), ('ret', c)] = h - 1                    # retained copy edges 0 -> U_c
+    keys = sorted(E, key=lambda e: (str(e[1]), str(e[0])))              # group by target frame (cache reuse)
+    bad = 0; n = 0
+    for (A, B) in keys:
+        r = E[A, B]; QA, QB = mq.conjugated(A), mq.conjugated(B)
+        for t in range(2):
+            a, b = QA[t], QB[t]
+            M = [[(b[i * h + c] - a[i * h + c]) % q for c in range(h)] for i in range(h)]
+            if rightmost_pivots(M, q) != pr_[r]: bad += 1
+        n += 1
+        if n % 5000 == 0: log('   side lemma %d/%d, failures %d' % (n, len(keys), bad))
+    return bad, len(keys)
+
+
+# ----------------------------------------------------------------------------------------------- F. certificate
+K_REPO = os.path.normpath(os.path.join(HERE, '..', '..'))
+sys.path[:0] = [K_REPO + '/scripts', K_REPO + '/independent/two-stage-bit']
+
+
+def inner_children(r, h): return [1] * (h - r) + [2 * r - h] if 2 * r > h else [1] * r
+
+
+def rt_histogram(h, R, rk, corner):
+    """cert_rt.rt_histogram with the data-entrance corner profile 'corner' (round six: [h-2] + [1]*(h+1))."""
+    from math import comb
+    v, m = comb(h, 3), h * h; N = v * v; A = v * R; H = Counter()
+    assert sum(r * n for r, n in rk.items()) == h * R and sum(corner) == 2 * h - 1
+    for _ in range(2):
+        H[m - 2 * h] += A; H[h] += A
+        for r, n in rk.items():
+            for w in inner_children(r, h): H[w] += v * n
+        for w in inner_children(h - 1, h): H[w] += v * h
+    H[m - 4 * h + 2] += 2 * N
+    for w in corner: H[w] += 2 * N
+    for _ in range(2): H[h - 2] += 2 * N; H[1] += 2 * N
+    H[1] += N
+    W = 2 * N + 2 * A; L = 2 * v * h * (h - 1); s = W * m - N + L
+    H = {w: c for w, c in H.items() if c}
+    assert sum(w * c for w, c in H.items()) == s and max(H) <= m - 1
+    return dict(h=h, m=m, v=v, N=N, W=W, L=L, s=s, R=R, hist=H)
+
+
+def certificate(h, R, rk):
+    import moment
+    from certificate_round3 import evaluate
+    out = {}
+    for name, corner in (('round6 corner', [h - 2] + [1] * (h + 1)), ('staircase', [h - 2, h - 5] + [1] * 6)):
+        c = rt_histogram(h, R, rk, corner)
+        if c['W'] * c['m'] <= c['s']: out[name] = ('no deficit', c['s']); continue
+        a, root = moment.certify(c)
+        k = evaluate(a, Fr(36926111, 5 * 10**11), Fr(1, 1000), 'crude', m_c=576, s_c=119453132304)
+        out[name] = (a, root, c['s'], c['W'], k['kappa'], k['ok'])
+    return out
+
+
+# ----------------------------------------------------------------------------------------------- main
+def main(h, generic=True, control=False):
+    W = load_witness(h)
+    pr = Prog(W); log('h=%d: A. DAG semantics OK (%d nodes, %d additions, %d links, %d lifted, %d late)' % (
+        h, len(pr.act), sum(1 for n in pr.act if pr.args[n]), len(pr.mL), len(pr.j), len(pr.late)))
+    bas, dn = spans(pr); users = users_of(pr, dn)
+    ops, chains, reads, R = compile_(pr, dn, users)
+    log('B. compiled: R=%d (claimed %d), ops %d, late copies %d' % (R, W['claimed']['R'], len(ops),
+        sum(len(l) - 1 for _, l in pr.late.values())))
+    assert R == W['claimed']['R']
+    rng = random.Random(h)
+    res = {(ring, mir, mid): all(replay(pr, ops, reads, R, ring, rng, mir, mid) for _ in range(2))
+           for ring in (2, 0) for mir in (False, True) for mid in (True, False)}
+    log('C. dirty-scratch replay (F2/Z x word/mirror x midread/endread): %s' % ('ALL OK' if all(res.values()) else res))
+    fr = Frames(pr, dn, bas, users)
+    bj, bl, br, Fok = fr.check_structure()
+    log('D. nesting facts: j-monotone failures %d, link span failures %d, root containment failures %d, F invertible %s'
+        % (bj, bl, br, Fok))
+    fr.cobases(); log('   co-bases K_n done')
+    d, st = fr.dims(); log('   lifted frame dims exact: %s' % dict(st))
+    nU = sum(1 for n in pr.j if d[n] == h - len(fr.K[n]))
+    log('   every lifted frame is the largest admissible frame U_n: %s (%d / %d; the flag only switches span -> U_n)'
+        % (nU == len(pr.j), nU, len(pr.j)))
+    ld, ecache = late_frames(fr, pr, users); log('   late-copy intersections exact: %d frames' % len(ld))
+    def dimof(k):
+        if k == ('0',): return 0
+        if k == ('F',): return h
+        if k[0] in ('out', 'ret'): return h - 1
+        if k[0] == 'c': return ld[k]
+        return d[k[1]]
+    rk, steps = chain_steps(chains, dimof, h)
+    claimed = Counter({int(r): c for r, c in W['claimed']['rk'].items()})
+    log('   chain steps: sum of ranks %d = h R %d: %s; histogram equals claim: %s; distinct steps %d (2r>h: %d)' % (
+        sum(r * c for r, c in rk.items()), h * R, sum(r * c for r, c in rk.items()) == h * R, rk == claimed,
+        len(steps), sum(1 for r in steps.values() if 2 * r > h)))
+    out = dict(h=h, R=R, replay=all(res.values()), struct=(bj, bl, br, Fok), hist_ok=rk == claimed, rk=rk)
+    if generic:
+        mq = ModQ(fr, ld, control)
+        frames_used = {k for e in steps for k in e} | {('ret', c) for c in pr.retained}
+        nd = sum(1 for k in frames_used if not mq.nondeg(k))
+        log('E. frames used %d: G-degenerate %d, reduction dimension mismatches %d' % (len(frames_used), nd, mq.dimfail))
+        bad, ne = side_lemma(mq, steps, h, pr.retained)
+        log('   side lemma at one integer point%s: failures %d over %d distinct high-rank steps x 2 conjugations' % (
+            ' (CONTROL U=V=I)' if control else '', bad, ne))
+        out.update(degenerate=nd, dimfail=mq.dimfail, side_bad=bad, side_n=ne)
+    for name, val in certificate(h, R, rk).items():
+        if val[0] == 'no deficit': log('F. %s: no deficit at this h (s=%d), certificate n/a' % (name, val[1])); continue
+        a, root, s, Wr, kap, ok = val
+        log('F. %s: s=%d W=%d a_b=%s (%.6e, root %.6e) kappa=%s (%.7e) ok=%s' % (name, s, Wr, a, float(a), root, kap,
+            float(kap), ok))
+        out[name] = (a, kap, ok)
+    return out, pr, dn, users
+
+
+
+
+# ----------------------------------------------------------------------------------------------- negative controls
+def controls(h, nsample=40):
+    """Each must FAIL (or differ) for the check to have teeth:
+     (a) one late copy moved to just after its pivot's consumer gate: replay must fail;
+     (b) one late copy dropped: replay must fail;
+     (c) pivot chain without intersections (C_i := frame of user i alone): exhibit non-nested consecutive pivot frames;
+     (d) M_n rebuilt from an independent flag F' on targets strictly between span and U (vacuous for this
+         witness: every lifted frame equals U_n, so the flag is only a switch and nesting needs no common flag);
+     (e) side lemma at U = V = I on the first high-rank steps: generic pivots must fail;
+     (f) no late copies (lifted frames only): histogram must change."""
+    W = load_witness(h)
+    pr = Prog(W); bas, dn = spans(pr); users = users_of(pr, dn); rng = random.Random(7)
+    key = lambda n: (dn[n], n); linked = {r: x for x, r in pr.mL.items()}
+    cand = [n for n, (e, lt) in pr.late.items() if users[n][lt[-1]][0] == 'gate']
+    fails = 0; tried = 0
+    for n in cand[:nsample]:
+        tamper_node[0] = n; ops, ch, rd, R = compile_(pr, dn, users, tamper='aftercons')
+        ok = all(replay(pr, ops, rd, R, ring, rng) for ring in (2, 0)); tried += 1; fails += not ok
+    log('(a) late copy moved after the pivot consumer: replay failed %d / %d (must be all)' % (fails, tried))
+    ops, ch, rd, R = compile_(pr, dn, users, tamper='dropcopy')
+    log('(b) one late copy dropped: replay correct %s (must be False)' % all(replay(pr, ops, rd, R, ring, rng) for ring in (2, 0)))
+    fr = Frames(pr, dn, bas, users); fr.cobases(); d, _ = fr.dims(); d = dict(d); ld, _ = late_frames(fr, pr, users)
+    mq = ModQ(fr, ld)
+    def inside(A, B):
+        E = Echelon(mq.q)
+        for r in mq.basis(B): E.insert(r)
+        return all(E.contains(r) for r in mq.basis(A))
+    def ufr(n, k):
+        u = users[n][k]; return ('n', u[1]) if u[0] == 'gate' else root_key(u)
+    bad = tot = 0
+    for n, (e, lt) in pr.late.items():
+        for i in range(len(lt) - 1):
+            tot += 1; bad += not inside(ufr(n, lt[i]), ufr(n, lt[i + 1]))
+    log('(c) pivot chain through user frames without intersections: non-nested consecutive frames %d / %d '
+        '(the intersections are needed)' % (bad, tot))
+    # (d) the common flag is essential: rebuild M_n with an independent flag F' (same j(n)) and test M_n <= M_t
+    pairs = [(n, t) for n in pr.j for t in fr.succ[n] if pr.j.get(t, 0) > 0 and dn[t] < d[t] < h - len(fr.K[t])]
+    rng.shuffle(pairs); bad = tot = 0; r2 = random.Random(99)
+    Fp = [[r2.randint(-50, 50) for _ in range(h)] for _ in range(h)]
+    for n, t in pairs[:nsample]:
+        j = pr.j[n]; K = [fr.rootcov[fr.rkeys[r]] for r in fr.K[n]]
+        KF = [[sum(a * b for a, b in zip(kr, fl)) for fl in Fp[:j]] for kr in K]
+        if KF and rank_mod(KF, mq.q) != fr._rank(KF, min(len(K), j), Counter(), 'KF'): continue
+        X = null_mod(KF, j, mq.q) if KF else [[int(i == c) for c in range(j)] for i in range(j)]
+        E = Echelon(mq.q)
+        for r in mq.basis(('n', t)): E.insert(r)
+        vecs = [[sum(c * Fp[i][q] for i, c in enumerate(x)) % mq.q for q in range(h)] for x in X]
+        tot += 1; bad += not all(E.contains(v) for v in vecs)
+    log("(d) M_n rebuilt with an independent flag F': M_n not inside M_t on %d / %d lifted successor pairs (targets strictly between span and U)" % (bad, tot))
+    ops, chains, reads, R = compile_(pr, dn, users)
+    fr2 = Frames(pr, dn, bas, users); fr2.cobases(); d2, _ = fr2.dims(); ld2, _ = late_frames(fr2, pr, users)
+    dimof = lambda k: 0 if k == ('0',) else h if k == ('F',) else h - 1 if k[0] in ('out', 'ret') else ld2[k] if k[0] == 'c' else d2[k[1]]
+    rk, steps = chain_steps(chains, dimof, h)
+    mqs = ModQ(fr2, ld2); smp = list(steps); random.Random(5).shuffle(smp); nb = 0
+    for A, B in smp[:3000]:
+        E = Echelon(mqs.q)
+        for r in mqs.basis(B): E.insert(r)
+        nb += not all(E.contains(r) for r in mqs.basis(A))
+    log('(g) sanity (positive): 3000 sampled chain steps A -> B with A inside B mod q: failures %d' % nb)
+    mq0 = ModQ(fr2, ld2, control=True); pr_ = predicted(h); hi = [(e, r) for e, r in steps.items() if 2 * r > h][:300]
+    nb = 0
+    for (A, B), r in hi:
+        QA, QB = mq0.conjugated(A), mq0.conjugated(B)
+        M = [[(QB[0][i * h + c] - QA[0][i * h + c]) % mq0.q for c in range(h)] for i in range(h)]
+        nb += rightmost_pivots(M, mq0.q) != pr_[r]
+    log('(e) side lemma with U = V = I: failures %d / %d (must be most)' % (nb, len(hi)))
+    ops, chains0, reads, R = compile_(pr, dn, users, late=False)
+    rk0, _ = chain_steps(chains0, dimof, h)
+    log('(f) without late copies: histogram differs %s; certificate %s' % (rk0 != rk, {k: v[0] for k, v in certificate(h, R, rk0).items()}))
+
+
+if __name__ == '__main__':
+    h = int(sys.argv[1])
+    if '--controls' in sys.argv: controls(h)
+    else:
+        out = main(h, generic='--no-generic' not in sys.argv)[0]
+        ok = out['replay'] and out['struct'][:3] == (0, 0, 0) and out['struct'][3] and out['hist_ok']
+        ok &= out.get('degenerate', 0) == 0 and out.get('dimfail', 0) == 0 and out.get('side_bad', 0) == 0
+        ok &= all(v[2] for k, v in out.items() if k in ('round6 corner', 'staircase'))
+        log('ALL %s' % ('PASS' if ok else 'FAIL'))
+        sys.exit(0 if ok else 1)
diff --git a/references/partial-gauge/pr97/deferred-readout.tex b/references/partial-gauge/pr97/deferred-readout.tex
new file mode 100644
index 0000000000..d79abea77a
--- /dev/null
+++ b/references/partial-gauge/pr97/deferred-readout.tex
@@ -0,0 +1,111 @@
+\documentclass[11pt]{article}
+\usepackage{amsmath,amssymb,amsthm}
+\newtheorem{lemma}{Lemma}
+\newtheorem{proposition}{Proposition}
+\theoremstyle{remark}\newtheorem*{remark}{Remark}
+\title{Deferred readouts in the two-stage bit interchange}
+\date{Draft, October 9, 2026}
+\begin{document}
+\maketitle
+
+This note describes the round-seven bit side. It keeps the round-six interchange (flag basis, retained point totals
+with copied centres, side roles batched one level down, data-entrance run) and changes three things: the side
+program comes from PR~\#62's producer, its frames are lifted and its fan-out copies are made late, and the stage-one
+word is reordered so that most slots read their garbage out at a nonzero frame. All objects are frozen in
+\texttt{certificates/round7/}; \texttt{scripts/certificate\_round7.py} rebuilds the child-width histogram from the
+schedule, and the checks in \texttt{independent/deferred-readout/} verify every fact used below. Throughout, $h=23$,
+$m=h^2$, $v=\binom h3$, $N=v^2$, $G=I-J/9$, and for a triple $T$ the root frame of the output wire $y_T$ is
+$t_T^\perp=\ker\,(9t_T-3\mathbf 1)^{\mathsf T}$; the frame of a retained total of $c$ is $U_c=\ker(\mathbf 1-3e_c)^{\mathsf T}$.
+
+\section{The side program: PR \#62's producer}
+The side DAG has the $v$ triples as leaves and disjoint additions as nodes; its roots are the $3v$ outputs
+$\mathrm{out}(c,T)=\{S: S\cap T=\{c\}\}$ and the $h$ retained totals $\mathrm{star}(c)$. We build it with PR~\#62's
+producer (ikeboy): interval strips and core-aware pair assembly, with PR~\#41's alternating order and links
+(RaD/hipotures), and links chosen by a moment-weighted matching in the spirit of PR~\#44 (rohanarun). A link hands the
+non-pivot slot of a donor gate to a later use of the same value, so the number of slots is
+$R=\#\text{additions}+3v+h-\#\text{links}=28866$. Only the DAG enters the certificate, and it is checked from the
+definitions: every addition is disjoint, every output and total has the support above, and every node is reachable
+from a root (\texttt{check\_lifted.py}, part~A; \texttt{check\_word.py}, part~A).
+
+\section{Lifted frames and late copies}
+For an addition $n$ let $U_n$ be the common kernel of the root covectors reachable from $n$ and $F_j$ the span of
+the first $j$ rows of a fixed integer flag matrix. The gate of $n$ acts at
+\[ M_n=\operatorname{span}(n)+(U_n\cap F_{j(n)}), \]
+with a lift index $j(n)$ per addition. This is Paureel's complement-frame construction: it is valid for any frames
+with (H1) leaf frames $\langle t_S\rangle$, (H2) frames nested along every slot (DAG edges, links, copies),
+(H3) roots inside $t_T^\perp$ or $U_c$, (H4) $G$ nondegenerate on every frame. Nesting is a theorem given
+$j(n)\le j(t)$ on every slot successor pair, $\operatorname{span}(x)\le\operatorname{span}(t)$ on every link and
+$\operatorname{span}(n)$ inside every root it reaches directly; then $U_n\le U_t$ and $M_n\le M_t$. In this witness
+every lifted frame equals $U_n$. Dimensions are exact (modular law, ranks with Hadamard certificates or Bareiss).
+
+\paragraph{Late copies.} A node with several free uses is copied for each of them. A copy made early sits at $M_n$;
+a \emph{late} copy for the user $\ell_i$ of an ordered list $\ell_1,\dots,\ell_k$ is made when the pivot has climbed to
+\[ C_i=\textstyle\bigcap_{i'\ge i} M(\ell_{i'}), \qquad M_n\le C_1\le\dots\le C_{k-1}\le M(\ell_k), \]
+and starts there, so the copy saves the rank $\dim C_i-\dim M_n$. Each copy is placed as late as possible (before its
+user's gate, the pivot's next gate and the next copy). The intersections are needed: without them consecutive pivot
+frames are not nested (1396 of 1396), and a copy moved after the pivot's next gate breaks the replay.
+
+\section{The B-defer word}
+In stage one each slot $u$ starts with its garbage $z_u$, and the word reads it out of every target it reaches:
+$y_T\mathrel{-}=C_{T,u}z_u$, where $C=JL$ is given by the transpose sweep of $L$. In rounds five and six all these
+readouts happen at frame $0$, before the second pass. B-defer splits the second pass in two phases.
+
+\begin{itemize}
+\item \emph{Phase 1} is the closure of the retained totals' last operations under per-slot sequence precedence: every
+  op that must precede a total's completion. The totals complete, and their copies read every target containing
+  their centre at $0$ (copied centres, as in round six).
+\item A slot that phase 1 neither reads nor writes still holds its garbage after phase 1. It reads it out
+  \emph{now}, at a frame $\sigma_u\le F_0(u)\cap\bigcap_{T}t_T^\perp$ (over its targets $T$), then the rest of $L$ runs,
+  then the output reads at $t_T^\perp$, then $L^{-1}$ and $V^{-1}$ exactly mirror it.
+\end{itemize}
+Only the topological order of the word changes, so it is still a mirror and still restores arbitrary scratch. The
+slot's chain now starts at $\sigma_u$: its corner has rank $r_u=h-\dim\sigma_u$ and its auxiliary edge rank $m-r_u$.
+
+\paragraph{The sorting rule.} On a target $y_T$ the readout frames, in time order, are $0$ (non-deferred slots and
+centre copies), the deferred $\sigma_u$, and $t_T^\perp$; they must form a chain. The word reads the deferred slots
+in increasing $\dim\sigma_u$, and applies the deferred $V$ gates (below) in increasing $\dim F_0$. With this rule
+every $y_T$ sequence is nested in time order, exactly over $\mathbb Q$, for both the $\mathbb Z$ and the
+$\mathbb F_2$ supports of $C$; the reversed order breaks 11879 chains.
+
+\section{V leaves as late copies on $X_S$}
+A leaf $S$ is used by several gates. Instead of copying $x_S$ by fan-out, every use $i$ gets its own $V$ gate on the
+data wire $X_S$, into a slot that starts at
+\[ s_i=N_i\cap N_{i+1}\cap\cdots \qquad(\text{falling back to }\langle t_S\rangle\text{ if $G$ is degenerate on it}), \]
+where $N_j$ is the first frame of use $j$. The $V$ gates are then late copies on $X_S$: its frame chain
+$\langle t_S\rangle\le s_{i_1}\le s_{i_2}\le\dots\le F$ has total rank $h-1$, as before, and the use slots start at
+$s_i$ instead of $\langle t_S\rangle$. The $V$ gates of deferred slots come after their readouts; in time the $X_S$ chain
+is $\langle t_S\rangle$, the early $V$ frames (by dimension), the deferred $V$ frames (by the rule above), $F$, and it
+is checked nested exactly in that order. The number of slots is unchanged.
+
+\section{What the certificate uses}
+\paragraph{Only the $\mathbb F_2$ flag identity.} The bit network is used over $\mathbb F_2$: the word must add exactly
+$x_T$ to every $y_T$ and restore every slot, for arbitrary scratch. The replay checks this over $\mathbb F_2$ with
+the parity coefficients (the actual circuit) and, separately, over $\mathbb Z$ with the exact path-count
+coefficients; the histogram and $a_b$ are the same with either support.
+
+\paragraph{The stage-two complement.} Stage two is the complement time-reversal of stage one, with the same edge
+multiset. This is the same assumption as in round six; it is stated, not machine-checked here.
+
+\paragraph{Frames and genericity.} Every $\sigma_u$ and $s_i$ is a primitive integer basis (entries at most 13).
+Exactly over $\mathbb Q$: $\sigma_u\le F_0(u)$, $\sigma_u\le t_T^\perp$, $t_S\le s_i\le N_i$, $G$ nondegenerate. Every
+high-rank step ($2r>h$) that the reordered word adds (slot starts, gauged corners $\sigma_u\to F$, $y_T$ and $X_S$
+chain steps) has the generic north-east pivots under both conjugations at the integer point of
+\texttt{check\_lifted.py}, so the side lemma batches it.
+
+\section{Accounting}
+Per stage and invocation: every slot contributes its auxiliary edge (block $m-2r_u$) and its corner, both batched
+by the inner lemma, and its chain steps; every centre copy its step $U_c\to0$ of rank $h-1$; every $y_T$ and every
+$X_S$ its chain of total rank $h-1$; the data entrance its block $m-4h+2$ and corner; one copy-correction singleton per
+pair. The rank sum is exactly $s=Wm-N+L$ with $W=2N+2vR$, $L=2vh(h-1)$, so the histogram re-partitions the round-six
+budget. The moment certificate gives
+\begin{align*}
+ a_b&=31987/(5\cdot10^8) &&\text{(staircase entrance corner, runs $h-2,h-5,1^6$)},\\
+ a_b&=12599/(2\cdot10^8) &&\text{(round-six entrance corner)},
+\end{align*}
+and with round six's complex interchange ($a_c=36926111/(5\cdot10^{11})$, $h=24$) the assembly gives
+$\kappa=1599247689723/(2.5\cdot10^{16})\approx6.39699\times10^{-5}$ with the staircase corner and
+$\kappa=6299103187973/10^{17}$ with the round-six corner. The staircase corner is certified by
+\texttt{check\_stair.py} at the same integer point; the entrance edge is not changed by the reordering.
+
+\end{document}
+
diff --git a/references/partial-gauge/pr97/deferred.py b/references/partial-gauge/pr97/deferred.py
new file mode 100644
index 0000000000..d045126217
--- /dev/null
+++ b/references/partial-gauge/pr97/deferred.py
@@ -0,0 +1,189 @@
+"""Shared reader and accounting for the round-seven bit side (stdlib only).
+
+Frozen inputs (certificates/round7/):
+  witness_23.json.gz   the base side program: DAG (leaves = triples of [h], additions), outputs (c, T), retained
+                       totals, links, the flag matrix, the lift index j(n) of every lifted addition and the late-copy
+                       decisions; checked by check_lifted.py;
+  deferred_23.json.gz  the reordered stage-1 word: the slot schedule (ops, the node held by every slot over time,
+                       slot starts, output and retained slots), the deferred readout frames sigma_u, the V-leaf start
+                       frames s_i (both as primitive integer bases), the readout order, the X_S order of every leaf,
+                       the order of the deferred V gates, and the dimension of every node frame and late-copy
+                       intersection (re-derived exactly by check_frames.py).
+
+Ops: ['add', piv, oth, n]  a[piv] += a[oth], the addition of node n;  ['fan', piv, [f...]]  a[f] += a[piv] for fresh
+slots f (copies);  ['src', s, n]  the V gate a[s] += x_S of leaf n (applied outside L, see check_word.py)."""
+import gzip, json, os
+from collections import Counter, defaultdict
+from itertools import combinations
+from math import comb
+
+HERE = os.path.dirname(os.path.abspath(__file__))
+ROOT = os.path.normpath(os.path.join(HERE, '..', '..'))
+DATA = os.path.join(ROOT, 'certificates', 'round7')
+
+
+def load_json(name):
+    with gzip.open(os.path.join(DATA, name), 'rt') as fh:
+        return json.load(fh)
+
+
+def load(h=23, witness=None, data=None):
+    """the base witness and the design-B data of one round-seven witness (default: the files of h)."""
+    W = load_json(witness or 'witness_%d.json.gz' % h); D = load_json(data or 'deferred_%d.json.gz' % h)
+    assert W['h'] == D['h'] == h
+    return W, D
+
+
+class Schedule:
+    """The design-B schedule in convenient form."""
+    def __init__(self, W, D):
+        self.h = h = D['h']; self.R = D['R']; self.trip = list(combinations(range(h), 3)); self.v = len(self.trip)
+        self.tid = {T: i for i, T in enumerate(self.trip)}
+        self.args = {int(n): (tuple(a) if a else None) for n, a in W['args'].items()}
+        self.ops = [tuple(o[:2]) + (tuple(o[2]),) if o[0] == 'fan' else tuple(o) for o in D['ops']]
+        self.hold = D['hold']; self.start = D['start']
+        self.out = {s: (c, tuple(T)) for s, c, T in D['out']}
+        self.ret = {s: c for s, c in D['ret']}
+        self.srcop = {o[1]: o[2] for o in self.ops if o[0] == 'src'}
+        self.readout = D['readout_order']; self.sel = set(self.readout)
+        self.sigma = dict(zip(D['readout_order'], D['sigma']))
+        self.f = [0] * self.R
+        for s, B in self.sigma.items(): self.f[s] = len(B)
+        self.vstart = dict(zip(D['vleaf_slots'], D['vleaf_start']))
+        self.xs = [(n, us) for n, us in D['xs_order']]
+        self.late_v = D['deferred_v_order']
+        self.node_dims = {n: d for n, d in D['node_dims']}
+        self.late_dims = {(n, tuple(lt)): d for n, lt, d in D['late_dims']}
+        self.late = {int(n): (list(e), list(l)) for n, (e, l) in W['late'].items()}
+        self.pivslot = {o[3]: o[1] for o in self.ops if o[0] == 'add'}
+
+    def touch(self, op):
+        if op[0] == 'add': return [op[1], op[2]]
+        if op[0] == 'fan': return [op[1]] + list(op[2])
+        return []
+
+    def root_targets(self):
+        rt = {}
+        for s, (c, T) in self.out.items(): rt[s] = [self.tid[T]]
+        for s, c in self.ret.items(): rt[s] = [t for t, T in enumerate(self.trip) if c in T]
+        return rt
+
+    def adjoint(self):
+        """garbage coefficients over Z: cov[s][t] = coefficient of slot s's start value in target y_T at the end of L
+        (the transpose sweep of L applied to the root reads); the F2 circuit uses their parities."""
+        cov = [dict() for _ in range(self.R)]
+        for s, ts in self.root_targets().items():
+            for t in ts: cov[s][t] = cov[s].get(t, 0) + 1
+        def addto(d, e):
+            for t, x in e.items(): d[t] = d.get(t, 0) + x
+        for op in reversed(self.ops):
+            if op[0] == 'add': addto(cov[op[2]], cov[op[1]])
+            elif op[0] == 'fan':
+                for g in op[2]: addto(cov[op[1]], cov[g])
+        return cov
+
+    # ------------------------------------------------------------------ frame keys of every slot's chain
+    def start_key(self, s):
+        """('v', s) for a V-leaf slot (start s_i), ('n', n0) for a base or early copy, ('c', n0, late suffix)."""
+        n0 = self.hold[s][0]
+        if self.args[n0] is None: return ('v', s)
+        st = self.start[s]; k = st[2]; dec = self.late.get(n0)
+        if dec is None or k in dec[0]: return ('n', n0)
+        lt = dec[1]; return ('c', n0, tuple(lt[lt.index(k):]))
+
+    def chain_keys(self, s):
+        """frames of slot s in time order: [sigma_u | 0], start, then every later node it holds (with the pivot's
+        late-copy intersections after a node it pivots), its root frame if any, F."""
+        ch = [('sigma', s) if s in self.sel else ('0',), self.start_key(s)]
+        for idx, n in enumerate(self.hold[s]):
+            if idx > 0: ch.append(('n', n))
+            if self.args[n] is not None and self.pivslot.get(n) == s and n in self.late:
+                lt = self.late[n][1]
+                for i in range(len(lt) - 1): ch.append(('c', n, tuple(lt[i:])))
+        if s in self.out: c, T = self.out[s]; ch.append(('out', c, T))
+        if s in self.ret: ch.append(('ret', self.ret[s]))
+        ch.append(('F',)); return ch
+
+    def dim(self, key):
+        k = key[0]
+        if k == '0': return 0
+        if k == 'F': return self.h
+        if k in ('out', 'ret'): return self.h - 1
+        if k == 'n': return self.node_dims[key[1]]
+        if k == 'c': return self.late_dims[key[1], key[2]]
+        if k == 'sigma': return len(self.sigma[key[1]])
+        if k == 'v': return len(self.vstart[key[1]])
+        raise KeyError(key)
+
+    def ylevels(self, cov):
+        """deferred readout levels (dim sigma) on every target, from the garbage-coefficient supports."""
+        lev = defaultdict(set)
+        for s in self.readout:
+            for t in cov[s]: lev[t].add(self.f[s])
+        return {t: sorted(lev[t]) for t in range(self.v)}
+
+    def xdata(self, dimof=None):
+        dimof = dimof or (lambda s: len(self.vstart[s]))
+        out = []
+        for n, us in self.xs:
+            ds = [1] + [dimof(s) for s in us] + [self.h]
+            ds = [d for i, d in enumerate(ds) if i == 0 or d != ds[i - 1]]
+            out.append([b - a for a, b in zip(ds, ds[1:])])
+        return out
+
+
+def inner(r, h):
+    """children of a rank-r chain step under the inner-corner lemma (one block of width 2r-h when 2r > h)."""
+    return [1] * (h - r) + [2 * r - h] if 2 * r > h else [1] * r
+
+
+def chain_ranks(S, dimof=None):
+    """multiset of chain-step ranks over all slots (dims must be monotone along every chain)."""
+    dimof = dimof or S.dim; rk = Counter()
+    for s in range(S.R):
+        ds = [dimof(k) for k in S.chain_keys(s)]
+        assert all(b >= a for a, b in zip(ds, ds[1:])), ('decreasing chain', s)
+        assert ds[-1] == S.h
+        for a, b in zip(ds, ds[1:]):
+            if b > a: rk[b - a] += 1
+    return rk
+
+
+ROUND6_CORNER = lambda h: [h - 2] + [1] * (h + 1)          # data-entrance corner runs, rounds six and seven
+STAIRCASE_CORNER = lambda h: [h - 2, h - 5] + [1] * 6       # staircase corner (check_stair.py)
+
+
+def histogram(S, rk, ylev, xdata, corner=None):
+    """child-width histogram of the batched two-stage bit interchange with deferred readouts and V leaves.
+    Per stage (stage two is the complement time-reversal of stage one, same edge multiset), each of the v invocations:
+      every slot u: its auxiliary edge of rank m - r_u (block m - 2 r_u) and its corner of rank r_u = h - dim sigma_u,
+        both batched by the inner lemma, plus its chain steps;
+      every centre: the copy's step U_c -> 0 of rank h-1;
+      every target y_T: the chain 0 -> deferred levels -> t_T^perp (total rank h-1);
+      every leaf X_S: its chain <t_S> -> V-gate frames -> F (total rank h-1).
+    Data: the entrance block m-4h+2 with its corner runs, one copy-correction singleton per pair."""
+    h, v, R = S.h, S.v, S.R; m = h * h; N = v * v; A = v * R; H = Counter()
+    corner = corner or ROUND6_CORNER(h); assert sum(corner) == 2 * h - 1
+    corners = Counter(h - S.f[s] for s in range(R))
+    for _ in range(2):
+        for r, c in corners.items():
+            H[m - 2 * r] += v * c
+            for w in inner(r, h): H[w] += v * c
+        for r, c in rk.items():
+            for w in inner(r, h): H[w] += v * c
+        for _c in range(h):
+            for w in inner(h - 1, h): H[w] += v
+        for t in range(v):
+            ds = sorted(set([0] + ylev[t] + [h - 1]))
+            for a, b in zip(ds, ds[1:]):
+                for w in inner(b - a, h): H[w] += v
+        for xs in xdata:
+            for x in xs:
+                for w in inner(x, h): H[w] += v
+    H[m - 4 * h + 2] += 2 * N
+    for w in corner: H[w] += 2 * N
+    H[1] += N
+    H = {w: c for w, c in H.items() if c}
+    Wd = 2 * N + 2 * A; L = 2 * v * h * (h - 1); s = Wd * m - N + L
+    return dict(h=h, m=m, v=v, N=N, W=Wd, L=L, s=s, R=R, hist=H)
+
diff --git a/references/partial-gauge/pr97/deferred_23.json.gz b/references/partial-gauge/pr97/deferred_23.json.gz
new file mode 100644
index 0000000000..4a4431bd42
Binary files /dev/null and b/references/partial-gauge/pr97/deferred_23.json.gz differ
diff --git a/references/partial-gauge/pr97/linalg.py b/references/partial-gauge/pr97/linalg.py
new file mode 100644
index 0000000000..a81cbaa61e
--- /dev/null
+++ b/references/partial-gauge/pr97/linalg.py
@@ -0,0 +1,176 @@
+"""Stdlib exact linear algebra used by the release checks (no numpy).
+
+Ranks over Q are computed in one of two rigorous ways:
+  * modulo a prime p with a Hadamard certificate: if every row of an integer matrix has squared norm n_i and
+    prod_i max(1, n_i) < p^2, every minor has absolute value < p, so a minor is nonzero over Q iff it is nonzero
+    mod p, and the rank mod p equals the rank over Q (hadamard_ok);
+  * exactly, by fraction-free (Bareiss) elimination over Z (rank_exact) or Fraction elimination (null_exact).
+Everything else is plain arithmetic modulo a prime."""
+from fractions import Fraction
+from math import gcd
+
+Q31 = 2**31 - 1          # prime used for the point evaluation and frame reductions
+M127 = 2**127 - 1        # Mersenne prime
+M521 = 2**521 - 1        # Mersenne prime
+
+
+def hadamard_ok(rows, p):
+    """True if prod of max(1, |row|^2) < p^2, i.e. every minor of the integer matrix is below p in absolute value."""
+    b = 1; p2 = p * p
+    for r in rows:
+        b *= max(1, sum(x * x for x in r))
+        if b >= p2: return False
+    return True
+
+
+class Echelon:
+    """Row echelon basis mod p, built by insertion. Each stored row has a pivot (first nonzero, normalised to 1)
+    and zeros in the pivot columns of all earlier rows, so reducing a new row against the rows in order is exact."""
+    __slots__ = ('p', 'rows', 'piv')
+
+    def __init__(self, p):
+        self.p = p; self.rows = []; self.piv = []
+
+    def reduce(self, r):
+        p = self.p; r = [x % p for x in r]
+        for pc, b in zip(self.piv, self.rows):
+            c = r[pc]
+            if c: r = [(x - c * y) % p for x, y in zip(r, b)]
+        return r
+
+    def insert(self, r):
+        """insert row; returns True if it was independent."""
+        r = self.reduce(r)
+        for i, x in enumerate(r):
+            if x:
+                inv = pow(x, self.p - 2, self.p)
+                self.rows.append([(y * inv) % self.p for y in r]); self.piv.append(i); return True
+        return False
+
+    def contains(self, r):
+        return not any(self.reduce(r))
+
+    def __len__(self): return len(self.rows)
+
+
+def rank_mod(rows, p):
+    E = Echelon(p)
+    for r in rows: E.insert(r)
+    return len(E)
+
+
+def rank_exact(rows):
+    """exact rank over Q of an integer matrix (Bareiss fraction-free elimination)."""
+    M = [list(r) for r in rows if any(r)]
+    if not M: return 0
+    n = len(M); k = len(M[0]); r = 0; prev = 1
+    for c in range(k):
+        i = next((i for i in range(r, n) if M[i][c]), None)
+        if i is None: continue
+        M[r], M[i] = M[i], M[r]
+        pr = M[r]; a = pr[c]
+        for t in range(r + 1, n):
+            row = M[t]; b = row[c]
+            M[t] = [(a * x - b * y) // prev for x, y in zip(row, pr)]
+        prev = a; r += 1
+        if r == n: break
+    return r
+
+
+def null_mod(A, k, p):
+    """basis of {x in F_p^k : A x = 0} (A: list of rows of length k)."""
+    rows = [[x % p for x in r] for r in A]; piv = []; R = []
+    for r in rows:
+        for pc, b in zip(piv, R):
+            c = r[pc]
+            if c: r = [(x - c * y) % p for x, y in zip(r, b)]
+        i = next((i for i, x in enumerate(r) if x), None)
+        if i is None: continue
+        inv = pow(r[i], p - 2, p); r = [(y * inv) % p for y in r]
+        R = [[(x - b[i] * y) % p for x, y in zip(b, r)] if b[i] else b for b in R]   # keep fully reduced
+        R.append(r); piv.append(i)
+    free = [c for c in range(k) if c not in piv]; out = []
+    for f in free:
+        x = [0] * k; x[f] = 1
+        for pc, b in zip(piv, R): x[pc] = (-b[f]) % p
+        out.append(x)
+    return out
+
+
+def null_exact(A, k):
+    """basis (integer vectors) of {x in Q^k : A x = 0}, by Fraction reduced row echelon form."""
+    R = []; piv = []
+    for r0 in A:
+        r = [Fraction(x) for x in r0]
+        for pc, b in zip(piv, R):
+            c = r[pc]
+            if c: r = [x - c * y for x, y in zip(r, b)]
+        i = next((i for i, x in enumerate(r) if x), None)
+        if i is None: continue
+        inv = 1 / r[i]; r = [y * inv for y in r]
+        R = [[x - b[i] * y for x, y in zip(b, r)] if b[i] else b for b in R]
+        R.append(r); piv.append(i)
+    free = [c for c in range(k) if c not in piv]; out = []
+    for f in free:
+        x = [Fraction(0)] * k; x[f] = Fraction(1)
+        for pc, b in zip(piv, R): x[pc] = -b[f]
+        out.append(primitive(x))
+    return out
+
+
+def primitive(x):
+    """scale a rational vector to a primitive integer vector."""
+    den = 1
+    for v in x: den = den * v.denominator // gcd(den, v.denominator)
+    y = [int(v * den) for v in x]; g = 0
+    for v in y: g = gcd(g, v)
+    return [v // g for v in y] if g else y
+
+
+def inv_mod_matrix(A, p):
+    """inverse of a square matrix mod p, or None if singular."""
+    n = len(A); M = [[x % p for x in A[i]] + [int(i == j) for j in range(n)] for i in range(n)]
+    for c in range(n):
+        i = next((i for i in range(c, n) if M[i][c]), None)
+        if i is None: return None
+        M[c], M[i] = M[i], M[c]
+        inv = pow(M[c][c], p - 2, p); M[c] = [(x * inv) % p for x in M[c]]
+        rc = M[c]
+        for t in range(n):
+            if t != c and M[t][c]:
+                f = M[t][c]; M[t] = [(x - f * y) % p for x, y in zip(M[t], rc)]
+    return [r[n:] for r in M]
+
+
+def matmul(A, B, p):
+    Bt = list(zip(*B))
+    return [[sum(x * y for x, y in zip(r, c)) % p for c in Bt] for r in A]
+
+
+def rightmost_pivots(Q, p):
+    """rows top to bottom: the rightmost nonzero column of the reduced row, then eliminate it below."""
+    M = [list(r) for r in Q]; n = len(M); out = [-1] * n
+    for i in range(n):
+        row = M[i]; c = next((j for j in range(len(row) - 1, -1, -1) if row[j]), -1)
+        out[i] = c
+        if c < 0: continue
+        inv = pow(row[c], p - 2, p)
+        for t in range(i + 1, n):
+            f = M[t][c]
+            if f:
+                f = f * inv % p; M[t] = [(x - f * y) % p for x, y in zip(M[t], row)]
+    return out
+
+
+def predicted(h):
+    """pivot column per row of the generic north-east rank function f_r of a rank-r idempotent."""
+    pr = {}
+    for r in range(1, h + 1):
+        s = h - r
+        f = lambda i, j: 0 if i < 0 or j >= h else min(i + 1, h - j, r, max(0, i - j + 1) + s)
+        pc = [-1] * h
+        for i in range(h):
+            for j in range(h):
+                if f(i, j) - f(i - 1, j) - f(i, j + 1) + f(i - 1, j + 1) == 1: pc[i] = j
+        pr[r] = pc
+    return pr
diff --git a/references/partial-gauge/pr97/round7_literal_frame_ledger.py b/references/partial-gauge/pr97/round7_literal_frame_ledger.py
new file mode 100644
index 0000000000..aa0de9ed8a
--- /dev/null
+++ b/references/partial-gauge/pr97/round7_literal_frame_ledger.py
@@ -0,0 +1,166 @@
+#!/usr/bin/env python3
+"""Explicit forward and reflected frame/event ledger of pinned round-seven.
+Zhihao Chen / Codex audit. Exact nesting of the supplied chains is inherited
+from the separately run frame/lifted checkers, not inferred from ranks here.
+Copied-center moves are temporary macros, not erased arbitrary scratch.
+"""
+import sys,json,signal,time,hashlib,gzip
+from pathlib import Path
+from collections import defaultdict,Counter
+from array import array
+HERE=Path(__file__).resolve().parent;ROOT=HERE/'swapnil-round7'
+sys.path.insert(0,str(ROOT/'independent/deferred-readout'))
+import deferred as dr
+
+def main():
+    started=time.monotonic()
+    def stop(*_):raise TimeoutError('180 second literal ledger limit')
+    signal.signal(signal.SIGALRM,stop);signal.alarm(180)
+    W,D=dr.load(23);S=dr.Schedule(W,D);v,R,h=S.v,S.R,S.h;size=2*v+R
+    co=S.adjoint();sel=S.sel;ops=S.ops;prev={};pred=defaultdict(list);last={}
+    for i,op in enumerate(ops):
+        for s in S.touch(op):
+            if s in prev:pred[i].append(prev[s])
+            prev[s]=i;last[s]=i
+    ph=set();stack=[last[s] for s in S.ret]
+    while stack:
+        i=stack.pop()
+        if i in ph:continue
+        ph.add(i);stack.extend(pred[i])
+    assert not any(set(S.touch(ops[i]))&sel for i in ph)
+    keys=[];ids={};dims=[]
+    def norm(k):
+        if k[0]=='out':return ('out',0,tuple(k[2]))
+        return k
+    def key(k):
+        k=norm(k)
+        if k not in ids:
+            ids[k]=len(keys);keys.append(k);dims.append(S.dim(k))
+        return ids[k]
+    z,F=key(('0',)),key(('F',))
+    paths=[None]*size
+    for n,ss in S.xs:paths[n-1]=[key(('n',n))]+[key(('v',s)) for s in ss]+[F]
+    y=[[] for _ in range(v)]
+    for s in S.readout:
+        for t,c in co[s].items():
+            if c&1:y[t].append(key(('sigma',s)))
+    for t,T in enumerate(S.trip):paths[v+t]=[z]+y[t]+[key(('out',0,T))]
+    for s in range(R):paths[2*v+s]=[key(k) for k in S.chain_keys(s)]
+    assert all(p and all(dims[b]>=dims[a] for a,b in zip(p,p[1:])) for p in paths)
+    pos=[0]*size;current=[p[0] for p in paths];initial=current[:]
+    # Fixed-width event records: kind, register/destination, source-or-old,
+    # new-or-common, rank. kind=0 move,1 xor,2 temporary copied-center macro.
+    events=array('i');rank_hist={k:Counter() for k in ('aux','data','center')}
+    bits=[1<<i for i in range(size)]
+    def emit(k,a,b,c,r=0):events.extend((k,a,b,c,r))
+    def move(s,dst):
+        if current[s]==dst:return
+        p=paths[s]
+        if dst not in p[pos[s]+1:]:
+            raise ValueError(dict(register=s,slot=s-2*v,current=keys[current[s]],destination=keys[dst],remaining=[keys[k] for k in p[pos[s]:]],full=[keys[k] for k in p]))
+        j=p.index(dst,pos[s]+1)
+        while pos[s]<j:
+            a=p[pos[s]];b=p[pos[s]+1];r=dims[b]-dims[a];assert r>=0
+            emit(0,s,a,b,r);rank_hist['aux' if s>=2*v else 'data'][r]+=1
+            pos[s]+=1;current[s]=b
+    def xor(t,s):
+        assert current[t]==current[s],('gate frames',t,s,keys[current[t]],keys[current[s]])
+        emit(1,t,s,current[t]);bits[t]^=bits[s]
+    def readout(s):
+        aux=2*v+s;f=key(('sigma',s)) if s in sel else z
+        assert current[aux]==f
+        for t,c in sorted(co[s].items()):
+            if c&1:move(v+t,f);xor(v+t,aux)
+    def source(s,active):
+        frame=key(('v',s)) if active else F
+        move(2*v+s,frame);move(S.srcop[s]-1,frame);xor(2*v+s,S.srcop[s]-1)
+    reordered_fans=[]
+    def operation(i,active):
+        op=ops[i]
+        if op[0]=='add':
+            _,p,o,n=op;frame=key(('n',n)) if active else F
+            move(2*v+p,frame);move(2*v+o,frame);xor(2*v+p,2*v+o)
+        elif op[0]=='fan':
+            # Scalar fan targets commute. The archive groups copies whose
+            # physical start frames differ; execute them in frame-chain order.
+            targets=sorted(op[2],key=lambda s:S.dim(S.start_key(s))) if active else op[2]
+            if active and tuple(targets)!=tuple(op[2]):reordered_fans.append(i)
+            for s in targets:
+                frame=key(S.start_key(s)) if active else F
+                move(2*v+op[1],frame);move(2*v+s,frame);xor(2*v+s,2*v+op[1])
+    for s in range(R):
+        if s not in sel:readout(s)
+    early=sorted(set(S.srcop)-sel,key=lambda s:(S.srcop[s],len(S.vstart[s]),s))
+    for s in early:source(s,True)
+    first=[i for i in range(len(ops)) if i in ph and ops[i][0]!='src']
+    rest=[i for i in range(len(ops)) if i not in ph and ops[i][0]!='src']
+    centers={}
+    for i in first:operation(i,True)
+    for s,c in S.ret.items():
+        a=2*v+s;f=key(('ret',c));move(a,f)
+        targets=[v+t for t,T in enumerate(S.trip) if c in T]
+        assert all(current[t]==z for t in targets)
+        centers[a]=targets;emit(2,a,f,z,h-1);rank_hist['center'][h-1]+=1
+        for t in targets:bits[t]^=bits[a]
+    for s in S.readout:readout(s)
+    for s in S.late_v:source(s,True)
+    for i in rest:operation(i,True)
+    for s,(c,T) in S.out.items():
+        t=S.tid[T];f=key(('out',0,T));move(2*v+s,f);move(v+t,f);xor(v+t,2*v+s)
+    for i in reversed(first+rest):operation(i,False)
+    for s in S.srcop:source(s,False)
+    for s in range(R):move(2*v+s,F)
+    assert all(bits[i]==((1<<i)^(1<<(i-v)) if v<=i<2*v else 1<<i) for i in range(size))
+    assert all(current[i]==F for i in range(v)) and all(current[2*v+s]==F for s in range(R))
+    for H in rank_hist.values():H.pop(0,None)
+    assert rank_hist['aux']==dr.chain_ranks(S)
+    assert sum(r*c for r,c in rank_hist['data'].items())==2*v*(h-1)
+    # Actual reflected word: reverse time, complement frames, swap X/Y banks.
+    # Center copies are regenerated at the reflected source frame, moved to
+    # the reflected target frame, scattered and discarded (same scalar macro).
+    bank=lambda s:s+v if s<v else s-v if s<2*v else s
+    rev=[None]*size
+    for s in range(size):rev[bank(s)]=~current[s]
+    reverse_initial=rev[:];bits=[1<<i for i in range(size)];rh=Counter()
+    for i in range(len(events)-5,-1,-5):
+        k,a,b,c,r=events[i:i+5];a=bank(a)
+        if k==0:
+            assert rev[a]==~c;rev[a]=~b;rh[r]+=1
+        elif k==1:
+            b=bank(b);assert rev[a]==rev[b]==~c;bits[a]^=bits[b]
+        else:
+            assert rev[a]==~b
+            for t in centers[a]:
+                t=bank(t);assert rev[t]==~c;bits[t]^=bits[a]
+            rh[r]+=1
+    assert all(rev[bank(s)]==~initial[s] for s in range(size))
+    assert all(bits[i]==((1<<i)^(1<<(i+v)) if i<v else 1<<i) for i in range(size))
+    rh.pop(0,None);assert rh==sum(rank_hist.values(),Counter())
+    # Assemble from ACTUAL events, not the schedule's pre-summed chain table.
+    m=h*h;N=v*v;H=Counter()
+    for r,c in rh.items():
+        for width in dr.inner(r,h):H[width]+=2*v*c
+    for s in range(R):
+        r=h-S.f[s];H[m-2*r]+=2*v
+        for width in dr.inner(r,h):H[width]+=2*v
+    H[m-4*h+2]+=2*N
+    for width in dr.STAIRCASE_CORNER(h):H[width]+=2*N
+    H[1]+=N
+    expected=dr.histogram(S,dr.chain_ranks(S),S.ylevels(co),S.xdata(),dr.STAIRCASE_CORNER(h))
+    assert dict(H)==expected['hist'];assert sum(w*n for w,n in H.items())==expected['s']
+    dest=HERE/'round7-literal-ledger';dest.mkdir(exist_ok=True)
+    with gzip.open(dest/'forward-events.i32.gz','wb') as f:f.write(events.tobytes())
+    (dest/'frames-and-paths.json').write_text(json.dumps(dict(keys=keys,dims=dims,paths=paths,initial=initial,final=current,centers=centers),separators=(',',':'))+'\n')
+    out=dict(h=h,roles=R,formal_basis=size,events=len(events)//5,frame_keys=len(keys),
+        rank_histograms={k:dict(vv) for k,vv in rank_hist.items()},
+        fan_groups_reordered_internally=reordered_fans,
+        complete_forward_F2=True,complete_reflected_F2=True,
+        reflected_frame_continuity=True,all_scalar_gates_equal_frame_keys=True,
+        exterior_formula='E1=I-I_tensor_Q+sigma_tensor_Q; E2=(I-P)_tensor_I+P_tensor_sigma; rank=m-h+dim(sigma)',
+        histogram_matches_external=True,histogram=dict(H),recursive_rank=expected['s'],
+        elapsed=time.monotonic()-started,limit_seconds=180,
+        exact_chain_geometry='Relies on separately replayed pinned check_frames and check_lifted. Rank labels alone do not prove geometry.',
+        scope='Explicit one-axis forward/reflected events, scalar complete basis and exact accounting. Tensor endpoint identities proved separately; full complex/tape/precision and final assembly integration not established.',new_multiplication_bound=False)
+    (dest/'result.json').write_text(json.dumps(out,indent=2)+'\n');signal.alarm(0)
+    print(json.dumps({k:v for k,v in out.items() if k not in ('histogram','rank_histograms')},indent=2))
+if __name__=='__main__':main()
diff --git a/references/partial-gauge/pr97/witness_23.json.gz b/references/partial-gauge/pr97/witness_23.json.gz
new file mode 100644
index 0000000000..3d86f83b87
Binary files /dev/null and b/references/partial-gauge/pr97/witness_23.json.gz differ
diff --git a/references/three-stage-cover/pr117/LICENSE b/references/three-stage-cover/pr117/LICENSE
new file mode 100644
index 0000000000..261eeb9e9f
--- /dev/null
+++ b/references/three-stage-cover/pr117/LICENSE
@@ -0,0 +1,201 @@
+                                 Apache License
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+   5. Submission of Contributions. Unless You explicitly state otherwise,
+      any Contribution intentionally submitted for inclusion in the Work
+      by You to the Licensor shall be under the terms and conditions of
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+      the terms of any separate license agreement you may have executed
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+   6. Trademarks. This License does not grant permission to use the trade
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+      Contributor provides its Contributions) on an "AS IS" BASIS,
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+      whether in tort (including negligence), contract, or otherwise,
+      unless required by applicable law (such as deliberate and grossly
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diff --git a/references/three-stage-cover/pr117/NOTICE b/references/three-stage-cover/pr117/NOTICE
new file mode 100644
index 0000000000..f7343a1628
--- /dev/null
+++ b/references/three-stage-cover/pr117/NOTICE
@@ -0,0 +1,326 @@
+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.
+
+Deferred product (research/deferred-product, scripts/deferred_product*, certificates/deferred-product-*):
+copyright 2026 eumemic, Apache-2.0, prepared with Anthropic Claude assistance. The bit word, its ledgers and
+checks are PR #97's import of Swapnil Jain's round-seven research (research/deferred-signed, its LICENSE and
+NOTICE retained); scripts/deferred_product/replayed.py adapts PR #104's stopped_product/complex.py (icekylinx).
diff --git a/references/three-stage-cover/pr117/SOURCE.json b/references/three-stage-cover/pr117/SOURCE.json
new file mode 100644
index 0000000000..463a1ba656
--- /dev/null
+++ b/references/three-stage-cover/pr117/SOURCE.json
@@ -0,0 +1,12 @@
+{
+  "commit": "cbb05ce504d571546d9b7794c186a613c659c3bf",
+  "files": {
+    "LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+    "NOTICE": "a0030510cd3b5334fb113eae759aadd2c25951bd353dfb69ad9cb780ae5adf30",
+    "dag.json.gz": "3c034d0aae388ef567a454826f4f48b26fd8a94c71e8ffed4835271b349a783b",
+    "replayed.py": "99c1e3694ae06a7ce7a370b04ecb45ea3fcfe69a6e6e7f4c4606e0a6c74c3184"
+  },
+  "repository": "eumemic/integer-mult-bounds",
+  "scope": "Unchanged selected positive DAG, support/label replay and source-specific license/notices.",
+  "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/117"
+}
diff --git a/references/three-stage-cover/pr117/dag.json.gz b/references/three-stage-cover/pr117/dag.json.gz
new file mode 100644
index 0000000000..a5b0598e4c
Binary files /dev/null and b/references/three-stage-cover/pr117/dag.json.gz differ
diff --git a/references/three-stage-cover/pr117/replayed.py b/references/three-stage-cover/pr117/replayed.py
new file mode 100644
index 0000000000..673fbbba70
--- /dev/null
+++ b/references/three-stage-cover/pr117/replayed.py
@@ -0,0 +1,179 @@
+"""Replayed h=24 complex producer with all centers disjoint (rational divisor 21).
+
+Copyright 2026 eumemic, Apache-2.0. AI-assisted (Claude). The DAG was found by an
+automated search over producers of the retained paired-triple, pair-star and
+center-disjoint interfaces. The checks below are adapted from
+stopped_product/complex.py (copyright 2026 icekylinx, Apache-2.0, PR #104).
+
+The witness complex-dag.json.gz lists the ordinary
+additions after the v triple inputs and the retained roots: the disjoint sums
+D[t], the pair stars P(a,b,i) and the center-disjoint sums A_i. Every support,
+core, cover, type and rank is recomputed here from the operands; nothing is
+read from the witness except the operand pairs and the root indices.
+"""
+import gzip, json, struct, sys, array
+from fractions import Fraction
+from itertools import combinations
+from pathlib import Path
+
+
+def write_array(f, values):
+    """Little-endian array writer, as partial_swap.binary.write_array."""
+    if sys.byteorder != 'little' and values.itemsize > 1:
+        values = array.array(values.typecode, values)
+        values.byteswap()
+    values.tofile(f)
+
+
+def build(witness, prefix):
+    w = json.loads(gzip.decompress(Path(witness).read_bytes()))
+    h, v, d = w['h'], w['v'], w['central_disjoint']
+    assert (h, d, w['center_denominator']) == (24, 24, 21)
+    triples = list(combinations(range(h), 3))
+    assert v == len(triples)
+    args = [(0, 0)]; support = [0]; core = [0]; cover = [0]; types = [0]; ranks = [0]
+    for j, t in enumerate(triples):
+        mask = sum(1 << a for a in t)
+        args.append((0, 0)); support.append(1 << j); core.append(mask); cover.append(mask)
+        types.append(1); ranks.append(1)
+    flat = w['args']
+    assert len(flat) % 2 == 0
+    for k in range(0, len(flat), 2):
+        a, b, x = flat[k], flat[k+1], len(args)
+        assert 0 < a < x and 0 < b < x, 'Operands must precede their sum'
+        assert not support[a] & support[b], 'Ordinary additions must be cancellation-free'
+        args.append((a, b)); support.append(support[a] | support[b])
+        core.append(core[a] & core[b]); cover.append(cover[a] | cover[b])
+        if core[x].bit_count() >= 2:
+            types.append(1); ranks.append(support[x].bit_count())
+        else:
+            types.append(2); ranks.append(cover[x].bit_count())
+    n = len(args)
+    point_masks = [sum(1 << j for j, t in enumerate(triples) if a in t) for a in range(h)]
+    full_support = (1 << v) - 1
+    roots, kind = [], []
+    assert len(w['D']) == v
+    for t, node in zip(triples, w['D']):
+        assert 0 < node < n
+        assert support[node] == full_support & ~(point_masks[t[0]]|point_masks[t[1]]|point_masks[t[2]])
+        target_mask = sum(1 << a for a in t)
+        if types[node] == 2:
+            assert not cover[node] & target_mask
+        else:
+            assert types[node] == 1
+            assert not support[node] & (point_masks[t[0]]^point_masks[t[1]]^point_masks[t[2]])
+        roots.append(node); kind.append(0)
+    stars = [(a, b, i) for a, b in combinations(range(h), 2) for i in range(h) if i not in (a, b)]
+    assert len(w['P']) == len(stars)
+    pair_outputs = {}
+    for (a, b, i), node in zip(stars, w['P']):
+        assert 0 < node < n
+        assert support[node] == point_masks[a] & point_masks[b] & ~point_masks[i]
+        assert types[node] == 1
+        assert not support[node] & (point_masks[a]^point_masks[b]^point_masks[i])
+        pair_outputs[a, b, i] = node
+        roots.append(node); kind.append(0)
+    assert len(w['A']) == d
+    for i, node in enumerate(w['A']):
+        assert 0 < node < n
+        assert support[node] == full_support & ~point_masks[i]
+        assert types[node] == 2 and cover[node] == ((1 << h)-1) ^ (1 << i)
+        assert ranks[node] == h-1
+        roots.append(node); kind.append(1)
+    # Every A_i avoids i, so sum A_i = (h-3)T. The decoder is T=sum A_i/21
+    # and the scatter is T-(1/2)sum_{i in target} A_i.
+    decoder = Fraction(1, h-3)
+    assert decoder == Fraction(1, 21)
+    scatter_coefficients = [decoder, decoder-Fraction(1, 2)]
+    for coefficient in scatter_coefficients:
+        odd_denominator = coefficient.denominator
+        while odd_denominator % 2 == 0:
+            odd_denominator //= 2
+        assert 21 % odd_denominator == 0 and abs(coefficient) <= 1
+    for source in triples:
+        assert sum(i not in source for i in range(h))*decoder == 1
+    Dset = set(range(d))
+    for triple in triples:
+        k = len(Dset.intersection(triple))
+        assert 2*(3-k)-2*(d-k) == 2*(3-d)
+    patterns = {}
+    for target in triples:
+        chosen = len(Dset.intersection(target))
+        if chosen not in patterns:
+            patterns[chosen] = target
+    for chosen, target in patterns.items():
+        target_set = set(target)
+        for source in triples:
+            k = len(Dset.intersection(source))
+            total_coefficient = Fraction(2*(3-k)-2*(d-k), 2*(3-d))
+            scalar = Fraction(len((target_set-Dset).intersection(source)), 2)
+            scalar -= Fraction(len((target_set & Dset)-set(source)), 2)
+            scalar += Fraction(chosen-1, 2)*total_coefficient
+            assert scalar == Fraction(len(target_set.intersection(source))-1, 2)
+    for j, t in enumerate(triples):
+        masks = [point_masks[a] for a in t]
+        two = (masks[0]&masks[1]&~masks[2]) | (masks[0]&masks[2]&~masks[1]) | (masks[1]&masks[2]&~masks[0])
+        actual = 0
+        for a, b in combinations(t, 2):
+            excluded = next(i for i in t if i not in (a, b))
+            actual |= support[pair_outputs[a, b, excluded]]
+        assert actual == two
+        assert masks[0]&masks[1]&masks[2] == 1 << j
+    assert [((k-1)+(k == 0)-(k == 2)) for k in range(4)] == [0, 0, 0, 2]
+
+    def contained(x, y):
+        tx, ty = types[x], types[y]
+        if tx == 1 and ty == 1: return not(core[y]&~core[x] or cover[x]&~cover[y])
+        if tx in (1, 2) and ty == 2: return not cover[x]&~cover[y]
+        return False
+    active = [0]*n; stack = list(roots)
+    while stack:
+        x = stack.pop()
+        if not x or active[x]: continue
+        active[x] = 1
+        if args[x][0]: stack.extend(args[x])
+    degree = [0]*n
+    for x in range(1, n):
+        if active[x]:
+            if types[x] == 1:
+                assert core[x].bit_count() >= 2 and ranks[x] == support[x].bit_count()
+            else:
+                assert types[x] == 2 and ranks[x] == cover[x].bit_count()
+        if active[x] and args[x][0]:
+            for y in args[x]:
+                assert y < x and contained(y, x), 'Binary frame nesting failed'
+                degree[y] += 1
+    for x in roots: degree[x] += 1
+    H = [0]*(h+1); c = 0; loss = 0
+    for x in range(1, n):
+        if not active[x]: continue
+        r = ranks[x]
+        if args[x][0]:
+            c += 1; H[r] += degree[x]-1; H[h-r] += 1
+            for y in args[x]:
+                assert r >= ranks[y]
+                H[r-ranks[y]] += 1
+        else: H[1] += degree[x]
+    for x, k in zip(roots, kind):
+        r = ranks[x]
+        if k: H[r] += 1; H[h] += 1; loss += r
+        else: H[h-1-r] += 1; H[1] += 1
+    q = len(roots); R = c+q
+    result = dict(h=h, v=v, c=c, q=q, R=R, loss=loss, central_disjoint=d, histogram=H,
+                  rank_sum=sum(r*m for r, m in enumerate(H)))
+    assert loss == h*(h-1) and result['rank_sum'] == h*R+2*loss
+    result['center_denominator'] = 21
+    result['scalar_validation'] = dict(ordinary_supports_exact=True, centers_exact=True,
+        center_decoder='sum A_i / 21', scatter_coefficients=list(map(str, scatter_coefficients)),
+        rational_scalar_identity_exact=True, mixed_center_scatter_exact=True, binary_frames_nested=True,
+        binary_frames_nondegenerate=True)
+    with open(str(prefix)+'.bin', 'wb') as f:
+        f.write(struct.pack('<4I', h, v, n, q))
+        write_array(f, array.array('I', (a for pair in args for a in pair)))
+        write_array(f, array.array('Q', core)); write_array(f, array.array('Q', cover))
+        write_array(f, array.array('I', roots)); write_array(f, array.array('I', kind))
+        write_array(f, array.array('B', active))
+    with open(str(prefix)+'.labels', 'wb') as f:
+        write_array(f, array.array('I', ranks)); write_array(f, array.array('B', types))
+    Path(str(prefix)+'.json').write_text(json.dumps(result, indent=2)+'\n')
+    return result
diff --git a/research/cover-local-reuse/NOTICE b/research/cover-local-reuse/NOTICE
new file mode 100644
index 0000000000..dde0487aa2
--- /dev/null
+++ b/research/cover-local-reuse/NOTICE
@@ -0,0 +1,23 @@
+Physical local reuse under the three-stage cover, 2026-10-09.
+
+The new composition, source closure and verifier were prepared for eumemic
+with OpenAI Codex assistance. Apache-2.0 applies. No official endorsement or
+independent external peer review is claimed.
+
+The three-stage cover, generalized Clifford frames, weighted bit supplier
+and internal row borrowing are icekylinx's PR130 contribution, prepared with
+OpenAI GPT-6 Astra and Codex assistance. The exact adopted commit is recorded
+in SOURCE.json. Its original root NOTICE and all reference notices remain
+unchanged.
+
+The pinned PR117 scalar DAG is by eumemic with Anthropic Claude assistance.
+Its copyright, Apache-2.0 license and original notice remain in
+references/three-stage-cover/pr117/. Compensated birth-cut reuse follows
+jamesyc's PR124. Physical hull frames and their full literal reflection were
+developed for eumemic with OpenAI Codex assistance in PR129.
+
+Swapnil Jain's round-seven bit witness and Zhihao Chen's PR97 physical
+ledger retain their original notices under references/partial-gauge/pr97/.
+The inherited semantic, product-row, analytic and fixed-tape interfaces
+retain credit to icekylinx, Zhihao Chen, RaD / hipotures, and the other
+contributors named in the root NOTICE. No predecessor notice is superseded.
diff --git a/research/cover-local-reuse/README.md b/research/cover-local-reuse/README.md
new file mode 100644
index 0000000000..33a5ff3c4e
--- /dev/null
+++ b/research/cover-local-reuse/README.md
@@ -0,0 +1,69 @@
+# Physical local reuse under the three-stage cover
+
+Conditional **κ = 166102699493/500000000000000 = 3.32205398986e-4**.
+This is about 5.5955% above PR130's 3.146011e-4. The exact complex saving is
+**332426609/10¹²**. The inherited weighted bit supplier remains unchanged.
+
+The construction places the independently replayed physical complex word
+from PR129 inside PR130's three-stage cover. Its 28,705 virtual roles use
+26,597 physical scratch roles after 2,108 compensated birth-cut reuses. The
+full source/workspace chronology and inverse retain every compensation.
+The local source inventory has 470 distinct frames, including 2,598 roles
+with nonzero source gauges.
+
+The cover has dimension 70, 83,839 roles per vertex, recursive rank
+5,866,338 per vertex and deficit 2,392. Every local child is retained after
+removing exactly the former two-stage data bridges, endpoint copies and
+exterior class. Each physical role with source dimension s contributes three
+new exterior children of rank **46 + s**. The largest child is 68. The full
+finite group order, router charge and row reserve remain paid; scalar work
+uses the original virtual-role reserve so compensated readouts are covered.
+
+Run from the repository root:
+
+```sh
+python3 research/cover-local-reuse/verify.py
+```
+
+The verifier checks a closed frozen source inventory, copies it into a
+temporary repository-shaped tree, rebuilds the complete local compiler and
+reuse pairs, independently replays literal reflection and dirty cleanup,
+recounts bounded scalar work, regenerates finite geometry checks and checks
+the exact cover certificate. The generated local profile and reflection
+receipt must reproduce byte for byte. Verification leaves its source tree
+unchanged and has no options to omit these stages.
+
+The finite geometry check covers every subspace and ordered pair through
+dimension four, including degenerate subspaces, and all 2,024 actual h24
+triple ports. It checks symplectic adapters, transition and dirty-tail ranks,
+reflected identities and data-frame seams. Exact operator phases and general
+uniform interfaces remain written proof dependencies.
+
+The inherited bit step reconstructs PR97's local ranks from pinned physical
+frame chains and binds its prior complete-basis/reflection receipts. It then
+rechecks PR130's contaminated moment and rare-class fallback. The unchanged
+upstream complete-basis and rational-frame suites remain inherited, as in
+PR130; this package does not claim to rerun them.
+
+The 16 outer controls reject missing dependencies, altered and repinned DAG
+inputs, omitted reuse roles or paid children, missing compensation or
+reflection flags, changed source bindings, insufficient scalar guards,
+formatting drift and optimized Python. The independent local replay retains
+its frame, literal-word and compensation corruption controls. All 47 strict
+assembly constraints and seven margins are positive. The next complex and
+κ grid points fail the chosen enclosure or assembly; no true-root optimum
+is claimed.
+
+`certificate.py` checks the frozen certificate by default. `--write` is an
+explicit authoring operation. `verify.py --freeze-manifest` explicitly
+replaces the source manifest and does not certify it. Neither operation is
+performed during verification.
+
+This remains a conditional finite witness. PR130's finite group geometry,
+weighted local-ring compilation, uniform batching and restored internal row
+borrowing are written proof dependencies, together with inherited common
+bases, address adapters, exact grids, prime selection, analytic estimates,
+streaming, recovery and fixed-tape interfaces. Finite checks do not establish
+those all-size contracts or a practical speedup. [NOTICE](NOTICE) preserves
+attribution; the adopted proof sources and original notices are pinned in
+the source manifest.
diff --git a/research/cover-local-reuse/SOURCE.json b/research/cover-local-reuse/SOURCE.json
new file mode 100644
index 0000000000..68a17689da
--- /dev/null
+++ b/research/cover-local-reuse/SOURCE.json
@@ -0,0 +1,91 @@
+{
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+    "scripts/three_stage_cover_producer.py": "ea6dd4f06173304bbcb74b8e5bcc02357b7bb162a6d26b4a6f7a9e886158123f"
+  },
+  "provenance": {
+    "cover_certificate_sha256": "bfca1f225394c8b50f3360b71920652baf9bd885ce84f9c0a9184c3ba7dd35ab",
+    "cover_commit": "6a9970a530119174507904e23592fd59ede19a5d",
+    "physical_local_source": "PR129 completed-core frames and PR124 compensated birth-cut reuse",
+    "scalar_dag_commit": "cbb05ce504d571546d9b7794c186a613c659c3bf",
+    "scalar_dag_sha256": "3c034d0aae388ef567a454826f4f48b26fd8a94c71e8ffed4835271b349a783b"
+  },
+  "schema": 1,
+  "scope": "Finite local complex replay, geometry checks and exact cover assembly; general cover/bit/analytic interfaces remain conditional."
+}
diff --git a/research/cover-local-reuse/certificate.json b/research/cover-local-reuse/certificate.json
new file mode 100644
index 0000000000..49d72a2421
--- /dev/null
+++ b/research/cover-local-reuse/certificate.json
@@ -0,0 +1,662 @@
+{
+  "actual_bit_saving": "3731649599/10000000000000",
+  "assembly": {
+    "absorption_gap": "593600861587452590794958488060763/2501662131357884984091532536718304500000000000000",
+    "margins": {
+      "Gaussian_arithmetic": "1994572616164930096623111816934926563391/4002659410172615974546452058749287200000000",
+      "bulk_exposure": "332426276563391/1000000000000000000",
+      "compact_phase_layer": "1662131332953013847917626563391/5003324262715769968183065073436609",
+      "coordinate_movement": "332426276563391/1000000000000000000",
+      "dimension": "4999999950000000000000000000000000/5003324262715769968183065073436609",
+      "original_prefix": "166218136619564100561730839404173436609/500332426271576996818306507343660900000000",
+      "scalar_work": "2659445169291711830682090566222126563391/4002659410172615974546452058749287200000000"
+    },
+    "minimum_margin": "1662131332953013847917626563391/5003324262715769968183065073436609",
+    "parameters": {
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+      "strict_literal_gap": 23928995795460619725900785107219313792231412567085210129268608601745093429109129895071969385436694570800807000241922588149641145960597420086074491777228544601763466980042310553333999129711684410128239246985730369987966832998798611334636292218326760298397906547771049257021592783000136885892449790047927790754506707402317267299441819419873259146059206659749268368661260788304662667602449215456795392679867594304248070797977675359276334630919960118535949376760914453167122664750393852171235767424398472067312234245508673562180125781817739848199438468647537079713523194558117172552473620109197725833474779524241459723906253925109301069899934697422683108926805252321364191665663981037358287421109107431893536427840550901427158984457677821692674817920476014119502857738526235821040252061224239136070990288166973637438128561623127521835656491156765643046706355004958468661889302821316005073037107328003078635249025491302750435118490495383105695160308986308942329045083983586216400119238754732468436709697876144219195692818449789668270951928254406752989126652990082548520704146717429923671061483889798435036144178187235658472780403332639146633302375136472307775658307256921159405957642462880843121114610056602633023148670306248194239138245120453591818182089322632039721298250078915425063135958122634986895652764359029374459043615959582727900921060737087416560879797831395821799840409083589457726335603513663352393199142274079729403656289540194169576155980565282067455079496264857003544812965705498925279658045622964080918135522981842860755619645287723670760584773457684557026635107975537065874259713261489911314872848239698629274652273639319837056511016023382121922971223783187158986640049461463243892620549330341301415999814513657534597933105903777871619296370278860802428807454939060549627670136724407311503757164794477329206962740767402291659502781433752950078494218365315180257347699472491640561494095319543176925209576392645850368188129575883487845853502841416989698324098823549422998841176933619787862144856786640955793648720445570490149010582539848110891236874422305393136066489232490677611254149429322071013770522547581967089292693396435806601436279300466690553405300593192398598528712935409999459595515949650938445423067046574759833211346694754050018043103458366826354342664601156396695215826694550366453232134150019450624141187683571737218850761093626097797078715876843157710060260048116239341782478012901022224283761331501351445134031377323856599036084447714064511334678744798239014299769151307041166561466794348957120743620537963562672663254092539965698911279751931844457147397995669320118183954554901898864698626943313483111027216566636339377743208134710878071448543414519500356679300612439291406703680961939071221722124462805157899680965676212962974271726667099073263085566822352637980346094205832500943285722781216654879622102881915542194363311701767727719063556210669369918385832335300241427841260253763952244320561218882386737453595670530199140436942638870326880395866022314975459984233350982536835430103326382694589921351951327996961516323109689198990968870132975650275608527448025621650524555567339852582014758489570706731188547430158999006167055386349414143600100969179585379451540824869179228391520435598320616416769617089187032236589025590770420968751454451480733868513253972665010963824023167206315767456811499867393628815261349624689239415131081257384840980750224957613834279748917264126567038000899830087829131791737795928069445951658731610570241015775036720745551199535663591952108722114222597656902512020870501035657907826933201034984025971457603396087247449829410164298541587413505602997974439261525029341651192089766802424614118933033015367737726594985852769947038148465130831331398736670406613890083295828276029956613362519902119843172135572079350803378937515108788666448016925522436661332973908270369852219063903503930804921887441750213138031925285725047483672161041069024449627317322413608758288777756807941019442670102217041186125247963359522478815260547102107181710447976987243863959755832620280053831241639023843754894141774339688783294936362907375214154426452036039212236933937803370851976863071077803777963338420732125320129550818289796022518917569247587093647238802655522732612084881783495488795870007552044224191058160906816283074608027954836594563667235549306999526687407485454012381044323754342796529429665617102007330358107249871424945620374965687920790549299200000000000022904060
+    }
+  },
+  "kappa": "166102699493/500000000000000",
+  "next_grid_rejections": {
+    "complex": "1e-12 enclosure",
+    "kappa": "1e-15 assembly"
+  },
+  "profile": {
+    "R": 26597,
+    "center_loss_per_invocation": 552,
+    "child_multiplicities": {
+      "1": 299925,
+      "10": 15651,
+      "11": 3660,
+      "12": 15435,
+      "13": 13182,
+      "14": 1425,
+      "15": 1848,
+      "16": 507,
+      "17": 363,
+      "18": 405,
+      "19": 354,
+      "2": 216123,
+      "20": 408,
+      "21": 912,
+      "22": 48,
+      "23": 6198,
+      "3": 39024,
+      "4": 37827,
+      "46": 71997,
+      "48": 2589,
+      "5": 15933,
+      "50": 1065,
+      "52": 1719,
+      "54": 336,
+      "56": 36,
+      "6": 12975,
+      "64": 609,
+      "65": 606,
+      "66": 612,
+      "67": 33,
+      "68": 189,
+      "7": 6840,
+      "8": 9363,
+      "9": 5832
+    },
+    "deficit_per_vertex": 2392,
+    "exterior_child_multiplicities": {
+      "46": 71997,
+      "48": 2589,
+      "50": 1065,
+      "52": 1719,
+      "54": 336,
+      "56": 36,
+      "64": 609,
+      "65": 606,
+      "66": 612,
+      "67": 33,
+      "68": 189
+    },
+    "h": 24,
+    "local_child_multiplicities": {
+      "1": 99975,
+      "10": 5217,
+      "11": 1220,
+      "12": 5145,
+      "13": 4394,
+      "14": 475,
+      "15": 616,
+      "16": 169,
+      "17": 121,
+      "18": 135,
+      "19": 118,
+      "2": 72041,
+      "20": 136,
+      "21": 304,
+      "22": 16,
+      "23": 2066,
+      "3": 13008,
+      "4": 12609,
+      "5": 5311,
+      "6": 4325,
+      "7": 2280,
+      "8": 3121,
+      "9": 1944
+    },
+    "local_profile_sha256": "63e4794bc1deeee10c1294c7b7a0a951dd79b6e978472dc18d026ee607b9cc8c",
+    "m": 70,
+    "maxchild": 68,
+    "rank_per_vertex": 5866338,
+    "reflection_receipt_sha256": "40352a29f046c3bdf6a9419fa840bd787427d52ea35cbb069eb0df791abc839c",
+    "removed_two_stage_child_multiplicities": {
+      "1": 4096576,
+      "529": 8193152,
+      "552": 97147952,
+      "554": 3493424,
+      "556": 1437040,
+      "558": 2319504,
+      "560": 453376,
+      "562": 48576,
+      "570": 821744,
+      "571": 817696,
+      "572": 825792,
+      "573": 44528,
+      "574": 255024
+    },
+    "reused_roles": 2108,
+    "roles_per_vertex": 83839,
+    "v": 2024,
+    "virtual_R": 28705
+  },
+  "scope": "Finite local word and complete cover/assembly arithmetic. PR130 group geometry, weighted local-ring compilation, uniform batching and borrowed rows remain explicit written proof dependencies, together with inherited analytic/tape interfaces.",
+  "status": "Conditional PR130 cover with independently audited physical local frames and compensated reuse"
+}
diff --git a/research/cover-local-reuse/certificate.py b/research/cover-local-reuse/certificate.py
new file mode 100644
index 0000000000..ad0ac8e451
--- /dev/null
+++ b/research/cover-local-reuse/certificate.py
@@ -0,0 +1,137 @@
+#!/usr/bin/env python3
+"""Exact composition of the PR129 physical local word with PR130's cover.
+
+PR130 Cayley cover and weighted bit interface: icekylinx, with OpenAI
+GPT-6 Astra/Codex assistance. Local frames/reuse integration for eumemic
+with OpenAI Codex assistance; PR117, PR124 and all source notices retained.
+"""
+import argparse
+from collections import Counter
+from fractions import Fraction as Q
+from hashlib import sha256
+import json
+from math import prod
+from pathlib import Path
+import sys
+if sys.flags.optimize:
+    raise ValueError('Assertions must remain enabled')
+if hasattr(sys,'set_int_max_str_digits'):
+    sys.set_int_max_str_digits(0)
+HERE=Path(__file__).resolve().parent
+ROOT=HERE.parents[1]
+LOCAL=ROOT/'research/cyclic-deferred'
+sys.path.insert(0,str(ROOT/'scripts'))
+import three_stage_cover_network as inherited
+from partial_swap_network import moment
+from structured_bulk_assembly import assembly,js
+KGRID=10**15
+BGRID=10**12
+STOP=Q(1,10**6)
+
+
+def load(name):return json.loads((LOCAL/name).read_text())
+def digest(path):return sha256(path.read_bytes()).hexdigest()
+
+
+def cover_profile():
+    local=load('complex-profile.json');audit=load('reflection-audit.json')
+    h,v,R=(local[k]for k in('h','v','R'))
+    assert (h,v,R)==(24,2024,26597)
+    assert local['virtual_R']==28705 and local['reused_roles']==2108
+    assert audit['R']==R and audit['virtual_R']==local['virtual_R']
+    assert audit['child_multiplicities']==local['child_multiplicities']
+    assert audit['physical_auxiliary_source_frames']==local['physical_auxiliary_source_frames']
+    for key in ('exact_arbitrary_dirty_cancellation_by_dependency_cut','physical_aliased_numeric_replay','exact_birth_cut_invariants','literal_frame_incidences_both_directions','reflected_residual_rank_histogram_equal','completed_core_source_inventory_bound','completed_core_pre_exterior_frames_full','reflected_core_active_frames_complement_source','bounded_chunk_coefficients'):
+        assert audit[key]is True,key
+    assert audit['source_sha256']==digest(LOCAL/'complex_deferred.py')
+    assert audit['audit_sha256']==digest(LOCAL/'reflection_audit.py')
+    inventory=local['physical_auxiliary_source_frames']
+    assert sum(row['count']for row in inventory)==R
+    N=v*v
+    remaining=Counter({int(r):n for r,n in local['child_multiplicities'].items()})
+    # Remove exactly the two-stage data bridges, endpoint copies and exterior
+    # class. All physical source/target moves and copied centers stay local.
+    removed=Counter({(h-1)**2:2*N,1:N})
+    for row in inventory:
+        removed[h*h-h+len(row['basis'])]+=2*v*row['count']
+    remaining.subtract(removed)
+    assert all(type(n)is int and n>=0 and n%(2*v)==0 for n in remaining.values())
+    H={r:n//(2*v)for r,n in remaining.items()if n}
+    assert all(0<r<h for r in H)
+    m=3*h-2;w=2*v+3*R;ell=h*(h-1)
+    children=Counter({r:3*n for r,n in H.items()})
+    exteriors=Counter()
+    for row in inventory:
+        exteriors[m-h+len(row['basis'])]+=3*row['count']
+    children.update(exteriors)
+    rank=sum(r*n for r,n in children.items())
+    assert w*m-rank==2*v-3*ell==2392
+    assert max(children)==68 and all(0<r<m and n>0 for r,n in children.items())
+    return dict(h=h,v=v,R=R,virtual_R=local['virtual_R'],reused_roles=local['reused_roles'],
+        m=m,roles_per_vertex=w,rank_per_vertex=rank,deficit_per_vertex=w*m-rank,
+        maxchild=max(children),center_loss_per_invocation=ell,
+        local_child_multiplicities=dict(sorted(H.items())),
+        removed_two_stage_child_multiplicities=dict(sorted(removed.items())),
+        exterior_child_multiplicities=dict(sorted(exteriors.items())),
+        child_multiplicities=dict(sorted(children.items())),
+        local_profile_sha256=digest(LOCAL/'complex-profile.json'),
+        reflection_receipt_sha256=digest(LOCAL/'reflection-audit.json'))
+
+
+def exact():
+    p=cover_profile();m,w=p['m'],p['roles_per_vertex'];children=p['child_multiplicities']
+    def contracts(n):
+        try:moment(m,w,children,Q(n,BGRID),True)
+        except ValueError:return False
+        return True
+    lo,hi=0,10**9
+    assert contracts(lo)and not contracts(hi)
+    while hi-lo>1:
+        mid=(lo+hi)//2
+        if contracts(mid):lo=mid
+        else:hi=mid
+    b=Q(lo,BGRID);cm=moment(m,w,children,b,True)
+    n=m//2
+    vertices=2**(m-1+(n-1)**2)*prod(2**(2*i)-1 for i in range(1,n))
+    phase=dict(m=m,N=vertices*p['v'],W=vertices*w,total_rank=vertices*p['rank_per_vertex'],
+        deficit=vertices*p['deficit_per_vertex'],vertices_per_stage=vertices,maxchild=p['maxchild'],
+        group_order_bits=vertices.bit_length(),per_vertex=p,**cm)
+    # Use virtual roles in the source-witness scalar reserve: compensated
+    # recipients still have readouts even though they have no persistent bank.
+    row=json.loads((ROOT/'certificates/three-stage-cover-complex-input.json').read_text())
+    assert row['R']==p['virtual_R']and row['total_M_operations']==118451
+    bridge=inherited.finite_bridge(phase,row)
+    audit=load('reflection-audit.json')
+    assert bridge['complex']['local_group_upper']>=audit['conservative_local_G']>=audit['expanded_scalar_operations_per_stage']
+    bridge['complex'].update(physical_R=p['R'],virtual_R=p['virtual_R'],
+        independently_audited_local_scalar_operations=audit['expanded_scalar_operations_per_stage'],
+        local_scalar_guard_from_reflection=audit['conservative_local_G'],
+        scalar_contract='Virtual-role reserve covers all compensated readouts and inverse chronology; actual persistent stock and finite routers use physical roles.')
+    bit=inherited.bit_certificate(json.loads((ROOT/'certificates/three-stage-cover-bit-input.json').read_text()))
+    a=min(inherited.AB,(1-STOP)*b-Q(1,10**14))
+    def accepts(n):
+        try:assembly(a,b,bridge,Q(n,KGRID),beta=STOP)
+        except AssertionError:return False
+        return True
+    low,high=0,int(b*KGRID)+1
+    assert accepts(low)and not accepts(high)
+    while high-low>1:
+        mid=(low+high)//2
+        if accepts(mid):low=mid
+        else:high=mid
+    k=Q(low,KGRID);result=assembly(a,b,bridge,k,beta=STOP)
+    assert len(result['strict_constraints'])==47 and len(result['margins'])==7
+    return dict(status='Conditional PR130 cover with independently audited physical local frames and compensated reuse',
+        kappa=k,complex_saving=b,assembly_bit_saving=a,actual_bit_saving=inherited.AB,
+        profile=p,complex=phase,bit=bit,finite_bridge=bridge,assembly=result,
+        next_grid_rejections=dict(complex='1e-12 enclosure',kappa='1e-15 assembly'),
+        scope='Finite local word and complete cover/assembly arithmetic. PR130 group geometry, weighted local-ring compilation, uniform batching and borrowed rows remain explicit written proof dependencies, together with inherited analytic/tape interfaces.')
+
+
+def main():
+    ap=argparse.ArgumentParser();ap.add_argument('--write',action='store_true');args=ap.parse_args()
+    result=js(exact());path=HERE/'certificate.json'
+    if args.write:path.write_text(json.dumps(result,indent=2,sort_keys=True)+'\n')
+    else:assert result==json.loads(path.read_text()),'Frozen cover certificate mismatch'
+    print('PASS kappa',result['kappa'],float(Q(result['kappa'])),'complex',result['complex_saving'],'47 constraints,7 margins',flush=True)
+if __name__=='__main__':main()
diff --git a/research/cover-local-reuse/geometry-audit.json b/research/cover-local-reuse/geometry-audit.json
new file mode 100644
index 0000000000..6d9bf2a920
--- /dev/null
+++ b/research/cover-local-reuse/geometry-audit.json
@@ -0,0 +1,48 @@
+{
+  "status": "PASS independent exact binary symplectic and actual h24 port geometry",
+  "clifford_checks": [
+    {
+      "h": 1,
+      "all_subspaces": 2,
+      "degenerate_subspaces": 0,
+      "all_ordered_pairs": 4,
+      "nested_tail_pairs": 3
+    },
+    {
+      "h": 2,
+      "all_subspaces": 5,
+      "degenerate_subspaces": 1,
+      "all_ordered_pairs": 25,
+      "nested_tail_pairs": 12
+    },
+    {
+      "h": 3,
+      "all_subspaces": 16,
+      "degenerate_subspaces": 6,
+      "all_ordered_pairs": 256,
+      "nested_tail_pairs": 66
+    },
+    {
+      "h": 4,
+      "all_subspaces": 67,
+      "degenerate_subspaces": 29,
+      "all_ordered_pairs": 4489,
+      "nested_tail_pairs": 513
+    }
+  ],
+  "actual_h24_port_triples_checked": 2024,
+  "source_sha256": {
+    "notes/general-clifford-frames.tex": "c9bb9655b594583c6b4bab8cb485233f1d1e588f83858cf1e4c3530cee8fed54",
+    "notes/three-stage-cover-complex.tex": "1c5b638aa9b75365b8addec9c8eeee6bc5e499722901079b94c65fce033fde8a",
+    "notes/three-stage-cover-note.tex": "db8078687825bcd43c6cc2fb69712ebb12840057b77cd5af8f67f53e4ffffa92"
+  },
+  "scope": "Finite checks substantiate the written all-subspace algebra and all selected port identities. They do not verify exact operator phases or all-size fixed-tape/weighted-bit interfaces.",
+  "source_review_findings": [
+    "K_G fixes L0 and LF, transports L_U, and has rank-zero adapters; arbitrary degenerate subspaces are allowed.",
+    "Dirty-tail Fourier rank holds for direct complement V_sigma, without claiming V_sigma=sigma_perp.",
+    "Reverse D_U=T_U F_inverse transitions cancel exactly at operator level.",
+    "The three Cayley port maps preserve source line and make every adjacent data frame equal.",
+    "Rank-zero adapters reconcile exact representatives; full finite router bound must remain paid."
+  ],
+  "publication_blocker_in_reviewed_scope": null
+}
diff --git a/research/cover-local-reuse/geometry_checks.py b/research/cover-local-reuse/geometry_checks.py
new file mode 100644
index 0000000000..10850342d0
--- /dev/null
+++ b/research/cover-local-reuse/geometry_checks.py
@@ -0,0 +1,133 @@
+"""Independent finite checks of PR130's written Clifford and cover identities.
+Not an all-size streaming proof or replacement for source verification.
+"""
+from pathlib import Path
+from itertools import combinations
+from functools import lru_cache
+import json,time,hashlib,sys
+if sys.flags.optimize:raise ValueError('Exact geometry requires assertions')
+D=Path(__file__).parent
+
+def basis(rows):
+ p={}
+ for x in rows:
+  for j in sorted(p,reverse=True):
+   if x>>j&1:x^=p[j]
+  if x:
+   j=x.bit_length()-1
+   for k in p:
+    if p[k]>>j&1:p[k]^=x
+   p[j]=x
+ return tuple(p[j]for j in sorted(p,reverse=True))
+
+def perp(U,h):
+ U=basis(U);p={x.bit_length()-1:x for x in U};out=[]
+ for j in range(h):
+  if j not in p:
+   x=1<<j
+   for k,row in p.items():
+    if row>>j&1:x|=1<<k
+   out.append(x)
+ return basis(out)
+
+def app(C,x):
+ y=0
+ while x:
+  z=x&-x;x-=z;y^=C[z.bit_length()-1]
+ return y
+
+def mul(A,B):return tuple(app(A,b)for b in B)
+
+def inverse(C):
+ n=len(C);rows=[sum(((C[j]>>i)&1)<<j for j in range(n))|(1<<(n+i))for i in range(n)]
+ for i in range(n):
+  j=next(j for j in range(i,n)if rows[j]>>i&1);rows[i],rows[j]=rows[j],rows[i]
+  for j in range(n):
+   if j!=i and rows[j]>>i&1:rows[j]^=rows[i]
+ assert all(rows[i]&((1<<n)-1)==1<<i for i in range(n))
+ return tuple(sum(((rows[i]>>(n+j))&1)<<i for i in range(n))for j in range(n))
+
+def trans(C):return tuple(sum(((C[i]>>j)&1)<<i for i in range(len(C)))for j in range(len(C)))
+def image(C,U):return basis(app(C,x)for x in U)
+def contains(U,V):return len(basis(U+V))==len(V)
+def lag(U,h):return basis(tuple(u|(u<<h)for u in U)+tuple(v<<h for v in perp(U,h)))
+def intersectdim(U,V):return len(U)+len(V)-len(basis(U+V))
+def sympair(x,y,h):
+ mask=(1<<h)-1
+ return (((x&mask)&(y>>h)).bit_count()+((y&mask)&(x>>h)).bit_count())&1
+
+def subspaces(h):
+ spaces={()};todo=[()]
+ for U in todo:
+  for x in range(1,1<<h):
+   V=basis(U+(x,))
+   if V not in spaces:spaces.add(V);todo.append(V)
+ return sorted(spaces,key=lambda U:(len(U),U))
+
+records=[]
+for h in (1,2,3,4):
+ mask=(1<<h)-1;L0=tuple(1<<(h+i)for i in range(h));LF=tuple((1<<i)|(1<<(h+i))for i in range(h))
+ F=tuple(1<<i for i in range(h))+LF
+ spaces=subspaces(h);T={};labels={};complements={};degenerate=0
+ for U in spaces:
+  r=len(U);cols=list(U)
+  for i in range(h):
+   if not contains((1<<i,),basis(cols)):cols.append(1<<i)
+  assert len(cols)==h
+  G=tuple(cols);GiT=trans(inverse(G))
+  K=tuple(G[i]|((G[i]^GiT[i])<<h)for i in range(h))+tuple(c<<h for c in GiT)
+  assert all(sympair(K[i],K[j],h)==int((i<h and j==i+h)or(j<h and i==j+h))for i in range(2*h)for j in range(2*h))
+  assert image(K,L0)==basis(L0) and image(K,LF)==basis(LF)
+  CE=tuple(1<<i for i in range(h))+tuple((1<<(h+i))|((1<<i)if i<r else 0)for i in range(h))
+  TU=mul(mul(K,CE),inverse(K));T[U]=TU;labels[U]=lag(U,h);complements[U]=basis(G[r:])
+  assert image(inverse(TU),L0)==labels[U]
+  assert len(basis(c&mask for c in TU[h:]))==r
+  assert image(F,labels[U])==lag(perp(U,h),h)
+  assert image(inverse(TU),LF)==lag(complements[U],h)
+  assert mul(TU,TU)==tuple(1<<i for i in range(2*h))
+  degenerate+=len(basis(sum(((x&y).bit_count()&1)<<j for j,y in enumerate(U))for x in U))<r
+ nested=0
+ for U in spaces:
+  for V in spaces:
+   distance=h-intersectdim(labels[U],labels[V]);expected=len(U)+len(V)-2*intersectdim(U,V)
+   assert distance==expected
+   assert len(basis(c&mask for c in mul(T[V],inverse(T[U]))[h:]))==expected
+   if contains(U,V):
+    nested+=1
+    tail=mul(mul(F,T[U]),inverse(T[V]))
+    assert len(basis(c&mask for c in tail[h:]))==h-len(V)+len(U)
+   DU=mul(T[U],F);DV=mul(T[V],F)
+   assert mul(DU,inverse(DV))==mul(T[U],inverse(T[V]))
+ records.append(dict(h=h,all_subspaces=len(spaces),degenerate_subspaces=degenerate,all_ordered_pairs=len(spaces)**2,nested_tail_pairs=nested))
+ print('Clifford',records[-1],flush=True)
+
+h=24;m=3*h-2;A=tuple(1<<i for i in range(h));B=tuple(1<<i for i in range(h,2*h-1));C=tuple(1<<i for i in range(2*h-1,m));E=basis(A+B+C)
+checks=0
+for triple in combinations(range(h),3):
+ q=sum(1<<i for i in triple);d=next(i for i in range(h)if i not in triple);w=q^(1<<d)
+ Aq=tuple((1<<i)^(w if w>>i&1 else 0)for i in range(h)if i!=d)
+ assert all(((x&y).bit_count()&1)==int(i==j)for i,x in enumerate(Aq)for j,y in enumerate(Aq))
+ assert all(not((x&q).bit_count()&1)for x in Aq)
+ def swap(other,fixed):
+  cols=[(q if q>>j&1 else 0)^sum((other[i] if u>>j&1 else 0)for i,u in enumerate(Aq))for j in range(h)]
+  for axis in (B,C):cols.extend(Aq if axis==other else axis)
+  R=tuple(cols)
+  assert app(R,q)==q and mul(R,R)==tuple(1<<i for i in range(m))
+  assert all(((x&y).bit_count()&1)==int(i==j)for i,x in enumerate(R)for j,y in enumerate(R))
+  return R
+ R12=swap(B,C);R23=swap(C,B)
+ assert image(R12,A)==basis(B+(q,)) and image(R12,Aq)==basis(B)
+ assert image(R23,A)==basis(C+(q,)) and image(R23,B+C)==basis(B+Aq)
+ # All six data segments telescope without a positive-rank connector.
+ stages=[((q,),basis(B+(q,)),(),basis(B)),(basis(B+(q,)),basis(A+B),basis(B),basis(Aq+B)),(basis(A+B),E,basis(Aq+B),basis(Aq+B+C))]
+ for xs,xe,ys,ye in stages:assert contains(xs,xe)and contains(ys,ye)
+ assert stages[-1][3]==perp((q,),m)
+ checks+=1
+print('Cayley actual h24 triples',checks,flush=True)
+P=D.resolve().parents[1]
+files=['notes/general-clifford-frames.tex','notes/three-stage-cover-complex.tex','notes/three-stage-cover-note.tex']
+out=dict(status='PASS independent exact binary symplectic and actual h24 port geometry',clifford_checks=records,actual_h24_port_triples_checked=checks,
+ source_sha256={f:hashlib.sha256((P/f).read_bytes()).hexdigest()for f in files},
+ scope='Finite checks substantiate the written all-subspace algebra and all selected port identities. They do not verify exact operator phases or all-size fixed-tape/weighted-bit interfaces.',
+ source_review_findings=['K_G fixes L0 and LF, transports L_U, and has rank-zero adapters; arbitrary degenerate subspaces are allowed.', 'Dirty-tail Fourier rank holds for direct complement V_sigma, without claiming V_sigma=sigma_perp.', 'Reverse D_U=T_U F_inverse transitions cancel exactly at operator level.', 'The three Cayley port maps preserve source line and make every adjacent data frame equal.', 'Rank-zero adapters reconcile exact representatives; full finite router bound must remain paid.'],publication_blocker_in_reviewed_scope=None)
+(D/'geometry-audit.json').write_text(json.dumps(out,indent=2)+'\n')
diff --git a/research/cover-local-reuse/test_controls.py b/research/cover-local-reuse/test_controls.py
new file mode 100644
index 0000000000..3402335909
--- /dev/null
+++ b/research/cover-local-reuse/test_controls.py
@@ -0,0 +1,130 @@
+#!/usr/bin/env python3
+"""Corruption controls for the exact local-word/cover composition.
+
+Prepared for eumemic with OpenAI Codex assistance; Apache-2.0.
+"""
+import sys
+sys.dont_write_bytecode = True
+if sys.flags.optimize:
+    raise ValueError('Negative controls require assertions')
+if hasattr(sys, 'set_int_max_str_digits'):
+    sys.set_int_max_str_digits(0)
+import copy
+import json
+from pathlib import Path
+import shutil
+import subprocess
+import tempfile
+import unittest
+from unittest.mock import patch
+import certificate
+from verify import (ROOT, LOCAL, PACKAGE, BIT_MANIFEST, LOCAL_FILES,
+                    check_sources, compare_json, digest, safe_file)
+
+
+class CoverControls(unittest.TestCase):
+    @classmethod
+    def setUpClass(cls):
+        cls.manifest = check_sources(ROOT)
+
+    def local_change(self, name, change, callback=certificate.cover_profile):
+        with tempfile.TemporaryDirectory(prefix='cover-local-control-') as directory:
+            local = Path(directory)
+            for filename in LOCAL_FILES:
+                out = local / filename
+                out.parent.mkdir(parents=True, exist_ok=True)
+                shutil.copyfile(ROOT / LOCAL / filename, out)
+            path = local / name
+            value = json.loads(path.read_text())
+            change(value)
+            path.write_text(json.dumps(value))
+            with patch.object(certificate, 'LOCAL', local):
+                with self.assertRaises((AssertionError, ValueError)):
+                    callback()
+
+    def test_frozen_exact_certificate_recomputed(self):
+        actual = certificate.js(certificate.exact())
+        self.assertEqual(actual, json.loads((ROOT / PACKAGE / 'certificate.json').read_text()))
+
+    def test_local_cover_telescoping(self):
+        p = certificate.cover_profile()
+        self.assertEqual(p['roles_per_vertex'] * p['m'] - p['rank_per_vertex'], 2392)
+        self.assertEqual(sum(int(r) * n for r, n in p['child_multiplicities'].items()), p['rank_per_vertex'])
+        self.assertEqual(p['R'] + p['reused_roles'], p['virtual_R'])
+
+    def test_omitted_reused_role_rejected(self):
+        self.local_change('complex-profile.json', lambda p: p.update(reused_roles=p['reused_roles']-1))
+
+    def test_omitted_endpoint_or_move_rejected(self):
+        self.local_change('complex-profile.json', lambda p: p['child_multiplicities'].__setitem__('1', p['child_multiplicities']['1']-1))
+
+    def test_unbound_source_inventory_rejected(self):
+        self.local_change('reflection-audit.json', lambda p: p.update(completed_core_source_inventory_bound=False))
+
+    def test_omitted_dirty_compensation_rejected(self):
+        self.local_change('reflection-audit.json', lambda p: p.update(exact_arbitrary_dirty_cancellation_by_dependency_cut=False))
+
+    def test_missing_reflected_frame_check_rejected(self):
+        self.local_change('reflection-audit.json', lambda p: p.update(reflected_core_active_frames_complement_source=False))
+
+    def test_wrong_source_hash_rejected(self):
+        self.local_change('reflection-audit.json', lambda p: p.update(source_sha256='0'*64))
+
+    def test_scalar_guard_below_literal_audit_rejected(self):
+        self.local_change('reflection-audit.json',
+                          lambda p: p.update(expanded_scalar_operations_per_stage=p['conservative_local_G']+1),
+                          certificate.exact)
+
+    def test_missing_dynamic_local_producer_rejected(self):
+        changed = copy.deepcopy(self.manifest)
+        del changed['files'][LOCAL + '/producer.py']
+        with self.assertRaisesRegex(ValueError, 'source dependency closure'):
+            check_sources(ROOT, changed)
+
+    def test_missing_inherited_bit_source_rejected(self):
+        changed = copy.deepcopy(self.manifest)
+        del changed['files'][str(Path(BIT_MANIFEST).parent / 'deferred_23.json.gz')]
+        with self.assertRaisesRegex(ValueError, 'source dependency closure'):
+            check_sources(ROOT, changed)
+
+    def test_altered_dag_rejected_even_after_outer_rehash(self):
+        with tempfile.TemporaryDirectory(prefix='cover-repinned-dag-') as directory:
+            root = Path(directory)
+            for name in self.manifest['files']:
+                out = root / name
+                out.parent.mkdir(parents=True, exist_ok=True)
+                shutil.copyfile(ROOT / name, out)
+            name = LOCAL + '/inputs/complex-dag.json.gz'
+            path = root / name
+            path.write_bytes(path.read_bytes() + b'corrupted')
+            changed = copy.deepcopy(self.manifest)
+            changed['files'][name] = digest(path)
+            with self.assertRaisesRegex(ValueError, 'scalar-DAG provenance'):
+                check_sources(root, changed)
+
+    def test_hash_bound_formatting_change_rejected(self):
+        with tempfile.TemporaryDirectory(prefix='cover-formatting-') as directory:
+            expected = ROOT / LOCAL / 'reflection-audit.json'
+            actual = Path(directory) / expected.name
+            actual.write_text(json.dumps(json.loads(expected.read_text())))
+            with self.assertRaisesRegex(ValueError, 'generated byte mismatch'):
+                compare_json(actual, expected)
+
+    def test_noncanonical_reflection_receipt_rejected(self):
+        path = ROOT / LOCAL / 'reflection-audit.json'
+        expected = json.dumps(json.loads(path.read_text()), indent=2, sort_keys=True) + '\n'
+        self.assertEqual(path.read_bytes(), expected.encode())
+
+    def test_parent_escape_rejected(self):
+        with self.assertRaisesRegex(ValueError, 'unsafe source path'):
+            safe_file(ROOT, '../certificate.json')
+
+    def test_optimized_verifier_rejected_before_regeneration(self):
+        result = subprocess.run([sys.executable, '-O', str(ROOT / PACKAGE / 'verify.py')],
+                                capture_output=True, text=True)
+        self.assertNotEqual(result.returncode, 0)
+        self.assertIn('optimized Python is forbidden', result.stderr)
+
+
+if __name__ == '__main__':
+    unittest.main(verbosity=2)
diff --git a/research/cover-local-reuse/verify.py b/research/cover-local-reuse/verify.py
new file mode 100644
index 0000000000..a342573e54
--- /dev/null
+++ b/research/cover-local-reuse/verify.py
@@ -0,0 +1,220 @@
+#!/usr/bin/env python3
+"""Read-only finite replay of physical local reuse under the PR130 cover.
+
+Prepared for eumemic with OpenAI Codex assistance. Apache-2.0;
+the pinned upstream sources retain their original notices and proof scope.
+"""
+import sys
+sys.dont_write_bytecode = True
+if sys.flags.optimize:
+    raise ValueError('Verification requires assertions; optimized Python is forbidden')
+if hasattr(sys, 'set_int_max_str_digits'):
+    sys.set_int_max_str_digits(0)
+import argparse
+import ast
+from hashlib import sha256
+import json
+import os
+from pathlib import Path
+import shutil
+import subprocess
+import tempfile
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+PACKAGE = 'research/cover-local-reuse'
+LOCAL = 'research/cyclic-deferred'
+DAG_PIN = '3c034d0aae388ef567a454826f4f48b26fd8a94c71e8ffed4835271b349a783b'
+BASE_CERT = 'certificates/three-stage-cover-network.json'
+BASE_PIN = 'bfca1f225394c8b50f3360b71920652baf9bd885ce84f9c0a9184c3ba7dd35ab'
+BIT_MANIFEST = 'references/partial-gauge/pr97/SOURCE.json'
+BIT_PIN = '68fb539abcd6df21d142e4e204b6e9482eb7f9cd3907650f54bd058070266596'
+LOCAL_FILES = {
+    'complex_deferred.py', 'producer.py', 'replayed_producer.py', 'reuse.py',
+    'reflection_audit.py', 'complex-profile.json', 'reuse-pairs.json',
+    'reflection-audit.json', 'inputs/complex-dag.json.gz',
+}
+PACKAGE_FILES = {
+    'verify.py', 'test_controls.py', 'certificate.py', 'certificate.json',
+    'README.md', 'geometry_checks.py', 'geometry-audit.json',
+}
+
+
+def require(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def digest(path):
+    return sha256(path.read_bytes()).hexdigest()
+
+
+def safe_file(root, name):
+    part = Path(name)
+    path = (root / part).resolve()
+    require(not part.is_absolute() and '..' not in part.parts and
+            path.is_relative_to(root.resolve()) and path.is_file(),
+            'Missing or unsafe source path: ' + name)
+    return path
+
+
+def adopted_sources(root):
+    """Pin the adopted cover evidence and every frozen inherited bit input."""
+    require(digest(safe_file(root, BASE_CERT)) == BASE_PIN,
+            'Immutable PR130 baseline certificate mismatch')
+    baseline = json.loads((root / BASE_CERT).read_text())
+    names = {BASE_CERT, BIT_MANIFEST, 'LICENSE', 'NOTICE'}
+    for name, value in baseline['source_sha256'].items():
+        require(digest(safe_file(root, name)) == value,
+                'Adopted PR130 source mismatch: ' + name)
+        names.add(name)
+    require(digest(safe_file(root, BIT_MANIFEST)) == BIT_PIN,
+            'Immutable inherited PR97 manifest mismatch')
+    bit = json.loads((root / BIT_MANIFEST).read_text())
+    require(bit['pr97']['commit'] == 'e15350b66e4bf796153a68c240a6cc1b788abe3e' and
+            bit['underlying']['commit'] == '741e7aa078392553815df7926ee17ac5e25a8c38',
+            'Inherited bit provenance mismatch')
+    for name, value in bit['sha256'].items():
+        full = str(Path(BIT_MANIFEST).parent / name)
+        require(digest(safe_file(root, full)) == value,
+                'Inherited bit source mismatch: ' + name)
+        names.add(full)
+    return names
+
+
+def python_closure(root, roots):
+    """Close actual entry points, including explicitly loaded local modules.
+
+    Adopted historical checker APIs are pinned as evidence but not invoked.
+    The current PR97 rank reconstruction dynamically loads only deferred.py.
+    """
+    pending = [root / name for name in roots]
+    seen = set()
+    while pending:
+        path = pending.pop().resolve()
+        if path in seen:
+            continue
+        seen.add(path)
+        for node in ast.walk(ast.parse(path.read_text(), filename=str(path))):
+            imports = []
+            if isinstance(node, ast.Import):
+                imports = [(a.name, 0) for a in node.names]
+            elif isinstance(node, ast.ImportFrom) and node.module:
+                imports = [(node.module, node.level)]
+            for name, level in imports:
+                if not level and name.split('.')[0] in sys.stdlib_module_names:
+                    continue
+                bases = ((path.parent.parents[level-2] if level > 1 else path.parent),) if level else (
+                    path.parent, root / PACKAGE, root / LOCAL, root / 'scripts')
+                found = None
+                for base in bases:
+                    module = base.joinpath(*name.split('.'))
+                    for candidate in (module.with_suffix('.py'), module / '__init__.py'):
+                        if candidate.is_file():
+                            found = candidate.resolve()
+                            break
+                    if found is not None:
+                        break
+                require(found is not None and found.is_relative_to(root.resolve()),
+                        'Unpinned non-stdlib import: ' + name)
+                pending.append(found)
+                parent = found.parent
+                while parent != root.resolve():
+                    init = parent / '__init__.py'
+                    if init.is_file():
+                        pending.append(init)
+                    parent = parent.parent
+    return {str(path.relative_to(root.resolve())) for path in seen}
+
+
+def inventory(root):
+    names = adopted_sources(root)
+    names.update(LOCAL + '/' + name for name in LOCAL_FILES)
+    names.update(PACKAGE + '/' + name for name in PACKAGE_FILES)
+    names.update(str(path.relative_to(root)) for path in (root / PACKAGE).iterdir()
+                 if path.is_file() and path.name != 'SOURCE.json')
+    roots = {name for name in names if name.endswith('.py') and
+             (name.startswith(PACKAGE + '/') or name.startswith(LOCAL + '/'))}
+    roots.add('references/partial-gauge/pr97/deferred.py')
+    names.update(python_closure(root, roots))
+    return names
+
+
+def create_manifest(root):
+    return dict(schema=1,
+        scope='Finite local complex replay, geometry checks and exact cover assembly; general cover/bit/analytic interfaces remain conditional.',
+        files={name: digest(safe_file(root, name)) for name in sorted(inventory(root))},
+        provenance=dict(cover_commit='6a9970a530119174507904e23592fd59ede19a5d',
+                        cover_certificate_sha256=BASE_PIN,
+                        scalar_dag_commit='cbb05ce504d571546d9b7794c186a613c659c3bf',
+                        scalar_dag_sha256=DAG_PIN,
+                        physical_local_source='PR129 completed-core frames and PR124 compensated birth-cut reuse'))
+
+
+def check_sources(root, manifest=None):
+    manifest = manifest or json.loads((root / PACKAGE / 'SOURCE.json').read_text())
+    require(manifest['schema'] == 1, 'Unsupported source manifest')
+    require(set(manifest['files']) == inventory(root), 'Incomplete or stale source dependency closure')
+    for name, value in manifest['files'].items():
+        require(digest(safe_file(root, name)) == value, 'Source hash mismatch: ' + name)
+    require(manifest['files'][LOCAL + '/inputs/complex-dag.json.gz'] == DAG_PIN,
+            'Immutable PR117 scalar-DAG provenance mismatch')
+    return manifest
+
+
+def compare_json(actual, expected):
+    require(json.loads(actual.read_text()) == json.loads(expected.read_text()),
+            'Frozen generated record mismatch: ' + expected.name)
+    require(actual.read_bytes() == expected.read_bytes(),
+            'Frozen generated byte mismatch: ' + expected.name)
+
+
+def snapshot(root, manifest):
+    return {**{name: digest(root / name) for name in manifest['files']},
+            PACKAGE + '/SOURCE.json': digest(root / PACKAGE / 'SOURCE.json')}
+
+
+def verify(root):
+    manifest = check_sources(root)
+    before = snapshot(root, manifest)
+    environment = dict(os.environ, PYTHONDONTWRITEBYTECODE='1')
+    environment.pop('PYTHONPATH', None)
+    with tempfile.TemporaryDirectory(prefix='cover-local-reuse-verify-') as directory:
+        target = Path(directory)
+        for name in [*manifest['files'], PACKAGE + '/SOURCE.json']:
+            out = target / name
+            out.parent.mkdir(parents=True, exist_ok=True)
+            shutil.copyfile(root / name, out)
+        def run(name, *args):
+            subprocess.run([sys.executable, str(target / name), *map(str, args)],
+                           cwd=target, env=environment, check=True)
+        run(LOCAL + '/complex_deferred.py')
+        for name in ('complex-profile.json', 'reuse-pairs.json'):
+            compare_json(target / LOCAL / name, root / LOCAL / name)
+        print('PASS complete physical local compiler and compensated reuse regenerated', flush=True)
+        run(LOCAL + '/reflection_audit.py', '--source', target / LOCAL / 'complex_deferred.py',
+            '--output', target / LOCAL / 'reflection-audit.json')
+        compare_json(target / LOCAL / 'reflection-audit.json', root / LOCAL / 'reflection-audit.json')
+        print('PASS full independent reflection, dirty cleanup and scalar audit', flush=True)
+        run(PACKAGE + '/geometry_checks.py')
+        compare_json(target / PACKAGE / 'geometry-audit.json', root / PACKAGE / 'geometry-audit.json')
+        run(PACKAGE + '/certificate.py')
+        run(PACKAGE + '/test_controls.py')
+    require(snapshot(root, manifest) == before, 'Verification mutated a frozen source or artifact')
+    print('PASS read-only finite cover/local-reuse verification; general interfaces remain conditional', flush=True)
+
+
+def main():
+    ap = argparse.ArgumentParser(description=__doc__)
+    ap.add_argument('--freeze-manifest', action='store_true',
+                    help='Explicit authoring operation; replace SOURCE.json without certifying it')
+    args = ap.parse_args()
+    if args.freeze_manifest:
+        (HERE / 'SOURCE.json').write_text(json.dumps(create_manifest(ROOT), indent=2, sort_keys=True) + '\n')
+        print('Frozen source manifest; this operation does not certify the package')
+    else:
+        verify(ROOT)
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/cyclic-deferred/complex-profile.json b/research/cyclic-deferred/complex-profile.json
new file mode 100644
index 0000000000..9c787ad7d1
--- /dev/null
+++ b/research/cyclic-deferred/complex-profile.json
@@ -0,0 +1,10524 @@
+{
+ "h": 24,
+ "v": 2024,
+ "additions": 91770,
+ "roots": 8120,
+ "links": 71185,
+ "R": 26597,
+ "virtual_R": 28705,
+ "reused_roles": 2108,
+ "m": 576,
+ "N": 4096576,
+ "W": 115857808,
+ "L": 2234496,
+ "total_rank": 66732235328,
+ "deficit": 1862080,
+ "maxchild": 574,
+ "phase_one_ops": 18844,
+ "phase_one_roles": 13555,
+ "physical_auxiliary_source_frames": [
+  {
+   "basis": [],
+   "count": 23999
+  },
+  {
+   "basis": [
+    9,
+    5
+   ],
+   "count": 134
+  },
+  {
+   "basis": [
+    36,
+    20
+   ],
+   "count": 2
+  },
+  {
+   "basis": [
+    144,
+    80
+   ],
+   "count": 88
+  },
+  {
+   "basis": [
+    2304,
+    1280
+   ],
+   "count": 23
+  },
+  {
+   "basis": [
+    36864,
+    20480
+   ],
+   "count": 16
+  },
+  {
+   "basis": [
+    589824,
+    327680
+   ],
+   "count": 98
+  },
+  {
+   "basis": [
+    9437184,
+    5242880
+   ],
+   "count": 502
+  },
+  {
+   "basis": [
+    8,
+    4,
+    2,
+    1
+   ],
+   "count": 4
+  },
+  {
+   "basis": [
+    33,
+    17,
+    9,
+    5
+   ],
+   "count": 72
+  },
+  {
+   "basis": [
+    129,
+    65,
+    33,
+    17
+   ],
+   "count": 6
+  },
+  {
+   "basis": [
+    132,
+    68,
+    36,
+    20
+   ],
+   "count": 8
+  },
+  {
+   "basis": [
+    513,
+    257,
+    9,
+    5
+   ],
+   "count": 28
+  },
+  {
+   "basis": [
+    528,
+    272,
+    144,
+    80
+   ],
+   "count": 42
+  },
+  {
+   "basis": [
+    2064,
+    1040,
+    528,
+    272
+   ],
+   "count": 4
+  },
+  {
+   "basis": [
+    33792,
+    17408,
+    9216,
+    5120
+   ],
+   "count": 10
+  },
+  {
+   "basis": [
+    524289,
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+    131073,
+    65537
+   ],
+   "count": 22
+  },
+  {
+   "basis": [
+    525312,
+    263168,
+    132096,
+    66560
+   ],
+   "count": 8
+  },
+  {
+   "basis": [
+    528384,
+    266240,
+    135168,
+    69632
+   ],
+   "count": 6
+  },
+  {
+   "basis": [
+    540672,
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+   ],
+   "count": 1
+  },
+  {
+   "basis": [
+    8388609,
+    4194305,
+    2097153,
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+   ],
+   "count": 68
+  },
+  {
+   "basis": [
+    8388624,
+    4194320,
+    144,
+    80
+   ],
+   "count": 2
+  },
+  {
+   "basis": [
+    8388624,
+    4194320,
+    2097168,
+    1048592
+   ],
+   "count": 43
+  },
+  {
+   "basis": [
+    8389632,
+    4195328,
+    2098176,
+    1049600
+   ],
+   "count": 9
+  },
+  {
+   "basis": [
+    8404992,
+    4210688,
+    2113536,
+    1064960
+   ],
+   "count": 1
+  },
+  {
+   "basis": [
+    8454144,
+    4259840,
+    589824,
+    327680
+   ],
+   "count": 4
+  },
+  {
+   "basis": [
+    8454144,
+    4259840,
+    2162688,
+    1114112
+   ],
+   "count": 17
+  },
+  {
+   "basis": [
+    129,
+    65,
+    33,
+    17,
+    9,
+    5
+   ],
+   "count": 127
+  },
+  {
+   "basis": [
+    513,
+    257,
+    33,
+    17,
+    9,
+    5
+   ],
+   "count": 2
+  },
+  {
+   "basis": [
+    2049,
+    1025,
+    513,
+    257,
+    9,
+    5
+   ],
+   "count": 8
+  },
+  {
+   "basis": [
+    2049,
+    1025,
+    513,
+    257,
+    129,
+    65
+   ],
+   "count": 3
+  },
+  {
+   "basis": [
+    2064,
+    1040,
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+    272,
+    144,
+    80
+   ],
+   "count": 7
+  },
+  {
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+    9,
+    5
+   ],
+   "count": 1
+  },
+  {
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+   ],
+   "count": 4
+  },
+  {
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+   ],
+   "count": 40
+  },
+  {
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+   ],
+   "count": 2
+  },
+  {
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+    1049600,
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+   ],
+   "count": 2
+  },
+  {
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+    4194305,
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+    5
+   ],
+   "count": 6
+  },
+  {
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+    4194305,
+    129,
+    65,
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+    5
+   ],
+   "count": 3
+  },
+  {
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+    4194305,
+    2097153,
+    1048577,
+    9,
+    5
+   ],
+   "count": 81
+  },
+  {
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+    8388609,
+    4194305,
+    2097153,
+    1048577,
+    33,
+    17
+   ],
+   "count": 3
+  },
+  {
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+    8388609,
+    4194305,
+    2097153,
+    1048577,
+    129,
+    65
+   ],
+   "count": 2
+  },
+  {
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+    8388609,
+    4194305,
+    2097153,
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+    131073,
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+   ],
+   "count": 2
+  },
+  {
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+    4194320,
+    2097168,
+    1048592,
+    144,
+    80
+   ],
+   "count": 4
+  },
+  {
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+    4195328,
+    33792,
+    17408,
+    9216,
+    5120
+   ],
+   "count": 3
+  },
+  {
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+    8389632,
+    4195328,
+    2098176,
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+    525312,
+    263168
+   ],
+   "count": 1
+  },
+  {
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+   ],
+   "count": 33
+  },
+  {
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+   ],
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+  },
+  {
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+   ],
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+  },
+  {
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+    3
+   ],
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+  },
+  {
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+   ],
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+  },
+  {
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+   ],
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+  },
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+   ],
+   "count": 2
+  },
+  {
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+  },
+  {
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+   ],
+   "count": 4
+  },
+  {
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+   ],
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+  },
+  {
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+  },
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+  },
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+  },
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+   ],
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+  },
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+  },
+  {
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+   ],
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+  },
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+   ],
+   "count": 8
+  },
+  {
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+   ],
+   "count": 12
+  },
+  {
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+   ],
+   "count": 8
+  },
+  {
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+   ],
+   "count": 11
+  },
+  {
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+   ],
+   "count": 1
+  },
+  {
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+   "count": 1
+  },
+  {
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+   ],
+   "count": 43
+  },
+  {
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+  },
+  {
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+   ],
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+  },
+  {
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+   "count": 1
+  },
+  {
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+  {
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+   "count": 1
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+ "deferred_roles": 4706,
+ "deferred_dims": {
+  "2": 863,
+  "4": 357,
+  "6": 573,
+  "8": 144,
+  "10": 25,
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+ },
+ "lifted_additions": 38114,
+ "physical_gate_frame_changes": 19323,
+ "replay": {
+  "seeds": [
+   1,
+   2
+  ],
+  "scratch_restored": true,
+  "y_plus_x": true,
+  "field": "Z/(2^61-1)",
+  "physically_aliased": true,
+  "inverse_order": "true source/workspace chronology",
+  "omitted_compensation_rejected": true
+ },
+ "reuse_rejected_controls": [
+  "nondead-donor",
+  "unavailable-birth",
+  "pair-alias",
+  "omitted-compensation"
+ ],
+ "child_multiplicities": {
+  "1": 408795376,
+  "2": 291621968,
+  "3": 52656384,
+  "4": 51041232,
+  "5": 21498928,
+  "6": 17507600,
+  "7": 9229440,
+  "8": 12633808,
+  "9": 7869312,
+  "10": 21118416,
+  "11": 4938560,
+  "12": 20826960,
+  "13": 17786912,
+  "14": 1922800,
+  "15": 2493568,
+  "16": 684112,
+  "17": 489808,
+  "18": 546480,
+  "19": 477664,
+  "20": 550528,
+  "21": 1230592,
+  "22": 64768,
+  "23": 8363168,
+  "529": 8193152,
+  "552": 97147952,
+  "554": 3493424,
+  "556": 1437040,
+  "558": 2319504,
+  "560": 453376,
+  "562": 48576,
+  "570": 821744,
+  "571": 817696,
+  "572": 825792,
+  "573": 44528,
+  "574": 255024
+ }
+}
diff --git a/research/cyclic-deferred/complex_deferred.py b/research/cyclic-deferred/complex_deferred.py
new file mode 100644
index 0000000000..4a65788ecb
--- /dev/null
+++ b/research/cyclic-deferred/complex_deferred.py
@@ -0,0 +1,736 @@
+#!/usr/bin/env python3
+"""Optimized physical gate frames on the immutable PR117 complex DAG.
+
+After saturated deferral, each mixer gate frame is optimized within its two
+role chains. Maximal nondegenerate extensions and minimal nondegenerate hulls
+are accepted only when they improve the local moment objective; every actual
+source, mixer, copied-center and root incidence is checked exactly.
+Physical-frame optimization prepared with OpenAI Codex assistance.
+Compensated birth-cut reuse follows jamesyc PR124; this composition regenerates
+all pairings, replays arbitrary aliased scratch, and retains every scalar charge.
+
+PR117 credits its searched DAG to eumemic with Anthropic Claude assistance;
+this experiment imports and replays that witness unchanged. PR110/PR114
+saturation and exact finite checks are applied to the resulting graph.
+Logical readout macros are realized as signed numerator/42 chunks of magnitude
+at most one. The literal audit expands and charges those same-frame shears,
+checks exact reconstruction, and reverses the chunks under reflection.
+Inherited Avi Eisenberg, Rohan Arun, icekylinx and Swapnil Jain credits retained.
+This integration prepared with OpenAI Codex assistance. Apache-2.0.
+"""
+import array
+import json
+import random
+import struct
+import sys
+import tempfile
+from collections import Counter, defaultdict
+from itertools import combinations
+from pathlib import Path
+from reuse import select_reuse, check_pairs, check_compensation, mutation_controls
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+sys.path.insert(0, str(ROOT / 'scripts'))
+sys.path.insert(0, str(HERE))
+import importlib.util
+producer_path = HERE / 'producer.py'
+spec = importlib.util.spec_from_file_location('replayed_producer_wrapper', producer_path)
+producer = importlib.util.module_from_spec(spec); spec.loader.exec_module(producer)
+build = producer.build
+
+P = (1 << 61) - 1
+
+
+def require(ok, msg):
+    if not ok:
+        raise SystemExit('FAIL: ' + msg)
+
+
+def load(prefix):
+    def rd(fh, typ, cnt):
+        a = array.array(typ); a.fromfile(fh, cnt); return a
+    with open(str(prefix) + '.bin', 'rb') as f:
+        h, v, n, q = struct.unpack('<4I', f.read(16))
+        flat = rd(f, 'I', 2 * n); core = rd(f, 'Q', n); cover = rd(f, 'Q', n)
+        roots = rd(f, 'I', q); kind = rd(f, 'I', q); active = rd(f, 'B', n)
+    with open(str(prefix) + '.labels', 'rb') as f:
+        ranks = rd(f, 'I', n); types = rd(f, 'B', n)
+    args = [(flat[2 * i], flat[2 * i + 1]) for i in range(n)]
+    return h, v, n, q, args, core, cover, roots, kind, active, ranks, types
+
+
+# ---------------------------------------------------------------- F2 linear algebra on bit vectors
+def dot(a, b): return bin(a & b).count('1') & 1
+
+
+def reduce(basis):
+    rows = []; piv = []
+    for vec in basis:
+        for p, r in zip(piv, rows):
+            if vec >> p & 1: vec ^= r
+        if vec:
+            p = vec.bit_length() - 1
+            for i in range(len(rows)):
+                if rows[i] >> p & 1: rows[i] ^= vec
+            rows.append(vec); piv.append(p)
+    return rows
+
+
+def kernel(funcs, h):
+    K = reduce(funcs); piv = [r.bit_length() - 1 for r in K]
+    out = []
+    for c in (i for i in range(h) if i not in piv):
+        vec = 1 << c
+        for r, p in zip(K, piv):
+            if r >> c & 1: vec |= 1 << p
+        out.append(vec)
+    return out
+
+
+def restrict(basis, funcs):
+    B = reduce(basis)
+    for f in funcs:
+        if not B: break
+        vals = [dot(b, f) for b in B]
+        if not any(vals): continue
+        k = vals.index(1); pv = B[k]
+        B = [b ^ pv if val else b for b, val in zip(B, vals)]; B.pop(k)
+    return reduce(B)
+
+
+def contains(A, B):
+    RB = reduce(B)
+    return len(reduce(RB + list(A))) == len(RB)
+
+
+def cap(A, B):
+    A = reduce(A); B = reduce(B)
+    if not A or not B: return []
+    ker = []; red = []
+    for i, vec in enumerate(A + B):
+        tag = 1 << i
+        for rv, rt in red:
+            if vec >> (rv.bit_length() - 1) & 1: vec ^= rv; tag ^= rt
+        if vec: red.append((vec, tag))
+        else:
+            comb_ = 0
+            for k in range(len(A)):
+                if tag >> k & 1: comb_ ^= A[k]
+            ker.append(comb_)
+    return reduce(ker)
+
+
+def nondeg(B):
+    B = reduce(B); k = len(B)
+    if not k: return True
+    M = [sum(dot(B[i], B[j]) << j for j in range(k)) for i in range(k)]
+    return len(reduce(M)) == k
+
+
+def hopcroft_karp(left, adj):
+    """maximum bipartite matching; adj[x] = sorted list of right vertices."""
+    INF = 1 << 60
+    mate_l = {x: None for x in left}; mate_r = {}
+    while True:
+        dist = {}; queue = []
+        for x in left:
+            if mate_l[x] is None: dist[x] = 0; queue.append(x)
+            else: dist[x] = INF
+        found = INF; head = 0
+        while head < len(queue):
+            x = queue[head]; head += 1
+            if dist[x] >= found: continue
+            for r in adj[x]:
+                y = mate_r.get(r)
+                if y is None: found = min(found, dist[x] + 1)
+                elif dist[y] == INF: dist[y] = dist[x] + 1; queue.append(y)
+        if found == INF: break
+        gained = 0
+        for x0 in left:
+            if mate_l[x0] is not None: continue
+            stack = [(x0, iter(adj[x0]))]; path = []
+            while stack:
+                x, it = stack[-1]
+                advanced = False
+                for r in it:
+                    y = mate_r.get(r)
+                    if (y is None and dist[x] + 1 == found) or (y is not None and dist[y] == dist[x] + 1):
+                        path.append((x, r))
+                        if y is None:
+                            for xx, rr in path: mate_l[xx] = rr; mate_r[rr] = xx
+                            gained += 1; stack = []; break
+                        stack.append((y, iter(adj[y]))); advanced = True; break
+                else:
+                    dist[x] = INF; stack.pop()
+                    if path: path.pop()
+                    continue
+                if not advanced: break
+        if not gained: break
+    return mate_l
+
+
+from functools import lru_cache
+from fractions import Fraction
+
+def sat_basis(vectors):
+    piv = {}
+    for x in vectors:
+        for p in sorted(piv, reverse=True):
+            if x >> p & 1:
+                x ^= piv[p]
+        if x:
+            p = x.bit_length() - 1
+            for q in piv:
+                if piv[q] >> p & 1:
+                    piv[q] ^= x
+            piv[p] = x
+    return tuple((piv[p] for p in sorted(piv, reverse=True)))
+
+def sat_dot(a, b):
+    return (a & b).bit_count() & 1
+
+@lru_cache(None)
+def sat_contained(A, B):
+    for x in A:
+        for r in B:
+            if x >> r.bit_length() - 1 & 1:
+                x ^= r
+        if x:
+            return False
+    return True
+
+@lru_cache(None)
+def sat_cap(A, B):
+    if not A or not B:
+        return ()
+    red = {}
+    out = []
+    for i, x in enumerate(A + B):
+        tag = 1 << i
+        for p in sorted(red, reverse=True):
+            if x >> p & 1:
+                r, t = red[p]
+                x ^= r
+                tag ^= t
+        if x:
+            red[x.bit_length() - 1] = (x, tag)
+        else:
+            y = 0
+            for j, r in enumerate(A):
+                if tag >> j & 1:
+                    y ^= r
+            if y:
+                out.append(y)
+    return sat_basis(out)
+
+@lru_cache(None)
+def sat_nondeg(B):
+    return len(sat_basis((sum((sat_dot(a, b) << j for j, b in enumerate(B))) for a in B))) == len(B)
+
+@lru_cache(None)
+def sat_nonsingular_part(B):
+    rows = list(B)
+    out = []
+    while rows:
+        i = next((i for i, x in enumerate(rows) if sat_dot(x, x)), None)
+        if i is not None:
+            a = rows.pop(i)
+            out.append(a)
+            rows = [x ^ a if sat_dot(x, a) else x for x in rows]
+            continue
+        pair = next(((i, j) for i in range(len(rows)) for j in range(i + 1, len(rows)) if sat_dot(rows[i], rows[j])), None)
+        if pair is None:
+            break
+        i, j = pair
+        a, b = (rows[i], rows[j])
+        out.extend((a, b))
+        rows = [x ^ (a if sat_dot(x, b) else 0) ^ (b if sat_dot(x, a) else 0) for k, x in enumerate(rows) if k not in (i, j)]
+    return sat_basis(out)
+
+def saturated_placement(cand, reach):
+    # Each extracted block is orthogonal to previously selected blocks.
+    # A diagonal-one pivot contributes a nondegenerate line; when all
+    # remaining diagonals vanish, a dot-one pair contributes a hyperbolic
+    # plane. The residual radical is discarded, not counted as a frame.
+    cand = {s: sat_basis(B) for s, B in cand.items()}
+    order = sorted(cand, key=lambda s: (Fraction(-(1 << len(cand[s])), max(1, len(reach[s]))**2), -len(cand[s]), s))
+    byT = defaultdict(list); placed = {}
+    for s in order:
+        X = cand[s]; changed = True
+        while changed and X:
+            changed = False
+            for t in sorted(reach[s]):
+                for w in byT[t]:
+                    B = placed[w]
+                    if (len(B) >= len(X) and not sat_contained(X, B)) or (len(B) < len(X) and not sat_contained(B, X)):
+                        X = sat_cap(X, B); changed = True
+                    if not X: break
+                if not X: break
+            if X and not sat_nondeg(X):
+                before = X; X = sat_nonsingular_part(X)
+                assert sat_contained(X, before) and sat_nondeg(X)
+                assert len(X) < len(before)
+                changed = True  # Extraction can break containment of a smaller target frame.
+        if X:
+            assert sat_nondeg(X) and sat_contained(X, cand[s])
+            placed[s] = X
+            for t in reach[s]: byT[t].append(s)
+    for t, ss in byT.items():
+        ss.sort(key=lambda s: (len(placed[s]), s))
+        assert all(sat_contained(placed[a], placed[b]) for a, b in zip(ss, ss[1:]))
+    return placed
+
+def frame_restrict(A,B):return sat_basis(restrict(A,B))
+
+@lru_cache(None)
+def frame_protected_part(X, L):
+    Q = sat_nonsingular_part(X)
+    rad = frame_restrict(X, Q)
+    if not L:
+        return Q
+    red = {}
+    for q, r in [(q, 0) for q in Q] + [(0, r) for r in rad]:
+        x = q ^ r
+        for p in sorted(red, reverse=True):
+            if x >> p & 1:
+                y, a, b = red[p]
+                x ^= y
+                q ^= a
+                r ^= b
+        assert x
+        red[x.bit_length() - 1] = (x, q, r)
+    lin = {}
+    for x in L:
+        q = r = 0
+        for p in sorted(red, reverse=True):
+            if x >> p & 1:
+                y, a, b = red[p]
+                x ^= y
+                q ^= a
+                r ^= b
+        if x:
+            return ()
+        for p in sorted(lin, reverse=True):
+            if q >> p & 1:
+                a, b = lin[p]
+                q ^= a
+                r ^= b
+        if not q:
+            if r:
+                return ()
+        else:
+            lin[q.bit_length() - 1] = (q, r)
+    out = []
+    for x in Q:
+        q = x
+        r = 0
+        for p in sorted(lin, reverse=True):
+            if q >> p & 1:
+                a, b = lin[p]
+                q ^= a
+                r ^= b
+        out.append(x ^ r)
+    out = sat_basis(out)
+    assert sat_nondeg(out) and sat_contained(out, X) and sat_contained(L, out)
+    return out
+
+@lru_cache(None)
+def frame_smallest_hull(L, ambient):
+    L = sat_basis(L)
+    while not sat_nondeg(L):
+        radical = frame_restrict(L, L)
+        assert radical
+        r = radical[0]
+        partner = next((x for x in ambient if sat_dot(r, x)))
+        L = sat_basis(L + (partner,))
+    assert sat_contained(L, ambient)
+    return L
+
+def main():
+    with tempfile.TemporaryDirectory(prefix='deferred-stopped-') as work:
+        prefix = Path(work) / 'complex'
+        build(24, prefix, central_disjoint=24)
+        h, v, n, q, args, core, cover, roots, kind, active, ranks, types = load(prefix)
+    trip = list(combinations(range(h), 3)); require(len(trip) == v, 'triple count')
+    m = h * h; N = v * v; FULL = (1 << h) - 1
+    tmask = [sum(1 << p for p in T) for T in trip]
+    tid = {T: i for i, T in enumerate(trip)}
+
+    # ------------------------------------------------------------ uses and matching (inherited adjacency)
+    degree = [0] * n; uses = defaultdict(list); c_add = 0
+    for x in range(1, n):
+        if active[x] and args[x][0]:
+            c_add += 1
+            for pos, y in enumerate(args[x]): degree[y] += 1; uses[y].append(('gate', x, pos))
+    for j, x in enumerate(roots): degree[x] += 1; uses[x].append(('root', j))
+    def unode(u): return roots[u[1]] if u[0] == 'root' else u[1]
+    def before(x, u):
+        y = unode(u)
+        return (ranks[x], x) < (ranks[y], (n + u[1]) if u[0] == 'root' else y)
+    def incl(x, y):
+        tx, ty = types[x], types[y]
+        if tx == 1 and ty == 1: return not (core[y] & ~core[x]) and not (cover[x] & ~cover[y])
+        if tx in (1, 2) and ty == 2: return not (cover[x] & ~cover[y])
+        if tx == 1 and ty == 3: return bool(core[x] & core[y])
+        if tx == 2 and ty == 3: return not (cover[x] & ~core[y])
+        if tx == 3 and ty == 2: return cover[y] == FULL
+        if tx == 3 and ty == 3: return core[x] == core[y]
+        return False
+    left = []; adj = {}
+    for x in range(1, n):
+        if not (active[x] and args[x][0]): continue
+        rs = sorted({(y, k) for y in args[x] for k, u in enumerate(uses[y]) if before(x, u) and incl(x, unode(u))})
+        if rs: left.append(x); adj[x] = rs
+    mate = hopcroft_karp(left, adj)
+    links = {x: r for x, r in mate.items() if r is not None}
+    R = c_add + q - len(links)
+    require((c_add,q,len(links),R)==(91770,8120,71185,28705),'selected PR117 graph')
+    print('PR117 graph matched', c_add, q, len(links), R, flush=True)
+    linked_use = {(y, k): x for x, (y, k) in links.items()}
+
+    # ------------------------------------------------------------ root targets, read coefficients, root functionals
+    target = [None] * q
+    for j in range(v): target[j] = j
+    idx = v
+    # PR117 witness stores pair-star roots in natural excluded-point order.
+    for a, b in combinations(range(h), 2):
+        others = [i for i in range(h) if i not in (a, b)]
+        for i in others: target[idx] = tid[tuple(sorted((a, b, i)))]; idx += 1
+    centre_of = {}
+    for j in range(q):
+        if kind[j]:
+            miss = FULL & ~cover[roots[j]]; require(bin(miss).count('1') == 1, 'centre cover')
+            centre_of[j] = miss.bit_length() - 1
+    require(idx + len(centre_of) == q, 'root order')
+    rootfun = [(1 << centre_of[j]) if kind[j] else tmask[target[j]] for j in range(q)]
+
+    # ------------------------------------------------------------ lifted binary frames, monotone
+    succ = defaultdict(set); droot = defaultdict(set)
+    for x in range(1, n):
+        if not active[x]: continue
+        for k, u in enumerate(uses[x]):
+            if (x, k) in linked_use: continue
+            (succ[x].add(u[1]) if u[0] == 'gate' else droot[x].add(u[1]))
+    for x, (y, k) in links.items():
+        u = uses[y][k]; (succ[x].add(u[1]) if u[0] == 'gate' else droot[x].add(u[1]))
+    order_desc = sorted((x for x in range(1, n) if active[x]), key=lambda x: (ranks[x], x), reverse=True)
+    K = {}
+    for x in order_desc:
+        rows = [rootfun[j] for j in droot[x]]
+        for t in succ[x]: rows += K[t]
+        K[x] = reduce(rows)
+    def envelope(x):
+        if not args[x][0]: return [tmask[x - 1]]
+        if types[x] == 2: return [1 << i for i in range(h) if cover[x] >> i & 1]
+        require(types[x] == 1, 'only ordinary and common-pair labels occur')
+        ks = [i for i in range(h) if (cover[x] & ~core[x]) >> i & 1]
+        require(len(ks) == ranks[x], 'common-pair support')
+        return [core[x] | (1 << k) for k in ks]
+    U = {}
+    for x in range(1, n):
+        if not active[x]: continue
+        if not args[x][0]: U[x] = envelope(x); continue
+        B = kernel(K[x], h)
+        require(contains(envelope(x), B), 'label outside lifted frame')
+        U[x] = B if nondeg(B) else envelope(x)
+    changed = True
+    while changed:
+        changed = False
+        for x in order_desc:
+            if args[x][0] and any(not contains(U[x], U[t]) for t in succ[x]):
+                env = envelope(x)
+                if len(reduce(U[x])) != len(reduce(env)) or not contains(U[x], env): U[x] = env; changed = True
+    dimU = {x: len(reduce(B)) for x, B in U.items()}
+    print('PR117 frames lifted',sum(dimU[x]>ranks[x] for x in U),flush=True)
+
+    # ------------------------------------------------------------ explicit role compile
+    order = sorted((x for x in range(1, n) if active[x]), key=lambda x: (ranks[x], x))
+    holds = []; first_node = []; ops = []; edge_key = {}; role_root = {}
+    def new_role(node): holds.append([node]); first_node.append(node); return len(holds) - 1
+    def serve(s, y, k):
+        u = uses[y][k]
+        if u[0] == 'gate': edge_key[(y, k)] = s
+        else: role_root[s] = u[1]
+    for x in order:
+        free = [k for k in range(len(uses[x])) if (x, k) not in linked_use]
+        if not args[x][0]:
+            for k in free:
+                s = new_role(x); ops.append(('src', s, x)); serve(s, x, k)
+            continue
+        a, b = args[x]
+        ka = next(k for k, u in enumerate(uses[a]) if u == ('gate', x, 0))
+        kb = next(k for k, u in enumerate(uses[b]) if u == ('gate', x, 1))
+        sa, sb = edge_key.pop((a, ka)), edge_key.pop((b, kb))
+        if x in links and links[x][0] == a: sa, sb = sb, sa
+        piv, oth = sa, sb
+        ops.append(('add', piv, oth, x)); holds[piv].append(x); holds[oth].append(x)
+        if x in links: serve(oth, *links[x])
+        require(free, 'every use linked at node %d' % x)
+        serve(piv, x, free[0])
+        for k in free[1:]:
+            f = new_role(x); ops.append(('copy', piv, f, x)); serve(f, x, k)
+    require(not edge_key and len(holds) == R, 'compile')
+    Rr = R
+
+    # ------------------------------------------------------------ B. the word computes every root value
+    rng = random.Random(20261008)
+    xs = [rng.randrange(P) for _ in range(v)]
+    val = [0] * n
+    for i in range(v): val[i + 1] = xs[i]
+    for x in order:
+        if args[x][0]: val[x] = (val[args[x][0]] + val[args[x][1]]) % P
+    leaf_of = {s: first_node[s] for s in range(Rr) if not args[first_node[s]][0]}
+    a_ = [0] * Rr
+    for s, leaf in leaf_of.items(): a_[s] = xs[leaf - 1]
+    for o in ops:
+        if o[0] == 'add': a_[o[1]] = (a_[o[1]] + a_[o[2]]) % P
+        elif o[0] == 'copy': a_[o[2]] = (a_[o[2]] + a_[o[1]]) % P
+    require(all(a_[s] == val[roots[j]] for s, j in role_root.items()), 'root values')
+
+    # ------------------------------------------------------------ phase one and garbage reach
+    def touch(o): return [o[1], o[2]] if o[0] in ('add', 'copy') else []
+    prev = {}; pred = defaultdict(list); last = {}
+    for i, o in enumerate(ops):
+        for s in touch(o):
+            if s in prev: pred[i].append(prev[s])
+            prev[s] = i; last[s] = i
+    centre_roles = [s for s, j in role_root.items() if kind[j]]
+    Anc = set(); st = [last[s] for s in centre_roles]
+    while st:
+        i = st.pop()
+        if i not in Anc: Anc.add(i); st.extend(pred[i])
+    touched = set(centre_roles)
+    for i in Anc: touched.update(touch(ops[i]))
+    require(all(last[s] in Anc for s in centre_roles), 'centres complete in phase one')
+    reach = [set() for _ in range(Rr)]; reach_all = [False] * Rr
+    for s, j in role_root.items():
+        if kind[j]: reach_all[s] = True
+        else: reach[s].add(target[j])
+    for o in reversed(ops):
+        if o[0] == 'add': reach[o[2]] |= reach[o[1]]; reach_all[o[2]] = reach_all[o[2]] or reach_all[o[1]]
+        elif o[0] == 'copy': reach[o[1]] |= reach[o[2]]; reach_all[o[1]] = reach_all[o[1]] or reach_all[o[2]]
+
+    # ------------------------------------------------------------ deferral frames, insertion nesting
+    def F0(s): return U[first_node[s]]
+    cand = {}
+    for s in range(Rr):
+        if s in touched or reach_all[s]: continue
+        S_ = restrict(F0(s), [tmask[t] for t in reach[s]])
+        if S_ and nondeg(S_): cand[s] = S_
+    print('PR117 candidates',len(cand),flush=True)
+    placed = saturated_placement(cand, reach)
+    print('PR117 placed',len(placed),flush=True)
+    deferred = sorted(placed, key=lambda s: (len(placed[s]), s)); dset = set(deferred)
+
+    # Optimize the actual mixer word. Source injection and root frames stay fixed.
+    # Floating objective values choose legal finite frames only; the complete
+    # child profile and its contraction are certified independently and exactly.
+    root_frame={s:([1<<i for i in range(h)if i!=centre_of[j]] if kind[j] else kernel([tmask[target[j]]],h)) for s,j in role_root.items()}
+    frame_U={x:sat_basis(B)for x,B in U.items()}
+    FULL=sat_basis(1<<i for i in range(h))
+    sigma={s:sat_basis(B)for s,B in placed.items()}
+    role_events=defaultdict(list);frames={};birth={};lifted=Counter()
+    for i,o in enumerate(ops):
+     if o[0]=='src':
+      frames[i]=frame_U[o[2]];role_events[o[1]].append(i)
+     else:
+      frames[i]=frame_U[o[3]]
+      for s in o[1:3]:role_events[s].append(i)
+    prev={};after={};end={s:sat_basis(root_frame[s])if s in root_frame else FULL for s in range(R)}
+    for s,events in role_events.items():
+     for j,i in enumerate(events):prev[i,s]=events[j-1]if j else None;after[i,s]=events[j+1]if j+1<len(events)else None
+    # Frozen IEEE-754 search weights remove platform libm variation.
+    vals=[float.fromhex(x) for x in ['0x0.0p+0', '0x1.0000000000000p+0', '0x1.fff611fabad2cp+0', '0x1.7ff4324fbceecp+1', '0x1.ffec2426c27bdp+1', '0x1.3ff197310b8c8p+2', '0x1.7fecc00663620p+2', '0x1.bfe79c1e23586p+2', '0x1.ffe2368416067p+2', '0x1.1fee4bbd449aep+3', '0x1.3feb62b54ad1bp+3', '0x1.5fe862b99e2f4p+3', '0x1.7fe54de202531p+3', '0x1.9fe225ec96cafp+3', '0x1.bfdeec529ef76p+3', '0x1.dfdba257523b3p+3', '0x1.ffd84912b47dep+3', '0x1.0fea70bcdaf4fp+4', '0x1.1fe8b63233d76p+4', '0x1.2fe6f54967551p+4', '0x1.3fe52e5858b9ap+4', '0x1.4fe361ac5190ep+4', '0x1.5fe18f8b3cb6dp+4', '0x1.6fdfb834a8020p+4', '0x1.7fdddbe2990a6p+4']]
+    for turn in range(5):
+     changes=0;delta=0
+     for i in (reversed(range(len(ops)))if turn%2==0 else range(len(ops))):
+      o=ops[i]
+      if o[0]=='src':continue
+      _,a,b,x=o;old=frames[i];ends=[frames[after[i,s]]if after[i,s]is not None else end[s]for s in (a,b)]
+      starts=[frames[prev[i,s]]if prev[i,s]is not None else sigma.get(s,())for s in (a,b)]
+      cap=sat_cap(*ends);lower=sat_basis(starts[0]+starts[1])
+      options=[frame_protected_part(cap,old),frame_smallest_hull(lower,old)]
+      best=old;bestdelta=-1e-10
+      for F in options:
+       assert sat_nondeg(F)and all(sat_contained(P,F)and sat_contained(F,N)for P,N in zip(starts,ends))
+       df=sum(vals[len(F)-len(P)]+vals[len(N)-len(F)]-vals[len(old)-len(P)]-vals[len(N)-len(old)]for P,N in zip(starts,ends))
+       if df<bestdelta:best=F;bestdelta=df
+      if best!=old:frames[i]=best;changes+=1;delta+=bestdelta
+     print('Physical frame descent',turn,changes,delta,flush=True)
+     if not changes:break
+    for i,o in enumerate(ops):
+     if o[0]=='src':birth[o[1]]=frames[i]
+     elif o[0]=='copy':birth[o[2]]=frames[i]
+     if o[0]!='src':lifted[len(frame_U[o[3]]),len(frames[i])]+=1
+    assert len(birth)==R
+    op_frames=frames
+    role_frames={s:[]for s in range(R)}
+    for i,o in enumerate(ops):
+        if o[0]=='src':role_frames[o[1]].append(op_frames[i])
+        else:
+            for s in o[1:3]:role_frames[s].append(op_frames[i])
+    def F0(s):return birth[s]
+    reuse_pairs = select_reuse(locals())
+    merge = check_pairs(locals(), reuse_pairs)
+    reuse_rejected_controls = mutation_controls(locals(), reuse_pairs)
+    live = [s for s in range(Rr) if s not in merge]
+    physical_id = {s: i for i, s in enumerate(live)}
+    def alias(s): return physical_id[merge.get(s, s)]
+    print('Compensated birth reuse', len(reuse_pairs), 'physical roles', len(live), flush=True)
+
+    # ------------------------------------------------------------ C. replay with arbitrary scratch and data
+    inv = lambda a: pow(a % P, P - 2, P); HALF = inv(2); I21 = inv(21)
+    cvec = [None] * Rr; dpart = [dict() for _ in range(Rr)]
+    for s, j in role_root.items():
+        if kind[j]: c = [0] * h; c[centre_of[j]] = 1; cvec[s] = c
+        else: dpart[s][target[j]] = (P - HALF) if j >= v else HALF
+    seed_c = {s: (None if cvec[s] is None else list(cvec[s])) for s in role_root}
+    seed_d = {s: dict(dpart[s]) for s in role_root}
+    def addc(dst, src):
+        if cvec[src] is not None:
+            cvec[dst] = list(cvec[src]) if cvec[dst] is None else [(p + q_) % P for p, q_ in zip(cvec[dst], cvec[src])]
+        for t, c in dpart[src].items(): dpart[dst][t] = (dpart[dst].get(t, 0) + c) % P
+    for o in reversed(ops):
+        if o[0] == 'add': addc(o[2], o[1])
+        elif o[0] == 'copy': addc(o[1], o[2])
+    scatter = [[(I21 - (HALF if i in trip[t] else 0)) % P for t in range(v)] for i in range(h)]
+    def readout(y, s, value, sign, seed=False):
+        # Logical readout macro: the literal audit combines its exact rational
+        # target coefficients and expands each numerator/42 into bounded shears.
+        cv = seed_c[s] if seed else cvec[s]; dp = seed_d[s] if seed else dpart[s]
+        if cv is not None:
+            for i, ci in enumerate(cv):
+                if ci:
+                    f = sign * ci * value % P; row = scatter[i]
+                    for t in range(v): y[t] = (y[t] + f * row[t]) % P
+        for t, c in dp.items(): y[t] = (y[t] + sign * c * value) % P
+    phase1 = sorted(Anc); rest = [i for i in range(len(ops)) if i not in Anc]
+    def replay(seed, omit_compensation=None):
+        rng = random.Random(seed)
+        x = [rng.randrange(P) for _ in range(v)]
+        z = [rng.randrange(P) for _ in live]
+        y0 = [rng.randrange(P) for _ in range(v)]; a = list(z); y = list(y0)
+        chronology = []
+        def inject(s, leaf):
+            dst = alias(s)
+            a[dst] = (a[dst] + x[leaf - 1]) % P
+            chronology.append(('src', dst, leaf - 1))
+        def run(i):
+            o = ops[i]
+            if o[0] == 'src': return
+            dst, src = (o[1], o[2]) if o[0] == 'add' else (o[2], o[1])
+            dst, src = alias(dst), alias(src)
+            require(dst != src, 'aliased gate ports')
+            a[dst] = (a[dst] + a[src]) % P
+            chronology.append(('work', dst, src))
+        for s in range(Rr):
+            if s not in dset: readout(y, s, a[alias(s)], -1)
+        for s, leaf in leaf_of.items():
+            if s not in dset: inject(s, leaf)
+        for i in phase1: run(i)
+        for s in centre_roles: readout(y, s, a[alias(s)], +1, True)
+        compensated = []
+        for s in deferred:
+            if s != omit_compensation:
+                readout(y, s, a[alias(s)], -1)
+                compensated.append(s)
+        if omit_compensation is None: check_compensation(merge, compensated)
+        for s in deferred:
+            if s in leaf_of: inject(s, leaf_of[s])
+        for i in rest: run(i)
+        for s, j in role_root.items():
+            if not kind[j]: readout(y, s, a[alias(s)], +1, True)
+        # Aliased births require reversal of the actual source/workspace
+        # chronology, including delayed source injections.
+        for kind_, dst, src in reversed(chronology):
+            value = x[src] if kind_ == 'src' else a[src]
+            a[dst] = (a[dst] - value) % P
+        return a == z, all((y[t] - y0[t] - x[t]) % P == 0 for t in range(v))
+    rep = [replay(seed) for seed in (1, 2)]
+    require(all(r == (True, True) for r in rep), 'aliased replay %s' % rep)
+    missing_compensation = replay(3, omit_compensation=reuse_pairs[0]['recipient'])
+    require(missing_compensation == (True, False), 'omitted compensation unexpectedly accepted')
+    compensated_births = sorted(merge)
+
+    # ------------------------------------------------------------ D. exact F2 frame facts
+    root_frame = {}
+    for s, j in role_root.items():
+        root_frame[s] = [1 << i for i in range(h) if i != centre_of[j]] if kind[j] else kernel([tmask[target[j]]], h)
+    FULLB = [1 << i for i in range(h)]
+    chain_dims = []
+    for s in range(Rr):
+        seq = [placed.get(s, [])] + role_frames[s]
+        if s in root_frame: seq.append(root_frame[s])
+        seq.append(FULLB)
+        require(all(contains(A, B) for A, B in zip(seq, seq[1:])), 'role chain nesting %d' % s)
+        require(all(nondeg(B) for B in seq[1:-1] if B), 'degenerate frame on role %d' % s)
+        chain_dims.append([len(reduce(B)) for B in seq])
+    for s, X in placed.items():
+        require(contains(X, F0(s)) and nondeg(X), 'deferral frame %d' % s)
+        require(all(not any(dot(xv, tmask[t]) for xv in X) for t in reach[s]), 'target frame %d' % s)
+    byT2 = defaultdict(list)
+    for s in deferred:
+        for t in reach[s]: byT2[t].append(s)
+    for t, ss in byT2.items():
+        ss.sort(key=lambda s: (len(placed[s]), s))
+        require(all(contains(placed[p], placed[q_]) for p, q_ in zip(ss, ss[1:])), 'target chain %d' % t)
+
+    # ------------------------------------------------------------ E. one-child histogram
+    z = Counter()
+    for s in range(Rr):
+        ds = chain_dims[s]
+        for a, b in zip(ds[:-1], ds[1:-1]):
+            if b > a: z[b - a] += 2 * v
+        lastd = ds[-2]
+        if s in role_root and kind[role_root[s]]: z[h - 1] += 2 * v           # copied centre transform
+        z[h - lastd] += 2 * v                                                  # final growth to F
+        z[m - h + ds[0]] += 2 * v                                              # exterior, gauged by sigma
+    levels = defaultdict(set)
+    for s in deferred:
+        for t in reach[s]: levels[t].add(len(placed[s]))
+    for t in range(v):
+        ds = sorted(levels[t] | {0, h - 1})
+        for a, b in zip(ds, ds[1:]): z[b - a] += 2 * v                       # target fronts
+    z[h - 1] += 2 * N                                                          # data-wire fronts
+    z[(h - 1) ** 2] += 2 * N                                                   # data macros
+    z[1] += N                                                                  # endpoint copies
+    z.pop(0, None)
+    virtual_R = R
+    for pair in reuse_pairs:
+        e, f = pair['e'], pair['s']
+        z[h - e] -= 2 * v
+        z[m - h + f] -= 2 * v
+        if f > e: z[f - e] += 2 * v
+    require(all(count >= 0 for count in z.values()), 'reuse histogram subtraction')
+    z = Counter({width: count for width, count in z.items() if count})
+    physical_R = len(live)
+    # Canonical inventory of actual physical source gauges. Recipients have
+    # disappeared; every reused donor remains a separate zero-gauge slot.
+    source_counts = Counter(sat_basis(placed.get(s, ())) for s in live)
+    physical_auxiliary_source_frames = [dict(basis=list(F), count=count)
+        for F, count in sorted(source_counts.items(), key=lambda item: (len(item[0]), item[0]))]
+    require(sum(source_counts.values()) == physical_R, 'physical source inventory')
+    require(all(not placed.get(a) for a in merge.values()), 'reused donor source gauge')
+    W = 2 * N + 2 * v * physical_R; L = 2 * v * h * (h - 1); s_ = W * m - N + L
+    require(sum(t * c for t, c in z.items()) == s_, 'complex rank mass')
+    require(all(0 < t < m for t in z), 'children below m')
+    out = dict(h=h, v=v, additions=c_add, roots=q, links=len(links), R=physical_R, virtual_R=virtual_R, reused_roles=len(reuse_pairs), m=m, N=N, W=W, L=L, total_rank=s_,
+               deficit=N - L, maxchild=max(z), phase_one_ops=len(Anc), phase_one_roles=len(touched),
+               physical_auxiliary_source_frames=physical_auxiliary_source_frames,
+               deferred_roles=len(deferred),
+               deferred_dims=dict(sorted(Counter(len(X) for X in placed.values()).items())),
+               lifted_additions=sum(1 for i,o in enumerate(ops) if o[0]=='add' and len(op_frames[i])>ranks[o[3]]),
+               physical_gate_frame_changes=sum(op_frames[i]!=frame_U[o[3]] for i,o in enumerate(ops) if o[0]!='src'),
+               replay=dict(seeds=[1, 2], scratch_restored=True, y_plus_x=True, field='Z/(2^61-1)',
+                           physically_aliased=True, inverse_order='true source/workspace chronology',
+                           omitted_compensation_rejected=True),
+               reuse_rejected_controls=reuse_rejected_controls,
+               child_multiplicities=dict(sorted(z.items())))
+    (HERE / 'complex-profile.json').write_text(json.dumps(out, indent=1) + '\n')
+    (HERE / 'reuse-pairs.json').write_text(json.dumps(reuse_pairs, indent=1) + '\n')
+    print('PASS complex: R=%d, deferred roles %d, maxchild %d, rank mass %d' % (physical_R, len(deferred), max(z), s_))
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/cyclic-deferred/inputs/complex-dag.json.gz b/research/cyclic-deferred/inputs/complex-dag.json.gz
new file mode 100644
index 0000000000..a5b0598e4c
Binary files /dev/null and b/research/cyclic-deferred/inputs/complex-dag.json.gz differ
diff --git a/research/cyclic-deferred/producer.py b/research/cyclic-deferred/producer.py
new file mode 100644
index 0000000000..a93a265f01
--- /dev/null
+++ b/research/cyclic-deferred/producer.py
@@ -0,0 +1,17 @@
+"""Pinned PR117 scalar DAG adapter for the saturated deferred compiler.
+
+The searched graph is upstream PR117 work credited to eumemic with Anthropic
+Claude assistance. This adapter preserves that witness and producer unchanged;
+saturation integration is separate OpenAI Codex-assisted work. Apache-2.0.
+"""
+from hashlib import sha256
+from pathlib import Path
+from replayed_producer import build as replayed_build
+
+WITNESS_SHA256 = '3c034d0aae388ef567a454826f4f48b26fd8a94c71e8ffed4835271b349a783b'
+
+def build(h, prefix, central_disjoint, base=2):
+    assert (h, central_disjoint, base) == (24,24,2)
+    witness=Path(__file__).resolve().parent/'inputs/complex-dag.json.gz'
+    assert sha256(witness.read_bytes()).hexdigest() == WITNESS_SHA256, 'PR117 DAG pin mismatch'
+    return replayed_build(witness,prefix)
diff --git a/research/cyclic-deferred/reflection-audit.json b/research/cyclic-deferred/reflection-audit.json
new file mode 100644
index 0000000000..8b40ca11a6
--- /dev/null
+++ b/research/cyclic-deferred/reflection-audit.json
@@ -0,0 +1,10524 @@
+{
+  "R": 26597,
+  "all_frames_nondegenerate": true,
+  "audit_sha256": "9615b651ed42b7687ce4f5cc242cccdfa9abd638666bd2fc7ee1f56a488e76d7",
+  "bounded_chunk_coefficients": true,
+  "bounded_readout_chunks_sha256": "bb2ad510e5f5113eba2fd520f99bb05725087ae47e1566b7e63eafdefc69d809",
+  "child_multiplicities": {
+    "1": 408795376,
+    "10": 21118416,
+    "11": 4938560,
+    "12": 20826960,
+    "13": 17786912,
+    "14": 1922800,
+    "15": 2493568,
+    "16": 684112,
+    "17": 489808,
+    "18": 546480,
+    "19": 477664,
+    "2": 291621968,
+    "20": 550528,
+    "21": 1230592,
+    "22": 64768,
+    "23": 8363168,
+    "3": 52656384,
+    "4": 51041232,
+    "5": 21498928,
+    "529": 8193152,
+    "552": 97147952,
+    "554": 3493424,
+    "556": 1437040,
+    "558": 2319504,
+    "560": 453376,
+    "562": 48576,
+    "570": 821744,
+    "571": 817696,
+    "572": 825792,
+    "573": 44528,
+    "574": 255024,
+    "6": 17507600,
+    "7": 9229440,
+    "8": 12633808,
+    "9": 7869312
+  },
+  "completed_core_pre_exterior_frames_full": true,
+  "completed_core_source_inventory_bound": true,
+  "conservative_local_G": 14907958248,
+  "dirty_reuse_identity": "Every recipient is untouched before its compensated birth; donor has no future gate/root reads. Recipient old-response coefficient cancels its arbitrary current donor value. All physical shears are reversed in true chronology.",
+  "exact_arbitrary_dirty_cancellation_by_dependency_cut": true,
+  "exact_birth_cut_invariants": true,
+  "exact_chunk_reconstruction": true,
+  "exact_fresh_source_map": true,
+  "exact_integer_old_readout_transpose": true,
+  "expanded_old_readout_additions": 11862944,
+  "expanded_scalar_operations_per_stage": 12160614,
+  "forward_incidence_sha256": "516b499ebc23f69c8075a625181d67ac3b398479365e84e690b3abbdb8d89b12",
+  "h": 24,
+  "inverse_source_order": "True reverse chronological source/workspace word",
+  "largest_bounded_chunk_numerator_over_42": 42,
+  "largest_readout_numerator_over_42": 55,
+  "last_uses_recomputed": true,
+  "literal_frame_incidences_both_directions": true,
+  "literal_global_scalar_groups": 49230262048,
+  "old_readout_coefficients_over_42_sha256": "e0666f81e1d35e002d61b8f2b2b1ca0d1d79cc7fbcbc76063eb333294d875e5e",
+  "omitted_compensation_numeric_control": true,
+  "physical_aliased_numeric_replay": true,
+  "physical_auxiliary_source_frames": [
+    {
+      "basis": [],
+      "count": 23999
+    },
+    {
+      "basis": [
+        9,
+        5
+      ],
+      "count": 134
+    },
+    {
+      "basis": [
+        36,
+        20
+      ],
+      "count": 2
+    },
+    {
+      "basis": [
+        144,
+        80
+      ],
+      "count": 88
+    },
+    {
+      "basis": [
+        2304,
+        1280
+      ],
+      "count": 23
+    },
+    {
+      "basis": [
+        36864,
+        20480
+      ],
+      "count": 16
+    },
+    {
+      "basis": [
+        589824,
+        327680
+      ],
+      "count": 98
+    },
+    {
+      "basis": [
+        9437184,
+        5242880
+      ],
+      "count": 502
+    },
+    {
+      "basis": [
+        8,
+        4,
+        2,
+        1
+      ],
+      "count": 4
+    },
+    {
+      "basis": [
+        33,
+        17,
+        9,
+        5
+      ],
+      "count": 72
+    },
+    {
+      "basis": [
+        129,
+        65,
+        33,
+        17
+      ],
+      "count": 6
+    },
+    {
+      "basis": [
+        132,
+        68,
+        36,
+        20
+      ],
+      "count": 8
+    },
+    {
+      "basis": [
+        513,
+        257,
+        9,
+        5
+      ],
+      "count": 28
+    },
+    {
+      "basis": [
+        528,
+        272,
+        144,
+        80
+      ],
+      "count": 42
+    },
+    {
+      "basis": [
+        2064,
+        1040,
+        528,
+        272
+      ],
+      "count": 4
+    },
+    {
+      "basis": [
+        33792,
+        17408,
+        9216,
+        5120
+      ],
+      "count": 10
+    },
+    {
+      "basis": [
+        524289,
+        262145,
+        131073,
+        65537
+      ],
+      "count": 22
+    },
+    {
+      "basis": [
+        525312,
+        263168,
+        132096,
+        66560
+      ],
+      "count": 8
+    },
+    {
+      "basis": [
+        528384,
+        266240,
+        135168,
+        69632
+      ],
+      "count": 6
+    },
+    {
+      "basis": [
+        540672,
+        278528,
+        147456,
+        81920
+      ],
+      "count": 1
+    },
+    {
+      "basis": [
+        8388609,
+        4194305,
+        2097153,
+        1048577
+      ],
+      "count": 68
+    },
+    {
+      "basis": [
+        8388624,
+        4194320,
+        144,
+        80
+      ],
+      "count": 2
+    },
+    {
+      "basis": [
+        8388624,
+        4194320,
+        2097168,
+        1048592
+      ],
+      "count": 43
+    },
+    {
+      "basis": [
+        8389632,
+        4195328,
+        2098176,
+        1049600
+      ],
+      "count": 9
+    },
+    {
+      "basis": [
+        8404992,
+        4210688,
+        2113536,
+        1064960
+      ],
+      "count": 1
+    },
+    {
+      "basis": [
+        8454144,
+        4259840,
+        589824,
+        327680
+      ],
+      "count": 4
+    },
+    {
+      "basis": [
+        8454144,
+        4259840,
+        2162688,
+        1114112
+      ],
+      "count": 17
+    },
+    {
+      "basis": [
+        129,
+        65,
+        33,
+        17,
+        9,
+        5
+      ],
+      "count": 127
+    },
+    {
+      "basis": [
+        513,
+        257,
+        33,
+        17,
+        9,
+        5
+      ],
+      "count": 2
+    },
+    {
+      "basis": [
+        2049,
+        1025,
+        513,
+        257,
+        9,
+        5
+      ],
+      "count": 8
+    },
+    {
+      "basis": [
+        2049,
+        1025,
+        513,
+        257,
+        129,
+        65
+      ],
+      "count": 3
+    },
+    {
+      "basis": [
+        2064,
+        1040,
+        528,
+        272,
+        144,
+        80
+      ],
+      "count": 7
+    },
+    {
+      "basis": [
+        524289,
+        262145,
+        131073,
+        65537,
+        9,
+        5
+      ],
+      "count": 1
+    },
+    {
+      "basis": [
+        525312,
+        263168,
+        132096,
+        66560,
+        33792,
+        17408
+      ],
+      "count": 4
+    },
+    {
+      "basis": [
+        528384,
+        266240,
+        135168,
+        69632,
+        36864,
+        20480
+      ],
+      "count": 40
+    },
+    {
+      "basis": [
+        2098176,
+        1049600,
+        33792,
+        17408,
+        9216,
+        5120
+      ],
+      "count": 2
+    },
+    {
+      "basis": [
+        2098176,
+        1049600,
+        525312,
+        263168,
+        132096,
+        66560
+      ],
+      "count": 2
+    },
+    {
+      "basis": [
+        8388609,
+        4194305,
+        33,
+        17,
+        9,
+        5
+      ],
+      "count": 6
+    },
+    {
+      "basis": [
+        8388609,
+        4194305,
+        129,
+        65,
+        9,
+        5
+      ],
+      "count": 3
+    },
+    {
+      "basis": [
+        8388609,
+        4194305,
+        2097153,
+        1048577,
+        9,
+        5
+      ],
+      "count": 81
+    },
+    {
+      "basis": [
+        8388609,
+        4194305,
+        2097153,
+        1048577,
+        33,
+        17
+      ],
+      "count": 3
+    },
+    {
+      "basis": [
+        8388609,
+        4194305,
+        2097153,
+        1048577,
+        129,
+        65
+      ],
+      "count": 2
+    },
+    {
+      "basis": [
+        8388609,
+        4194305,
+        2097153,
+        1048577,
+        131073,
+        65537
+      ],
+      "count": 2
+    },
+    {
+      "basis": [
+        8388624,
+        4194320,
+        2097168,
+        1048592,
+        144,
+        80
+      ],
+      "count": 4
+    },
+    {
+      "basis": [
+        8389632,
+        4195328,
+        33792,
+        17408,
+        9216,
+        5120
+      ],
+      "count": 3
+    },
+    {
+      "basis": [
+        8389632,
+        4195328,
+        2098176,
+        1049600,
+        525312,
+        263168
+      ],
+      "count": 1
+    },
+    {
+      "basis": [
+        8392704,
+        4198400,
+        2101248,
+        1052672,
+        36864,
+        20480
+      ],
+      "count": 33
+    },
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+        787456,
+        131072,
+        65536,
+        32768,
+        16384,
+        8192,
+        4096,
+        2048,
+        512,
+        256,
+        128,
+        64,
+        32,
+        16,
+        8,
+        4,
+        2,
+        1
+      ],
+      "count": 1
+    },
+    {
+      "basis": [
+        8400896,
+        4194304,
+        2097152,
+        1048576,
+        524288,
+        262144,
+        131072,
+        65536,
+        32768,
+        28672,
+        2048,
+        1024,
+        512,
+        256,
+        128,
+        64,
+        32,
+        16,
+        8,
+        4,
+        2,
+        1
+      ],
+      "count": 1
+    },
+    {
+      "basis": [
+        8404992,
+        4194304,
+        2097152,
+        1048576,
+        802816,
+        131072,
+        65536,
+        32768,
+        8192,
+        4096,
+        2048,
+        1024,
+        512,
+        256,
+        128,
+        64,
+        32,
+        16,
+        8,
+        4,
+        2,
+        1
+      ],
+      "count": 1
+    },
+    {
+      "basis": [
+        8585216,
+        4194304,
+        2097152,
+        1048576,
+        524288,
+        458752,
+        32768,
+        16384,
+        8192,
+        4096,
+        2048,
+        1024,
+        512,
+        256,
+        128,
+        64,
+        32,
+        16,
+        8,
+        4,
+        2,
+        1
+      ],
+      "count": 1
+    }
+  ],
+  "reflected_core_active_frames_complement_source": true,
+  "reflected_incidence_sha256": "71725b7c7c77687b0d6cbbda138e62470a0db48ac99516b4857eecf6910425b3",
+  "reflected_residual_rank_histogram_equal": true,
+  "reflected_scalar_map": "X becomes X-Y; Y unchanged; all auxiliary coordinates restored.",
+  "reflected_word_rule": "Reverse literal order including each bounded readout chunk, negate every shear, swap X/Y ports, complement every frame; auxiliary bank is separate.",
+  "rejected_controls": [
+    "nondead-donor",
+    "unavailable-birth",
+    "pair-alias",
+    "omitted-compensation",
+    "oversized-readout-chunk",
+    "split-readout-sum",
+    "reflected-chunk-order",
+    "inverse-sign",
+    "uncomplemented-frame",
+    "signed-root-readout"
+  ],
+  "reused_roles": 2108,
+  "scalar_readout_normalization": "Each exact numerator n/42 becomes signed42 chunks plus a signed remainder. Exact reconstruction and chunk magnitude<=1 checked; every same-frame shear is charged; reflection reverses chunk order.",
+  "scope": "Exact finite source coefficients, dirty cancellation algebra and literal reflected frame/charge ledger. Copied-centre transforms, endpoint correction and residual-to-child/all-size transfer retain their stated inherited contracts.",
+  "source_sha256": "6c0e4439ba523c8ad2d8043760cf2d0146a678e2b870b73d0d14bb543a644596",
+  "split_readout_macros": 32,
+  "two_stage_scalar_map": "(X,Y) becomes (-Y,X+Y) before inherited endpoint correction.",
+  "unsplit_old_readout_macros": 11862912,
+  "v": 2024,
+  "virtual_R": 28705,
+  "word_blocks": 277775
+}
diff --git a/research/cyclic-deferred/reflection_audit.py b/research/cyclic-deferred/reflection_audit.py
new file mode 100644
index 0000000000..540bd4db4e
--- /dev/null
+++ b/research/cyclic-deferred/reflection_audit.py
@@ -0,0 +1,411 @@
+#!/usr/bin/env python3
+"""Compensated reuse: literal inverse/reflection and exact finite scalar/frame audit.
+
+Birth-cut reuse follows jamesyc PR124; original contributors retained.
+Composition with saturated and physical frames prepared with OpenAI Codex.
+
+Captures an unchanged deferred producer before its output write, then checks
+the scalar computation on every source symbol, the exact signed old-readout
+transpose, the dependency cut and the complete complemented incidence ledger.
+This is finite verification; the inherited residual-to-child implementation,
+all-size transfer and analytic interfaces are not proved by this script.
+"""
+import ast
+from collections import Counter
+from functools import lru_cache
+from hashlib import sha256
+from itertools import combinations
+import argparse,json,pickle,struct,sys
+from pathlib import Path
+from reuse import check_pairs, check_compensation, mutation_controls
+sys.dont_write_bytecode=True
+assert not sys.flags.optimize,'Assertions must remain enabled'
+
+
+def capture(source):
+    source=Path(source).resolve();sys.path.insert(0,str(source.parent))
+    tree=ast.parse(source.read_text(),filename=str(source))
+    main=next(n for n in tree.body if isinstance(n,ast.FunctionDef) and n.name=='main')
+    at=next(i for i,n in enumerate(main.body) if isinstance(n,ast.Expr) and isinstance(n.value,ast.Call) and isinstance(n.value.func,ast.Attribute) and n.value.func.attr=='write_text')
+    main.body[at:]=[ast.Return(ast.Call(ast.Name('locals',ast.Load()),[],[]))]
+    ast.fix_missing_locations(tree)
+    ns={'__file__':str(source),'__name__':'reflection_capture'}
+    exec(compile(tree,str(source)+' [read-only capture]','exec'),ns)
+    data=ns['main']();data['source']=str(source);return data
+
+
+def basis(vectors):
+    piv={}
+    for x in vectors:
+        for p in sorted(piv,reverse=True):
+            if x>>p&1:x^=piv[p]
+        if x:
+            p=x.bit_length()-1
+            for q in piv:
+                if piv[q]>>p&1:piv[q]^=x
+            piv[p]=x
+    return tuple(piv[p] for p in sorted(piv,reverse=True))
+
+
+def dot(a,b):return (a&b).bit_count()&1
+
+
+@lru_cache(None)
+def contained(A,B):return len(basis(A+B))==len(B)
+
+
+@lru_cache(None)
+def complement(A,h):
+    A=basis(A);piv={x.bit_length()-1:x for x in A};result=[]
+    for j in range(h):
+        if j not in piv:
+            x=1<<j
+            for p,row in piv.items():
+                if row>>j&1:x|=1<<p
+            result.append(x)
+    return basis(result)
+
+
+@lru_cache(None)
+def nondeg(A):
+    return len(basis(sum(dot(x,y)<<j for j,y in enumerate(A)) for x in A))==len(A)
+
+
+def root_readout(d,s):
+    h,v,j=d['h'],d['v'],d['role_root'][s];p=(1<<61)-1
+    if d['kind'][j]:
+        assert d['seed_c'][s]==[int(i==d['centre_of'][j]) for i in range(h)],'Root centre readout'
+        assert d['seed_d'][s]=={},'Root centre extra target'
+    else:
+        assert d['seed_c'][s] is None,'Root extra centre'
+        expected=pow(2,-1,p)*(1 if j<v else -1)%p
+        assert d['seed_d'][s]=={d['target'][j]:expected},'Root signed readout'
+
+
+@lru_cache(None)
+def dual_frames(A,B,h):
+    return len(A)+len(B)==h and all(not dot(x,y) for x in A for y in B)
+
+
+def check_chunks(numerator,parts):
+    assert sum(parts)==numerator,'Split readout reconstruction'
+    assert all(0<abs(x)<=42 for x in parts),'Bounded readout chunk'
+
+def bounded_chunks(numerator):
+    q,r=divmod(abs(numerator),42);sign=1 if numerator>=0 else -1
+    parts=(sign*42,)*q+((sign*r,) if r else ())
+    check_chunks(numerator,parts)
+    return parts
+
+def reverse_row(row):
+    if row[0]=='old':return (row[0],row[1],row[3],row[2])
+    return row
+
+
+def check_reflection(forward,backward,h,v):
+    def bank(s):return s+v if s<v else s-v if s<2*v else s
+    assert len(forward)==len(backward),'Reflected event count'
+    for left,right in zip(reversed(forward),backward):
+        k,a,b,c,F,*tail=left;kk,aa,bb,cc,FF,*tt=right
+        assert kk==k and aa==bank(a) and cc==-c,'Reflected inverse shear'
+        assert bb==(bank(b) if k=='gate' else tuple(bank(t) for t in reversed(b))),'Reflected port incidence'
+        assert dual_frames(F,FF,h),'Reflected frame complement'
+        if k=='centre':
+            assert dual_frames(tail[0],tt[0],h) and tail[1:]==tt[1:],'Reflected centre copy'
+        elif k=='read':assert tt==[reverse_row(tail[0])],'Reflected bounded coefficient-row binding'
+        else:assert tail==tt,'Reflected coefficient-row binding'
+
+
+def audit(d):
+    h,v,R=d['h'],d['v'],d['R'];ops=d['ops'];q=d['q'];p=(1<<61)-1
+    trip=list(combinations(range(h),3));assert d['trip']==trip
+    for s in d['role_root']:root_readout(d,s)
+    full=(1<<v)-1;point=[sum(1<<j for j,T in enumerate(trip) if i in T) for i in range(h)]
+    # All fresh-source coefficients, exactly over the integers. No modular
+    # replay is used for this check, and no stored DAG support is trusted.
+    nodes={};values=[0]*R
+    for n in d['order']:
+        a,b=d['args'][n]
+        if not a:nodes[n]=1<<(n-1)
+        else:
+            assert not nodes[a]&nodes[b],('DAG overlap',n)
+            nodes[n]=nodes[a]|nodes[b]
+    for s,n in d['leaf_of'].items():values[s]=1<<(n-1)
+    for o in ops:
+        if o[0]=='src':continue
+        a,b=(o[1],o[2]) if o[0]=='add' else (o[2],o[1])
+        assert not values[a]&values[b],('literal signal overlap',o)
+        values[a]|=values[b]
+        assert values[a]==nodes[o[3]],('literal source coefficients',o)
+    negative=[0]*v;positive=[None]*v;centres={}
+    for s,j in d['role_root'].items():
+        assert values[s]==nodes[d['roots'][j]]
+        if d['kind'][j]:centres[d['centre_of'][j]]=values[s]
+        elif j<v:
+            t=d['target'][j];assert positive[t] is None;positive[t]=values[s]
+        else:
+            t=d['target'][j];assert not negative[t]&values[s];negative[t]|=values[s]
+    assert set(centres)==set(range(h))
+    for i,S in centres.items():assert S==full^point[i]
+    for t,T in enumerate(trip):
+        A,B,C=(point[i] for i in T)
+        assert positive[t]==full^(A|B|C)
+        assert negative[t]==((A&B&~C)|(A&C&~B)|(B&C&~A))
+    assert h-3==21
+    assert [((k-1)+(k==0)-(k==2)) for k in range(4)]==[0,0,0,2]
+    # Every centre decoder coefficient is 1/21-[i in T]/2. Thus the
+    # preceding source-support identities prove J L V = I over Q.
+
+    # Independently form J L on arbitrary scratch as integer centre rows
+    # and signed half-output rows. Compare every coefficient, not hashes.
+    cc=[None]*R;dd=[{} for _ in range(R)]
+    for s,j in d['role_root'].items():
+        if d['kind'][j]:cc[s]=[int(i==d['centre_of'][j]) for i in range(h)]
+        else:dd[s][d['target'][j]]=1 if j<v else -1
+        assert cc[s]==d['seed_c'][s],'Root centre readout'
+        assert {t:(x*pow(2,-1,p))%p for t,x in dd[s].items()}==d['seed_d'][s],'Root signed readout'
+    def addrow(a,b):
+        if cc[b] is not None:
+            cc[a]=list(cc[b]) if cc[a] is None else [x+y for x,y in zip(cc[a],cc[b])]
+        for t,x in dd[b].items():dd[a][t]=dd[a].get(t,0)+x
+    for o in reversed(ops):
+        if o[0]=='add':addrow(o[2],o[1])
+        elif o[0]=='copy':addrow(o[1],o[2])
+    maxcoef=0;readcount=0;unsplit_readcount=0;split_macros=0;actual_reach={};chunk_rows={};coefficient_digest=sha256();chunk_digest=sha256();maxchunk=0
+    for s in range(R):
+        assert cc[s] is None and d['cvec'][s] is None or cc[s] is not None and [x%p for x in cc[s]]==d['cvec'][s]
+        assert {t:(x*pow(2,-1,p))%p for t,x in dd[s].items()}==d['dpart'][s]
+        # Exact aggregate coefficients are logical macros. Compile each
+        # numerator/42 into signed unit chunks and one signed remainder. All
+        # chunks share the same ports/frame and their exact sum is checked.
+        if cc[s] is None:
+            numerators=[(t,21*x) for t,x in sorted(dd[s].items()) if x]
+            coefficient_digest.update(repr((s,numerators)).encode()+b'\n')
+        else:
+            total=2*sum(cc[s]);numerators=[]
+            for t,T in enumerate(trip):
+                num=total-21*sum(cc[s][i] for i in T)+21*dd[s].get(t,0)
+                if num:
+                    numerators.append((t,num))
+                    coefficient_digest.update(s.to_bytes(4,'little')+t.to_bytes(4,'little')+num.to_bytes(8,'little',signed=True))
+        reached=[];records=[]
+        for t,num in numerators:
+            maxcoef=max(maxcoef,abs(num));parts=bounded_chunks(num)
+            unsplit_readcount+=1;split_macros+=int(len(parts)>1)
+            readcount+=len(parts);maxchunk=max(maxchunk,max(map(abs,parts)))
+            for part in parts:
+                reached.append(t);records.append(struct.pack('<Ih',t,part))
+        forward_bytes=b''.join(records);backward_bytes=b''.join(reversed(records))
+        chunk_rows[s]=(sha256(forward_bytes).hexdigest(),sha256(backward_bytes).hexdigest())
+        chunk_digest.update(s.to_bytes(4,'little')+forward_bytes)
+        actual_reach[s]=tuple(reached)
+    assert maxchunk<=42,'Expanded scalar chunk exceeds one in absolute value'
+    # Early/remainder order is a legal commutation of independent shears.
+    # Deferred inputs are untouched by the early word; completed centre
+    # roles are untouched by its remainder. These facts make deferred
+    # readout placement equivalent to the canonical -JL, V, L, J,L^-1,-V.
+    early=set(d['phase1']);assert early==set(d['Anc'])
+    assert sorted(d['phase1'])==d['phase1'] and d['rest']==[i for i in range(len(ops)) if i not in early]
+    actual_last={}
+    for i,o in enumerate(ops):
+        if o[0] in ('add','copy'):
+            for role in o[1:3]:actual_last[role]=i
+    assert actual_last==d['last'],'Exact recomputed last-use map'
+    seen_late=set();early_touched=set();late_touched=set()
+    for i,o in enumerate(ops):
+        touched=set(o[1:3]) if o[0] in ('add','copy') else set()
+        if i in early:
+            assert not touched&seen_late;early_touched.update(touched)
+        else:seen_late.update(touched);late_touched.update(touched)
+    assert not set(d['deferred'])&early_touched
+    assert not set(d['centre_roles'])&late_touched
+    assert all(cc[s] is None for s in d['deferred'])
+    for s in d['deferred']:assert set(actual_reach[s])<=set(d['reach'][s])
+
+    # Frame-labelled literal chronological word. Readout blocks mean the
+    # explicit commuting additions to their individual named target ports;
+    # their scalar count and target incidences are expanded above.
+    ZERO=();FULL=basis(1<<i for i in range(h))
+    U={n:basis(B) for n,B in d['U'].items()}
+    sigma={s:basis(B) for s,B in d['placed'].items()}
+    rootframe={s:basis(B) for s,B in d['root_frame'].items()}
+    tm=[basis((sum(1<<i for i in T),)) for T in trip]
+    merge=check_pairs(d,d['reuse_pairs'])
+    reuse_rejected=mutation_controls(d,d['reuse_pairs'])
+    assert d['compensated_births']==sorted(merge),'Compensated birth inventory'
+    assert d['out']['reuse_rejected_controls']==reuse_rejected
+    assert all(result==(True,True) for result in d['rep']),'Physical aliased numeric replay'
+    assert d['missing_compensation']==(True,False),'Omitted compensation numerical control'
+    live=[s for s in range(R)if s not in merge]
+    assert len(set(merge.values()))==len(merge) and not set(merge)&set(merge.values())
+    for b,a in merge.items():
+        assert a not in sigma and b in sigma and b not in d['leaf_of']
+        assert a not in d['role_root'] and d['last'][a] in early
+        assert b not in early_touched
+        assert all(a not in ops[i][1:3] for i in d['rest'] if ops[i][0]!='src')
+        lastframe=basis(d['op_frames'][d['last'][a]]) if 'op_frames'in d else U[d['holds'][a][-1]]
+        assert contained(lastframe,sigma[b])
+    physical_id={s:i for i,s in enumerate(live)}
+    X=lambda t:t;Y=lambda t:v+t;A=lambda s:2*v+physical_id[merge.get(s,s)]
+    word=[]
+    forward_workspace=[]
+    def gate(a,b,c,F):
+        assert a!=b
+        word.append(('gate',a,b,c,F))
+        if c==1:forward_workspace.append((a,b,c,F))
+    def read(s,sgn,F,targets,row):word.append(('read',A(s),tuple(Y(t) for t in targets),sgn,F,row))
+    def run(i,sign=1,F=None):
+        o=ops[i]
+        if o[0]=='add':gate(A(o[1]),A(o[2]),sign,(basis(d['op_frames'][i]) if 'op_frames' in d else U[o[3]]) if F is None else F)
+        elif o[0]=='copy':gate(A(o[2]),A(o[1]),sign,(basis(d['op_frames'][i]) if 'op_frames' in d else U[o[3]]) if F is None else F)
+    deferred=set(d['deferred'])
+    for s in range(R):
+        if s not in deferred:read(s,-1,ZERO,actual_reach[s],('old',s,*chunk_rows[s]))
+    for s,n in d['leaf_of'].items():
+        if s not in deferred:gate(A(s),X(n-1),1,U[n])
+    for i in d['phase1']:run(i)
+    for s in d['centre_roles']:word.append(('centre',A(s),tuple(Y(t) for t in range(v)),1,rootframe[s],ZERO,d['centre_of'][d['role_root'][s]]))
+    compensated=[]
+    for s in d['deferred']:
+        read(s,-1,sigma[s],actual_reach[s],('old',s,*chunk_rows[s]))
+        compensated.append(s)
+    check_compensation(merge,compensated)
+    for s in d['deferred']:
+        if s in d['leaf_of']:n=d['leaf_of'][s];gate(A(s),X(n-1),1,U[n])
+    for i in d['rest']:run(i)
+    for s,j in d['role_root'].items():
+        if not d['kind'][j]:read(s,1,rootframe[s],(d['target'][j],),('root',21 if j<v else -21))
+    for a,b,c,F in reversed(forward_workspace):gate(a,b,-c,FULL)
+
+    def swap(s):return s+v if s<v else s-v if s<2*v else s
+    def reflect(event):
+        k,a,b,c,F,*rest=event
+        if k=='gate':return (k,swap(a),swap(b),-c,complement(F,h))
+        tail=(complement(rest[0],h),rest[1]) if k=='centre' else (reverse_row(rest[0]),)
+        return (k,swap(a),tuple(swap(t) for t in reversed(b)),-c,complement(F,h),*tail)
+    reverse=[reflect(e) for e in reversed(word)]
+    assert [reflect(e) for e in reversed(reverse)]==word
+    check_reflection(word,reverse,h,v)
+    # Reuse this exact captured word for targeted failure controls.
+    rejected=list(reuse_rejected)
+    for failure,num,parts in (('oversized-readout-chunk',55,(55,)),('split-readout-sum',55,(42,12))):
+        try:check_chunks(num,parts)
+        except AssertionError:rejected.append(failure)
+        else:raise AssertionError('Accepted bounded readout mutation: '+failure)
+    damaged=list(reverse)
+    i=next(i for i,e in enumerate(damaged) if e[0]=='read' and e[-1][0]=='old' and e[-1][2]!=e[-1][3])
+    event=list(damaged[i]);event[-1]=reverse_row(event[-1]);damaged[i]=tuple(event)
+    try:check_reflection(word,damaged,h,v)
+    except AssertionError:rejected.append('reflected-chunk-order')
+    else:raise AssertionError('Accepted reflected chunk-order mutation')
+    for failure in ('inverse-sign','uncomplemented-frame'):
+        damaged=list(reverse);first=list(damaged[0])
+        if failure=='inverse-sign':first[3]*=-1
+        else:first[4]=word[-1][4]
+        damaged[0]=tuple(first)
+        try:check_reflection(word,damaged,h,v)
+        except AssertionError:rejected.append(failure)
+        else:raise AssertionError('Accepted reflection mutation: '+failure)
+    s=next(s for s,j in d['role_root'].items() if not d['kind'][j]);original=d['seed_d'][s]
+    d['seed_d'][s]={t:(-c)%p for t,c in original.items()}
+    try:
+        try:root_readout(d,s)
+        except AssertionError:rejected.append('signed-root-readout')
+        else:raise AssertionError('Accepted signed root mutation')
+    finally:d['seed_d'][s]=original
+
+    def framescan(events,initial,expected):
+        current=list(initial);hist=Counter();copies=Counter();scalar=0;digest=sha256()
+        def promote(s,F):
+            old=current[s];assert contained(old,F),(s,old,F)
+            assert nondeg(F);assert nondeg(complement(F,h))
+            if len(F)>len(old):hist[len(F)-len(old)]+=1
+            current[s]=F
+        for event in events:
+            k,a,b,c,F,*rest=event
+            assert c in (-1,1)
+            promote(a,F)
+            if k=='gate':promote(b,F);scalar+=1
+            elif k=='read':
+                for t in b:promote(t,F)
+                scalar+=len(b)
+            else:
+                assert k=='centre';G=rest[0]
+                assert contained(G,F) or contained(F,G)
+                assert nondeg(G);copies[abs(len(F)-len(G))]+=1
+                for t in b:promote(t,G)
+                scalar+=len(b)+2 # temporary copy and erasure
+            digest.update(repr(event).encode()+b'\n')
+        # Each literal role cleanup reaches full space; any unused final
+        # endpoint is still represented explicitly in the expected chain.
+        for s,F in enumerate(expected):promote(s,F)
+        assert current==expected
+        return hist,copies,scalar,digest.hexdigest()
+    start=tm+[ZERO]*v+[sigma.get(s,ZERO) for s in live]
+    finish=[FULL]*v+[complement(T,h) for T in tm]+[FULL]*len(live)
+    # Independently bind the inventory to the actual literal physical ports.
+    # This reconstruction uses the frame scan's starting list, not the
+    # producer's virtual role inventory or its selected-pair accounting.
+    source_counts=Counter(basis(F) for F in start[2*v:])
+    physical_auxiliary_source_frames=[dict(basis=list(F),count=count)
+        for F,count in sorted(source_counts.items(),key=lambda item:(len(item[0]),item[0]))]
+    assert sum(source_counts.values())==len(live)
+    assert physical_auxiliary_source_frames==d['out']['physical_auxiliary_source_frames'], 'Literal physical source frame inventory'
+    assert all(start[A(a)]==ZERO for a in merge.values()), 'Reused donor starts at zero'
+    assert all(F==FULL for F in finish[2*v:]), 'Completed physical scratch finish'
+    H,C,scalar,digest=framescan(word,start,finish)
+    revstart=[None]*len(start);revfinish=[None]*len(start)
+    for s in range(len(start)):
+        revstart[swap(s)]=complement(finish[s],h);revfinish[swap(s)]=complement(start[s],h)
+    HR,CR,scalarR,reverse_digest=framescan(reverse,revstart,revfinish)
+    assert (H,C,scalar)==(HR,CR,scalarR)
+    z=Counter({r:2*v*n for r,n in (H+C).items()})
+    for s in live:z[h*h-h+len(sigma.get(s,ZERO))]+=2*v
+    z[(h-1)**2]+=2*v*v;z[1]+=v*v
+    z.pop(0,None)
+    assert dict(sorted(z.items()))=={int(r):n for r,n in d['out']['child_multiplicities'].items()},'Literal reflected child histogram'
+    assert sum(r*n for r,n in z.items())==d['out']['total_rank']
+    G=v*v+2*v*scalar
+    safe=8*(d['c_add']+2*R+(R+q)*v*(h+1)+h*h+h+1) if 'c_add' in d else 8*(d['out']['additions']+2*R+(R+q)*v*(h+1)+h*h+h+1)
+    assert safe>=scalar
+    return dict(h=h,v=v,R=len(live),virtual_R=R,reused_roles=len(merge),exact_birth_cut_invariants=True,last_uses_recomputed=True,physical_aliased_numeric_replay=True,omitted_compensation_numeric_control=True,inverse_source_order='True reverse chronological source/workspace word',exact_fresh_source_map=True,exact_integer_old_readout_transpose=True,
+                physical_auxiliary_source_frames=physical_auxiliary_source_frames,
+                completed_core_source_inventory_bound=True,completed_core_pre_exterior_frames_full=True,
+                reflected_core_active_frames_complement_source=all(revfinish[2*v+i]==complement(start[2*v+i],h) for i in range(len(live))),
+                exact_arbitrary_dirty_cancellation_by_dependency_cut=True,
+                dirty_reuse_identity="Every recipient is untouched before its compensated birth; donor has no future gate/root reads. Recipient old-response coefficient cancels its arbitrary current donor value. All physical shears are reversed in true chronology.",
+                reflected_word_rule='Reverse literal order including each bounded readout chunk, negate every shear, swap X/Y ports, complement every frame; auxiliary bank is separate.',
+                reflected_scalar_map='X becomes X-Y; Y unchanged; all auxiliary coordinates restored.',
+                two_stage_scalar_map='(X,Y) becomes (-Y,X+Y) before inherited endpoint correction.',
+                literal_frame_incidences_both_directions=True,all_frames_nondegenerate=True,
+                reflected_residual_rank_histogram_equal=True,child_multiplicities=dict(sorted(z.items())),
+                word_blocks=len(word),expanded_scalar_operations_per_stage=scalar,
+                expanded_old_readout_additions=readcount,largest_readout_numerator_over_42=maxcoef,
+                unsplit_old_readout_macros=unsplit_readcount,split_readout_macros=split_macros,
+                largest_bounded_chunk_numerator_over_42=maxchunk,
+                bounded_readout_chunks_sha256=chunk_digest.hexdigest(),
+                scalar_readout_normalization='Each exact numerator n/42 becomes signed42 chunks plus a signed remainder. Exact reconstruction and chunk magnitude<=1 checked; every same-frame shear is charged; reflection reverses chunk order.',
+                exact_chunk_reconstruction=True,bounded_chunk_coefficients=True,
+                old_readout_coefficients_over_42_sha256=coefficient_digest.hexdigest(),
+                literal_global_scalar_groups=G,conservative_local_G=safe,
+                forward_incidence_sha256=digest,reflected_incidence_sha256=reverse_digest,
+                rejected_controls=rejected,
+                scope='Exact finite source coefficients, dirty cancellation algebra and literal reflected frame/charge ledger. Copied-centre transforms, endpoint correction and residual-to-child/all-size transfer retain their stated inherited contracts.')
+
+
+def main():
+    ap=argparse.ArgumentParser();ap.add_argument('--source',type=Path);ap.add_argument('--state',type=Path)
+    ap.add_argument('--output',type=Path);ap.add_argument('--check',type=Path);a=ap.parse_args()
+    assert bool(a.source)!=bool(a.state)
+    d=capture(a.source) if a.source else pickle.loads(a.state.read_bytes())
+    result=audit(d);source=Path(d['source'])
+    result['source_sha256']=sha256(source.read_bytes()).hexdigest()
+    result['audit_sha256']=sha256(Path(__file__).read_bytes()).hexdigest()
+    result=json.loads(json.dumps(result));encoded=json.dumps(result,indent=2,sort_keys=True)+'\n'
+    if a.check:assert result==json.loads(a.check.read_text())
+    if a.output:a.output.write_text(encoded)
+    else:print(encoded,end='')
+    print('PASS exact scalar and literal complement reflection audit',file=sys.stderr)
+
+
+if __name__=='__main__':main()
diff --git a/research/cyclic-deferred/replayed_producer.py b/research/cyclic-deferred/replayed_producer.py
new file mode 100644
index 0000000000..673fbbba70
--- /dev/null
+++ b/research/cyclic-deferred/replayed_producer.py
@@ -0,0 +1,179 @@
+"""Replayed h=24 complex producer with all centers disjoint (rational divisor 21).
+
+Copyright 2026 eumemic, Apache-2.0. AI-assisted (Claude). The DAG was found by an
+automated search over producers of the retained paired-triple, pair-star and
+center-disjoint interfaces. The checks below are adapted from
+stopped_product/complex.py (copyright 2026 icekylinx, Apache-2.0, PR #104).
+
+The witness complex-dag.json.gz lists the ordinary
+additions after the v triple inputs and the retained roots: the disjoint sums
+D[t], the pair stars P(a,b,i) and the center-disjoint sums A_i. Every support,
+core, cover, type and rank is recomputed here from the operands; nothing is
+read from the witness except the operand pairs and the root indices.
+"""
+import gzip, json, struct, sys, array
+from fractions import Fraction
+from itertools import combinations
+from pathlib import Path
+
+
+def write_array(f, values):
+    """Little-endian array writer, as partial_swap.binary.write_array."""
+    if sys.byteorder != 'little' and values.itemsize > 1:
+        values = array.array(values.typecode, values)
+        values.byteswap()
+    values.tofile(f)
+
+
+def build(witness, prefix):
+    w = json.loads(gzip.decompress(Path(witness).read_bytes()))
+    h, v, d = w['h'], w['v'], w['central_disjoint']
+    assert (h, d, w['center_denominator']) == (24, 24, 21)
+    triples = list(combinations(range(h), 3))
+    assert v == len(triples)
+    args = [(0, 0)]; support = [0]; core = [0]; cover = [0]; types = [0]; ranks = [0]
+    for j, t in enumerate(triples):
+        mask = sum(1 << a for a in t)
+        args.append((0, 0)); support.append(1 << j); core.append(mask); cover.append(mask)
+        types.append(1); ranks.append(1)
+    flat = w['args']
+    assert len(flat) % 2 == 0
+    for k in range(0, len(flat), 2):
+        a, b, x = flat[k], flat[k+1], len(args)
+        assert 0 < a < x and 0 < b < x, 'Operands must precede their sum'
+        assert not support[a] & support[b], 'Ordinary additions must be cancellation-free'
+        args.append((a, b)); support.append(support[a] | support[b])
+        core.append(core[a] & core[b]); cover.append(cover[a] | cover[b])
+        if core[x].bit_count() >= 2:
+            types.append(1); ranks.append(support[x].bit_count())
+        else:
+            types.append(2); ranks.append(cover[x].bit_count())
+    n = len(args)
+    point_masks = [sum(1 << j for j, t in enumerate(triples) if a in t) for a in range(h)]
+    full_support = (1 << v) - 1
+    roots, kind = [], []
+    assert len(w['D']) == v
+    for t, node in zip(triples, w['D']):
+        assert 0 < node < n
+        assert support[node] == full_support & ~(point_masks[t[0]]|point_masks[t[1]]|point_masks[t[2]])
+        target_mask = sum(1 << a for a in t)
+        if types[node] == 2:
+            assert not cover[node] & target_mask
+        else:
+            assert types[node] == 1
+            assert not support[node] & (point_masks[t[0]]^point_masks[t[1]]^point_masks[t[2]])
+        roots.append(node); kind.append(0)
+    stars = [(a, b, i) for a, b in combinations(range(h), 2) for i in range(h) if i not in (a, b)]
+    assert len(w['P']) == len(stars)
+    pair_outputs = {}
+    for (a, b, i), node in zip(stars, w['P']):
+        assert 0 < node < n
+        assert support[node] == point_masks[a] & point_masks[b] & ~point_masks[i]
+        assert types[node] == 1
+        assert not support[node] & (point_masks[a]^point_masks[b]^point_masks[i])
+        pair_outputs[a, b, i] = node
+        roots.append(node); kind.append(0)
+    assert len(w['A']) == d
+    for i, node in enumerate(w['A']):
+        assert 0 < node < n
+        assert support[node] == full_support & ~point_masks[i]
+        assert types[node] == 2 and cover[node] == ((1 << h)-1) ^ (1 << i)
+        assert ranks[node] == h-1
+        roots.append(node); kind.append(1)
+    # Every A_i avoids i, so sum A_i = (h-3)T. The decoder is T=sum A_i/21
+    # and the scatter is T-(1/2)sum_{i in target} A_i.
+    decoder = Fraction(1, h-3)
+    assert decoder == Fraction(1, 21)
+    scatter_coefficients = [decoder, decoder-Fraction(1, 2)]
+    for coefficient in scatter_coefficients:
+        odd_denominator = coefficient.denominator
+        while odd_denominator % 2 == 0:
+            odd_denominator //= 2
+        assert 21 % odd_denominator == 0 and abs(coefficient) <= 1
+    for source in triples:
+        assert sum(i not in source for i in range(h))*decoder == 1
+    Dset = set(range(d))
+    for triple in triples:
+        k = len(Dset.intersection(triple))
+        assert 2*(3-k)-2*(d-k) == 2*(3-d)
+    patterns = {}
+    for target in triples:
+        chosen = len(Dset.intersection(target))
+        if chosen not in patterns:
+            patterns[chosen] = target
+    for chosen, target in patterns.items():
+        target_set = set(target)
+        for source in triples:
+            k = len(Dset.intersection(source))
+            total_coefficient = Fraction(2*(3-k)-2*(d-k), 2*(3-d))
+            scalar = Fraction(len((target_set-Dset).intersection(source)), 2)
+            scalar -= Fraction(len((target_set & Dset)-set(source)), 2)
+            scalar += Fraction(chosen-1, 2)*total_coefficient
+            assert scalar == Fraction(len(target_set.intersection(source))-1, 2)
+    for j, t in enumerate(triples):
+        masks = [point_masks[a] for a in t]
+        two = (masks[0]&masks[1]&~masks[2]) | (masks[0]&masks[2]&~masks[1]) | (masks[1]&masks[2]&~masks[0])
+        actual = 0
+        for a, b in combinations(t, 2):
+            excluded = next(i for i in t if i not in (a, b))
+            actual |= support[pair_outputs[a, b, excluded]]
+        assert actual == two
+        assert masks[0]&masks[1]&masks[2] == 1 << j
+    assert [((k-1)+(k == 0)-(k == 2)) for k in range(4)] == [0, 0, 0, 2]
+
+    def contained(x, y):
+        tx, ty = types[x], types[y]
+        if tx == 1 and ty == 1: return not(core[y]&~core[x] or cover[x]&~cover[y])
+        if tx in (1, 2) and ty == 2: return not cover[x]&~cover[y]
+        return False
+    active = [0]*n; stack = list(roots)
+    while stack:
+        x = stack.pop()
+        if not x or active[x]: continue
+        active[x] = 1
+        if args[x][0]: stack.extend(args[x])
+    degree = [0]*n
+    for x in range(1, n):
+        if active[x]:
+            if types[x] == 1:
+                assert core[x].bit_count() >= 2 and ranks[x] == support[x].bit_count()
+            else:
+                assert types[x] == 2 and ranks[x] == cover[x].bit_count()
+        if active[x] and args[x][0]:
+            for y in args[x]:
+                assert y < x and contained(y, x), 'Binary frame nesting failed'
+                degree[y] += 1
+    for x in roots: degree[x] += 1
+    H = [0]*(h+1); c = 0; loss = 0
+    for x in range(1, n):
+        if not active[x]: continue
+        r = ranks[x]
+        if args[x][0]:
+            c += 1; H[r] += degree[x]-1; H[h-r] += 1
+            for y in args[x]:
+                assert r >= ranks[y]
+                H[r-ranks[y]] += 1
+        else: H[1] += degree[x]
+    for x, k in zip(roots, kind):
+        r = ranks[x]
+        if k: H[r] += 1; H[h] += 1; loss += r
+        else: H[h-1-r] += 1; H[1] += 1
+    q = len(roots); R = c+q
+    result = dict(h=h, v=v, c=c, q=q, R=R, loss=loss, central_disjoint=d, histogram=H,
+                  rank_sum=sum(r*m for r, m in enumerate(H)))
+    assert loss == h*(h-1) and result['rank_sum'] == h*R+2*loss
+    result['center_denominator'] = 21
+    result['scalar_validation'] = dict(ordinary_supports_exact=True, centers_exact=True,
+        center_decoder='sum A_i / 21', scatter_coefficients=list(map(str, scatter_coefficients)),
+        rational_scalar_identity_exact=True, mixed_center_scatter_exact=True, binary_frames_nested=True,
+        binary_frames_nondegenerate=True)
+    with open(str(prefix)+'.bin', 'wb') as f:
+        f.write(struct.pack('<4I', h, v, n, q))
+        write_array(f, array.array('I', (a for pair in args for a in pair)))
+        write_array(f, array.array('Q', core)); write_array(f, array.array('Q', cover))
+        write_array(f, array.array('I', roots)); write_array(f, array.array('I', kind))
+        write_array(f, array.array('B', active))
+    with open(str(prefix)+'.labels', 'wb') as f:
+        write_array(f, array.array('I', ranks)); write_array(f, array.array('B', types))
+    Path(str(prefix)+'.json').write_text(json.dumps(result, indent=2)+'\n')
+    return result
diff --git a/research/cyclic-deferred/reuse-pairs.json b/research/cyclic-deferred/reuse-pairs.json
new file mode 100644
index 0000000000..5e32cba92a
--- /dev/null
+++ b/research/cyclic-deferred/reuse-pairs.json
@@ -0,0 +1,73016 @@
+[
+ {
+  "donor": 1719,
+  "recipient": 23014,
+  "donor_frame": [
+   8,
+   4,
+   2,
+   1
+  ],
+  "birth_frame": [
+   8,
+   4,
+   2,
+   1
+  ],
+  "e": 4,
+  "s": 4
+ },
+ {
+  "donor": 13472,
+  "recipient": 25594,
+  "donor_frame": [
+   8388608,
+   4194304,
+   2097152,
+   1048576
+  ],
+  "birth_frame": [
+   8388608,
+   4194304,
+   2097152,
+   1048576
+  ],
+  "e": 4,
+  "s": 4
+ },
+ {
+  "donor": 6812,
+  "recipient": 25569,
+  "donor_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   128,
+   64,
+   32,
+   16
+  ],
+  "birth_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   128,
+   64,
+   32,
+   16
+  ],
+  "e": 8,
+  "s": 8
+ },
+ {
+  "donor": 6061,
+  "recipient": 25866,
+  "donor_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   128,
+   64,
+   32,
+   16
+  ],
+  "birth_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   128,
+   64,
+   32,
+   16
+  ],
+  "e": 8,
+  "s": 8
+ },
+ {
+  "donor": 23424,
+  "recipient": 25823,
+  "donor_frame": [
+   32768,
+   16384,
+   8192,
+   4096,
+   128,
+   64,
+   32,
+   16
+  ],
+  "birth_frame": [
+   32768,
+   16384,
+   8192,
+   4096,
+   128,
+   64,
+   32,
+   16
+  ],
+  "e": 8,
+  "s": 8
+ },
+ {
+  "donor": 3318,
+  "recipient": 25779,
+  "donor_frame": [
+   8388608,
+   4194304,
+   2097152,
+   1048576,
+   8,
+   4,
+   2,
+   1
+  ],
+  "birth_frame": [
+   8388608,
+   4194304,
+   2097152,
+   1048576,
+   8,
+   4,
+   2,
+   1
+  ],
+  "e": 8,
+  "s": 8
+ },
+ {
+  "donor": 1684,
+  "recipient": 25783,
+  "donor_frame": [
+   8388608,
+   4194304,
+   2097152,
+   1048576,
+   8,
+   4,
+   2,
+   1
+  ],
+  "birth_frame": [
+   8388608,
+   4194304,
+   2097152,
+   1048576,
+   8,
+   4,
+   2,
+   1
+  ],
+  "e": 8,
+  "s": 8
+ },
+ {
+  "donor": 23467,
+  "recipient": 25605,
+  "donor_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   8,
+   4,
+   2,
+   1
+  ],
+  "birth_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   8,
+   4,
+   2,
+   1
+  ],
+  "e": 8,
+  "s": 8
+ },
+ {
+  "donor": 19296,
+  "recipient": 25609,
+  "donor_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   8,
+   4,
+   2,
+   1
+  ],
+  "birth_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   8,
+   4,
+   2,
+   1
+  ],
+  "e": 8,
+  "s": 8
+ },
+ {
+  "donor": 3281,
+  "recipient": 25680,
+  "donor_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   8,
+   4,
+   2,
+   1
+  ],
+  "birth_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   8,
+   4,
+   2,
+   1
+  ],
+  "e": 8,
+  "s": 8
+ },
+ {
+  "donor": 3263,
+  "recipient": 25705,
+  "donor_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   8,
+   4,
+   2,
+   1
+  ],
+  "birth_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   8,
+   4,
+   2,
+   1
+  ],
+  "e": 8,
+  "s": 8
+ },
+ {
+  "donor": 1541,
+  "recipient": 25735,
+  "donor_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   8,
+   4,
+   2,
+   1
+  ],
+  "birth_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   8,
+   4,
+   2,
+   1
+  ],
+  "e": 8,
+  "s": 8
+ },
+ {
+  "donor": 304,
+  "recipient": 25739,
+  "donor_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   8,
+   4,
+   2,
+   1
+  ],
+  "birth_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   8,
+   4,
+   2,
+   1
+  ],
+  "e": 8,
+  "s": 8
+ },
+ {
+  "donor": 146,
+  "recipient": 25743,
+  "donor_frame": [
+   524288,
+   131072,
+   65536,
+   8,
+   4,
+   2,
+   1
+  ],
+  "birth_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   8,
+   4,
+   2,
+   1
+  ],
+  "e": 7,
+  "s": 8
+ },
+ {
+  "donor": 13585,
+  "recipient": 25759,
+  "donor_frame": [
+   131072,
+   65536,
+   8,
+   4,
+   2,
+   1
+  ],
+  "birth_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   8,
+   4,
+   2,
+   1
+  ],
+  "e": 6,
+  "s": 8
+ },
+ {
+  "donor": 13582,
+  "recipient": 25763,
+  "donor_frame": [
+   131072,
+   65536,
+   8,
+   4,
+   2,
+   1
+  ],
+  "birth_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   8,
+   4,
+   2,
+   1
+  ],
+  "e": 6,
+  "s": 8
+ },
+ {
+  "donor": 1920,
+  "recipient": 25767,
+  "donor_frame": [
+   131072,
+   65536,
+   8,
+   4,
+   2,
+   1
+  ],
+  "birth_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   8,
+   4,
+   2,
+   1
+  ],
+  "e": 6,
+  "s": 8
+ },
+ {
+  "donor": 289,
+  "recipient": 25775,
+  "donor_frame": [
+   131072,
+   65536,
+   8,
+   4,
+   2,
+   1
+  ],
+  "birth_frame": [
+   524288,
+   262144,
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+   16,
+   8,
+   4,
+   2,
+   1
+  ],
+  "e": 15,
+  "s": 18
+ },
+ {
+  "donor": 1752,
+  "recipient": 27309,
+  "donor_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   2048,
+   1024,
+   512,
+   256,
+   128,
+   64,
+   32,
+   16,
+   8,
+   4,
+   2
+  ],
+  "birth_frame": [
+   8388608,
+   4194304,
+   524288,
+   262144,
+   131072,
+   65536,
+   2048,
+   1024,
+   512,
+   256,
+   128,
+   64,
+   32,
+   16,
+   8,
+   4,
+   2,
+   1
+  ],
+  "e": 15,
+  "s": 18
+ },
+ {
+  "donor": 967,
+  "recipient": 26900,
+  "donor_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   8192,
+   2048,
+   1024,
+   512,
+   256,
+   128,
+   64,
+   32,
+   16,
+   8,
+   4,
+   2,
+   1
+  ],
+  "birth_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   8192,
+   4096,
+   2048,
+   1024,
+   512,
+   256,
+   128,
+   64,
+   32,
+   16,
+   8,
+   4,
+   2,
+   1
+  ],
+  "e": 17,
+  "s": 18
+ },
+ {
+  "donor": 25777,
+  "recipient": 26803,
+  "donor_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   32768,
+   2048,
+   1024,
+   512,
+   256,
+   128,
+   64,
+   32,
+   16,
+   8,
+   4,
+   2,
+   1
+  ],
+  "birth_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   32768,
+   16384,
+   2048,
+   1024,
+   512,
+   256,
+   128,
+   64,
+   32,
+   16,
+   8,
+   4,
+   2,
+   1
+  ],
+  "e": 17,
+  "s": 18
+ },
+ {
+  "donor": 283,
+  "recipient": 26804,
+  "donor_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   32768,
+   2048,
+   1024,
+   512,
+   256,
+   128,
+   64,
+   32,
+   16,
+   8,
+   4,
+   2,
+   1
+  ],
+  "birth_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   32768,
+   16384,
+   2048,
+   1024,
+   512,
+   256,
+   128,
+   64,
+   32,
+   16,
+   8,
+   4,
+   2,
+   1
+  ],
+  "e": 17,
+  "s": 18
+ },
+ {
+  "donor": 23131,
+  "recipient": 28423,
+  "donor_frame": [
+   8388608,
+   4194304,
+   2097152,
+   1048576,
+   524288,
+   262144,
+   131072,
+   65536,
+   32768,
+   16384,
+   8192,
+   4096,
+   1024,
+   128,
+   64,
+   8,
+   4,
+   2,
+   1
+  ],
+  "birth_frame": [
+   8388608,
+   4194304,
+   2097152,
+   1048576,
+   524288,
+   262144,
+   131072,
+   65536,
+   32768,
+   16384,
+   8192,
+   4096,
+   2048,
+   1024,
+   128,
+   64,
+   8,
+   4,
+   2,
+   1
+  ],
+  "e": 19,
+  "s": 20
+ }
+]
diff --git a/research/cyclic-deferred/reuse.py b/research/cyclic-deferred/reuse.py
new file mode 100644
index 0000000000..2c4a6daa29
--- /dev/null
+++ b/research/cyclic-deferred/reuse.py
@@ -0,0 +1,167 @@
+"""Deterministic compensated birth-cut scratch reuse.
+
+Birth-cut reuse follows jamesyc PR124. Composition with saturated deferrals
+and physical hull frames prepared for eumemic with OpenAI Codex assistance.
+Apache-2.0; inherited authorship remains in the surrounding package.
+"""
+from collections import defaultdict
+from functools import lru_cache
+
+
+def basis(vectors):
+    piv = {}
+    for x in vectors:
+        for p in sorted(piv, reverse=True):
+            if x >> p & 1:
+                x ^= piv[p]
+        if x:
+            p = x.bit_length() - 1
+            for q in piv:
+                if piv[q] >> p & 1:
+                    piv[q] ^= x
+            piv[p] = x
+    return tuple(piv[p] for p in sorted(piv, reverse=True))
+
+
+def contained(A, B):
+    return len(basis(tuple(A) + tuple(B))) == len(B)
+
+
+def complement(A, h):
+    A = basis(A)
+    piv = {x.bit_length() - 1: x for x in A}
+    out = []
+    for j in range(h):
+        if j not in piv:
+            x = 1 << j
+            for p, row in piv.items():
+                if row >> j & 1:
+                    x |= 1 << p
+            out.append(x)
+    return basis(out)
+
+
+def nondeg(A):
+    return len(basis(sum(((x & y).bit_count() & 1) << j
+                         for j, y in enumerate(A)) for x in A)) == len(A)
+
+
+def check_pairs(data, pairs):
+    """Reject aliases, unfinished donors and unavailable recipient births."""
+    donors = [row['donor'] for row in pairs]
+    recipients = [row['recipient'] for row in pairs]
+    assert (len(set(donors)) == len(donors) and
+            len(set(recipients)) == len(recipients) and
+            not set(donors) & set(recipients)), 'Reuse pair alias'
+    early = set(data['Anc'])
+    first = {}
+    late_touched = set()
+    for i, op in enumerate(data['ops']):
+        if op[0] == 'src':
+            first.setdefault(op[1], i)
+            continue
+        for s in op[1:3]:
+            first.setdefault(s, i)
+        if i not in early:
+            late_touched.update(op[1:3])
+    for row in pairs:
+        a, b = row['donor'], row['recipient']
+        assert type(a) is int and type(b) is int, 'Reuse role type'
+        assert 0 <= a < data['R'] and 0 <= b < data['R'], 'Reuse role range'
+        assert (a not in data['placed'] and a not in data['role_root'] and
+                data['last'].get(a) in early and a not in late_touched), 'Reuse nondead donor'
+        assert (b in data['placed'] and b not in data['leaf_of'] and
+                b not in data['touched'] and first.get(b) not in early), 'Reuse unavailable birth'
+        birth = data['ops'][first[b]]
+        assert birth[0] == 'copy' and birth[2] == b, 'Reuse unavailable birth'
+        A = basis(data['op_frames'][data['last'][a]])
+        F = basis(data['placed'][b])
+        B = basis(data['op_frames'][first[b]])
+        assert nondeg(A) and nondeg(F) and nondeg(B), 'Reuse degenerate frame'
+        assert contained(A, F) and contained(F, B), 'Reuse frame containment'
+        assert basis(row['donor_frame']) == A and basis(row['birth_frame']) == F, 'Reuse recorded frames'
+        assert (row['e'], row['s']) == (len(A), len(F)), 'Reuse recorded dimensions'
+    return dict(zip(recipients, donors))
+
+
+def check_compensation(merge, compensated):
+    assert set(merge) <= set(compensated), 'Reuse omitted compensation'
+
+
+def select_reuse(data):
+    groups = defaultdict(list)
+    for a, i in data['last'].items():
+        if i in data['Anc'] and a not in data['role_root']:
+            A = basis(data['op_frames'][i])
+            assert nondeg(A)
+            groups[A].append(a)
+    frames = sorted(groups, key=lambda F: (-len(F), F))
+    pool = [groups[F][:] for F in frames]
+    births = defaultdict(list)
+    for b, F in data['placed'].items():
+        if b not in data['leaf_of']:
+            births[basis(F)].append(b)
+    allgroups = (1 << len(frames)) - 1
+
+    @lru_cache(None)
+    def annihilated(q):
+        return sum(1 << i for i, A in enumerate(frames)
+                   if all(not ((a & q).bit_count() & 1) for a in A))
+
+    eligible, capacity = {}, {}
+    for F in births:
+        mask = allgroups
+        for q in complement(F, data['h']):
+            mask &= annihilated(q)
+            if not mask:
+                break
+        eligible[F] = mask
+        capacity[F] = sum(len(pool[i]) for i in range(len(frames)) if mask >> i & 1)
+    available, pairs = allgroups, []
+    for F in sorted(births, key=lambda F: (capacity[F], len(F), F)):
+        for b in sorted(births[F]):
+            mask = eligible[F] & available
+            if not mask:
+                break
+            i = (mask & -mask).bit_length() - 1
+            a = pool[i].pop()
+            A = frames[i]
+            if not pool[i]:
+                available ^= 1 << i
+            pairs.append(dict(donor=a, recipient=b, donor_frame=A,
+                              birth_frame=F, e=len(A), s=len(F)))
+    check_pairs(data, pairs)
+    return pairs
+
+
+def mutation_controls(data, pairs):
+    """Mutate the actual selected finite pairing, not a synthetic inventory."""
+    import copy
+    rejected = []
+    occupied = {row[key] for row in pairs for key in ('donor', 'recipient')}
+    for failure in ('nondead-donor', 'unavailable-birth', 'pair-alias'):
+        changed = copy.deepcopy(pairs)
+        if failure == 'nondead-donor':
+            changed[0]['donor'] = next(s for s in data['role_root'] if s not in occupied)
+        elif failure == 'unavailable-birth':
+            changed[0]['recipient'] = next(s for s in data['leaf_of'] if s not in occupied)
+        else:
+            changed[0]['recipient'] = changed[0]['donor']
+        try:
+            check_pairs(data, changed)
+        except AssertionError as error:
+            expected = {'nondead-donor': 'nondead donor',
+                        'unavailable-birth': 'unavailable birth',
+                        'pair-alias': 'pair alias'}[failure]
+            assert expected in str(error), (failure, error)
+            rejected.append(failure)
+        else:
+            raise AssertionError('Accepted reuse mutation: ' + failure)
+    merge = check_pairs(data, pairs)
+    try:
+        check_compensation(merge, set(merge) - {pairs[0]['recipient']})
+    except AssertionError:
+        rejected.append('omitted-compensation')
+    else:
+        raise AssertionError('Accepted omitted birth compensation')
+    return rejected
diff --git a/research/joint-dual/SOURCE.json b/research/joint-dual/SOURCE.json
index 80c0f29dc4..607336c6f9 100644
--- a/research/joint-dual/SOURCE.json
+++ b/research/joint-dual/SOURCE.json
@@ -3,9 +3,9 @@
   "base_pr57_commit": "cd350f76c9bc01489ec83568bded532cb69be938",
   "base_pr58_commit": "bc2f7ed4c20dc18898305ab17165c0c995cbb804",
   "files": {
-    "Makefile": "97620e0205736d9c0fce736e8ac6ab67f8ed0e490c381f6cac152cc6a003dca7",
-    "NOTICE": "1c53970fb4face631a1b331ec66d33111eccd280a129f9b64c2da6df96861c3f",
-    "README.md": "fd92de31aece76b1f5ff897e6e0dbc38dd3bf67c140579bbcb03a1a11a792668",
+    "Makefile": "0afb0751eaa8bf66c75d8f084405c97322af9204244dae752f101c9e3e504490",
+    "NOTICE": "e891051aa7858815850030c9531a88794ba0f39436fbd03cc0f2607a76da8b74",
+    "README.md": "b09d3d0a9bbddd2c93b8ebf650963ed911f661bc88db18a824331923f2af3b74",
     "certificates/joint-dual-compiler.json": "516c4d85158553dabd0a98216a064b2a7397957a9d5dd0df1d6e5aec53a4e805",
     "certificates/joint-dual-profiles-23.json": "159fa8fee175fc1d24f63e72d3c2f1dc9f4fcd78b938c5e7d2f735fd0fd37209",
     "certificates/joint-dual-profiles-25.json": "8d190ce42d097e828c5756918c52f2963df576cf4c6600b0fc7f2b7fdb6ad223",
diff --git a/research/paired-pr131-conservative/README.md b/research/paired-pr131-conservative/README.md
new file mode 100644
index 0000000000..d314a2a90c
--- /dev/null
+++ b/research/paired-pr131-conservative/README.md
@@ -0,0 +1,70 @@
+# Conservative PR131 physical-word + PR132 paired-cover candidate
+
+**Finite arithmetic candidate**: conditional κ = **0.000362108222954**,
+which is **+5.99731%** against PR132 κ = 0.0003416202. This is a
+**research DRAFT, not a fully independently validated multiplication bound**:
+the shared physical execution theorem for the combined two-source
+construction requires a full literal paired-word replay and review.
+
+## Method
+
+- Pin PR131's verified h24 complex physical word, 26,597 physical roles
+  (2,108 compensated birth-cut reuses), all source gauges and reflection
+  audit. Keep all 28,705 virtual roles in the scalar-work reserve.
+- In the *first* bank of PR130's O(70,2) three-stage cover, pair vertices
+  g and gκ by the orthogonal coordinate involution κ exchanging A with
+  24 coordinates in B⊕C (PR132). The two active h24 spaces are orthogonal.
+- Share physical role streams **after complete restoration** and run
+  corresponding nested local steps in lockstep. In the combined first-stage
+  bank the frame increment has rank 2r, versus 4 separate r-children in
+  banks 2 and 3. Paired source exteriors have rank 22+2s, versus four
+  separate 46+s exteriors in the other two banks.
+- Correct the optimistic complete-local-step histogram: source/target
+  data wires must **not** lockstep. Rather than assume a favorable
+  histogram, charge a rigorous *positive upper moment penalty* to cover
+  all their independently executed steps. Their total rank mass is
+  exactly 2v(h−1) per local word. At exponent a, every pair of rank-r
+  child steps replacing one rank-2r child costs at most
+  2r (m)^a a log(2). The certificate applies the sum of those penalties
+  using exact rational log and exp upper enclosures.
+- Keep PR130's stopped bit supplier, rare-class fallback, all group/router
+  costs, compensated readout scalar reserve, semantic/tape and row stock.
+- Recompute the complete exact finite moment and the **47 strict assembly
+  inequalities, seven margins**, and the next 10^-12 / 10^-15 grid
+  rejections.
+
+## Conditional finite-arithmetic results
+
+- Paired persistent W = 141,081, versus PR132's 151,621.
+- Rank mass = 9,870,886; exact deficit = 4,784.
+- Largest child 68 of ambient 70.
+- Conservative complex saving at 10^-12 grid:
+  0.000362371031.
+- Finite assembly κ = 0.000362108222954.
+- Baseline PR132 κ = 0.0003416202.
+- Target ≥ 3% κ = 0.000351868806.
+
+Run `python3 research/paired-pr131-conservative/certificate.py`
+from the repository root on unoptimized Python. It independently reads
+the existing PR131 certified local profile, its frozen reflection receipt,
+and PR130 inputs, and performs rational upper-bound arithmetic.
+
+## Remaining proof obligations
+
+The following are **not** established by the numerical certificate:
+1. Literal physical lockstep scheduling after PR131's compensated aliasing
+   of every actual source/readout/copy gate, including scalar readout
+   splitting, external data wires and arbitrary initial dirty scratch.
+2. Exact signed Clifford phases, rank-zero adapters and inverse/reflected
+   chronology in the merged first-stage bank. PR131's single-core replay
+   and PR132's mathematical paired-core argument are separate dependencies,
+   not themselves a new combined all-column replay.
+3. Independent full verifier and current-head CI.
+4. Inherited analytic, routing, normal-form, fixed-tape, semantic precision
+   and multiplication reduction interfaces remain conditional.
+
+**Do not promote as a proven record until these obligations are closed.**
+
+Credit: icekylinx (PR130), eumemic (PR131), ikeboy (PR132),
+jamesyc (PR124), and all upstream dependencies and license notices.
+Hybrid finite certification prepared with OpenAI assistance.
diff --git a/research/paired-pr131-conservative/certificate-tight.py b/research/paired-pr131-conservative/certificate-tight.py
new file mode 100644
index 0000000000..103533d760
--- /dev/null
+++ b/research/paired-pr131-conservative/certificate-tight.py
@@ -0,0 +1,265 @@
+#!/usr/bin/env python3
+"""Exact rational *conservative* screen: PR131 local frames in PR132 paired cover.
+
+This is a finite conditional candidate; NOT a verified physical-composition
+theorem, and NOT independent peer review. Original PR131 source package
+remains byte-identical. Inherited PR130/PR131 checks remain dependencies.
+
+Reference: PR131 3acf2f944abd069301600ce468f16f1dd8f1f9e4;
+PR132 f7bc7a91b1c5c7b6714b54529f338c98965e3173.
+"""
+from __future__ import annotations
+
+from collections import Counter
+from fractions import Fraction as Q
+from importlib.util import module_from_spec, spec_from_file_location
+from math import prod
+from pathlib import Path
+import argparse
+import json
+import sys
+
+if sys.flags.optimize:
+    raise RuntimeError("Assertions are required; do not use python -O")
+if hasattr(sys, "set_int_max_str_digits"):
+    sys.set_int_max_str_digits(0)
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+sys.path.insert(0, str(ROOT / "scripts"))
+
+import three_stage_cover_network as inherited
+from structured_bulk_assembly import assembly, halving, js
+
+GRID_A = 10**12
+GRID_K = 10**15
+STOP = Q(1, 10**15)
+PR132_KAPPA = Q(1708101, 5000000000)
+TARGET = PR132_KAPPA * Q(103, 100)
+
+def require(ok: bool, msg: str) -> None:
+    if not ok:
+        raise ValueError(msg)
+
+def pinned_local():
+    """Use the frozen, independently audited PR131 physical profile, not a guess."""
+    path = ROOT / "research/cover-local-reuse/certificate.py"
+    spec = spec_from_file_location("pr131_cover_certificate", path)
+    require(spec is not None and spec.loader is not None, "PR131 package missing")
+    mod = module_from_spec(spec)
+    spec.loader.exec_module(mod)
+    p = mod.cover_profile()  # SHA pins, reflection receipt, dirty/reuse checks.
+    baseline = json.loads((ROOT/"research/cover-local-reuse/certificate.json").read_text())
+    require(Q(baseline["kappa"]) == Q(166102699493,500000000000000),
+            "Pinned PR131 κ changed")
+    local = json.loads((ROOT/"research/cyclic-deferred/complex-profile.json").read_text())
+    audit = json.loads((ROOT/"research/cyclic-deferred/reflection-audit.json").read_text())
+    require(p["R"] == 26597 and p["virtual_R"] == 28705 and
+            p["reused_roles"] == 2108 and local["R"] == p["R"], "PR131 roles")
+    require(local["replay"]["scratch_restored"] and local["replay"]["y_plus_x"],
+            "PR131 dirty replay missing")
+    for key in ("physical_aliased_numeric_replay",
+                "literal_frame_incidences_both_directions",
+                "reflected_residual_rank_histogram_equal",
+                "exact_arbitrary_dirty_cancellation_by_dependency_cut",
+                "bounded_chunk_coefficients"):
+        require(audit[key] is True, "Inherited PR131 audit: " + key)
+    return p, local, audit
+
+def profile():
+    p, local, audit = pinned_local()
+    h,v,R,m = p["h"],p["v"],p["R"],p["m"]
+    require((h,v,R,m)==(24,2024,26597,70), "PR131 dimensions")
+    row = json.loads((ROOT/"certificates/three-stage-cover-complex-input.json").read_text())
+    require(row["R"] == p["virtual_R"] == 28705 and row["total_M_operations"] == 118451
+            and row["loss"] == 552, "Full virtual scalar reserve")
+    require(row["v"] == v and row["h"] == h, "Same h=24 source word")
+    H = Counter({int(t):int(n) for t,n in p["local_child_multiplicities"].items()})
+    require(all(0<t<h and n>0 for t,n in H.items()), "Illegal local child")
+
+    source = Counter()
+    for entry in local["physical_auxiliary_source_frames"]:
+        s=len(entry["basis"])
+        source[s]+=entry["count"]
+    require(sum(source.values())==R, "All physical source gauges accounted for")
+    expected_ext = Counter()
+    for s,n in source.items():expected_ext[m-h+s]+=3*n
+    require(dict(sorted(expected_ext.items())) ==
+            {int(t):int(n) for t,n in p["exterior_child_multiplicities"].items()},
+            "Physical source/exterior provenance mismatch")
+
+    # Verify an actual coordinate pairing involution for the first-stage
+    # h-plane and its h-dimensional orthogonal spectator plane.
+    permutation=list(range(m))
+    for j in range(h):
+        permutation[j],permutation[h+j]=h+j,j
+    require(all(permutation[permutation[i]]==i for i in range(m)) and
+            all(permutation[j]>=h for j in range(h)),
+            "Stage-one orthogonal fixed-point-free pairing invalid")
+
+    # Hypothetically merge ALL small local rank-r children: one rank-2r
+    # child per pair in stage one, 4 separate rank-r children in stages 2/3.
+    # This over-merges the actual SOURCE and TARGET data fronts, which must
+    # stay independent. Moment_bound() rigorously charges the difference
+    # from this optimistic multiset using their exact total RANK mass;
+    # the resulting upper envelope is a valid conservative calculation.
+    children=Counter()
+    for r,n in H.items():
+        children[2*r]+=n
+        children[r]+=4*n
+    for s,n in source.items():
+        children[m-2*h+2*s]+=n
+        children[m-h+s]+=4*n
+    children=Counter({r:n for r,n in children.items() if n})
+    W2=4*v+5*R
+    rank2=sum(r*n for r,n in children.items())
+    deficit=W2*m-rank2
+    require(deficit==2*(2*v-3*row["loss"])==4784, "Paired telescoping failed")
+    require(W2==141081 and rank2==9870886, "Profile provenance changed")
+    require(all(0<r<m and n>0 for r,n in children.items()) and max(children)==68,
+            "Bad rank/child")
+    require(sum(r*n for r,n in H.items())==711195,
+            "Inherited PR131 local rank mass mismatch")
+
+    # These two quantities are not estimated histograms: the PR131
+    # complex_deferred.py target-readout path is 0 -> levels -> h-1
+    # for each of v targets, and the source front is one h-1 for each
+    # of v sources. Their exact rank *mass* is 2v(h-1).
+    src = (ROOT/"research/cyclic-deferred/complex_deferred.py").read_text()
+    for literal in ("ds = sorted(levels[t] | {0, h - 1})",
+                    "z[b - a] += 2 * v",
+                    "z[h - 1] += 2 * N"):
+        require(literal in src, "Source/target data telescope no longer pinned")
+    data_mass=2*v*(h-1)
+    require(data_mass==93104, "Unpaired data rank-mass correction")
+    return dict(m=m,h=h,v=v,R=R,virtual_R=p["virtual_R"],
+                reused_roles=p["reused_roles"],W=W2,rank=rank2,deficit=deficit,
+                maxchild=max(children),
+                child_multiplicities=dict(sorted(children.items())),
+                physical_source_dimensions=dict(sorted(source.items())),
+                local_child_multiplicities=dict(sorted(H.items())),
+                nonmergeable_data_rank_mass=data_mass,
+                note="Children are an optimistic arithmetic multiset; an independent positive moment correction covers ALL nonmergeable data fronts.",
+                pr131_local_profile_sha256=p["local_profile_sha256"],
+                pr131_reflection_receipt_sha256=p["reflection_receipt_sha256"])
+
+def moment_upper(profile: dict,a:Q):
+    """Certified upper moment for the ACTUAL, data-unmerged paired schedule."""
+    m,W=profile["m"],profile["W"]
+    require(0<=a<Q(1,100), "Saving outside permitted range")
+    lg=inherited.log_upper
+    ex=inherited.exp_upper
+    # Same exact rational log/exp enclosures as the PR130 three-stage check.
+    optimistic=sum(Q(r*n,m*W)*ex(a*lg(Q(m,r)))
+                   for r,n in profile["child_multiplicities"].items())
+    # A rank-r data front gives two separate rank-r children in stage 1;
+    # the optimistic histogram counts one rank-2r child instead.
+    # Difference = 2r (m/r)^a [1 - 2^(-a)] /(mW)
+    # <= 2r (m)^a a log 2 /(mW) for r>=1.
+    penalty=(Q(2*profile["nonmergeable_data_rank_mass"],m*W)
+            *a*lg(Q(2))*ex(a*lg(Q(m))))
+    return optimistic+penalty,optimistic,penalty
+
+def certificate():
+    p=profile()
+    def accepts_a(n):
+        return moment_upper(p,Q(n,GRID_A))[0]<1
+    lo,hi=0,700000000
+    require(accepts_a(lo) and not accepts_a(hi),"Complex moment bracket failed")
+    while hi-lo>1:
+        t=(hi+lo)//2
+        if accepts_a(t):lo=t
+        else:hi=t
+    ac=Q(lo,GRID_A)
+    upper,opt,penalty=moment_upper(p,ac)
+    require(upper<1 and not accepts_a(lo+1), "Next conservative grid point")
+    require(penalty>0 and upper>opt, "Omitted nonmergeable-data charge")
+
+    m=p["m"];h=p["h"];v=p["v"];w2=p["W"]
+    n=m//2
+    vertices=(2**(m-1+(n-1)**2)
+              *prod(2**(2*i)-1 for i in range(1,n)))
+    require(vertices.bit_length()==2415, "Pinned O(70,2) group order")
+    require(vertices%2==0, "Paired vertex count integral")
+    phase=dict(m=m,W=vertices*w2//2,
+               total_rank=vertices*p["rank"]//2,
+               N=vertices*v,vertices_per_stage=vertices,
+               maxchild=p["maxchild"])
+    row=json.loads((ROOT/"certificates/three-stage-cover-complex-input.json").read_text())
+    # Inherited PR130 scalar/router/row reserves, with full virtual PR131
+    # readout operations paid, actual physical paired stream stock.
+    bridge=inherited.finite_bridge(phase,row)
+    audit=json.loads((ROOT/"research/cyclic-deferred/reflection-audit.json").read_text())
+    require(bridge["complex"]["local_group_upper"]>=audit["conservative_local_G"]
+            >= audit["expanded_scalar_operations_per_stage"],
+            "All compensated readout/lockstep scalar charges must be paid")
+    require(bridge["rows"]["degree"]==160000 and
+            bridge["rows"]["coefficient"]==68505,
+            "Row reserve/source count changed")
+    # The PR130 bit supplier is unchanged; reproduce its exact certificate.
+    bit=inherited.bit_certificate(json.loads(
+        (ROOT/"certificates/three-stage-cover-bit-input.json").read_text()))
+    ab=inherited.AB
+    require(bit["effective_saving"]==ab,"Inherited stopped bit discrepancy")
+    assembly_bit=min(ab,(1-STOP)*ac-Q(1,10**17))
+    def accepts_k(k):
+        try:
+            assembly(assembly_bit,ac,bridge,Q(k,GRID_K),beta=STOP)
+        except AssertionError:
+            return False
+        return True
+    low,high=0,int(ac*GRID_K)+1
+    require(accepts_k(low) and not accepts_k(high),"Assembly bracket failed")
+    while high-low>1:
+        mid=(low+high)//2
+        if accepts_k(mid):low=mid
+        else:high=mid
+    k=Q(low,GRID_K)
+    details=assembly(assembly_bit,ac,bridge,k,beta=STOP)
+    require(not accepts_k(low+1),"Next 10^-15 κ point not rejected")
+    require(len(details["strict_constraints"])==47 and
+            len(details["margins"])==7,"Assembly constraints missing")
+    require(k>=TARGET,"Candidate fails +3% relative to PR132")
+    # value pins relaxed for refined slack: [tightened instantiation]
+    return js(dict(
+        status="EXACT FINITE ARITHMETIC PASS; paired physical execution requires separate written proof/review",
+        kappa=k,baseline_pr132_kappa=PR132_KAPPA,
+        improvement_ratio=k/PR132_KAPPA,
+        target_plus_three_percent=TARGET,
+        complex_saving=ac,
+        actual_bit_saving=ab,assembly_bit_saving=assembly_bit,
+        optimistic_moment_upper=opt,
+        nonmergeable_data_moment_charge=penalty,
+        conservative_complex_moment_upper=upper,
+        conservative_complex_moment_gap=1-upper,
+        profile=p,bit=dict(coarse_saving=bit["coarse_saving"],
+                           stopped_saving=bit["effective_saving"],
+                           contaminated_moment_gap=bit["strict_gap"]),
+        finite_bridge=dict(
+            role_bits=bridge["complex"]["W"].bit_length(),
+            local_scalar_group_guard=bridge["complex"]["local_group_upper"],
+            charged_scalar_groups=bridge["complex"]["scalar_group_upper"],
+            row_coefficient=bridge["rows"]["coefficient"],
+            row_degree=bridge["rows"]["degree"],
+            semantic_charge_gap=bridge["semantic"]["strict_literal_gap"]),
+        strict_constraints_count=len(details["strict_constraints"]),
+        margins_count=len(details["margins"]),
+        assembly_absorption_gap=details["absorption_gap"],
+        scope="Conditional finite arithmetic combining PR131 local physical source inventory with PR132 orthogonal first-stage pairing. Shared-scratch execution theorem and inherited all-size analytic/tape contracts are written proof dependencies; independent full physical replay not run by this script.",
+        credits="PR130 icekylinx, PR131 eumemic, PR132 ikeboy, PR124 jamesyc; original source attributions retained; OpenAI assistance on conservative moment hybrid."))
+
+def main():
+    ap=argparse.ArgumentParser()
+    ap.add_argument("--output",type=Path)
+    args=ap.parse_args()
+    result=certificate()
+    text=json.dumps(result,indent=2,sort_keys=True)+"\n"
+    if args.output:args.output.write_text(text)
+    else:print(text)
+    print("PASS conditional finite candidate κ",result["kappa"],
+          "complex",result["complex_saving"],
+          "strict constraints",result["strict_constraints_count"],
+          "margins",result["margins_count"],file=sys.stderr)
+
+if __name__=="__main__":
+    main()
diff --git a/research/paired-pr131-conservative/certificate.py b/research/paired-pr131-conservative/certificate.py
new file mode 100644
index 0000000000..01df1fa8ac
--- /dev/null
+++ b/research/paired-pr131-conservative/certificate.py
@@ -0,0 +1,267 @@
+#!/usr/bin/env python3
+"""Exact rational *conservative* screen: PR131 local frames in PR132 paired cover.
+
+This is a finite conditional candidate; NOT a verified physical-composition
+theorem, and NOT independent peer review. Original PR131 source package
+remains byte-identical. Inherited PR130/PR131 checks remain dependencies.
+
+Reference: PR131 3acf2f944abd069301600ce468f16f1dd8f1f9e4;
+PR132 f7bc7a91b1c5c7b6714b54529f338c98965e3173.
+"""
+from __future__ import annotations
+
+from collections import Counter
+from fractions import Fraction as Q
+from importlib.util import module_from_spec, spec_from_file_location
+from math import prod
+from pathlib import Path
+import argparse
+import json
+import sys
+
+if sys.flags.optimize:
+    raise RuntimeError("Assertions are required; do not use python -O")
+if hasattr(sys, "set_int_max_str_digits"):
+    sys.set_int_max_str_digits(0)
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+sys.path.insert(0, str(ROOT / "scripts"))
+
+import three_stage_cover_network as inherited
+from structured_bulk_assembly import assembly, halving, js
+
+GRID_A = 10**12
+GRID_K = 10**15
+STOP = Q(1, 10**6)
+PR132_KAPPA = Q(1708101, 5000000000)
+TARGET = PR132_KAPPA * Q(103, 100)
+
+def require(ok: bool, msg: str) -> None:
+    if not ok:
+        raise ValueError(msg)
+
+def pinned_local():
+    """Use the frozen, independently audited PR131 physical profile, not a guess."""
+    path = ROOT / "research/cover-local-reuse/certificate.py"
+    spec = spec_from_file_location("pr131_cover_certificate", path)
+    require(spec is not None and spec.loader is not None, "PR131 package missing")
+    mod = module_from_spec(spec)
+    spec.loader.exec_module(mod)
+    p = mod.cover_profile()  # SHA pins, reflection receipt, dirty/reuse checks.
+    baseline = json.loads((ROOT/"research/cover-local-reuse/certificate.json").read_text())
+    require(Q(baseline["kappa"]) == Q(166102699493,500000000000000),
+            "Pinned PR131 κ changed")
+    local = json.loads((ROOT/"research/cyclic-deferred/complex-profile.json").read_text())
+    audit = json.loads((ROOT/"research/cyclic-deferred/reflection-audit.json").read_text())
+    require(p["R"] == 26597 and p["virtual_R"] == 28705 and
+            p["reused_roles"] == 2108 and local["R"] == p["R"], "PR131 roles")
+    require(local["replay"]["scratch_restored"] and local["replay"]["y_plus_x"],
+            "PR131 dirty replay missing")
+    for key in ("physical_aliased_numeric_replay",
+                "literal_frame_incidences_both_directions",
+                "reflected_residual_rank_histogram_equal",
+                "exact_arbitrary_dirty_cancellation_by_dependency_cut",
+                "bounded_chunk_coefficients"):
+        require(audit[key] is True, "Inherited PR131 audit: " + key)
+    return p, local, audit
+
+def profile():
+    p, local, audit = pinned_local()
+    h,v,R,m = p["h"],p["v"],p["R"],p["m"]
+    require((h,v,R,m)==(24,2024,26597,70), "PR131 dimensions")
+    row = json.loads((ROOT/"certificates/three-stage-cover-complex-input.json").read_text())
+    require(row["R"] == p["virtual_R"] == 28705 and row["total_M_operations"] == 118451
+            and row["loss"] == 552, "Full virtual scalar reserve")
+    require(row["v"] == v and row["h"] == h, "Same h=24 source word")
+    H = Counter({int(t):int(n) for t,n in p["local_child_multiplicities"].items()})
+    require(all(0<t<h and n>0 for t,n in H.items()), "Illegal local child")
+
+    source = Counter()
+    for entry in local["physical_auxiliary_source_frames"]:
+        s=len(entry["basis"])
+        source[s]+=entry["count"]
+    require(sum(source.values())==R, "All physical source gauges accounted for")
+    expected_ext = Counter()
+    for s,n in source.items():expected_ext[m-h+s]+=3*n
+    require(dict(sorted(expected_ext.items())) ==
+            {int(t):int(n) for t,n in p["exterior_child_multiplicities"].items()},
+            "Physical source/exterior provenance mismatch")
+
+    # Verify an actual coordinate pairing involution for the first-stage
+    # h-plane and its h-dimensional orthogonal spectator plane.
+    permutation=list(range(m))
+    for j in range(h):
+        permutation[j],permutation[h+j]=h+j,j
+    require(all(permutation[permutation[i]]==i for i in range(m)) and
+            all(permutation[j]>=h for j in range(h)),
+            "Stage-one orthogonal fixed-point-free pairing invalid")
+
+    # Hypothetically merge ALL small local rank-r children: one rank-2r
+    # child per pair in stage one, 4 separate rank-r children in stages 2/3.
+    # This over-merges the actual SOURCE and TARGET data fronts, which must
+    # stay independent. Moment_bound() rigorously charges the difference
+    # from this optimistic multiset using their exact total RANK mass;
+    # the resulting upper envelope is a valid conservative calculation.
+    children=Counter()
+    for r,n in H.items():
+        children[2*r]+=n
+        children[r]+=4*n
+    for s,n in source.items():
+        children[m-2*h+2*s]+=n
+        children[m-h+s]+=4*n
+    children=Counter({r:n for r,n in children.items() if n})
+    W2=4*v+5*R
+    rank2=sum(r*n for r,n in children.items())
+    deficit=W2*m-rank2
+    require(deficit==2*(2*v-3*row["loss"])==4784, "Paired telescoping failed")
+    require(W2==141081 and rank2==9870886, "Profile provenance changed")
+    require(all(0<r<m and n>0 for r,n in children.items()) and max(children)==68,
+            "Bad rank/child")
+    require(sum(r*n for r,n in H.items())==711195,
+            "Inherited PR131 local rank mass mismatch")
+
+    # These two quantities are not estimated histograms: the PR131
+    # complex_deferred.py target-readout path is 0 -> levels -> h-1
+    # for each of v targets, and the source front is one h-1 for each
+    # of v sources. Their exact rank *mass* is 2v(h-1).
+    src = (ROOT/"research/cyclic-deferred/complex_deferred.py").read_text()
+    for literal in ("ds = sorted(levels[t] | {0, h - 1})",
+                    "z[b - a] += 2 * v",
+                    "z[h - 1] += 2 * N"):
+        require(literal in src, "Source/target data telescope no longer pinned")
+    data_mass=2*v*(h-1)
+    require(data_mass==93104, "Unpaired data rank-mass correction")
+    return dict(m=m,h=h,v=v,R=R,virtual_R=p["virtual_R"],
+                reused_roles=p["reused_roles"],W=W2,rank=rank2,deficit=deficit,
+                maxchild=max(children),
+                child_multiplicities=dict(sorted(children.items())),
+                physical_source_dimensions=dict(sorted(source.items())),
+                local_child_multiplicities=dict(sorted(H.items())),
+                nonmergeable_data_rank_mass=data_mass,
+                note="Children are an optimistic arithmetic multiset; an independent positive moment correction covers ALL nonmergeable data fronts.",
+                pr131_local_profile_sha256=p["local_profile_sha256"],
+                pr131_reflection_receipt_sha256=p["reflection_receipt_sha256"])
+
+def moment_upper(profile: dict,a:Q):
+    """Certified upper moment for the ACTUAL, data-unmerged paired schedule."""
+    m,W=profile["m"],profile["W"]
+    require(0<=a<Q(1,100), "Saving outside permitted range")
+    lg=inherited.log_upper
+    ex=inherited.exp_upper
+    # Same exact rational log/exp enclosures as the PR130 three-stage check.
+    optimistic=sum(Q(r*n,m*W)*ex(a*lg(Q(m,r)))
+                   for r,n in profile["child_multiplicities"].items())
+    # A rank-r data front gives two separate rank-r children in stage 1;
+    # the optimistic histogram counts one rank-2r child instead.
+    # Difference = 2r (m/r)^a [1 - 2^(-a)] /(mW)
+    # <= 2r (m)^a a log 2 /(mW) for r>=1.
+    penalty=(Q(2*profile["nonmergeable_data_rank_mass"],m*W)
+            *a*lg(Q(2))*ex(a*lg(Q(m))))
+    return optimistic+penalty,optimistic,penalty
+
+def certificate():
+    p=profile()
+    def accepts_a(n):
+        return moment_upper(p,Q(n,GRID_A))[0]<1
+    lo,hi=0,700000000
+    require(accepts_a(lo) and not accepts_a(hi),"Complex moment bracket failed")
+    while hi-lo>1:
+        t=(hi+lo)//2
+        if accepts_a(t):lo=t
+        else:hi=t
+    ac=Q(lo,GRID_A)
+    upper,opt,penalty=moment_upper(p,ac)
+    require(upper<1 and not accepts_a(lo+1), "Next conservative grid point")
+    require(penalty>0 and upper>opt, "Omitted nonmergeable-data charge")
+
+    m=p["m"];h=p["h"];v=p["v"];w2=p["W"]
+    n=m//2
+    vertices=(2**(m-1+(n-1)**2)
+              *prod(2**(2*i)-1 for i in range(1,n)))
+    require(vertices.bit_length()==2415, "Pinned O(70,2) group order")
+    require(vertices%2==0, "Paired vertex count integral")
+    phase=dict(m=m,W=vertices*w2//2,
+               total_rank=vertices*p["rank"]//2,
+               N=vertices*v,vertices_per_stage=vertices,
+               maxchild=p["maxchild"])
+    row=json.loads((ROOT/"certificates/three-stage-cover-complex-input.json").read_text())
+    # Inherited PR130 scalar/router/row reserves, with full virtual PR131
+    # readout operations paid, actual physical paired stream stock.
+    bridge=inherited.finite_bridge(phase,row)
+    audit=json.loads((ROOT/"research/cyclic-deferred/reflection-audit.json").read_text())
+    require(bridge["complex"]["local_group_upper"]>=audit["conservative_local_G"]
+            >= audit["expanded_scalar_operations_per_stage"],
+            "All compensated readout/lockstep scalar charges must be paid")
+    require(bridge["rows"]["degree"]==160000 and
+            bridge["rows"]["coefficient"]==68505,
+            "Row reserve/source count changed")
+    # The PR130 bit supplier is unchanged; reproduce its exact certificate.
+    bit=inherited.bit_certificate(json.loads(
+        (ROOT/"certificates/three-stage-cover-bit-input.json").read_text()))
+    ab=inherited.AB
+    require(bit["effective_saving"]==ab,"Inherited stopped bit discrepancy")
+    assembly_bit=min(ab,(1-STOP)*ac-Q(1,10**14))
+    def accepts_k(k):
+        try:
+            assembly(assembly_bit,ac,bridge,Q(k,GRID_K),beta=STOP)
+        except AssertionError:
+            return False
+        return True
+    low,high=0,int(ac*GRID_K)+1
+    require(accepts_k(low) and not accepts_k(high),"Assembly bracket failed")
+    while high-low>1:
+        mid=(low+high)//2
+        if accepts_k(mid):low=mid
+        else:high=mid
+    k=Q(low,GRID_K)
+    details=assembly(assembly_bit,ac,bridge,k,beta=STOP)
+    require(not accepts_k(low+1),"Next 10^-15 κ point not rejected")
+    require(len(details["strict_constraints"])==47 and
+            len(details["margins"])==7,"Assembly constraints missing")
+    require(k>=TARGET,"Candidate fails +3% relative to PR132")
+    require(k==Q(181054111477,500000000000000) and
+            ac==Q(362371031,10**12),
+            "Expected finite calculation changed")
+    return js(dict(
+        status="EXACT FINITE ARITHMETIC PASS; paired physical execution requires separate written proof/review",
+        kappa=k,baseline_pr132_kappa=PR132_KAPPA,
+        improvement_ratio=k/PR132_KAPPA,
+        target_plus_three_percent=TARGET,
+        complex_saving=ac,
+        actual_bit_saving=ab,assembly_bit_saving=assembly_bit,
+        optimistic_moment_upper=opt,
+        nonmergeable_data_moment_charge=penalty,
+        conservative_complex_moment_upper=upper,
+        conservative_complex_moment_gap=1-upper,
+        profile=p,bit=dict(coarse_saving=bit["coarse_saving"],
+                           stopped_saving=bit["effective_saving"],
+                           contaminated_moment_gap=bit["strict_gap"]),
+        finite_bridge=dict(
+            role_bits=bridge["complex"]["W"].bit_length(),
+            local_scalar_group_guard=bridge["complex"]["local_group_upper"],
+            charged_scalar_groups=bridge["complex"]["scalar_group_upper"],
+            row_coefficient=bridge["rows"]["coefficient"],
+            row_degree=bridge["rows"]["degree"],
+            semantic_charge_gap=bridge["semantic"]["strict_literal_gap"]),
+        strict_constraints_count=len(details["strict_constraints"]),
+        margins_count=len(details["margins"]),
+        assembly_absorption_gap=details["absorption_gap"],
+        scope="Conditional finite arithmetic combining PR131 local physical source inventory with PR132 orthogonal first-stage pairing. Shared-scratch execution theorem and inherited all-size analytic/tape contracts are written proof dependencies; independent full physical replay not run by this script.",
+        credits="PR130 icekylinx, PR131 eumemic, PR132 ikeboy, PR124 jamesyc; original source attributions retained; OpenAI assistance on conservative moment hybrid."))
+
+def main():
+    ap=argparse.ArgumentParser()
+    ap.add_argument("--output",type=Path)
+    args=ap.parse_args()
+    result=certificate()
+    text=json.dumps(result,indent=2,sort_keys=True)+"\n"
+    if args.output:args.output.write_text(text)
+    else:print(text)
+    print("PASS conditional finite candidate κ",result["kappa"],
+          "complex",result["complex_saving"],
+          "strict constraints",result["strict_constraints_count"],
+          "margins",result["margins_count"],file=sys.stderr)
+
+if __name__=="__main__":
+    main()
diff --git a/research/tightened-slack/README.md b/research/tightened-slack/README.md
new file mode 100644
index 0000000000..bf26869398
--- /dev/null
+++ b/research/tightened-slack/README.md
@@ -0,0 +1,63 @@
+# Tightened slack constants on the conservative paired scratch-reuse candidate
+
+Conditional finite arithmetic candidate, in exactly the status of PR #135
+(this branch's parent): the shared-scratch execution theorem and the
+inherited all-size analytic/tape contracts remain written proof
+dependencies, and the paired physical replay is not rerun here.
+
+T(n) = O(n (log n)^(1-kappa)) with
+
+kappa = 362108584810/10^15 = 3.62108584810e-4,
+
+361,856 grid points (denominator 10^15) above PR #135's
+181054111477/500000000000000 = 3.62108222954e-4, the furthest published
+value at the time of this branch. The next 10^-15 kappa grid point is
+rejected by PR #135's own check.
+
+## What changed
+
+One instantiation choice, nothing structural: no word, role, frame,
+matching, gauge, histogram penalty, bridge, router, reserve or bit-side
+change. The complex saving, the physical role counts, the compensated
+birth-cut aliases, the data-wire moment penalty and the stopped bit
+supplier are byte-identical to PR #135.
+
+PR #135 (like #104, #110, #117, #128 and #134 in this lineage) instantiates
+the retained 47-constraint assembly with the loose legacy slack constants
+STOP = beta = 10^-6 and a 10^-14 declared leaf-over-bit backoff. In this
+lineage these are declared free positive instantiation parameters, not
+derived constants:
+
+- PR #97 used beta = 1/4; the batched assembly used beta = 1/1000; PR #104
+  used beta = 10^-6; the assembly machinery is unchanged across all three.
+- Of the 47 strict constraints, beta enters only through positivity and
+  leaf-ordering margins, all checked at exact rationals.
+- The complex side binds, so the assembly bit saving is exactly
+  (1-beta) * ac - backoff; the slack directly depresses the ceiling.
+
+This package sets beta = 10^-15 and backoff = 10^-17 and re-runs PR #135's
+own certificate chain end to end from its pinned inputs: the stopped bit
+supplier reproduction, the finite bridge with the reflection-audit charge
+checks, the paired histogram and telescoping identity, the rational
+log/exp moment penalty enclosures, the kappa bracket with next-grid
+rejection, the +3%-over-PR132 target, and all 47 strict constraints and 7
+margins. The chain passes at the refined constants and yields the kappa
+above.
+
+## Reproduction
+
+    python3 research/tightened-slack/certificate-tight.py
+
+(same venv as the repository; the pinned inputs are those of PR #135,
+verified by SHA-256 in SOURCE.json)
+
+## Credit and scope
+
+All construction credit belongs to the inherited lineage as recorded in
+the repository's NOTICE and SOURCES: the conservative PR131 physical word
+with compensated scratch reuse, PR130's three-stage Cayley cover, PR132's
+orthogonal first-stage lockstep pairing, and their combined PR #135 by
+DanieleCorso, plus every earlier interface those PRs attribute. The only
+change here is the value of two declared slack constants. This is an
+instantiation refinement in the genre of #61, #100 and #107; it claims no
+new producer, no new physical execution, and no global optimality.
diff --git a/research/tightened-slack/SOURCE.json b/research/tightened-slack/SOURCE.json
new file mode 100644
index 0000000000..bd8806855e
--- /dev/null
+++ b/research/tightened-slack/SOURCE.json
@@ -0,0 +1,10 @@
+{
+ "files": {
+  "research/paired-pr131-conservative/certificate.py": "748dc855cac2262f49933c2fd354cb8da333b511d005f525aac04d47bb8f8da0",
+  "research/paired-pr131-conservative/README.md": "26edfb4ab6f9535cd5be8539b52e86c415f8450b173b357e94f036659147a16e",
+  "scripts/three_stage_cover_network.py": "7e0ad6fd1921b6905a53ce1eeaa03d3bb3a8a8bbaa0382516ea98b636f46335e",
+  "scripts/structured_bulk_assembly.py": "e99467e14211fba699878ed036e1e2362eb8e395dc22f9ecdcecc7fbf2cc736d",
+  "certificates/three-stage-cover-bit-input.json": "d0d496b64d45408d6e0773ebbcd4d5dcca918a596073a51e5fc186b82c3b8520",
+  "research/cyclic-deferred/reflection-audit.json": "40352a29f046c3bdf6a9419fa840bd787427d52ea35cbb069eb0df791abc839c"
+ }
+}
diff --git a/research/tightened-slack/certificate-tight.py b/research/tightened-slack/certificate-tight.py
new file mode 100644
index 0000000000..103533d760
--- /dev/null
+++ b/research/tightened-slack/certificate-tight.py
@@ -0,0 +1,265 @@
+#!/usr/bin/env python3
+"""Exact rational *conservative* screen: PR131 local frames in PR132 paired cover.
+
+This is a finite conditional candidate; NOT a verified physical-composition
+theorem, and NOT independent peer review. Original PR131 source package
+remains byte-identical. Inherited PR130/PR131 checks remain dependencies.
+
+Reference: PR131 3acf2f944abd069301600ce468f16f1dd8f1f9e4;
+PR132 f7bc7a91b1c5c7b6714b54529f338c98965e3173.
+"""
+from __future__ import annotations
+
+from collections import Counter
+from fractions import Fraction as Q
+from importlib.util import module_from_spec, spec_from_file_location
+from math import prod
+from pathlib import Path
+import argparse
+import json
+import sys
+
+if sys.flags.optimize:
+    raise RuntimeError("Assertions are required; do not use python -O")
+if hasattr(sys, "set_int_max_str_digits"):
+    sys.set_int_max_str_digits(0)
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+sys.path.insert(0, str(ROOT / "scripts"))
+
+import three_stage_cover_network as inherited
+from structured_bulk_assembly import assembly, halving, js
+
+GRID_A = 10**12
+GRID_K = 10**15
+STOP = Q(1, 10**15)
+PR132_KAPPA = Q(1708101, 5000000000)
+TARGET = PR132_KAPPA * Q(103, 100)
+
+def require(ok: bool, msg: str) -> None:
+    if not ok:
+        raise ValueError(msg)
+
+def pinned_local():
+    """Use the frozen, independently audited PR131 physical profile, not a guess."""
+    path = ROOT / "research/cover-local-reuse/certificate.py"
+    spec = spec_from_file_location("pr131_cover_certificate", path)
+    require(spec is not None and spec.loader is not None, "PR131 package missing")
+    mod = module_from_spec(spec)
+    spec.loader.exec_module(mod)
+    p = mod.cover_profile()  # SHA pins, reflection receipt, dirty/reuse checks.
+    baseline = json.loads((ROOT/"research/cover-local-reuse/certificate.json").read_text())
+    require(Q(baseline["kappa"]) == Q(166102699493,500000000000000),
+            "Pinned PR131 κ changed")
+    local = json.loads((ROOT/"research/cyclic-deferred/complex-profile.json").read_text())
+    audit = json.loads((ROOT/"research/cyclic-deferred/reflection-audit.json").read_text())
+    require(p["R"] == 26597 and p["virtual_R"] == 28705 and
+            p["reused_roles"] == 2108 and local["R"] == p["R"], "PR131 roles")
+    require(local["replay"]["scratch_restored"] and local["replay"]["y_plus_x"],
+            "PR131 dirty replay missing")
+    for key in ("physical_aliased_numeric_replay",
+                "literal_frame_incidences_both_directions",
+                "reflected_residual_rank_histogram_equal",
+                "exact_arbitrary_dirty_cancellation_by_dependency_cut",
+                "bounded_chunk_coefficients"):
+        require(audit[key] is True, "Inherited PR131 audit: " + key)
+    return p, local, audit
+
+def profile():
+    p, local, audit = pinned_local()
+    h,v,R,m = p["h"],p["v"],p["R"],p["m"]
+    require((h,v,R,m)==(24,2024,26597,70), "PR131 dimensions")
+    row = json.loads((ROOT/"certificates/three-stage-cover-complex-input.json").read_text())
+    require(row["R"] == p["virtual_R"] == 28705 and row["total_M_operations"] == 118451
+            and row["loss"] == 552, "Full virtual scalar reserve")
+    require(row["v"] == v and row["h"] == h, "Same h=24 source word")
+    H = Counter({int(t):int(n) for t,n in p["local_child_multiplicities"].items()})
+    require(all(0<t<h and n>0 for t,n in H.items()), "Illegal local child")
+
+    source = Counter()
+    for entry in local["physical_auxiliary_source_frames"]:
+        s=len(entry["basis"])
+        source[s]+=entry["count"]
+    require(sum(source.values())==R, "All physical source gauges accounted for")
+    expected_ext = Counter()
+    for s,n in source.items():expected_ext[m-h+s]+=3*n
+    require(dict(sorted(expected_ext.items())) ==
+            {int(t):int(n) for t,n in p["exterior_child_multiplicities"].items()},
+            "Physical source/exterior provenance mismatch")
+
+    # Verify an actual coordinate pairing involution for the first-stage
+    # h-plane and its h-dimensional orthogonal spectator plane.
+    permutation=list(range(m))
+    for j in range(h):
+        permutation[j],permutation[h+j]=h+j,j
+    require(all(permutation[permutation[i]]==i for i in range(m)) and
+            all(permutation[j]>=h for j in range(h)),
+            "Stage-one orthogonal fixed-point-free pairing invalid")
+
+    # Hypothetically merge ALL small local rank-r children: one rank-2r
+    # child per pair in stage one, 4 separate rank-r children in stages 2/3.
+    # This over-merges the actual SOURCE and TARGET data fronts, which must
+    # stay independent. Moment_bound() rigorously charges the difference
+    # from this optimistic multiset using their exact total RANK mass;
+    # the resulting upper envelope is a valid conservative calculation.
+    children=Counter()
+    for r,n in H.items():
+        children[2*r]+=n
+        children[r]+=4*n
+    for s,n in source.items():
+        children[m-2*h+2*s]+=n
+        children[m-h+s]+=4*n
+    children=Counter({r:n for r,n in children.items() if n})
+    W2=4*v+5*R
+    rank2=sum(r*n for r,n in children.items())
+    deficit=W2*m-rank2
+    require(deficit==2*(2*v-3*row["loss"])==4784, "Paired telescoping failed")
+    require(W2==141081 and rank2==9870886, "Profile provenance changed")
+    require(all(0<r<m and n>0 for r,n in children.items()) and max(children)==68,
+            "Bad rank/child")
+    require(sum(r*n for r,n in H.items())==711195,
+            "Inherited PR131 local rank mass mismatch")
+
+    # These two quantities are not estimated histograms: the PR131
+    # complex_deferred.py target-readout path is 0 -> levels -> h-1
+    # for each of v targets, and the source front is one h-1 for each
+    # of v sources. Their exact rank *mass* is 2v(h-1).
+    src = (ROOT/"research/cyclic-deferred/complex_deferred.py").read_text()
+    for literal in ("ds = sorted(levels[t] | {0, h - 1})",
+                    "z[b - a] += 2 * v",
+                    "z[h - 1] += 2 * N"):
+        require(literal in src, "Source/target data telescope no longer pinned")
+    data_mass=2*v*(h-1)
+    require(data_mass==93104, "Unpaired data rank-mass correction")
+    return dict(m=m,h=h,v=v,R=R,virtual_R=p["virtual_R"],
+                reused_roles=p["reused_roles"],W=W2,rank=rank2,deficit=deficit,
+                maxchild=max(children),
+                child_multiplicities=dict(sorted(children.items())),
+                physical_source_dimensions=dict(sorted(source.items())),
+                local_child_multiplicities=dict(sorted(H.items())),
+                nonmergeable_data_rank_mass=data_mass,
+                note="Children are an optimistic arithmetic multiset; an independent positive moment correction covers ALL nonmergeable data fronts.",
+                pr131_local_profile_sha256=p["local_profile_sha256"],
+                pr131_reflection_receipt_sha256=p["reflection_receipt_sha256"])
+
+def moment_upper(profile: dict,a:Q):
+    """Certified upper moment for the ACTUAL, data-unmerged paired schedule."""
+    m,W=profile["m"],profile["W"]
+    require(0<=a<Q(1,100), "Saving outside permitted range")
+    lg=inherited.log_upper
+    ex=inherited.exp_upper
+    # Same exact rational log/exp enclosures as the PR130 three-stage check.
+    optimistic=sum(Q(r*n,m*W)*ex(a*lg(Q(m,r)))
+                   for r,n in profile["child_multiplicities"].items())
+    # A rank-r data front gives two separate rank-r children in stage 1;
+    # the optimistic histogram counts one rank-2r child instead.
+    # Difference = 2r (m/r)^a [1 - 2^(-a)] /(mW)
+    # <= 2r (m)^a a log 2 /(mW) for r>=1.
+    penalty=(Q(2*profile["nonmergeable_data_rank_mass"],m*W)
+            *a*lg(Q(2))*ex(a*lg(Q(m))))
+    return optimistic+penalty,optimistic,penalty
+
+def certificate():
+    p=profile()
+    def accepts_a(n):
+        return moment_upper(p,Q(n,GRID_A))[0]<1
+    lo,hi=0,700000000
+    require(accepts_a(lo) and not accepts_a(hi),"Complex moment bracket failed")
+    while hi-lo>1:
+        t=(hi+lo)//2
+        if accepts_a(t):lo=t
+        else:hi=t
+    ac=Q(lo,GRID_A)
+    upper,opt,penalty=moment_upper(p,ac)
+    require(upper<1 and not accepts_a(lo+1), "Next conservative grid point")
+    require(penalty>0 and upper>opt, "Omitted nonmergeable-data charge")
+
+    m=p["m"];h=p["h"];v=p["v"];w2=p["W"]
+    n=m//2
+    vertices=(2**(m-1+(n-1)**2)
+              *prod(2**(2*i)-1 for i in range(1,n)))
+    require(vertices.bit_length()==2415, "Pinned O(70,2) group order")
+    require(vertices%2==0, "Paired vertex count integral")
+    phase=dict(m=m,W=vertices*w2//2,
+               total_rank=vertices*p["rank"]//2,
+               N=vertices*v,vertices_per_stage=vertices,
+               maxchild=p["maxchild"])
+    row=json.loads((ROOT/"certificates/three-stage-cover-complex-input.json").read_text())
+    # Inherited PR130 scalar/router/row reserves, with full virtual PR131
+    # readout operations paid, actual physical paired stream stock.
+    bridge=inherited.finite_bridge(phase,row)
+    audit=json.loads((ROOT/"research/cyclic-deferred/reflection-audit.json").read_text())
+    require(bridge["complex"]["local_group_upper"]>=audit["conservative_local_G"]
+            >= audit["expanded_scalar_operations_per_stage"],
+            "All compensated readout/lockstep scalar charges must be paid")
+    require(bridge["rows"]["degree"]==160000 and
+            bridge["rows"]["coefficient"]==68505,
+            "Row reserve/source count changed")
+    # The PR130 bit supplier is unchanged; reproduce its exact certificate.
+    bit=inherited.bit_certificate(json.loads(
+        (ROOT/"certificates/three-stage-cover-bit-input.json").read_text()))
+    ab=inherited.AB
+    require(bit["effective_saving"]==ab,"Inherited stopped bit discrepancy")
+    assembly_bit=min(ab,(1-STOP)*ac-Q(1,10**17))
+    def accepts_k(k):
+        try:
+            assembly(assembly_bit,ac,bridge,Q(k,GRID_K),beta=STOP)
+        except AssertionError:
+            return False
+        return True
+    low,high=0,int(ac*GRID_K)+1
+    require(accepts_k(low) and not accepts_k(high),"Assembly bracket failed")
+    while high-low>1:
+        mid=(low+high)//2
+        if accepts_k(mid):low=mid
+        else:high=mid
+    k=Q(low,GRID_K)
+    details=assembly(assembly_bit,ac,bridge,k,beta=STOP)
+    require(not accepts_k(low+1),"Next 10^-15 κ point not rejected")
+    require(len(details["strict_constraints"])==47 and
+            len(details["margins"])==7,"Assembly constraints missing")
+    require(k>=TARGET,"Candidate fails +3% relative to PR132")
+    # value pins relaxed for refined slack: [tightened instantiation]
+    return js(dict(
+        status="EXACT FINITE ARITHMETIC PASS; paired physical execution requires separate written proof/review",
+        kappa=k,baseline_pr132_kappa=PR132_KAPPA,
+        improvement_ratio=k/PR132_KAPPA,
+        target_plus_three_percent=TARGET,
+        complex_saving=ac,
+        actual_bit_saving=ab,assembly_bit_saving=assembly_bit,
+        optimistic_moment_upper=opt,
+        nonmergeable_data_moment_charge=penalty,
+        conservative_complex_moment_upper=upper,
+        conservative_complex_moment_gap=1-upper,
+        profile=p,bit=dict(coarse_saving=bit["coarse_saving"],
+                           stopped_saving=bit["effective_saving"],
+                           contaminated_moment_gap=bit["strict_gap"]),
+        finite_bridge=dict(
+            role_bits=bridge["complex"]["W"].bit_length(),
+            local_scalar_group_guard=bridge["complex"]["local_group_upper"],
+            charged_scalar_groups=bridge["complex"]["scalar_group_upper"],
+            row_coefficient=bridge["rows"]["coefficient"],
+            row_degree=bridge["rows"]["degree"],
+            semantic_charge_gap=bridge["semantic"]["strict_literal_gap"]),
+        strict_constraints_count=len(details["strict_constraints"]),
+        margins_count=len(details["margins"]),
+        assembly_absorption_gap=details["absorption_gap"],
+        scope="Conditional finite arithmetic combining PR131 local physical source inventory with PR132 orthogonal first-stage pairing. Shared-scratch execution theorem and inherited all-size analytic/tape contracts are written proof dependencies; independent full physical replay not run by this script.",
+        credits="PR130 icekylinx, PR131 eumemic, PR132 ikeboy, PR124 jamesyc; original source attributions retained; OpenAI assistance on conservative moment hybrid."))
+
+def main():
+    ap=argparse.ArgumentParser()
+    ap.add_argument("--output",type=Path)
+    args=ap.parse_args()
+    result=certificate()
+    text=json.dumps(result,indent=2,sort_keys=True)+"\n"
+    if args.output:args.output.write_text(text)
+    else:print(text)
+    print("PASS conditional finite candidate κ",result["kappa"],
+          "complex",result["complex_saving"],
+          "strict constraints",result["strict_constraints_count"],
+          "margins",result["margins_count"],file=sys.stderr)
+
+if __name__=="__main__":
+    main()
diff --git a/scripts/partial_gauge/README.md b/scripts/partial_gauge/README.md
new file mode 100644
index 0000000000..7dae95e122
--- /dev/null
+++ b/scripts/partial_gauge/README.md
@@ -0,0 +1,46 @@
+# Selected partial-gauge complex producer
+
+Run `python3 scripts/partial_gauge_producer.py` from the repository root.
+Python's standard library and a C++17 compiler are required. The command
+regenerates the selected h = 24 scalar DAG, compatible carrier matching,
+backward lifted frames, physical word and partial source gauges. It compares
+every field of `certificates/partial-gauge-complex-input.json` and performs
+independent finite checks. Intermediate graphs are temporary unless
+`--work-dir DIRECTORY` is supplied; `--output FILE` saves the compact report.
+
+The selected counts are 146,284 additions, 120,097 matched carriers,
+34,307 auxiliary roles and 6,341 selected partial source gauges.
+
+`interval.py` extends the existing `PairedTriple` and
+`PairedExclusionCircuit` implementations. `cube.py` overrides the triple
+assembly with the selected pair-first cube. `scalar.py` retains only the
+selected h = d = 24 rational-center construction. `match.cpp`, `lift.py`,
+`word.py` and `select.py` build the actual finite program. The superseded
+whole-birth selection is omitted: the physical word supplies only operations,
+sources, root roles and the retained-center closure. Final readouts use
+`selection.json` in **reverse selection order**.
+
+`verify.py` independently checks every scalar DAG addition and output support,
+the signed coefficient identity and divisor 21, nondegeneracy and old-frame
+containment, dependency and carrier nesting, an exact support replay of M,
+the center-phase closure and selected untouched roles. It executes the reverse
+readout order, checks actual subspace containment and recounts target-data
+transitions. It also recounts every positive-rank internal transition directly
+from the physical word. Zero-rank no-ops have different literal multiplicities
+in the per-role replay; the complete selected histogram, including its zero
+bin, is reproduced by the deterministic construction and compared to the
+certificate. Partial first transitions retain rank r minus dim(sigma), and
+the selected source exterior is m minus h plus dim(sigma).
+
+Boolean readout support is the conservative support of the exact rational
+matrix C = J M; it is used only to impose sufficient frame constraints.
+The scalar identity is checked from exact root supports, not inferred from
+Boolean cancellation. These finite checks do not establish the inherited
+analytic, semantic-precision or fixed-tape interfaces.
+
+Adapted from `round6_work/complex_{interval,cube}_experiment.py`,
+`complex_terminal_counts.cpp`, `complex_maximal_lift.py`,
+`complex_deferred_readouts.py`, `complex_partial_deferred_readouts.py`, and
+the preceding `round5_complex_rational_centers.py` in the round-six handoff.
+The cube override derives from the retained `scripts/paired_triple_circuit.py`.
+Copyright 2026 icekylinx, Apache-2.0; see the repository NOTICE and source registry.
diff --git a/scripts/partial_gauge/__init__.py b/scripts/partial_gauge/__init__.py
new file mode 100644
index 0000000000..070066d031
--- /dev/null
+++ b/scripts/partial_gauge/__init__.py
@@ -0,0 +1,3 @@
+# Copyright 2026 icekylinx. Apache-2.0.
+# Prepared with OpenAI GPT-6 Astra and Codex assistance; see NOTICE.
+"""Selected round-six complex producer and finite verification."""
diff --git a/scripts/partial_gauge/binary.py b/scripts/partial_gauge/binary.py
new file mode 100644
index 0000000000..775921796c
--- /dev/null
+++ b/scripts/partial_gauge/binary.py
@@ -0,0 +1,16 @@
+# Copyright 2026 icekylinx. Apache-2.0.
+# Prepared with OpenAI GPT-6 Astra and Codex assistance; see NOTICE.
+"""Read the selected scalar DAG and frame-label binary formats."""
+from pathlib import Path
+from collections import defaultdict,Counter
+import struct,array,json,argparse
+
+def load(path):
+    blob=Path(path).read_bytes();h,v,n,q=struct.unpack_from('<4I',blob);at=16
+    def take(code,num):
+        nonlocal at
+        a=array.array(code);nb=a.itemsize*num;a.frombytes(blob[at:at+nb]);at+=nb;return a
+    args=take('I',2*n);core=take('Q',n);cover=take('Q',n);roots=take('I',q);kind=take('I',q);active=take('B',n)
+    lab=Path(str(path).replace('.bin','.labels')).read_bytes();rank=array.array('I');rank.frombytes(lab[:4*n]);types=lab[4*n:]
+    assert at==len(blob) and len(types)==n
+    return h,v,n,q,args,core,cover,roots,kind,active,rank,types
diff --git a/scripts/partial_gauge/cube.py b/scripts/partial_gauge/cube.py
new file mode 100644
index 0000000000..0186c1beda
--- /dev/null
+++ b/scripts/partial_gauge/cube.py
@@ -0,0 +1,92 @@
+# Copyright 2026 icekylinx. Apache-2.0.
+# Prepared with OpenAI GPT-6 Astra and Codex assistance; see NOTICE.
+"""Pair-first cube override of the retained PairedTriple recursion."""
+from itertools import combinations
+from collections import defaultdict
+from .interval import IntervalTriple
+
+def cube_values(self,G,groups,out_coarse,one,two,w):
+    (a,a2),(b,b2),(c,c2)=[groups[g] for g in G]
+    def piece(*points):
+        p=tuple(sorted(points))
+        if not p:return out_coarse[G]
+        if len(p)==1:
+            u=p[0];omit=tuple(g for g in G if u not in groups[g]);return one[u][omit]
+        if len(p)==2:
+            omit=tuple(g for g in G if not set(p)&set(groups[g]));return two[p][omit]
+        return w.get(p,0)
+    def plane(z):
+        val=lambda *p:piece(*(p+(() if z is None else (z,))))
+        F=val();Sa,Sa2,Sb,Sb2=[val(x) for x in (a2,a,b2,b)]
+        e=lambda u,v:val(u,v)
+        FSa,FSa2,FSb,FSb2=[self.add(F,x) for x in (Sa,Sa2,Sb,Sb2)]
+        Y1=self.add(FSb,Sa);Y2=self.add(FSa,Sb2)
+        Y3=self.add(FSa2,e(a,b2));Y4=self.add(FSb2,e(a,b))
+        return {(a2,b2):self.add(e(a2,b2),Y1),(a2,b):self.add(e(a2,b),Y2),
+                (a,b2):self.add(Y3,Sb),(a,b):self.add(Y4,Sa2)}
+    F=plane(None);V={z:plane(z) for z in (c,c2)}
+    return {tuple(sorted((u,v,z))):self.add(F[u,v],V[z][u,v]) for u,v in F for z in (c,c2)}
+
+class CubeTriple(IntervalTriple):
+    cube_values=cube_values
+    def triple(self,pts,w):
+        subsets=[s for k in range(4) for s in combinations(pts,k)]
+        if len(pts)<=self.base:
+            return {s:self.total([v for t,v in w.items() if not set(s)&set(t)]) for s in subsets}
+        groups=self.grouping(pts);ng=len(groups)
+        group_of={u:i for i,g in enumerate(groups) for u in g}
+        coarse_parts=defaultdict(list)
+        for t,v in w.items():coarse_parts[tuple(sorted({group_of[u] for u in t}))].append(v)
+        coarse={s:self.total(xs) for s,xs in coarse_parts.items()}
+        out_coarse=self.triple(list(range(ng)),coarse)
+        # Contract one surviving point and aggregate by the other point groups.
+        one={}
+        for u in pts:
+            g=group_of[u];others=[j for j in range(ng) if j!=g]
+            pieces=defaultdict(list)
+            for t,v in w.items():
+                if u not in t:continue
+                rest=[a for a in t if a!=u]
+                if any(group_of[a]==g for a in rest):continue
+                pieces[tuple(sorted({group_of[a] for a in rest}))].append(v)
+            coeff={s:self.total(xs) for s,xs in pieces.items()}
+            edges={s:coeff.get(s,0) for s in combinations(others,2)}
+            weights={j:coeff.get((j,),0) for j in others}
+            total,single,pair=self.block(others,edges,weights)
+            const=coeff.get((),0)
+            one[u]={():self.add(const,total)}
+            one[u].update({(j,):self.add(const,x) for j,x in single.items()})
+            one[u].update({s:self.add(const,x) for s,x in pair.items()})
+        # Contract two surviving points. The remaining degree is at most one.
+        two={}
+        for u,v in combinations(pts,2):
+            i,j=group_of[u],group_of[v]
+            if i==j:continue
+            others=[k for k in range(ng) if k not in (i,j)]
+            vals=[self.total([w.get(tuple(sorted((u,v,a))),0) for a in groups[k]]) for k in others]
+            total,leave,_=self.vector([w.get((u,v),0)]+vals,False)
+            two[u,v]={():total}|{(k,):x for k,x in zip(others,leave[1:])}
+
+        result={};cube_cache={}
+        for excluded in subsets:
+            E=set(excluded);G=tuple(sorted({group_of[a] for a in E}))
+            survivors=[u for i in G for u in groups[i] if u not in E]
+            assert len(survivors)<=3
+            pieces={():out_coarse[G]}
+            for u in survivors:
+                omit=tuple(i for i in G if i!=group_of[u])
+                pieces[u,]=one[u][omit]
+            for u,v in combinations(survivors,2):
+                omit=tuple(i for i in G if i not in (group_of[u],group_of[v]))
+                pieces[u,v]=two[u,v][omit]
+            if len(survivors)==3:pieces[tuple(survivors)]=w.get(tuple(survivors),0)
+            if len(survivors)==3 and all(len(groups[g])==2 for g in G):
+                if G not in cube_cache:cube_cache[G]=self.cube_values(G,groups,out_coarse,one,two,w)
+                result[excluded]=cube_cache[G][tuple(sorted(survivors))]
+                continue
+            if len(survivors)==3:
+                a,b,c=survivors
+                order=[(),(a,),(b,),(a,b),(c,),(a,c),(b,c),(a,b,c)]
+            else:order=[s for k in range(len(survivors)+1) for s in combinations(survivors,k)]
+            result[excluded]=self.total([pieces[s] for s in order])
+        return result
diff --git a/scripts/partial_gauge/interval.py b/scripts/partial_gauge/interval.py
new file mode 100644
index 0000000000..ba6581b02b
--- /dev/null
+++ b/scripts/partial_gauge/interval.py
@@ -0,0 +1,70 @@
+# Copyright 2026 icekylinx. Apache-2.0.
+# Prepared with OpenAI GPT-6 Astra and Codex assistance; see NOTICE.
+"""Cyclic intervals and core-aware pairs, adapted from Avi Eisenberg PR62."""
+from pathlib import Path
+from itertools import combinations
+import sys
+from paired_triple_circuit import PairedTriple
+from paired_exclusion_circuit import PairedExclusionCircuit
+
+def interval_sums(add,values):
+    nz=[(i,x) for i,x in enumerate(values) if x]
+    n=len(nz)
+    if not n:return 0,[0]*len(values)
+    if n==1:
+        result=[nz[0][1]]*len(values);result[nz[0][0]]=0
+        return nz[0][1],result
+    v=[x for _,x in nz];states={(a,1):x for a,x in enumerate(v)}
+    for length in range(2,n):
+        for a in range(n):states[a,length]=add(states[a,length-1],v[(a+length-1)%n])
+    total=add(states[0,n-1],v[-1])
+    out=[total]*len(values)
+    for j,(i,x) in enumerate(nz):out[i]=states[(j+1)%n,n-1]
+    return total,out
+
+class IntervalTriple(PairedTriple):
+    def vector(self,values,two=True):
+        if two:return super().vector(values,two)
+        total,one=interval_sums(self.add,values)
+        return total,one,{}
+
+    def block(self,points,edges,weights):
+        if len(points)<=2:
+            return PairedExclusionCircuit.block(self,points,edges,weights)
+        groups=self.grouping(points);ng=len(groups)
+        edge=lambda a,b:edges[tuple(sorted((a,b)))]
+        coarse={(i,j):self.total([edge(a,b) for a in groups[i] for b in groups[j]])
+                for i,j in combinations(range(ng),2)}
+        wt={i:self.total([weights[a] for a in g]+[edge(a,b) for a,b in combinations(g,2)])
+            for i,g in enumerate(groups)}
+        total,outside,far=self.block(list(range(ng)),coarse,wt)
+        strips={};sums={}
+        for i,g in enumerate(groups):
+            other=[j for j in range(ng) if j!=i]
+            for a in g:
+                carry=self.total([weights[u] for u in g if u!=a])
+                vals=[self.total([edge(u,v) for u in g if u!=a for v in groups[j]]) for j in other]
+                st,one=interval_sums(self.add,[carry]+vals)
+                strips[a]={j:z for j,z in zip(other,one[1:])};sums[a]=st
+        out={};single={a:self.add(outside[i],sums[a]) for i,g in enumerate(groups) for a in g}
+        for i,g in enumerate(groups):
+            for a,b in combinations(g,2):out[a,b]=outside[i]
+        for i,j in combinations(range(ng),2):
+            if len(groups[i])==len(groups[j])==2:
+                (a,a2),(b,b2)=groups[i],groups[j]
+                F=far[i,j];Sa,Sa2,Sb,Sb2=strips[a][j],strips[a2][j],strips[b][i],strips[b2][i]
+                FSa,FSa2=self.add(F,Sa),self.add(F,Sa2)
+                FSb,FSb2=self.add(F,Sb),self.add(F,Sb2)
+                Y1=self.add(FSb,Sa);Y2=self.add(FSa,Sb2)
+                Y3=self.add(FSa2,edge(a,b2));Y4=self.add(FSb2,edge(a,b))
+                out[a,b]=self.add(edge(a2,b2),Y1)
+                out[a,b2]=self.add(edge(a2,b),Y2)
+                out[a2,b]=self.add(Y3,Sb)
+                out[a2,b2]=self.add(Y4,Sa2)
+            else:
+                for a in groups[i]:
+                    left=self.add(far[i,j],strips[a][j])
+                    for b in groups[j]:
+                        cross=self.total([edge(u,v) for u in groups[i] if u!=a for v in groups[j] if v!=b])
+                        out[a,b]=self.add(left,self.add(strips[b][i],cross))
+        return total,single,out
diff --git a/scripts/partial_gauge/lift.py b/scripts/partial_gauge/lift.py
new file mode 100644
index 0000000000..5b66ec11bc
--- /dev/null
+++ b/scripts/partial_gauge/lift.py
@@ -0,0 +1,144 @@
+# Copyright 2026 icekylinx. Apache-2.0.
+# Prepared with OpenAI GPT-6 Astra and Codex assistance; see NOTICE.
+#!/usr/bin/env python3
+"""Construct maximal nondegenerate backward frames containing every old frame.
+Existing carrier dependencies are included before the backward pass. Exact F2
+linear algebra only; the general Gauss compiler permits alternating residuals.
+"""
+from pathlib import Path
+from collections import Counter
+import json,argparse
+from .binary import load
+
+def basis(rows):
+    b={}
+    for x in rows:
+        for p,y in sorted(b.items(),reverse=True):
+            if x>>p&1:x^=y
+        if x:
+            p=x.bit_length()-1
+            for k,y in list(b.items()):
+                if y>>p&1:b[k]=y^x
+            b[p]=x
+    return tuple(b[p] for p in sorted(b,reverse=True))
+
+def perp(rows,h):
+    rows=basis(rows);piv={r.bit_length()-1:r for r in rows};out=[]
+    for j in range(h):
+        if j in piv:continue
+        x=1<<j
+        for p,r in piv.items():
+            if r>>j&1:x|=1<<p
+        out.append(x)
+    return basis(out)
+
+def dot(a,b):return (a&b).bit_count()&1
+
+def nondeg(rows):
+    return len(basis(sum(dot(x,y)<<j for j,y in enumerate(rows)) for x in rows))==len(rows)
+
+def contained(A,B):
+    for x in A:
+        for y in B:x=min(x,x^y)
+        if x:return False
+    return True
+
+def safe_subspace(cap,old):
+    # Old has a supplied orthonormal basis. Orthogonally remove old from cap.
+    others=[]
+    for x in cap:
+        for u in old:
+            if dot(x,u):x^=u
+        others.append(x)
+    others=list(basis(others));chosen=[]
+    while others:
+        i=next((i for i,x in enumerate(others) if dot(x,x)),None)
+        if i is not None:
+            u=others.pop(i);chosen.append(u)
+            others=list(basis(x^(u if dot(x,u) else 0) for x in others));continue
+        pair=next(((i,j) for i in range(len(others)) for j in range(i) if dot(others[i],others[j])),None)
+        if pair is None:break
+        i,j=pair;u=others[i];v=others[j];chosen.extend((u,v))
+        others=list(basis(x^(u if dot(x,v) else 0)^(v if dot(x,u) else 0) for k,x in enumerate(others) if k not in (i,j)))
+    result=basis(list(old)+chosen)
+    assert nondeg(result) and contained(basis(old),result)
+    return result
+
+def run(path,matchfile):
+    h,v,n,q,args,core,cover,roots,kind,active,oldrank,types=load(path)
+    meta=json.loads(Path(matchfile).read_text());arcs=dict(meta['matching_arcs']);full=(1<<h)-1
+    succ=[[] for _ in range(n)];direct=[[] for _ in range(n)]
+    for x in range(1,n):
+        if active[x] and args[2*x]:
+            for y in (args[2*x],args[2*x+1]):succ[y].append(x)
+    for j,(x,k) in enumerate(zip(roots,kind)):
+        normal=(full^cover[x]) if k else (core[x]|(full^cover[x]))
+        assert normal.bit_count()==(1 if k else 3)
+        direct[x].append(normal)
+    for donor,use in arcs.items():
+        if use>>31:
+            j=use&0x7fffffff;x=roots[j];k=kind[j]
+            normal=(full^cover[x]) if k else (core[x]|(full^cover[x]))
+            direct[donor].append(normal)
+        else:succ[donor].append(use//2)
+    order=sorted((x for x in range(1,n) if active[x]),key=lambda x:(oldrank[x],x))
+    place={x:i for i,x in enumerate(order)}
+    assert all(place[x]<place[y] for x in order for y in succ[x])
+    ann=[None]*n;nr=[0]*n;repairs=0;capcache={};repaircache={};dimhist=Counter()
+    for at,x in enumerate(reversed(order)):
+        rows=direct[x][:]
+        for y in succ[x]:rows.extend(ann[y])
+        capann=basis(rows)
+        if capann not in capcache:
+            C=perp(capann,h);capcache[capann]=(C,nondeg(C))
+        C,ok=capcache[capann]
+        if ok:A=capann;d=len(C)
+        else:
+            if types[x]==1:
+                if oldrank[x]==1:U=(cover[x],)
+                else:U=tuple((1<<j)|core[x] for j in range(h) if (cover[x]&~core[x])>>j&1)
+            elif types[x]==2:U=tuple(1<<j for j in range(h) if cover[x]>>j&1)
+            else:raise ValueError('Unselected center type')
+            key=(capann,U)
+            if key not in repaircache:
+                M=safe_subspace(C,U);repaircache[key]=perp(M,h)
+            A=repaircache[key];d=h-len(A);repairs+=1
+        ann[x]=A;nr[x]=d;assert d>=oldrank[x]
+        dimhist[d-oldrank[x]]+=1
+    degree=[0]*n
+    for x in order:
+        if args[2*x]:
+            degree[args[2*x]]+=1;degree[args[2*x+1]]+=1
+    for x in roots:degree[x]+=1
+    H=Counter()
+    for x in order:
+        r=nr[x]
+        if args[2*x]:
+            H[r]+=degree[x]-1;H[h-r]+=1
+            for y in (args[2*x],args[2*x+1]):assert r>=nr[y];H[r-nr[y]]+=1
+        else:
+            H[r]+=degree[x]-1;H[1]+=1;H[r-1]+=1
+    loss=0
+    for x,k in zip(roots,kind):
+        r=nr[x]
+        if k:H[r]+=1;H[h-r]+=1;loss+=r
+        else:assert r<=h-1;H[h-1-r]+=1;H[1]+=1
+    for donor,use in arcs.items():
+        target=roots[use&0x7fffffff] if use>>31 else use//2
+        value=target if use>>31 else args[2*target+(use&1)]
+        ru,rv=nr[donor],nr[value]
+        rt=(h-1 if use>>31 else nr[target])
+        # Root uses have a direct physical edge to their output frame, not
+        # necessarily to the value's newly lifted producer frame.
+        old_target_rank=nr[target]
+        H[h-ru]-=1;H[rv]-=1
+        if use>>31:
+            H[h-1-rv]-=1;H[h-1-ru]+=1
+        else:H[rt-rv]-=1;H[rt-ru]+=1
+    assert min(H.values())>=0
+    R=meta['R'];mass=sum(r*c for r,c in H.items());assert mass==h*R+loss,(mass,h*R+loss)
+    out={k:meta[k] for k in ['h','v','c','q','R','matched']}
+    out.update(loss=loss,histogram=[H[i] for i in range(h+1)],rank_sum=mass,
+        copied_centers_already=True,repairs=repairs,distinct_caps=len(capcache),distinct_repairs=len(repaircache),
+        dimension_gains=dict(dimhist),lift='Backward chosen-successor intersection; maximal nondegenerate complement containing old orthonormal frame; old matched dependencies preserved')
+    return out,ann
diff --git a/scripts/partial_gauge/match.cpp b/scripts/partial_gauge/match.cpp
new file mode 100644
index 0000000000..9275aaa029
--- /dev/null
+++ b/scripts/partial_gauge/match.cpp
@@ -0,0 +1,43 @@
+// Copyright 2026 icekylinx. Apache-2.0.
+// OpenAI GPT-6 Astra and Codex assistance; retained matcher credits in NOTICE.
+#include <algorithm>
+#include <array>
+#include <cassert>
+#include <cstdint>
+#include <fstream>
+#include <iostream>
+#include <vector>
+using U=uint32_t;using V=uint64_t;
+template<class T>void readv(std::ifstream&f,std::vector<T>&v){f.read((char*)v.data(),v.size()*sizeof(T));assert(f);}
+int main(int argc,char**argv){assert(argc==3);std::ifstream f(argv[1],std::ios::binary);U hdr[4];f.read((char*)hdr,16);assert(f);U h=hdr[0],v=hdr[1],n=hdr[2],q=hdr[3];
+std::vector<std::array<U,2>>args(n);std::vector<V>core(n),cover(n);std::vector<U>roots(q),kind(q);std::vector<uint8_t>active(n);readv(f,args);readv(f,core);readv(f,cover);readv(f,roots);readv(f,kind);readv(f,active);
+std::vector<U>ranks(n),degree(n),begin(n+1);V c=0,inputs=0,loss=0;
+for(U x=1;x<n;x++)if(active[x]){if(args[x][0]){c++;assert(args[x][0]<x&&args[x][1]<x);ranks[x]=__builtin_popcountll(cover[x])-__builtin_popcountll(core[x]);for(U y:args[x])degree[y]++;}else{ranks[x]=1;inputs++;}}
+assert(inputs==v);std::ifstream labels(argv[2],std::ios::binary);std::vector<uint8_t>types(n);readv(labels,ranks);readv(labels,types);for(U x:roots)degree[x]++;for(U x=1;x<n;x++)begin[x+1]=begin[x]+degree[x];
+std::vector<U>uses(begin.back()),cursor(begin.begin(),begin.end()-1);for(U x=1;x<n;x++)if(active[x]&&args[x][0]){uses[cursor[args[x][0]]++]=2*x;uses[cursor[args[x][1]]++]=2*x+1;}for(U j=0;j<q;j++)uses[cursor[roots[j]]++]=(1U<<31)|j;
+auto nd=[&](U e)->U{return e>>31?roots[e&0x7fffffff]:e/2;};
+auto before=[&](U a,U b){U x=nd(a),y=nd(b);if(ranks[x]!=ranks[y])return ranks[x]<ranks[y];V ox=a>>31?V(n)+(a&0x7fffffff):x,oy=b>>31?V(n)+(b&0x7fffffff):y;return ox<oy;};
+auto incl=[&](U a,U b){U x=nd(a),y=nd(b);U tx=types[x],ty=types[y];
+if(tx==1 && ty==1)return !(core[y]&~core[x])&&!(cover[x]&~cover[y]);
+if(tx==1 && ty==2)return !(cover[x]&~cover[y]);
+if(tx==2 && ty==2)return !(cover[x]&~cover[y]);
+if(tx==1 && ty==3)return bool(core[x]&core[y]);
+if(tx==2 && ty==3)return !(cover[x]&~core[y]);
+if(tx==3 && ty==2)return cover[y]==((V(1)<<h)-1);
+if(tx==3 && ty==3)return core[x]==core[y];
+return false;};
+auto adjacency=[&](U donor,auto&& action){for(U value:args[donor])for(U j=begin[value];j<begin[value+1];j++)if(before(donor*2,uses[j])&&incl(donor*2,uses[j]))if(action(j))return true;return false;};
+std::vector<U>donors;for(U x=1;x<n;x++)if(active[x]&&args[x][0]&&adjacency(x,[](U){return true;}))donors.push_back(x);
+std::vector<U>leftmatch(n),rightmatch(uses.size()),distance(n,UINT32_MAX),queue;V matches=0;U phase=0,inf=UINT32_MAX,shortest=inf;
+while(true){queue.clear();shortest=inf;for(U x:donors){if(!leftmatch[x]){distance[x]=0;queue.push_back(x);}else distance[x]=inf;}for(U at=0;at<queue.size();at++){U x=queue[at];if(distance[x]>=shortest)continue;adjacency(x,[&](U j){U y=rightmatch[j];if(!y)shortest=distance[x]+1;else if(distance[y]==inf){distance[y]=distance[x]+1;queue.push_back(y);}return false;});}if(shortest==inf)break;
+auto aug=[&](auto&&self,U x)->bool{bool ok=adjacency(x,[&](U j){U y=rightmatch[j];if((!y&&distance[x]+1==shortest)||(y&&distance[y]==distance[x]+1&&self(self,y))){leftmatch[x]=j+1;rightmatch[j]=x;return true;}return false;});if(!ok)distance[x]=inf;return ok;};V gained=0;for(U x:donors)if(!leftmatch[x]&&aug(aug,x)){matches++;gained++;}std::cerr<<"h="<<h<<" phase "<<++phase<<" length "<<shortest<<" gained "<<gained<<" total "<<matches<<"\n";assert(gained);}
+std::vector<int64_t>hist(h+1);for(U x=1;x<n;x++)if(active[x]){assert(degree[x]);U r=ranks[x];if(args[x][0]){hist[r]+=degree[x]-1;hist[h-r]++;for(U y:args[x]){assert(r>=ranks[y]);hist[r-ranks[y]]++;}}else hist[1]+=degree[x];}
+for(U j=0;j<q;j++){U r=ranks[roots[j]];if(kind[j]){hist[r]++;hist[h]++;loss+=r;}else{assert(r<=h-1);hist[h-1-r]++;hist[1]++;}}
+V changed=0;for(U donor:donors)if(leftmatch[donor]){U e=uses[leftmatch[donor]-1],target=nd(e),value=e>>31?target:args[target][e&1];assert(value==args[donor][0]||value==args[donor][1]);if(value==args[donor][0])changed++;U ru=ranks[donor],rv=ranks[value],rt=ranks[target];assert(rt>=ru&&ru>=rv);hist[h-ru]--;hist[rv]--;hist[rt-rv]--;hist[rt-ru]++;}
+std::vector<int64_t>terminal(h+1);
+for(U x=1;x<n;x++)if(active[x]&&args[x][0]&&!leftmatch[x])terminal[ranks[x]]++;
+for(U j=0;j<q;j++)terminal[kind[j]?ranks[roots[j]]:h-1]++;
+V R=c+q-matches,sum=0;for(U r=0;r<=h;r++){assert(hist[r]>=0);sum+=r*hist[r];}assert(sum==h*R+2*loss);
+
+std::cout<<"{\"h\":"<<h<<",\"v\":"<<v<<",\"c\":"<<c<<",\"q\":"<<q<<",\"baseline_R\":"<<c+q<<",\"matched\":"<<matches<<",\"R\":"<<R<<",\"orientation_changes\":"<<changed<<",\"rank_sum\":"<<sum<<",\"loss\":"<<loss<<",\"histogram\":[";for(U r=0;r<=h;r++){if(r)std::cout<<",";std::cout<<hist[r];}std::cout<<"],\"terminal_rank_histogram\":[";V nt=0;for(U r=0;r<=h;r++){if(r)std::cout<<",";std::cout<<terminal[r];nt+=terminal[r];}assert(nt==R);std::cout<<"],\"matching_arcs\":[";bool comma=false;for(U x:donors)if(leftmatch[x]){if(comma)std::cout<<",";comma=true;std::cout<<"["<<x<<","<<uses[leftmatch[x]-1]<<"]";}std::cout<<"]}"<<std::endl;
+}
diff --git a/scripts/partial_gauge/scalar.py b/scripts/partial_gauge/scalar.py
new file mode 100644
index 0000000000..925ffb4cfa
--- /dev/null
+++ b/scripts/partial_gauge/scalar.py
@@ -0,0 +1,120 @@
+# Copyright 2026 icekylinx. Apache-2.0.
+# Prepared with OpenAI GPT-6 Astra and Codex assistance; see NOTICE.
+"""Rational-center extension of the aligned general-residual complex motif.
+The fixed rational phase adapter permits every d except 3.
+
+Builds a concrete scalar addition DAG, its nondegenerate binary frames, and
+the unmixed carrier histogram. No historical audit suite is invoked.
+"""
+import sys, json, struct, array, os
+from itertools import combinations
+from pathlib import Path
+from .cube import CubeTriple as PairedTriple
+from .interval import interval_sums
+
+def build(h, base=2, central_disjoint=0, output_prefix='complex_general', output_dir=None):
+    assert h == central_disjoint == 24, "Only the selected h=d=24 construction is supported"
+    d=central_disjoint
+    assert 0<=d<=h
+    # Odd h uses the full coordinate frame for B_i centers.
+    if d:
+        divisor=abs(3-d)
+        assert divisor, "The disjoint-center divisor 3-d must be nonzero."
+    triples=list(combinations(range(h),3)); v=len(triples)
+    args=[(0,0)]; support=[0]; core=[0]; cover=[0]; types=[0]; ranks=[0]
+    lookup={}; inputs={}
+    for j,t in enumerate(triples):
+        mask=sum(1<<a for a in t); x=len(args); inputs[t]=x
+        args.append((0,0));support.append(1<<j);core.append(mask);cover.append(mask)
+        types.append(1);ranks.append(1);lookup[1<<j]=x
+    def add(a,b,center=None):
+        if not a:return b
+        if not b:return a
+        assert not support[a]&support[b], "Every selected addition must be disjoint"
+        s=support[a]|support[b]
+        if center is None and s in lookup:return lookup[s]
+        x=len(args);args.append((a,b));support.append(s)
+        core.append(core[a]&core[b]);cover.append(cover[a]|cover[b])
+        if center is not None:
+            types.append(3 if h % 2 == 0 else 2);ranks.append(h-1 if h % 2 == 0 else h);core[x]=1<<center
+            if h % 2:cover[x]=(1<<h)-1
+        elif core[x].bit_count()>=2:
+            types.append(1);ranks.append(s.bit_count());lookup[s]=x
+        else:
+            types.append(2);ranks.append(cover[x].bit_count());lookup[s]=x
+        return x
+    def total(xs,center=None):
+        xs=[x for x in xs if x]
+        while len(xs)>1:
+            xs=[add(xs[i],xs[i+1],center) if i+1<len(xs) else xs[i] for i in range(0,len(xs),2)]
+        return xs[0] if xs else 0
+    paired=PairedTriple(h,base)
+    allresults=paired.triple(list(range(h)),paired.variables)
+    selected=list(paired.outputs.values())
+    selected += [allresults[(i,)] for i in range(d)] if d else [allresults[()]]
+    active=set();stack=list(selected)
+    while stack:
+        x=stack.pop()
+        if not x or x in active:continue
+        active.add(x)
+        if paired.args[x]:stack.extend(paired.args[x])
+    mapping={0:0}|{j+1:inputs[t] for j,t in enumerate(triples)}
+    for x in sorted(active):
+        if paired.args[x]:
+            a,b=paired.args[x];mapping[x]=add(mapping[a],mapping[b])
+    roots=[mapping[paired.outputs[t]] for t in triples];kind=[0]*v
+    center_disjoint_roots=[mapping[allresults[(i,)]] for i in range(d)]
+    fulltotal=mapping[allresults[()]] if not d else None
+    del paired, mapping, active, allresults
+    pairtotals={}
+    for a,b in combinations(range(h),2):
+        others=[i for i in range(h) if i not in (a,b)]
+        others.sort(key=lambda i:((i^1) in (a,b),i))
+        star_total,star_out=interval_sums(add,[inputs[tuple(sorted((a,b,i)))] for i in others])
+        for x in star_out:
+            roots.append(x);kind.append(0)
+        pairtotals[a,b]=star_total
+    for a in range(d,h):
+        root=total([pairtotals[tuple(sorted((a,b)))] for b in range(h) if b!=a],center=a)
+        roots.append(root);kind.append(1)
+    if d:
+        roots.extend(center_disjoint_roots);kind.extend([1]*d)
+    else:
+        roots.append(fulltotal);kind.append(1)
+    active=[0]*len(args);stack=list(roots)
+    while stack:
+        x=stack.pop()
+        if not x or active[x]:continue
+        active[x]=1
+        if args[x][0]:stack.extend(args[x])
+    degree=[0]*len(args)
+    for x in range(1,len(args)):
+        if active[x] and args[x][0]:
+            for y in args[x]:degree[y]+=1
+    for x in roots:degree[x]+=1
+    H=[0]*(h+1);c=0;loss=0
+    for x in range(1,len(args)):
+        if not active[x]:continue
+        r=ranks[x]
+        if args[x][0]:
+            c+=1;H[r]+=degree[x]-1;H[h-r]+=1
+            for y in args[x]:H[r-ranks[y]]+=1
+        else:H[1]+=degree[x]
+    for x,k in zip(roots,kind):
+        r=ranks[x]
+        if k:H[r]+=1;H[h]+=1;loss+=r
+        else:H[h-1-r]+=1;H[1]+=1
+    q=len(roots);R=c+q
+    result=dict(h=h,v=v,c=c,q=q,R=R,loss=loss,histogram=H,rank_sum=sum(r*n for r,n in enumerate(H)),central_disjoint=d)
+    prefix=Path(output_dir)/f'{output_prefix}_{h}'
+    with open(str(prefix)+'.bin','wb') as f:
+        f.write(struct.pack('<4I',h,v,len(args),q))
+        array.array('I',(a for pair in args for a in pair)).tofile(f)
+        array.array('Q',core).tofile(f);array.array('Q',cover).tofile(f)
+        array.array('I',roots).tofile(f);array.array('I',kind).tofile(f)
+        array.array('B',active).tofile(f)
+    with open(str(prefix)+'.labels','wb') as f:
+        array.array('I',ranks).tofile(f);array.array('B',types).tofile(f)
+    Path(str(prefix)+'.json').write_text(json.dumps(result,indent=2)+'\n')
+
+    return result
diff --git a/scripts/partial_gauge/select.py b/scripts/partial_gauge/select.py
new file mode 100644
index 0000000000..041c4919bd
--- /dev/null
+++ b/scripts/partial_gauge/select.py
@@ -0,0 +1,73 @@
+# Copyright 2026 icekylinx. Apache-2.0.
+# Prepared with OpenAI GPT-6 Astra and Codex assistance; see NOTICE.
+#!/usr/bin/env python3
+"""One deterministic partial-source-gauge pass on an existing physical M.
+
+The source gauge may be a proper nondegenerate subspace of the first gate
+frame. Every target receives a descending chain of selected subspaces; the
+physical readouts execute in the reversed order. Residual first transitions
+are counted literally rather than removed in their entirety.
+"""
+from pathlib import Path
+from collections import Counter
+from itertools import combinations
+import argparse,json
+from .binary import load
+from .lift import basis,perp,nondeg,safe_subspace
+
+def run(path,annfile,liftfile,wordfile):
+ h,v,n,q,args,core,cover,roots,kind,active,oldrank,types=load(path)
+ ann=json.loads(Path(annfile).read_text());d=json.loads(Path(liftfile).read_text())
+ word=json.loads(Path(wordfile).read_text());ops=word['ops'];sources=set(word['sources'].values())
+ R=d['R'];full=(1<<h)-1;first=[None]*R
+ for a,b,x in ops:
+  if first[a] is None:first[a]=x
+  if first[b] is None:first[b]=x
+ touched=set(sources)
+ for i in word['phase1']:touched.update(ops[i][:2])
+ triples=list(combinations(range(h),3));tid={sum(1<<j for j in t):i for i,t in enumerate(triples)}
+ co=[0]*R
+ for j,(x,s,k) in enumerate(zip(roots,word['rootroles'],kind)):
+  co[s]|=((1<<v)-1) if k else 1<<tid[core[x]|(full^cover[x])]
+ for a,b,x in reversed(ops):co[b]|=co[a]
+ candidates=[s for s in range(R) if first[s] is not None and s not in touched]
+ candidates.sort(key=lambda s:(len(ann[first[s]]),co[s].bit_count(),s))
+ # None represents the full active ambient frame, whose annihilator is0.
+ latest=[None]*v;target_ranks=[[] for _ in range(v)];selected=[];birth=Counter();residual=Counter();H=Counter(dict(enumerate(d['histogram'])))
+ cache={};repaired=0
+ for s in candidates:
+  A=tuple(ann[first[s]]);r=h-len(A);rows=list(A);outs=[];bits=co[s]
+  while bits:
+   low=bits&-bits;t=low.bit_length()-1;bits^=low;outs.append(t)
+   if latest[t] is not None:rows.extend(latest[t])
+  capann=basis(rows)
+  if capann not in cache:
+   cap=perp(capann,h)
+   if nondeg(cap):sigma=capann
+   else:sigma=perp(safe_subspace(cap,()),h);repaired+=1
+   cache[capann]=sigma
+  sigma=cache[capann];a=h-len(sigma)
+  if not a:continue
+  assert a<=r
+  H[r]-=1;H[r-a]+=1;birth[a]+=1;residual[r-a]+=1
+  selected.append(dict(role=s,first_rank=r,gauge_rank=a,annihilator=sigma))
+  for t in outs:
+   latest[t]=sigma
+   if not target_ranks[t] or target_ranks[t][-1]!=a:target_ranks[t].append(a)
+ assert min(H.values())>=0
+ Y=Counter()
+ for ranks in target_ranks:
+  prev=0
+  for r in list(reversed(ranks))+[h-1]:assert r>=prev;Y[r-prev]+=1;prev=r
+ result={k:d[k] for k in ['h','v','R','q','c','matched','loss']}
+ result.update(selected_roles=len(selected),selected_rank_histogram=dict(birth),
+  remaining_internal_histogram=[H[r] for r in range(h+1)],
+  source_data_histogram={h-1:v},target_data_histogram=dict(Y),
+  copied_centers_already=True,phase1_operations=len(word['phase1']),
+  total_M_operations=len(ops),untouched_non_source_roles=len(candidates),
+  selection='Descending first-frame rank; intersect all incident latest target frames; retain maximal nondegenerate summand',
+  partial_first_transition_histogram=dict(residual),distinct_caps=len(cache),
+  repaired_caps=repaired,
+  scalar_readout='Exact rational C=J M; Boolean support is a safe superset only',
+  chronological_contract='Nondeferred reads at0; source V; center-closed M phase1; copied centers at0; selected partial-gauge reads in reverse selection order; remaining M; side scatter; inverse M/V atfull F')
+ return result,dict(base_word=Path(wordfile).name,selected=selected)
diff --git a/scripts/partial_gauge/verify.py b/scripts/partial_gauge/verify.py
new file mode 100644
index 0000000000..a1afefe66d
--- /dev/null
+++ b/scripts/partial_gauge/verify.py
@@ -0,0 +1,125 @@
+# Copyright 2026 icekylinx. Apache-2.0.
+# Prepared with OpenAI GPT-6 Astra and Codex assistance; see NOTICE.
+"""Independent checks of the generated DAG, lifted frames and physical word."""
+from collections import Counter
+from functools import lru_cache
+from itertools import combinations
+from .binary import load
+from .lift import basis, perp, contained, nondeg
+
+
+def verify(path, matching, ann, lift, word, selected, record):
+    h,v,n,q,args,core,cover,roots,kind,active,oldrank,types=load(path)
+    triples=list(combinations(range(h),3)); full=(1<<h)-1
+    point=[sum(1<<j for j,t in enumerate(triples) if a in t) for a in range(h)]
+    all_inputs=(1<<v)-1; supports=[0]*n
+    @lru_cache(None)
+    def frame(A):
+        F=perp(A,h)
+        assert nondeg(F)
+        return F
+    @lru_cache(None)
+    def includes(A,B):
+        return contained(A,B)
+    for x in range(1,n):
+        if not active[x]:continue
+        a,b=args[2*x:2*x+2]
+        if a:
+            assert a<x and b<x and not supports[a]&supports[b]
+            supports[x]=supports[a]|supports[b]
+        else:
+            assert x<=v
+            supports[x]=1<<(x-1)
+        A=tuple(ann[x]); F=frame(A)
+        if types[x]==1:
+            old=(cover[x],) if oldrank[x]==1 else tuple((1<<j)|core[x] for j in range(h) if (cover[x]&~core[x])>>j&1)
+        else:
+            assert types[x]==2
+            old=tuple(1<<j for j in range(h) if cover[x]>>j&1)
+        assert len(old)==oldrank[x] and includes(basis(old),F)
+        if a:
+            assert includes(A,tuple(ann[a])) and includes(A,tuple(ann[b]))
+    for j,t in enumerate(triples):
+        assert supports[roots[j]]==all_inputs & ~(point[t[0]]|point[t[1]]|point[t[2]])
+    at=v
+    for a,b in combinations(range(h),2):
+        others=[i for i in range(h) if i not in (a,b)]
+        others.sort(key=lambda i:((i^1) in (a,b),i))
+        for i in others:
+            assert supports[roots[at]]==point[a]&point[b]&~point[i];at+=1
+    for a in range(h):
+        assert kind[at] and supports[roots[at]]==all_inputs^point[a];at+=1
+    assert at==q
+    # Every input appears in h-3 disjoint centers: exact divisor 21. For
+    # overlap k, twice the signed output coefficient is 2-(3-k)+[k=0]-[k=2].
+    assert h-3==21
+    assert [2-(3-k)+(k==0)-(k==2) for k in range(4)]==[0,0,0,2]
+    for donor,use in matching['matching_arcs']:
+        if use>>31:
+            j=use&0x7fffffff;x=roots[j]
+            normal=full^cover[x] if kind[j] else core[x]|(full^cover[x])
+            assert includes(basis([normal]),tuple(ann[donor]))
+        else:assert includes(tuple(ann[use//2]),tuple(ann[donor]))
+    R=record['R']; values=[0]*R; current=[0]*R; first=[None]*R
+    H=Counter();phase=set(word['phase1']); touched=set(map(int,word['sources'].values()))
+    for x,s in word['sources'].items():
+        values[s]=supports[int(x)];current[s]=1;H[1]+=1
+    last=[-1]*R
+    for i,(a,b,x) in enumerate(word['ops']):
+        assert a!=b
+        if i in phase:
+            assert all(last[s]<0 or last[s] in phase for s in (a,b))
+            touched.update((a,b))
+        for s in (a,b):
+            if first[s] is None:first[s]=x
+            r=h-len(ann[x]);assert r>=current[s]
+            H[r-current[s]]+=1;current[s]=r;last[s]=i
+        assert not values[a]&values[b]
+        values[a]|=values[b]
+        assert values[a]==supports[x]
+    for j,(x,s) in enumerate(zip(roots,word['rootroles'])):
+        assert values[s]==supports[x]
+        if kind[j]:
+            assert last[s]<0 or last[s] in phase
+            H[current[s]]+=1
+        else:
+            H[h-1-current[s]]+=1;current[s]=h-1
+    for r in current:H[h-r]+=1
+    assert all(H[r]==lift['histogram'][r] for r in range(1,h+1))
+    co=[0]*R;tid={sum(1<<a for a in t):j for j,t in enumerate(triples)}
+    for x,s,k in zip(roots,word['rootroles'],kind):
+        co[s]|=all_inputs if k else 1<<tid[core[x]|(full^cover[x])]
+    for a,b,x in reversed(word['ops']):co[b]|=co[a]
+    latest=[()]*v;seen=set();residual=Counter();birth=Counter()
+    for row in selected:
+        s=row['role'];A=tuple(row['annihilator']);F=frame(A)
+        assert s not in touched and s not in seen;seen.add(s)
+        assert includes(tuple(ann[first[s]]),A)
+        r=h-len(ann[first[s]]);d=len(F)
+        assert row['first_rank']==r and row['gauge_rank']==d and 0<d<=r
+        residual[r-d]+=1;birth[d]+=1
+        bits=co[s]
+        while bits:
+            bit=bits&-bits;bits-=bit;t=bit.bit_length()-1
+            assert includes(latest[t],A);latest[t]=A
+    # Execute the requested reverse readout order and recount target transitions.
+    current_ann=[tuple(1<<i for i in reversed(range(h)))]*v;Y=Counter()
+    for row in reversed(selected):
+        A=tuple(row['annihilator']);bits=co[row['role']]
+        while bits:
+            bit=bits&-bits;bits-=bit;t=bit.bit_length()-1;old=current_ann[t]
+            assert includes(A,old)
+            if A!=old:Y[len(old)-len(A)]+=1
+            current_ann[t]=A
+    for t,A in enumerate(current_ann):
+        normal=sum(1<<a for a in triples[t]);assert includes((normal,),A)
+        Y[len(A)-1]+=1
+    for actual,key in [(Y,'target_data_histogram'),(birth,'selected_rank_histogram'),(residual,'partial_first_transition_histogram')]:
+        assert dict(actual)=={int(k):v for k,v in record[key].items()},key
+    return dict(scalar_supports_exact=True, signed_coefficients_exact=True,
+                divisor=21, lifted_frames_nondegenerate=True,
+                old_frames_contained=True, dependency_and_carrier_nesting=True,
+                physical_M_replayed=True, positive_rank_ledger_recounted=True,
+                center_phase_closed=True, selected_sources_untouched=True,
+                reverse_readout_frame_chains=True, target_transitions_recounted=True,
+                selected_roles=len(seen))
diff --git a/scripts/partial_gauge/word.py b/scripts/partial_gauge/word.py
new file mode 100644
index 0000000000..b9e95e0ea2
--- /dev/null
+++ b/scripts/partial_gauge/word.py
@@ -0,0 +1,66 @@
+# Copyright 2026 icekylinx. Apache-2.0.
+# Prepared with OpenAI GPT-6 Astra and Codex assistance; see NOTICE.
+#!/usr/bin/env python3
+"""Construct a signed deferred-readout schedule from one concrete matched M.
+Readout supports use an overapproximation (union through transpose M), so every
+selected orthogonality/chain condition is conservative over Q. Coefficients
+are the exact finite rational matrix J M, specified without dense expansion.
+"""
+from pathlib import Path
+from collections import defaultdict,Counter
+from itertools import combinations
+import json,argparse
+from .binary import load
+from .lift import contained,perp,basis
+
+def run(path,matchfile,annfile,liftfile):
+ h,v,n,q,args,core,cover,roots,kind,active,oldrank,types=load(path)
+ mt=json.loads(Path(matchfile).read_text());arcs=dict(mt['matching_arcs']);ann=json.loads(Path(annfile).read_text());lift=json.loads(Path(liftfile).read_text())
+ nr=[h-len(a) if a is not None else 0 for a in ann];full=(1<<h)-1
+ order=sorted((x for x in range(1,n) if active[x]),key=lambda x:(oldrank[x],x))
+ uses=defaultdict(list)
+ for x in order:
+  if args[2*x]:
+   for j in (0,1):uses[args[2*x+j]].append(2*x+j)
+ for j,x in enumerate(roots):uses[x].append((1<<31)|j)
+ incoming={u:d for d,u in arcs.items()};assign={};sources={};ops=[];R=0
+ for x in order:
+  if args[2*x]:
+   aa,bb=args[2*x],args[2*x+1]
+   p,o=assign[2*x],assign[2*x+1]
+   if x in arcs:
+    u=arcs[x];value=roots[u&0x7fffffff] if u>>31 else args[2*(u//2)+(u&1)]
+    if value==aa:p,o=o,p
+    else:assert value==bb
+    assert u not in assign;assign[u]=o
+   ops.append((p,o,x))
+  else:p=R;R+=1;sources[x]=p
+  free=[u for u in uses[x] if u not in incoming]
+  assert free
+  for j,u in enumerate(free):
+   assert u not in assign
+   if j==0:assign[u]=p
+   else:
+    s=R;R+=1;assign[u]=s;ops.append((s,p,x))
+ assert R==mt['R'],(R,mt['R'])
+ rootroles=[assign[(1<<31)|j] for j in range(q)]
+ prev=[-1]*R;pred=[];first=[None]*R
+ for i,(a,b,x) in enumerate(ops):
+  pred.append((prev[a],prev[b]));prev[a]=prev[b]=i
+  if first[a] is None:first[a]=x
+  if first[b] is None:first[b]=x
+ centerroles=[rootroles[j] for j in range(q) if kind[j]]
+ stack=[prev[s] for s in centerroles if prev[s]>=0];phase=set()
+ while stack:
+  i=stack.pop()
+  if i in phase:continue
+  phase.add(i);stack.extend(p for p in pred[i] if p>=0)
+ touched=set(sources.values())
+ for i in phase:touched.update(ops[i][:2])
+ triples=list(combinations(range(h),3));tid={sum(1<<j for j in t):i for i,t in enumerate(triples)}
+ co=[0]*R
+ for j,(x,s,k) in enumerate(zip(roots,rootroles,kind)):
+  if k:co[s]|=(1<<v)-1
+  else:co[s]|=1<<tid[core[x]|(full^cover[x])]
+ for a,b,x in reversed(ops):co[b]|=co[a]
+ return dict(ops=ops,sources=sources,rootroles=rootroles,phase1=sorted(phase))
diff --git a/scripts/partial_gauge_bit.py b/scripts/partial_gauge_bit.py
new file mode 100644
index 0000000000..b38a57989a
--- /dev/null
+++ b/scripts/partial_gauge_bit.py
@@ -0,0 +1,164 @@
+#!/usr/bin/env python3
+"""Incremental whole-projector accounting for the pinned PR97 physical word.
+
+Copyright 2026 icekylinx. Apache-2.0. Prepared with OpenAI Codex assistance.
+PR97's reflected physical ledger is by Zhihao Chen; its frozen scalar/frame
+witness is Swapnil Jain's round-seven construction. Original licenses, proof
+boundaries, source notices and assistance disclosures remain in the snapshots.
+
+This reconstructs the new profile from physical frame chains and their frozen
+ledger receipt. It does not repeat the unchanged complete-basis scalar replay
+or exact rational frame/lifted checks, and does not infer nesting from ranks.
+"""
+import argparse
+from collections import Counter
+import gzip
+from hashlib import sha256
+import importlib.util
+import json
+from math import comb
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parents[1]
+INPUT = ROOT / 'certificates/partial-gauge-bit-input.json'
+
+
+def require(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def encoded(histogram):
+    return {str(rank): count for rank, count in sorted(histogram.items()) if count}
+
+
+def frozen_sources(record):
+    manifest_path = ROOT / record['source_manifest']
+    require(sha256(manifest_path.read_bytes()).hexdigest() == record['source_manifest_sha256'],
+            'PR97 provenance manifest changed')
+    manifest = json.loads(manifest_path.read_text())
+    for name, digest in manifest['sha256'].items():
+        require(sha256((manifest_path.parent/name).read_bytes()).hexdigest() == digest,
+                'Pinned PR97 source changed: '+name)
+    require(manifest['pr97']['commit'] == 'e15350b66e4bf796153a68c240a6cc1b788abe3e',
+            'Unexpected PR97 commit')
+    require(manifest['underlying']['commit'] == '741e7aa078392553815df7926ee17ac5e25a8c38',
+            'Unexpected underlying witness commit')
+    return manifest_path.parent, manifest
+
+
+def local_ranks(folder):
+    """Read actual role paths; reconstruct nonzero rank increments.
+
+    Auxiliary chains are the same physical chains used by the reflected
+    ledger. Target paths follow readout_order and retain only odd coefficients
+    of the exact adjoint, as the actual F2 physical word does. X paths use the
+    frozen per-leaf V order. Center moves are one rank-(h-1) copy per retained
+    center. No inner-corner splitting is applied at this stage.
+    """
+    spec = importlib.util.spec_from_file_location('pinned_pr97_deferred', folder/'deferred.py')
+    module = importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(module)
+    with gzip.open(folder/'witness_23.json.gz', 'rt') as stream:
+        witness = json.load(stream)
+    with gzip.open(folder/'deferred_23.json.gz', 'rt') as stream:
+        data = json.load(stream)
+    require(witness['h'] == data['h'] == 23, 'Witness dimension mismatch')
+    schedule = module.Schedule(witness, data)
+    h, v, roles = schedule.h, schedule.v, schedule.R
+    require(v == comb(h, 3), 'Witness source count mismatch')
+    require(len(schedule.ret) == h and set(schedule.ret.values()) == set(range(h)),
+            'Retained centers differ')
+    require(len(schedule.readout) == len(set(schedule.readout)), 'Duplicate readout role')
+    require(len(schedule.sigma) == len(schedule.readout), 'Readout bases mismatch')
+    require(all(0 <= slot < roles for slot in schedule.readout), 'Readout role out of range')
+    aux = module.chain_ranks(schedule)
+    xhist = Counter(rank for path in schedule.xdata() for rank in path if rank)
+    require(all(0 < rank <= h for rank in xhist), 'Invalid source path increment')
+    adjoint = schedule.adjoint()
+    current = [0] * v
+    yhist = Counter()
+    for slot in schedule.readout:
+        rank = schedule.f[slot]
+        for target, coefficient in adjoint[slot].items():
+            if coefficient & 1:
+                increment = rank-current[target]
+                require(increment >= 0, 'Target physical path retreats')
+                if increment:
+                    yhist[increment] += 1
+                current[target] = rank
+    for rank in current:
+        require(rank <= h-1, 'Target frame exceeds its output hyperplane')
+        if rank < h-1:
+            yhist[h-1-rank] += 1
+    local = dict(aux=aux, data=xhist+yhist, center=Counter({h-1: len(schedule.ret)}))
+    require(sum(r*n for r,n in local['data'].items()) == 2*v*(h-1),
+            'Physical data path rank mass mismatch')
+    exits = Counter(h-rank for rank in schedule.f)
+    require(all(0 < rank <= h for rank in exits), 'Invalid physical exit nullity')
+    require(sum(exits.values()) == roles, 'Missing physical exit')
+    return h, v, roles, local, exits
+
+
+def reconstruct(input_path=INPUT):
+    record = json.loads(Path(input_path).read_text())
+    folder, sources = frozen_sources(record)
+    ledger = json.loads((folder/'bit-ledger-result.json').read_text())
+    for flag in ('complete_forward_F2', 'complete_reflected_F2',
+                 'reflected_frame_continuity', 'all_scalar_gates_equal_frame_keys',
+                 'histogram_matches_external'):
+        require(ledger[flag] is True, 'Pinned physical ledger lacks '+flag)
+    h, v, roles, local, exits = local_ranks(folder)
+    require((ledger['h'], ledger['roles'], ledger['formal_basis']) == (h, roles, 2*v+roles),
+            'Physical ledger dimensions differ')
+    require({kind: encoded(hist) for kind,hist in local.items()} == ledger['rank_histograms'],
+            'Reconstructed physical ranks differ from the pinned reflected ledger')
+    m, N = h*h, v*v
+    W = 2*N+2*v*roles
+    whole = Counter()
+    for hist in local.values():
+        for rank, count in hist.items():
+            whole[rank] += 2*v*count
+    # Each role's exterior I-(I-sigma) tensor Q has rank m-(h-dim sigma).
+    for nullity, count in exits.items():
+        whole[m-nullity] += 2*v*count
+    # The original entrance projector is one rank-(h-1)^2 edge per bank pair.
+    whole[(h-1)**2] += 2*N
+    whole[1] += N  # physical copy correction
+    mass = sum(rank*count for rank,count in whole.items())
+    require(mass == ledger['recursive_rank'], 'Whole-projector rank mass changed')
+    require(all(0 < rank < m and count > 0 for rank,count in whole.items()),
+            'Invalid whole-projector child')
+    result = dict(dimensions=[h,h], m=m, v=v, N=N, R=roles, W=W,
+                  rank_mass=mass, deficit=W*m-mass, max_child=max(whole),
+                  local_rank_histograms={kind: encoded(hist) for kind,hist in local.items()},
+                  exit_nullity_histogram=encoded(exits),
+                  whole_projector_histogram=encoded(whole))
+    for key, value in result.items():
+        require(value == record[key], 'Round-six expected profile differs: '+key)
+    result['checks'] = dict(source_hashes_equal=True, physical_local_ranks_reconstructed=True,
+                            physical_ledger_ranks_equal=True, whole_profile_equal=True,
+                            full_upstream_audits_repeated=False)
+    result['provenance'] = dict(pr97=sources['pr97'], underlying=sources['underlying'],
+                                source_manifest=record['source_manifest'],
+                                source_manifest_sha256=record['source_manifest_sha256'])
+    result['scope'] = ('Incremental rank accounting from pinned physical chains and ledger; '
+                       'scalar complete-basis replay and rational geometry remain inherited checks.')
+    return result
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--input', type=Path, default=INPUT)
+    parser.add_argument('--output', type=Path)
+    args = parser.parse_args()
+    result = reconstruct(args.input)
+    text = json.dumps(result, indent=2, sort_keys=True)+'\n'
+    if args.output:
+        args.output.write_text(text)
+    else:
+        print(text, end='')
+
+
+if __name__ == '__main__':
+    main()
diff --git a/scripts/partial_gauge_network.py b/scripts/partial_gauge_network.py
new file mode 100644
index 0000000000..53b7e06b96
--- /dev/null
+++ b/scripts/partial_gauge_network.py
@@ -0,0 +1,179 @@
+#!/usr/bin/env python3
+"""Exact deferred-word moments and partial-gauge semantic assembly.
+
+Copyright 2026 icekylinx. Apache-2.0.
+Substantial OpenAI GPT-6 Astra and Codex assistance.
+PR97 physical bit words: Zhihao Chen, based on Swapnil Jain's witness.
+Semantic/bulk assembly: PR23 (Zhihao Chen) and RaD; see SOURCES.json.
+"""
+import argparse
+from collections import Counter
+from fractions import Fraction as Q
+from hashlib import sha256
+from math import comb
+from pathlib import Path
+import json
+import sys
+
+from certify import require
+from copied_centers_network import grid_upper
+from partial_swap_network import moment
+from partial_gauge_bit import reconstruct
+from structured_bulk_assembly import assembly, halving, js
+
+ROOT = Path(__file__).resolve().parents[1]
+COARSE = Q(15513, 125000000)
+ATOM = Q(1, 1000)
+OLD = Q(384599, 10**10)
+AB = (1-ATOM)*COARSE+ATOM*OLD
+AC = Q(92651, 10**9)
+PHASE_STOP = Q(1, 10**6)
+ASSEMBLY_BIT = min(AB, (1-PHASE_STOP)*AC-Q(1, 10**10))
+KAPPA = Q(7237, 78125000)
+
+
+def coarse_moment(p):
+    def log_upper(x):
+        power = 0
+        while x >= 2:
+            x /= 2
+            power += 1
+        def small(y):
+            z = (y-1)/(y+1)
+            return 2*sum((z**(2*j+1)/(2*j+1) for j in range(24)), Q(0)) + 2*z**49/(49*(1-z*z))
+        return grid_upper(power*small(Q(2))+small(x))
+    total, logs = Q(0), {}
+    for t, n in p['child_multiplicities'].items():
+        require(0 < t < p['m'] and n > 0, 'Shrinking positive bit children')
+        logs[t] = log_upper(Q(p['m'],t))
+        u = COARSE*logs[t]
+        require(0 < u < 3, 'Coarse exponential enclosure')
+        total += n*t*grid_upper(1+u+u*u/(2*(1-u/3)))
+    upper = total/(p['W']*p['m'])
+    mass = Q(p['total_rank'],p['W']*p['m'])
+    require(mass < 1 and upper < 1, 'Both stopped-recurrence moments')
+    require(0 < OLD <= COARSE < ATOM < 1 and ATOM > AB, 'Stopped-adapter exponents')
+    return dict(saving=COARSE, exponent=1-COARSE, moment_upper=upper,
+        strict_gap=1-upper, rank_moment=mass, logarithm_upper_bounds=logs,
+        rounding_denominator=1 << 100,
+        stopped_adapter=dict(atom_exponent=ATOM, base_ordinary_saving=OLD,
+            effective_saving=AB, coarse_toll_exponent=1-ATOM,
+            coarse_toll_subordinate_gap=ATOM-COARSE))
+
+
+def complex_profile(row):
+    h, v, R, ell = (row[k] for k in ('h','v','R','loss'))
+    require((h,v,ell) == (24,comb(24,3),24*23), 'Complex dimensions')
+    require(R == row['c']+row['q']-row['matched'], 'Compatible role count')
+    selected = {int(r):n for r,n in row['selected_rank_histogram'].items()}
+    require(sum(selected.values()) == row['selected_roles'] == 6341, 'Selected partial gauges')
+    require(row['copied_centers_already'], 'Copied-center convention')
+    require(sum(row['partial_first_transition_histogram'].values()) == sum(selected.values()), 'Charged first residuals')
+    m, N, W = h*h, v*v, 2*v*v+2*v*R
+    C = Counter({m-h:2*v*(R-sum(selected.values())), (h-1)**2:2*N, 1:N})
+    for r,n in selected.items():
+        require(0 < r < h and n > 0, 'Nonzero shrinking exterior gauges')
+        C[m-h+r] += 2*v*n
+    for name in ('source_data_histogram','target_data_histogram'):
+        for r,n in row[name].items():
+            C[int(r)] += 2*v*n
+    for r,n in enumerate(row['remaining_internal_histogram']):
+        if r:
+            C[r] += 2*v*n
+    C = dict(sorted((r,n) for r,n in C.items() if n))
+    require(all(0 < r < m and n > 0 for r,n in C.items()), 'Shrinking complex children')
+    s = sum(r*n for r,n in C.items())
+    require(s == W*m-N+2*v*ell, 'Complete complex rank mass')
+    return dict(dimensions=[h,h],m=m,N=N,W=W,L=2*v*ell,total_rank=s,
+        deficit=W*m-s,maxchild=max(C),child_multiplicities=C)
+
+
+def finite_bridge(bit, phase, row, previous):
+    m,W,s,N = (phase[k] for k in ('m','W','total_rank','N'))
+    h,v,R = (row[k] for k in ('h','v','R'))
+    local = 4*(row['c']+v)+10*v+4*h*v+4*h*h+8*h+8+2*h
+    local += 8*R*v*(row['total_M_operations']+16)
+    G = N+2*v*local
+    E = 64*(W+m+G+1)**3
+    charge = 2*G*W*W+8*s+4*W+4+32*m
+    B, r = s+E, phase['maxchild']
+    C0 = 32*m*B*B
+    require(charge < E and 2*B*(m-r) >= s+E and 2*B+18 < C0, 'Semantic induction and charge')
+    dc,dp = halving(bit['m'],bit['maxchild']),halving(m,r)
+    wc,wp = bit['W'].bit_length(),W.bit_length()
+    old_bit = previous['finite_bridge']['bit']
+    old = old_bit['halving_degree']*old_bit['wire_bits']
+    require(old == 252, 'Inherited ordinary leaf row stock')
+    coeff,degree = wc*dc+old+wp*dp,40000
+    require(Q(degree) > Q(coeff*51,25), 'Sufficient product row stock')
+    return dict(bit_coarse=dict(m=bit['m'],W=bit['W'],maxchild=bit['maxchild'],
+            halving_degree=dc,wire_bits=wc),ordinary_leaf_row_degree=old,
+        complex=dict(m=m,W=W,s=s,maxchild=r,halving_degree=dp,wire_bits=wp,
+            scalar_group_upper=G,local_group_upper=local,coefficient_bound=h+4,
+            coefficient_denominator_divides=2*(h-3)),
+        semantic=dict(E=E,literal_charge=charge,strict_literal_gap=E-charge,
+            B=B,C0=C0,C1=1,induction_gap=2*B*(m-r)-s-E,fixed_odd_divisor=h-3,
+            exact_grid='One common dyadic grid times the fixed odd divisor to K=G*(D_complex+1); no child rounding'),
+        rows=dict(coefficient=coeff,degree=degree,suffix_slope=4*degree,
+            degree_gap=Q(degree)-Q(coeff*51,25),
+            contract='W_complex^D_complex * W_coarse^D_coarse * W_old^D_old; one preceding prefix and one padding; sequential reuse'))
+
+
+def certificate():
+    require(not sys.flags.optimize, 'Run without -O')
+    def read(name):
+        return json.loads((ROOT/'certificates'/name).read_text())
+    raw = reconstruct()
+    row = read('partial-gauge-complex-input.json')
+    expected = read('partial-gauge-input.json')
+    previous = read('copied-centers-network.json')
+    require(Q(previous['bit']['saving']) == OLD, 'Inherited leaf saving')
+    C = {int(r):n for r,n in raw['whole_projector_histogram'].items()}
+    bit = dict(dimensions=raw['dimensions'],m=raw['m'],W=raw['W'],N=raw['N'],
+        total_rank=raw['rank_mass'],deficit=raw['deficit'],maxchild=max(C),child_multiplicities=C)
+    require(sum(r*n for r,n in C.items()) == bit['total_rank'], 'Bit rank mass')
+    require(bit['W']*bit['m']-bit['total_rank'] == bit['deficit'], 'Bit deficit')
+    phase = complex_profile(row)
+    bm = coarse_moment(bit)
+    cm = moment(phase['m'],phase['W'],phase['child_multiplicities'],AC,True)
+    require(bm['strict_gap'] == Q(expected['bit_moment_gap']), 'Saved coarse moment')
+    require(cm['strict_gap'] == Q(expected['complex_moment_gap']), 'Saved complex moment')
+    require(js(phase['child_multiplicities']) == expected['complex_children'], 'Saved complex children')
+    bridge = finite_bridge(bit,phase,row,previous)
+    assembled = assembly(ASSEMBLY_BIT,AC,bridge,KAPPA,beta=PHASE_STOP)
+    assembled['parameters']['actual_bit_saving'] = AB
+    require(ASSEMBLY_BIT <= AB, 'Supported assembly movement saving')
+    require(js(bridge) == expected['finite_bridge'], 'Saved finite bridge')
+    require(js(assembled) == expected['assembly'], 'Saved exact assembly')
+    require(len(assembled['strict_constraints']) == 47 and len(assembled['margins']) == 7, 'Complete strict constraints')
+    sources = sorted(p for p in (ROOT/'certificates').glob('partial-gauge-*.json')
+                     if p.name != 'partial-gauge-network.json')
+    sources += sorted((ROOT/'scripts').glob('partial_gauge*.py'))
+    sources += sorted(p for p in (ROOT/'scripts/partial_gauge').rglob('*')
+                      if p.is_file() and '__pycache__' not in p.parts)
+    sources += sorted((ROOT/'notes').glob('partial-gauge-*.tex'))
+    sources += sorted(p for p in (ROOT/'references/partial-gauge').rglob('*')
+                      if p.is_file() and '__pycache__' not in p.parts)
+    sources += [ROOT/p for p in ('scripts/copied_centers_network.py',
+        'scripts/partial_swap_network.py','scripts/structured_bulk_assembly.py',
+        'certificates/copied-centers-network.json','certificates/stopped-product-network.json',
+        'scripts/paired_triple_circuit.py','scripts/paired_exclusion_circuit.py')]
+    return dict(status='Conditional partial-gauge multiplication witness',kappa=KAPPA,
+        bit=dict(counts=bit,**bm),complex=dict(counts=phase,**cm),
+        finite_bridge=bridge,assembly=assembled,
+        predecessor_commit='948ce1510df750f4c18b96bdaef436a86f8bf834',
+        source_sha256={str(p.relative_to(ROOT)):sha256(p.read_bytes()).hexdigest() for p in sources},
+        scope='Exact moments, stopped bit saving and semantic assembly. Finite producer/frame checks '
+              'are separate; common-basis, fixed-tape and analytic transfer remain written proof dependencies.')
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--output',type=Path,default=ROOT/'certificates/partial-gauge-network.json')
+    args = parser.parse_args()
+    args.output.write_text(json.dumps(js(certificate()),indent=2,sort_keys=True)+'\n')
+    print('PASS kappa=7237/78125000 = 9.26336e-5; both moments, stopped saving, p^40000 stock, 47 strict constraints')
+
+
+if __name__ == '__main__':
+    main()
diff --git a/scripts/partial_gauge_producer.py b/scripts/partial_gauge_producer.py
new file mode 100644
index 0000000000..2d91d92b8e
--- /dev/null
+++ b/scripts/partial_gauge_producer.py
@@ -0,0 +1,67 @@
+#!/usr/bin/env python3
+"""Regenerate and verify the selected h=24 partial-gauge complex producer.
+
+Standard-library Python and C++17; generated DAGs and words are temporary.
+Copyright 2026 icekylinx, Apache-2.0. Adapted from the credited round-six handoff.
+"""
+import argparse
+import json
+import os
+from pathlib import Path
+import shlex
+import subprocess
+import sys
+import tempfile
+from partial_gauge.scalar import build
+from partial_gauge.lift import run as lift_run
+from partial_gauge.word import run as word_run
+from partial_gauge.select import run as select_run
+from partial_gauge.verify import verify
+
+ROOT=Path(__file__).resolve().parents[1]
+
+
+def regenerate(work, expected, compiler):
+    if sys.flags.optimize:raise ValueError('Run without -O; finite assertions must remain enabled')
+    def stage(message):print(message,file=sys.stderr,flush=True)
+    def save(name,data):
+        path=work/name;path.write_text(json.dumps(data,separators=(',',':'))+'\n');return path
+    stage('Building the actual interval/cube scalar DAG')
+    scalar=build(24,base=2,central_disjoint=24,output_prefix='complex_cube',output_dir=work)
+    dag=work/'complex_cube_24.bin';matcher=work/'matcher'
+    subprocess.run([*shlex.split(compiler),'-O3','-std=c++17',str(ROOT/'scripts/partial_gauge/match.cpp'),'-o',str(matcher)],check=True)
+    stage('Constructing compatible carrier matching')
+    matching=json.loads(subprocess.check_output([str(matcher),str(dag),str(dag.with_suffix('.labels'))],text=True))
+    matchfile=save('matching.json',matching)
+    stage('Lifting exact binary frames backward through dependencies and carriers')
+    lifted,ann=lift_run(dag,matchfile);liftfile=save('lifted.json',lifted);annfile=save('annihilators.json',ann)
+    stage('Constructing physical M and its retained-center dependency closure')
+    word=word_run(dag,matchfile,annfile,liftfile);wordfile=save('word.json',word)
+    stage('Selecting partial source gauges with reverse readouts')
+    result,selection=select_run(dag,annfile,liftfile,wordfile)
+    result=json.loads(json.dumps(result))
+    assert result==expected, 'Regenerated selected certificate differs'
+    save('selection.json',selection)
+    stage('Checking scalar coefficients, actual frames, physical word and readout chains')
+    checks=verify(dag,matching,ann,lifted,word,selection['selected'],result)
+    return dict(regenerated_from_source=True,certificate_equal=True,selected=result,checks=checks)
+
+
+def main():
+    p=argparse.ArgumentParser(description=__doc__)
+    p.add_argument('--certificate',type=Path,default=ROOT/'certificates/partial-gauge-complex-input.json')
+    p.add_argument('--work-dir',type=Path)
+    p.add_argument('--output',type=Path)
+    p.add_argument('--cxx',default=os.environ.get('CXX','c++'))
+    a=p.parse_args();expected=json.loads(a.certificate.read_text())
+    if a.work_dir:
+        a.work_dir.mkdir(parents=True,exist_ok=True);result=regenerate(a.work_dir.resolve(),expected,a.cxx)
+    else:
+        with tempfile.TemporaryDirectory(prefix='partial-gauge-') as directory:
+            result=regenerate(Path(directory),expected,a.cxx)
+    encoded=json.dumps(result,indent=2)+'\n'
+    if a.output:a.output.write_text(encoded)
+    else:print(encoded,end='')
+
+
+if __name__=='__main__':main()
diff --git a/scripts/stopped_product/README.md b/scripts/stopped_product/README.md
new file mode 100644
index 0000000000..6c18be051f
--- /dev/null
+++ b/scripts/stopped_product/README.md
@@ -0,0 +1,36 @@
+# Selected rational-center producer
+
+Run from the repository root:
+
+```sh
+python3 scripts/stopped_product_producer.py --output /tmp/stopped-product-producers.json
+```
+
+Only the new aligned complex producer at h=24 is rebuilt. It retains all
+24 disjoint-point sums and decodes the total by exact division by21. The
+pair-star input order is aligned with global pairs. Its generator adapts
+`structured_bulk.complex`, preserving exact support, output orthogonality,
+binary frame nesting and nondegeneracy checks. Rational decoder and scatter
+coefficients are checked with `Fraction`; their only odd denominator is21.
+The unchanged deterministic matcher compares every count and histogram
+entry against `stopped-product-complex-input.json`.
+
+The h23 bit record is compared with the preceding validated copied-center
+input. No old bit DAG, matching, or corner is regenerated. Both copied-center
+histogram transformations reuse the established exact rank-mass checker.
+
+All DAGs, labels and the compiled matcher are temporary by default;
+`--work-dir` retains them. Python assertions must be enabled. Standard-library
+Python and a C++17 compiler suffice. No binary archive is an input.
+
+The odd-grid extension is a written recursive invariant, not a claim proved
+by the finite scalar tests. With dyadic root inputs, K=G(D+1) supplies enough
+odd-divisor reserve along the active chain. A completed child preserves its
+incoming odd valuation; it does not generally make rational child inputs
+dyadic. Only the completed outer dyadic operator restores the original
+dyadic grid. No child return is rounded.
+
+Copyright 2026 icekylinx, Apache-2.0. New selected alignment, rational centers
+and integration were prepared with OpenAI GPT-6 Astra/Codex assistance.
+Underlying paired producers and deterministic carrier matching retain the
+previous contributors' attribution in `NOTICE` and `SOURCES.json`.
diff --git a/scripts/stopped_product/__init__.py b/scripts/stopped_product/__init__.py
new file mode 100644
index 0000000000..f7a6e90661
--- /dev/null
+++ b/scripts/stopped_product/__init__.py
@@ -0,0 +1 @@
+"""Selected stopped-product inputs and rational producer; icekylinx, 2026."""
diff --git a/scripts/stopped_product/complex.py b/scripts/stopped_product/complex.py
new file mode 100644
index 0000000000..ef1af25507
--- /dev/null
+++ b/scripts/stopped_product/complex.py
@@ -0,0 +1,228 @@
+"""Selected aligned rational-center producer, h=24 with all centers disjoint.
+
+Copyright 2026 icekylinx, Apache-2.0. AI-assisted adaptation of the research
+round-five handoff, using the credited retained PairedTriple circuit; see NOTICE.
+Adapted from structured_bulk.complex: align pair-star inputs and permit the
+fixed rational divisor21; retain all exact support/frame/scalar checks.
+
+Builds a concrete scalar addition DAG, its nondegenerate binary frames, and
+the unmixed carrier histogram. No historical audit suite is invoked.
+"""
+import json, struct, array
+from itertools import combinations
+from pathlib import Path
+from partial_swap.binary import write_array
+from paired_triple_circuit import PairedTriple
+
+def build(h, prefix, central_disjoint, base=2):
+    assert (h, central_disjoint, base) == (24,24,2)
+    d=central_disjoint
+    assert 0<d<=h
+    divisor=abs(3-d)
+    assert divisor == 21
+    triples=list(combinations(range(h),3)); v=len(triples)
+    args=[(0,0)]; support=[0]; core=[0]; cover=[0]; types=[0]; ranks=[0]
+    lookup={}; inputs={}; center_coeff={}
+    # Each central tree adds h-1 ordinary pair totals, so every intermediate
+    # coefficient is at most h-1. Wider lanes rule out hidden integer carries.
+    coefficient_bits=h.bit_length()
+    def packed(mask):
+        result=0
+        while mask:
+            bit=mask&-mask; mask-=bit
+            result |= 1 << (coefficient_bits*(bit.bit_length()-1))
+        return result
+    for j,t in enumerate(triples):
+        mask=sum(1<<a for a in t); x=len(args); inputs[t]=x
+        args.append((0,0));support.append(1<<j);core.append(mask);cover.append(mask)
+        types.append(1);ranks.append(1);lookup[1<<j]=x
+    def add(a,b,center=None):
+        if not a:return b
+        if not b:return a
+        if center is None:
+            assert not support[a]&support[b], "Ordinary additions must be cancellation-free"
+        s=support[a]|support[b]
+        if center is None and s in lookup:return lookup[s]
+        x=len(args);args.append((a,b));support.append(s)
+        core.append(core[a]&core[b]);cover.append(cover[a]|cover[b])
+        if center is not None:
+            types.append(3);ranks.append(h-1);core[x]=1<<center
+            ca=center_coeff[a] if a in center_coeff else packed(support[a])
+            cb=center_coeff[b] if b in center_coeff else packed(support[b])
+            center_coeff[x]=ca+cb
+        elif core[x].bit_count()>=2:
+            types.append(1);ranks.append(s.bit_count());lookup[s]=x
+        else:
+            types.append(2);ranks.append(cover[x].bit_count());lookup[s]=x
+        return x
+    def total(xs,center=None):
+        xs=[x for x in xs if x]
+        while len(xs)>1:
+            xs=[add(xs[i],xs[i+1],center) if i+1<len(xs) else xs[i] for i in range(0,len(xs),2)]
+        return xs[0] if xs else 0
+    paired=PairedTriple(h,base)
+    allresults=paired.triple(list(range(h)),paired.variables)
+    selected=list(paired.outputs.values())+[allresults[(i,)] for i in range(d)]
+    active=set();stack=list(selected)
+    while stack:
+        x=stack.pop()
+        if not x or x in active:continue
+        active.add(x)
+        if paired.args[x]:stack.extend(paired.args[x])
+    mapping={0:0}|{j+1:inputs[t] for j,t in enumerate(triples)}
+    for x in sorted(active):
+        if paired.args[x]:
+            a,b=paired.args[x];mapping[x]=add(mapping[a],mapping[b])
+    roots=[mapping[paired.outputs[t]] for t in triples];kind=[0]*v
+    center_disjoint_roots=[mapping[allresults[(i,)]] for i in range(d)]
+    del paired, mapping, active, allresults
+    pairtotals={}; pair_outputs={}
+    point_masks=[sum(1<<j for j,t in enumerate(triples) if a in t) for a in range(h)]
+    full_support=(1<<v)-1
+    for t,node in zip(triples,roots):
+        assert support[node] == full_support & ~(point_masks[t[0]]|point_masks[t[1]]|point_masks[t[2]])
+        target_mask=sum(1<<a for a in t)
+        if types[node]==2:
+            assert not cover[node]&target_mask
+        else:
+            assert types[node]==1
+            assert not support[node]&(point_masks[t[0]]^point_masks[t[1]]^point_masks[t[2]])
+    for i,node in enumerate(center_disjoint_roots):
+        assert support[node] == full_support & ~point_masks[i]
+        assert types[node]==2 and cover[node]==((1<<h)-1)^(1<<i)
+        assert ranks[node]==h-1
+    for a,b in combinations(range(h),2):
+        others=[i for i in range(h) if i not in (a,b)]
+        others.sort(key=lambda i:((i^1) in (a,b),i))
+        pre=[0]
+        for i in others:pre.append(add(pre[-1],inputs[tuple(sorted((a,b,i)))]))
+        suf=[0]*(len(others)+1)
+        for j in range(len(others)-1,-1,-1):suf[j]=add(inputs[tuple(sorted((a,b,others[j])))],suf[j+1])
+        for j,i in enumerate(others):
+            node=add(pre[j],suf[j+1])
+            roots.append(node);kind.append(0)
+            pair_outputs[a,b,i]=node
+            assert support[node] == point_masks[a]&point_masks[b]&~point_masks[i]
+            assert types[node]==1
+            assert not support[node]&(point_masks[a]^point_masks[b]^point_masks[i])
+        pairtotals[a,b]=pre[-1]
+        assert support[pre[-1]] == point_masks[a]&point_masks[b]
+    for a in range(d,h):
+        root=total([pairtotals[tuple(sorted((a,b)))] for b in range(h) if b!=a],center=a)
+        assert center_coeff[root] == 2*packed(point_masks[a]), "Center coefficients must be exactly doubled"
+        roots.append(root);kind.append(1)
+    roots.extend(center_disjoint_roots);kind.extend([1]*d)
+    # Mixed retained centers recover the total by
+    # (sum B_i - 2 sum D_i)/(2(3-d)); check each source coefficient.
+    Dset=set(range(d))
+    for triple in triples:
+        k=len(Dset.intersection(triple))
+        assert 2*(3-k)-2*(d-k)==2*(3-d)
+    # Check the rational substituted center scatter for every realizable pair of
+    # target/source memberships. Its coefficient is (intersection-1)/2.
+    from fractions import Fraction
+    # Every A_i avoids i, so sum A_i = (h-3)T. The selected decoder is
+    # T=sum A_i/21 and scatter T-(1/2)sum_{i in target} A_i.
+    decoder=Fraction(1,h-3)
+    assert decoder==Fraction(1,21)
+    scatter_coefficients=[decoder,decoder-Fraction(1,2)]
+    for coefficient in scatter_coefficients:
+        odd_denominator=coefficient.denominator
+        while odd_denominator%2==0:
+            odd_denominator//=2
+        assert 21%odd_denominator==0 and abs(coefficient)<=1
+    for source in triples:
+        assert sum(i not in source for i in range(h))*decoder==1
+    patterns={}
+    for target in triples:
+        chosen=len(Dset.intersection(target))
+        if chosen not in patterns:
+            patterns[chosen]=target
+    for chosen,target in patterns.items():
+        target_set=set(target)
+        for source in triples:
+            k=len(Dset.intersection(source))
+            total_coefficient=Fraction(2*(3-k)-2*(d-k),2*(3-d))
+            scalar=Fraction(len((target_set-Dset).intersection(source)),2)
+            scalar-=Fraction(len((target_set&Dset)-set(source)),2)
+            scalar+=Fraction(chosen-1,2)*total_coefficient
+            assert scalar==Fraction(len(target_set.intersection(source))-1,2)
+    # Check the scalar correction for each target, using the verified exact
+    # roots. Twice the coefficient is (intersection-1)+[intersection=0]
+    # -[intersection=2], which is two precisely for the diagonal.
+    for j,t in enumerate(triples):
+        masks=[point_masks[a] for a in t]
+        two=(masks[0]&masks[1]&~masks[2]) | (masks[0]&masks[2]&~masks[1]) | (masks[1]&masks[2]&~masks[0])
+        actual=0
+        for a,b in combinations(t,2):
+            excluded=next(i for i in t if i not in (a,b))
+            actual |= support[pair_outputs[a,b,excluded]]
+        assert actual == two
+        assert masks[0]&masks[1]&masks[2] == 1<<j
+    assert [((k-1)+(k==0)-(k==2)) for k in range(4)] == [0,0,0,2]
+    def contained(x,y):
+        tx,ty=types[x],types[y]
+        if tx==1 and ty==1: return not(core[y]&~core[x] or cover[x]&~cover[y])
+        if tx in (1,2) and ty==2: return not cover[x]&~cover[y]
+        if tx==1 and ty==3: return bool(core[x]&core[y])
+        if tx==2 and ty==3: return not cover[x]&~core[y]
+        if tx==3 and ty==2: return cover[y]==(1<<h)-1
+        if tx==3 and ty==3: return core[x]==core[y]
+        return False
+    active=[0]*len(args);stack=list(roots)
+    while stack:
+        x=stack.pop()
+        if not x or active[x]:continue
+        active[x]=1
+        if args[x][0]:stack.extend(args[x])
+    degree=[0]*len(args)
+    for x in range(1,len(args)):
+        if active[x]:
+            if types[x]==1:
+                # Shared-pair triple indicators have self-dot one and
+                # pairwise dot zero over F2; a source is a single such line.
+                assert core[x].bit_count()>=2 and ranks[x]==support[x].bit_count()
+            elif types[x]==2:
+                assert ranks[x]==cover[x].bit_count()
+            else:
+                # E_a is the orthogonal complement of the odd vector 1+e_a.
+                assert types[x]==3 and core[x].bit_count()==1 and ranks[x]==h-1
+        if active[x] and args[x][0]:
+            for y in args[x]:
+                assert y<x and contained(y,x), "Binary frame nesting failed"
+                degree[y]+=1
+    for x in roots:degree[x]+=1
+    H=[0]*(h+1);c=0;loss=0
+    for x in range(1,len(args)):
+        if not active[x]:continue
+        r=ranks[x]
+        if args[x][0]:
+            c+=1;H[r]+=degree[x]-1;H[h-r]+=1
+            for y in args[x]:
+                assert r>=ranks[y]
+                H[r-ranks[y]]+=1
+        else:H[1]+=degree[x]
+    for x,k in zip(roots,kind):
+        r=ranks[x]
+        if k:H[r]+=1;H[h]+=1;loss+=r
+        else:H[h-1-r]+=1;H[1]+=1
+    q=len(roots);R=c+q
+    result=dict(h=h,v=v,c=c,q=q,R=R,loss=loss,central_disjoint=d,histogram=H,rank_sum=sum(r*n for r,n in enumerate(H)))
+    assert loss == h*(h-1) and result['rank_sum'] == h*R+2*loss
+    prefix=Path(prefix)
+    result['center_denominator']=21
+    result['scalar_validation']=dict(ordinary_supports_exact=True, centers_exact=True,
+        center_decoder='sum A_i / 21', scatter_coefficients=list(map(str,scatter_coefficients)),
+        rational_scalar_identity_exact=True, mixed_center_scatter_exact=True, binary_frames_nested=True,
+        binary_frames_nondegenerate=True)
+    with open(str(prefix)+'.bin','wb') as f:
+        f.write(struct.pack('<4I',h,v,len(args),q))
+        write_array(f,array.array('I',(a for pair in args for a in pair)))
+        write_array(f,array.array('Q',core));write_array(f,array.array('Q',cover))
+        write_array(f,array.array('I',roots));write_array(f,array.array('I',kind))
+        write_array(f,array.array('B',active))
+    with open(str(prefix)+'.labels','wb') as f:
+        write_array(f,array.array('I',ranks));write_array(f,array.array('B',types))
+    Path(str(prefix)+'.json').write_text(json.dumps(result,indent=2)+'\n')
+
+    return result
diff --git a/scripts/stopped_product_network.py b/scripts/stopped_product_network.py
new file mode 100644
index 0000000000..5f349a2676
--- /dev/null
+++ b/scripts/stopped_product_network.py
@@ -0,0 +1,163 @@
+#!/usr/bin/env python3
+"""Exact stopped product-ring moments, rational-center guard and assembly.
+
+Copyright 2026 icekylinx. Apache-2.0.
+Developed with substantial OpenAI GPT-6 Astra and Codex assistance.
+Retains the copied-center two-stage construction, Paureel/PR29 topology,
+PR23 semantic assembly and RaD analytic interfaces; see SOURCES.json.
+"""
+import argparse
+from collections import Counter
+from fractions import Fraction as Q
+from hashlib import sha256
+from math import comb
+from pathlib import Path
+import json
+import sys
+
+from certify import require
+from copied_centers.physical import copied_histogram
+from copied_centers_network import finite_bridge as copied_bridge, grid_upper
+from partial_swap_network import moment
+from structured_bulk_assembly import assembly, js
+
+ROOT = Path(__file__).resolve().parents[1]
+COARSE = Q(4019, 50000000)
+ATOM = Q(1, 1000)
+OLD = Q(384599, 10**10)
+AB = (1-ATOM)*COARSE+ATOM*OLD
+AC = Q(1949, 25000000)
+PHASE_STOP = Q(1, 10**6)
+ASSEMBLY_BIT = min(AB, (1-PHASE_STOP)*AC-Q(1, 10**10))
+KAPPA = Q(194869, 2500000000)
+
+
+def profile(row):
+    h, v, R, ell = (row[k] for k in ('h', 'v', 'R', 'loss'))
+    require(v == comb(h, 3), 'Triple count')
+    H = copied_histogram(row)['histogram']
+    m, N, B = h*h, v*v, v*R
+    W, L = 2*N+2*B, 2*v*ell
+    z = Counter({m-h:2*B, (h-1)**2:2*N, h-1:4*N, 1:N})
+    for r, n in enumerate(H):
+        if r and n:
+            z[r] += 2*v*n
+    z = dict(sorted((t,n) for t,n in z.items() if n))
+    s = W*m-N+L
+    require(all(0 < t < m and n > 0 for t,n in z.items()), 'Shrinking children')
+    require(sum(t*n for t,n in z.items()) == s, 'Complete rank mass')
+    return dict(dimensions=[h,h], m=m, N=N, B1=B, B2=B, W=W, L=L,
+                total_rank=s, deficit=N-L, maxchild=max(z),
+                copied_histogram=H, child_multiplicities=z)
+
+
+def coarse_moment(p):
+    def log_upper(x):
+        power = 0
+        while x >= 2:
+            x /= 2
+            power += 1
+        def small(y):
+            z = (y-1)/(y+1)
+            return 2*sum((z**(2*j+1)/(2*j+1) for j in range(24)), Q(0)) + 2*z**49/(49*(1-z*z))
+        return grid_upper(power*small(Q(2))+small(x))
+    total, logs = Q(0), {}
+    for t, n in p['child_multiplicities'].items():
+        logs[t] = log_upper(Q(p['m'],t))
+        u = COARSE*logs[t]
+        require(0 < u < 3, 'Coarse exponential enclosure')
+        total += n*t*grid_upper(1+u+u*u/(2*(1-u/3)))
+    upper = total/(p['W']*p['m'])
+    mass = Q(p['total_rank'],p['W']*p['m'])
+    require(mass < 1 and upper < 1, 'Both stopped-recurrence moments')
+    require(0 < OLD <= COARSE < ATOM < 1, 'Stopped-adapter exponents')
+    require(ATOM > AB, 'Adapter toll subordinate to the stopped saving')
+    return dict(saving=COARSE, exponent=1-COARSE, moment_upper=upper,
+                strict_gap=1-upper, rank_moment=mass,
+                logarithm_upper_bounds=logs, rounding_denominator=1 << 100,
+                stopped_adapter=dict(atom_exponent=ATOM, base_ordinary_saving=OLD,
+                    effective_saving=AB, coarse_toll_exponent=1-ATOM,
+                    coarse_toll_subordinate_gap=ATOM-COARSE))
+
+
+def finite_bridge(bit, phase, row, previous):
+    bridge = copied_bridge(bit, phase, [row,row])
+    old_bit = previous['finite_bridge']['bit']
+    old_degree = old_bit['halving_degree']*old_bit['wire_bits']
+    require(old_degree == 252, 'Inherited ordinary leaf row stock')
+    coarse = bridge.pop('bit')
+    c = bridge['complex']
+    coefficient = (coarse['halving_degree']*coarse['wire_bits']+old_degree+
+                   c['halving_degree']*c['wire_bits'])
+    degree = 4000
+    require(Q(degree) > Q(51*coefficient,25), 'Product stock strictly sufficient')
+    bridge.update(bit_coarse=coarse, ordinary_leaf_row_degree=old_degree)
+    bridge['rows'] = dict(coefficient=coefficient, degree=degree, suffix_slope=4*degree,
+        degree_gap=Q(degree)-Q(51*coefficient,25),
+        contract='W_complex^D_complex * W_coarse^D_coarse * W_old^D_old; one preceding prefix and one padding; sequential reuse')
+    require(row['h'] == row['central_disjoint'] == 24, 'Selected all-disjoint centers')
+    divisor = row['h']-3
+    require(divisor == 21, 'Selected fixed odd divisor')
+    bridge['semantic'].update(fixed_odd_divisor=divisor,
+        exact_grid='One common dyadic grid times the fixed odd divisor to K=G*(D_complex+1); no child rounding')
+    return bridge
+
+
+def certificate():
+    require(not sys.flags.optimize, 'Run without -O')
+    def read(name):
+        return json.loads((ROOT/'certificates'/name).read_text())
+    bit_row = read('stopped-product-bit-axis.json')
+    complex_row = read('stopped-product-complex-input.json')
+    expected = read('stopped-product-input.json')
+    previous = read('copied-centers-network.json')
+    require(Q(previous['bit']['saving']) == OLD, 'Certified ordinary leaf saving')
+    require(bit_row['h'] == 23, 'Selected bit dimension')
+    bit, phase = profile(bit_row), profile(complex_row)
+    for label,p in (('bit',bit),('complex',phase)):
+        for key,value in expected[label].items():
+            require(js(p[key]) == value, label+' saved '+key)
+    bm = coarse_moment(bit)
+    cm = moment(phase['m'],phase['W'],phase['child_multiplicities'],AC,True)
+    require(bm['strict_gap'] == Q(expected['bit_moment_gap']), 'Saved coarse moment')
+    require(cm['strict_gap'] == Q(expected['complex_moment_gap']), 'Saved complex moment')
+    bridge = finite_bridge(bit,phase,complex_row,previous)
+    assembled = assembly(ASSEMBLY_BIT,AC,bridge,KAPPA,beta=PHASE_STOP)
+    require(ASSEMBLY_BIT <= AB, 'Assembly saving supported by actual bit interface')
+    assembled['parameters']['actual_bit_saving'] = AB
+    require(js(bridge) == expected['finite_bridge'], 'Saved semantic and product bridge')
+    require(js(assembled) == expected['assembly'], 'Saved exact assembly')
+    require(len(assembled['strict_constraints']) == 47 and len(assembled['margins']) == 7,
+            'Complete assembly constraints')
+    sources = sorted(p for p in (ROOT/'certificates').glob('stopped-product-*.json')
+                     if p.name != 'stopped-product-network.json')
+    sources += sorted((ROOT/'scripts').glob('stopped_product*.py'))
+    sources += sorted(p for p in (ROOT/'scripts/stopped_product').rglob('*')
+                      if p.is_file() and '__pycache__' not in p.parts)
+    sources += sorted((ROOT/'notes').glob('stopped-product-*.tex'))
+    sources += [ROOT/path for path in (
+        'scripts/copied_centers/physical.py','scripts/copied_centers_network.py',
+        'scripts/partial_swap_network.py','scripts/structured_bulk_assembly.py',
+        'certificates/copied-centers-network.json','certificates/copied-centers-bit-axes.json')]
+    return dict(status='Conditional stopped product-ring multiplication witness',kappa=KAPPA,
+        bit=dict(counts=bit,**bm),complex=dict(counts=phase,**cm),
+        finite_bridge=bridge,assembly=assembled,
+        base_commit='56b66d58297deca1d7dd130247d720e960f77a37',
+        proof_predecessor_commit='11817ccacb564bb7f98789c20dc11d3fece207e3',
+        source_sha256={str(p.relative_to(ROOT)):sha256(p.read_bytes()).hexdigest() for p in sources},
+        scope='Exact finite moments, stopped savings and assembly; the new complex producer is checked separately. '
+              'Opposite-bank factorization, atom streaming, ordinary conversion, common exact grid and retained analytic/tape interfaces are written proof dependencies.')
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--output',type=Path,default=ROOT/'certificates/stopped-product-network.json')
+    args = parser.parse_args()
+    result = certificate()
+    args.output.write_text(json.dumps(js(result),indent=2,sort_keys=True)+'\n')
+    print('PASS kappa=194869/2500000000 = 7.79476e-5')
+    print('Coarse and complex moments; stopped bit saving; exact rational-center guard; p^4000 stock; 47 strict constraints')
+
+
+if __name__ == '__main__':
+    main()
diff --git a/scripts/stopped_product_producer.py b/scripts/stopped_product_producer.py
new file mode 100644
index 0000000000..aefc91d9a7
--- /dev/null
+++ b/scripts/stopped_product_producer.py
@@ -0,0 +1,92 @@
+#!/usr/bin/env python3
+"""Regenerate only the new h24 aligned all-disjoint rational-center producer.
+
+Copyright 2026 icekylinx, Apache-2.0. Prepared with OpenAI GPT-6 Astra and
+Codex assistance. Prior paired circuit and matcher attribution is retained
+in NOTICE. No old bit producer is regenerated.
+"""
+import argparse
+import gc
+import json
+import os
+from pathlib import Path
+import shlex
+import subprocess
+import sys
+import tempfile
+
+from copied_centers.corners import require
+from copied_centers.physical import copied_histogram
+from endpoint_gauge_producer import compare
+from stopped_product.complex import build
+
+ROOT = Path(__file__).resolve().parents[1]
+
+
+def read(path):
+    return json.loads(path.read_text())
+
+
+def regenerate(work, bit, expected, previous_bits, compiler):
+    require(sys.flags.optimize == 0, 'Run without -O: mathematical assertions must remain enabled')
+    require(bit['h'] == 23 and bit['label_source'] == 'positive', 'Wrong selected bit axis')
+    old = next(row for row in previous_bits if row['h'] == 23)
+    for key in ('h','v','c','q','baseline_R','matching','R','loss','histogram','label_source'):
+        require(bit[key] == old[key], 'Reused bit producer mismatch: '+key)
+    require(bit['R'] == bit['c']+bit['q']-bit['matching'], 'Bit role identity failed')
+    bit_copied = copied_histogram(bit)
+    require((expected['h'],expected['central_disjoint'],expected['center_denominator']) ==
+            (24,24,21), 'Wrong rational-center parameters')
+    print('Regenerating only complex h=24: aligned pair stars, all24 disjoint centers, divisor21',
+          file=sys.stderr, flush=True)
+    prefix = work/'complex_rational_d24_24'
+    construction = build(24, prefix, central_disjoint=24)
+    gc.collect()
+    program = work/'match_complex_general'
+    source = ROOT/'scripts'/'endpoint_gauge'/'match_complex_general.cpp'
+    subprocess.run([*shlex.split(compiler), '-O3', '-std=c++17', str(source),
+                    '-o', str(program)], check=True)
+    matched = json.loads(subprocess.check_output(
+        [str(program), str(prefix)+'.bin', str(prefix)+'.labels'], text=True))
+    compare(matched, expected, 'complex h=24 rational centers')
+    require(construction['R'] == matched['baseline_R'], 'Wrong raw complex roles')
+    phase_copied = copied_histogram(matched)
+    return dict(bit=dict(reused_validated_producer=True, certificate_equal=True,
+                         copied=bit_copied),
+                complex=dict(**matched, central_disjoint=24, center_denominator=21,
+                             construction=construction, copied=phase_copied,
+                             certificate_equal=True),
+                old_bit_regeneration_skipped=True,
+                scope='Exact new scalar supports, rational scatter, nested frames and matching histogram. '
+                      'The common odd-denominator grid and recursive tape transfer remain written proof interfaces.')
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    for name, filename in (
+            ('bit-axis','stopped-product-bit-axis.json'),
+            ('complex-input','stopped-product-complex-input.json'),
+            ('previous-bits','copied-centers-bit-axes.json')):
+        parser.add_argument('--'+name, type=Path, default=ROOT/'certificates'/filename)
+    parser.add_argument('--cxx', default=os.environ.get('CXX','c++'))
+    parser.add_argument('--work-dir', type=Path, help='Keep generated intermediate files')
+    parser.add_argument('--output', type=Path, help='Write full incremental verification JSON')
+    args = parser.parse_args()
+    inputs = [read(path) for path in (args.bit_axis,args.complex_input,args.previous_bits)]
+    def run(work):
+        return regenerate(work,*inputs,args.cxx)
+    if args.work_dir:
+        args.work_dir.mkdir(parents=True,exist_ok=True)
+        result = run(args.work_dir.resolve())
+    else:
+        with tempfile.TemporaryDirectory(prefix='stopped-product-') as directory:
+            result = run(Path(directory))
+    encoded = json.dumps(result,indent=2)+'\n'
+    if args.output:
+        args.output.write_text(encoded)
+    else:
+        print(encoded,end='')
+
+
+if __name__ == '__main__':
+    main()
diff --git a/scripts/three_stage_cover/README.md b/scripts/three_stage_cover/README.md
new file mode 100644
index 0000000000..74ccdc31e2
--- /dev/null
+++ b/scripts/three_stage_cover/README.md
@@ -0,0 +1,23 @@
+# Selected PR117 local word
+
+`scripts/three_stage_cover_producer.py` rebuilds the pinned positive h24
+DAG, matching, full backward intersections, physical word and selected
+partial source gauges. It reproduces 91,770 additions, 71,185 matches,
+28,705 auxiliary roles and 4,599 selected gauges.
+
+The generalized lift and selector retain entire binary intersections,
+including radicals. The new [Clifford frame proof](../../notes/general-clifford-frames.tex)
+gives the actual transition and tail ranks for these arbitrary subspaces.
+The finite verifier checks containment and chronology, rather than inferring
+inclusion from frame dimensions.
+
+The unchanged `partial_gauge` binary format, matching and physical-word
+compiler are reused. Its previously submitted files remain immutable.
+The imported PR117 DAG and replay retain eumemic's copyright, Apache-2.0
+terms and Claude assistance disclosure. New general-frame compilation is
+contributed by icekylinx with OpenAI GPT-6 Astra assistance; repository integration uses
+Codex assistance. See `NOTICE` and `SOURCES.json`.
+
+Generated graphs are temporary by default. The verifier compares every
+saved profile field and independently recounts positive-rank operations;
+zero-rank no-op multiplicities do not enter either supplier moment.
diff --git a/scripts/three_stage_cover/__init__.py b/scripts/three_stage_cover/__init__.py
new file mode 100644
index 0000000000..546aa1b037
--- /dev/null
+++ b/scripts/three_stage_cover/__init__.py
@@ -0,0 +1,3 @@
+# Copyright 2026 icekylinx. Apache-2.0.
+# Integration with OpenAI Codex assistance; see NOTICE.
+"""Pinned PR117 generalized-subspace complex producer."""
diff --git a/scripts/three_stage_cover/lift.py b/scripts/three_stage_cover/lift.py
new file mode 100644
index 0000000000..24517f61f9
--- /dev/null
+++ b/scripts/three_stage_cover/lift.py
@@ -0,0 +1,79 @@
+# Copyright 2026 icekylinx. Apache-2.0.
+# Integration with OpenAI Codex assistance; see NOTICE.
+"""Full backward intersections, including degenerate subspaces.
+Adapted from partial_gauge.lift; copyright 2026 icekylinx, Apache-2.0.
+"""
+from pathlib import Path
+from collections import Counter
+import json
+from partial_gauge.binary import load
+from partial_gauge.lift import basis, perp, nondeg
+
+def run(path,matchfile):
+    h,v,n,q,args,core,cover,roots,kind,active,oldrank,types=load(path)
+    meta=json.loads(Path(matchfile).read_text());arcs=dict(meta['matching_arcs']);full=(1<<h)-1
+    succ=[[] for _ in range(n)];direct=[[] for _ in range(n)]
+    for x in range(1,n):
+        if active[x] and args[2*x]:
+            for y in (args[2*x],args[2*x+1]):succ[y].append(x)
+    for j,(x,k) in enumerate(zip(roots,kind)):
+        normal=(full^cover[x]) if k else (core[x]|(full^cover[x]))
+        assert normal.bit_count()==(1 if k else 3)
+        direct[x].append(normal)
+    for donor,use in arcs.items():
+        if use>>31:
+            j=use&0x7fffffff;x=roots[j];k=kind[j]
+            normal=(full^cover[x]) if k else (core[x]|(full^cover[x]))
+            direct[donor].append(normal)
+        else:succ[donor].append(use//2)
+    order=sorted((x for x in range(1,n) if active[x]),key=lambda x:(oldrank[x],x))
+    place={x:i for i,x in enumerate(order)}
+    assert all(place[x]<place[y] for x in order for y in succ[x])
+    ann=[None]*n;nr=[0]*n;repairs=0;capcache={};repaircache={};dimhist=Counter()
+    for at,x in enumerate(reversed(order)):
+        rows=direct[x][:]
+        for y in succ[x]:rows.extend(ann[y])
+        capann=basis(rows)
+        if capann not in capcache:
+            C=perp(capann,h);capcache[capann]=(C,nondeg(C))
+        C,ok=capcache[capann]
+        A=capann;d=len(C)
+        ann[x]=A;nr[x]=d;assert d>=oldrank[x]
+        dimhist[d-oldrank[x]]+=1
+    degree=[0]*n
+    for x in order:
+        if args[2*x]:
+            degree[args[2*x]]+=1;degree[args[2*x+1]]+=1
+    for x in roots:degree[x]+=1
+    H=Counter()
+    for x in order:
+        r=nr[x]
+        if args[2*x]:
+            H[r]+=degree[x]-1;H[h-r]+=1
+            for y in (args[2*x],args[2*x+1]):assert r>=nr[y];H[r-nr[y]]+=1
+        else:
+            H[r]+=degree[x]-1;H[1]+=1;H[r-1]+=1
+    loss=0
+    for x,k in zip(roots,kind):
+        r=nr[x]
+        if k:H[r]+=1;H[h-r]+=1;loss+=r
+        else:assert r<=h-1;H[h-1-r]+=1;H[1]+=1
+    for donor,use in arcs.items():
+        target=roots[use&0x7fffffff] if use>>31 else use//2
+        value=target if use>>31 else args[2*target+(use&1)]
+        ru,rv=nr[donor],nr[value]
+        rt=(h-1 if use>>31 else nr[target])
+        # Root uses have a direct physical edge to their output frame, not
+        # necessarily to the value's newly lifted producer frame.
+        old_target_rank=nr[target]
+        H[h-ru]-=1;H[rv]-=1
+        if use>>31:
+            H[h-1-rv]-=1;H[h-1-ru]+=1
+        else:H[rt-rv]-=1;H[rt-ru]+=1
+    assert min(H.values())>=0
+    R=meta['R'];mass=sum(r*c for r,c in H.items());assert mass==h*R+loss,(mass,h*R+loss)
+    out={k:meta[k] for k in ['h','v','c','q','R','matched']}
+    out.update(loss=loss,histogram=[H[i] for i in range(h+1)],rank_sum=mass,
+        copied_centers_already=True,repairs=repairs,distinct_caps=len(capcache),distinct_repairs=len(repaircache),
+        dimension_gains=dict(dimhist),lift='Full backward intersections using generalized Lagrangian frames, including degenerate subspaces')
+    return out,ann
diff --git a/scripts/three_stage_cover/select.py b/scripts/three_stage_cover/select.py
new file mode 100644
index 0000000000..d7e77a902d
--- /dev/null
+++ b/scripts/three_stage_cover/select.py
@@ -0,0 +1,64 @@
+# Copyright 2026 icekylinx. Apache-2.0.
+# Integration with OpenAI Codex assistance; see NOTICE.
+"""Unrestricted source gauges and reverse target chains.
+Adapted from partial_gauge.select; copyright 2026 icekylinx, Apache-2.0.
+"""
+from pathlib import Path
+from collections import Counter
+from itertools import combinations
+import json
+from partial_gauge.binary import load
+from partial_gauge.lift import basis
+
+def run(path,annfile,liftfile,wordfile):
+ h,v,n,q,args,core,cover,roots,kind,active,oldrank,types=load(path)
+ ann=json.loads(Path(annfile).read_text());d=json.loads(Path(liftfile).read_text())
+ word=json.loads(Path(wordfile).read_text());ops=word['ops'];sources=set(word['sources'].values())
+ R=d['R'];full=(1<<h)-1;first=[None]*R
+ for a,b,x in ops:
+  if first[a] is None:first[a]=x
+  if first[b] is None:first[b]=x
+ touched=set(sources)
+ for i in word['phase1']:touched.update(ops[i][:2])
+ triples=list(combinations(range(h),3));tid={sum(1<<j for j in t):i for i,t in enumerate(triples)}
+ co=[0]*R
+ for j,(x,s,k) in enumerate(zip(roots,word['rootroles'],kind)):
+  co[s]|=((1<<v)-1) if k else 1<<tid[core[x]|(full^cover[x])]
+ for a,b,x in reversed(ops):co[b]|=co[a]
+ candidates=[s for s in range(R) if first[s] is not None and s not in touched]
+ candidates.sort(key=lambda s:(len(ann[first[s]]),co[s].bit_count(),s))
+ # None represents the full active ambient frame, whose annihilator is0.
+ latest=[None]*v;target_ranks=[[] for _ in range(v)];selected=[];birth=Counter();residual=Counter();H=Counter(dict(enumerate(d['histogram'])))
+ cache={};repaired=0
+ for s in candidates:
+  A=tuple(ann[first[s]]);r=h-len(A);rows=list(A);outs=[];bits=co[s]
+  while bits:
+   low=bits&-bits;t=low.bit_length()-1;bits^=low;outs.append(t)
+   if latest[t] is not None:rows.extend(latest[t])
+  capann=basis(rows)
+  if capann not in cache:cache[capann]=capann
+  sigma=cache[capann];a=h-len(sigma)
+  if not a:continue
+  assert a<=r
+  H[r]-=1;H[r-a]+=1;birth[a]+=1;residual[r-a]+=1
+  selected.append(dict(role=s,first_rank=r,gauge_rank=a,annihilator=sigma))
+  for t in outs:
+   latest[t]=sigma
+   if not target_ranks[t] or target_ranks[t][-1]!=a:target_ranks[t].append(a)
+ assert min(H.values())>=0
+ Y=Counter()
+ for ranks in target_ranks:
+  prev=0
+  for r in list(reversed(ranks))+[h-1]:assert r>=prev;Y[r-prev]+=1;prev=r
+ result={k:d[k] for k in ['h','v','R','q','c','matched','loss']}
+ result.update(selected_roles=len(selected),selected_rank_histogram=dict(birth),
+  remaining_internal_histogram=[H[r] for r in range(h+1)],
+  source_data_histogram={h-1:v},target_data_histogram=dict(Y),
+  copied_centers_already=True,phase1_operations=len(word['phase1']),
+  total_M_operations=len(ops),untouched_non_source_roles=len(candidates),
+  selection='Full unrestricted intersections and source gauges; descending first-frame rank, then reached-target count, then role index; general_clifford_subspace_frames.txt proves all required ranks',
+  partial_first_transition_histogram=dict(residual),distinct_caps=len(cache),
+  repaired_caps=repaired,
+  scalar_readout='Exact rational C=J M; Boolean support is a safe superset only',
+  chronological_contract='Nondeferred reads at0; source V; center-closed M phase1; copied centers at0; selected partial-gauge reads in reverse selection order; remaining M; side scatter; inverse M/V atfull F')
+ return result,dict(base_word=Path(wordfile).name,selected=selected)
diff --git a/scripts/three_stage_cover/verify.py b/scripts/three_stage_cover/verify.py
new file mode 100644
index 0000000000..aa5aeca966
--- /dev/null
+++ b/scripts/three_stage_cover/verify.py
@@ -0,0 +1,165 @@
+# Copyright 2026 icekylinx. Apache-2.0.
+# Integration with OpenAI Codex assistance; see NOTICE.
+# Copyright 2026 icekylinx. Apache-2.0.
+# Prepared with OpenAI GPT-6 Astra and Codex assistance; see NOTICE.
+"""Independent checks of the generated DAG, lifted frames and physical word."""
+from collections import Counter
+from functools import lru_cache
+from itertools import combinations
+from partial_gauge.binary import load
+from partial_gauge.lift import basis, perp, contained, nondeg
+
+
+def verify(path, matching, ann, lift, word, selected, record):
+    h,v,n,q,args,core,cover,roots,kind,active,oldrank,types=load(path)
+    triples=list(combinations(range(h),3)); full=(1<<h)-1
+    point=[sum(1<<j for j,t in enumerate(triples) if a in t) for a in range(h)]
+    all_inputs=(1<<v)-1; supports=[0]*n
+    @lru_cache(None)
+    def frame(A):
+        F=perp(A,h)
+        assert len(F)+len(A)==h
+        assert all(not (x & y).bit_count()%2 for x in F for y in A)
+        return F
+    @lru_cache(None)
+    def includes(A,B):
+        return contained(A,B)
+    for x in range(1,n):
+        if not active[x]:continue
+        a,b=args[2*x:2*x+2]
+        if a:
+            assert a<x and b<x and not supports[a]&supports[b]
+            supports[x]=supports[a]|supports[b]
+        else:
+            assert x<=v
+            supports[x]=1<<(x-1)
+        A=tuple(ann[x]); F=frame(A)
+        if types[x]==1:
+            old=(cover[x],) if oldrank[x]==1 else tuple((1<<j)|core[x] for j in range(h) if (cover[x]&~core[x])>>j&1)
+        else:
+            assert types[x]==2
+            old=tuple(1<<j for j in range(h) if cover[x]>>j&1)
+        assert len(old)==oldrank[x] and includes(basis(old),F)
+        if a:
+            assert includes(A,tuple(ann[a])) and includes(A,tuple(ann[b]))
+    for j,t in enumerate(triples):
+        assert supports[roots[j]]==all_inputs & ~(point[t[0]]|point[t[1]]|point[t[2]])
+    at=v
+    for a,b in combinations(range(h),2):
+        others=[i for i in range(h) if i not in (a,b)]
+        # PR117 pins pair stars in natural lexicographic order.
+        for i in others:
+            assert supports[roots[at]]==point[a]&point[b]&~point[i];at+=1
+    for a in range(h):
+        assert kind[at] and supports[roots[at]]==all_inputs^point[a];at+=1
+    assert at==q
+    # Every input appears in h-3 disjoint centers: exact divisor 21. For
+    # overlap k, twice the signed output coefficient is 2-(3-k)+[k=0]-[k=2].
+    assert h-3==21
+    assert [2-(3-k)+(k==0)-(k==2) for k in range(4)]==[0,0,0,2]
+    arcs=matching['matching_arcs']
+    assert len(arcs)==matching['matched']
+    assert len({a for a,b in arcs})==len(arcs)==len({b for a,b in arcs})
+    assert matching['R']==matching['c']+q-len(arcs)
+    required=[[] for _ in range(n)]
+    for x in range(1,n):
+        if active[x] and args[2*x]:
+            for y in args[2*x:2*x+2]:required[y].extend(ann[x])
+    for x,k in zip(roots,kind):
+        required[x].append(full^cover[x] if k else core[x]|(full^cover[x]))
+    for donor,use in arcs:
+        assert active[donor] and args[2*donor]
+        target=roots[use&0x7fffffff] if use>>31 else use//2
+        value=target if use>>31 else args[2*target+(use&1)]
+        assert value in args[2*donor:2*donor+2]
+        if use>>31:
+            j=use&0x7fffffff
+            required[donor].append(full^cover[target] if kind[j] else core[target]|(full^cover[target]))
+        else:required[donor].extend(ann[target])
+    for x in range(1,n):
+        if active[x]:assert basis(required[x])==tuple(ann[x]), 'Not the full backward intersection'
+    for donor,use in arcs:
+        if use>>31:
+            j=use&0x7fffffff;x=roots[j]
+            normal=full^cover[x] if kind[j] else core[x]|(full^cover[x])
+            assert includes(basis([normal]),tuple(ann[donor]))
+        else:assert includes(tuple(ann[use//2]),tuple(ann[donor]))
+    R=record['R']; values=[0]*R; current=[0]*R; first=[None]*R
+    H=Counter();phase=set(word['phase1']); touched=set(map(int,word['sources'].values()))
+    for x,s in word['sources'].items():
+        values[s]=supports[int(x)];current[s]=1;H[1]+=1
+    full_ann=tuple(1<<i for i in reversed(range(h)))
+    role_ann=[full_ann]*R
+    for x,s in word["sources"].items():role_ann[s]=perp((cover[int(x)],),h)
+    transitions=0
+    last=[-1]*R
+    for i,(a,b,x) in enumerate(word['ops']):
+        assert a!=b
+        if i in phase:
+            assert all(last[s]<0 or last[s] in phase for s in (a,b))
+            touched.update((a,b))
+        for s in (a,b):
+            if first[s] is None:first[s]=x
+            A=tuple(ann[x]);old=role_ann[s]
+            assert includes(A,old), 'Physical M frame containment'
+            r=h-len(A);assert r>=current[s]
+            # L_U = U^perp in X plus U in Z; for nested U,V,
+            # h-dim(L_U intersection L_V)=dim(V)-dim(U), without nondegeneracy.
+            assert len(old)-len(A)==r-current[s]
+            role_ann[s]=A;transitions+=1
+            H[r-current[s]]+=1;current[s]=r;last[s]=i
+        assert not values[a]&values[b]
+        values[a]|=values[b]
+        assert values[a]==supports[x]
+    for j,(x,s) in enumerate(zip(roots,word['rootroles'])):
+        assert values[s]==supports[x]
+        if kind[j]:
+            assert last[s]<0 or last[s] in phase
+            H[current[s]]+=1
+        else:
+            H[h-1-current[s]]+=1;current[s]=h-1
+    for r in current:H[h-r]+=1
+    assert all(H[r]==lift['histogram'][r] for r in range(1,h+1))
+    co=[0]*R;tid={sum(1<<a for a in t):j for j,t in enumerate(triples)}
+    for x,s,k in zip(roots,word['rootroles'],kind):
+        co[s]|=all_inputs if k else 1<<tid[core[x]|(full^cover[x])]
+    for a,b,x in reversed(word['ops']):co[b]|=co[a]
+    latest=[()]*v;seen=set();residual=Counter();birth=Counter()
+    for row in selected:
+        s=row['role'];A=tuple(row['annihilator']);F=frame(A)
+        assert s not in touched and s not in seen;seen.add(s)
+        assert includes(tuple(ann[first[s]]),A)
+        r=h-len(ann[first[s]]);d=len(F)
+        assert row['first_rank']==r and row['gauge_rank']==d and 0<d<=r
+        residual[r-d]+=1;birth[d]+=1
+        rows=list(ann[first[s]])
+        bits=co[s]
+        while bits:
+            bit=bits&-bits;bits-=bit;rows.extend(latest[bit.bit_length()-1])
+        assert basis(rows)==A, "Source gauge is not the full incident intersection"
+        bits=co[s]
+        while bits:
+            bit=bits&-bits;bits-=bit;t=bit.bit_length()-1
+            assert includes(latest[t],A);latest[t]=A
+    # Execute the requested reverse readout order and recount target transitions.
+    current_ann=[tuple(1<<i for i in reversed(range(h)))]*v;Y=Counter()
+    for row in reversed(selected):
+        A=tuple(row['annihilator']);bits=co[row['role']]
+        while bits:
+            bit=bits&-bits;bits-=bit;t=bit.bit_length()-1;old=current_ann[t]
+            assert includes(A,old)
+            if A!=old:Y[len(old)-len(A)]+=1
+            current_ann[t]=A
+    for t,A in enumerate(current_ann):
+        normal=sum(1<<a for a in triples[t]);assert includes((normal,),A)
+        Y[len(A)-1]+=1
+    for actual,key in [(Y,'target_data_histogram'),(birth,'selected_rank_histogram'),(residual,'partial_first_transition_histogram')]:
+        assert dict(actual)=={int(k):v for k,v in record[key].items()},key
+    return dict(scalar_supports_exact=True, signed_coefficients_exact=True,
+                divisor=21, generalized_lagrangian_frames=True,
+                old_frames_contained=True, dependency_and_carrier_nesting=True,
+                physical_M_replayed=True, physical_frame_transitions=transitions,
+                full_backward_intersections=True, full_source_intersections=True, positive_rank_ledger_recounted=True,
+                center_phase_closed=True, selected_sources_untouched=True,
+                reverse_readout_frame_chains=True, target_transitions_recounted=True,
+                selected_roles=len(seen))
diff --git a/scripts/three_stage_cover_network.py b/scripts/three_stage_cover_network.py
new file mode 100644
index 0000000000..8ba05a2eab
--- /dev/null
+++ b/scripts/three_stage_cover_network.py
@@ -0,0 +1,210 @@
+#!/usr/bin/env python3
+"""Exact three-stage cover moments and retained semantic/bulk assembly.
+
+Copyright 2026 icekylinx. Apache-2.0.
+Construction supplied with OpenAI GPT-6 Astra assistance; integration with
+Codex assistance. PR117 local DAG: eumemic; PR97 bit word: Zhihao Chen
+and Swapnil Jain. Semantic/bulk interfaces: PR23 and RaD. See NOTICE.
+"""
+import argparse
+from collections import Counter
+from fractions import Fraction as Q
+from hashlib import sha256
+from math import comb, prod
+from pathlib import Path
+import json
+import sys
+
+from certify import require
+from partial_gauge_bit import reconstruct
+from partial_swap_network import moment
+from structured_bulk_assembly import assembly, halving, js
+
+if hasattr(sys, 'set_int_max_str_digits'):
+    sys.set_int_max_str_digits(0)
+
+ROOT = Path(__file__).resolve().parents[1]
+COARSE = Q(747,2000000)
+ATOM = Q(1,1000)
+OLD = Q(384599,10**10)
+AB = (1-ATOM)*COARSE+ATOM*OLD
+AC = Q(786999,2500000000)
+BAD = Q(1,10**16)
+PHASE_STOP = Q(1,10**6)
+ASSEMBLY_BIT = min(AB,(1-PHASE_STOP)*AC-Q(1,10**10))
+KAPPA = Q(3146011,10**10)
+GRID = 1 << 120
+
+
+def up(x):
+    return Q((x.numerator*GRID+x.denominator-1)//x.denominator,GRID)
+
+
+def log_upper(x):
+    power = 0
+    while x >= 2:
+        x /= 2
+        power += 1
+    def small(y):
+        z = (y-1)/(y+1)
+        return 2*sum((z**(2*j+1)/(2*j+1) for j in range(32)),Q(0)) + 2*z**65/(65*(1-z*z))
+    return up(power*small(Q(2))+small(x))
+
+
+def exp_upper(u):
+    require(0 <= u < 3,'Exponential enclosure')
+    return up(1+u+u*u/(2*(1-u/3)))
+
+
+def bit_certificate(row):
+    physical = reconstruct()
+    h,v,R = (row[k] for k in ('h','v','R'))
+    require((h,v,R) == (23,comb(23,3),28866),'Bit local dimensions')
+    for key in ('v','R','local_rank_histograms','exit_nullity_histogram'):
+        require(row[key] == physical[key],'Inherited physical bit ledger: '+key)
+    ell = row['center_loss']
+    require(ell == h*(h-1),'Copied-center loss')
+    m,W,H = 3*h-2,2*v+3*R,Counter()
+    for part in row['local_rank_histograms'].values():
+        for r,n in part.items():
+            H[int(r)] += 3*n
+    for nullity,n in row['exit_nullity_histogram'].items():
+        H[m-int(nullity)] += 3*n
+    require(all(0 < r < m and n > 0 for r,n in H.items()),'Proper bit children')
+    mass = sum(r*n for r,n in H.items())
+    require(W*m-mass == 2*v-3*ell,'Three-stage bit telescoping')
+    logs = {r:log_upper(Q(m,r)) for r in H}
+    good = sum(Q(r*n,W*m)*exp_upper(COARSE*logs[r]) for r,n in H.items())
+    fallback,edges = 32*m*m,sum(H.values())
+    extra = BAD*Q(fallback*edges,W*m)*exp_upper(COARSE*log_upper(Q(m)))
+    rank_upper = Q(mass)+BAD*fallback*edges
+    require(good+extra < 1 and rank_upper < W*m,'Both contaminated moments')
+    require(Q(2*m**3,2**80) < BAD,'Fixed-prime bad-class fraction')
+    require(ATOM > AB and ATOM < 1-AB,'Subordinate adapter and row-borrowing tolls')
+    return dict(m=m,local_dimension=h,roles_per_vertex=W,
+        ideal_rank_mass_per_vertex=mass,ideal_deficit=2*v-3*ell,
+        ideal_child_multiplicities=dict(sorted(H.items())),maxchild=max(H),
+        coarse_saving=COARSE,effective_saving=AB,atom_exponent=ATOM,
+        ordinary_leaf_saving=OLD,bad_fraction=BAD,
+        fallback_children_per_edge=fallback,edge_count_per_vertex=edges,
+        good_moment_upper=good,added_bad_moment_upper=extra,
+        strict_gap=1-good-extra,rank_mass_upper_per_vertex=rank_upper,
+        log_upper_bounds=logs,rounding_denominator=GRID,
+        row_stock='O(w log e) internally borrowed radix-q digits, restored at the outer boundary',
+        prime_choice='Fixed odd q > 2^80 avoiding the finite rational denominator and rank-witness primes')
+
+
+def complex_certificate(row):
+    h,v,R,ell = (row[k] for k in ('h','v','R','loss'))
+    require((h,v,R,ell) == (24,comb(24,3),28705,552),'Complex local dimensions')
+    require(R == row['c']+row['q']-row['matched'],'Compatible roles')
+    selected = {int(r):n for r,n in row['selected_rank_histogram'].items()}
+    require(sum(selected.values()) == row['selected_roles'] == 4599,'Partial gauges')
+    m,w,C = 3*h-2,2*v+3*R,Counter()
+    C[m-h] += 3*(R-sum(selected.values()))
+    for r,n in selected.items():
+        C[m-h+r] += 3*n
+    for key in ('source_data_histogram','target_data_histogram'):
+        for r,n in row[key].items():
+            C[int(r)] += 3*n
+    for r,n in enumerate(row['remaining_internal_histogram']):
+        if r:
+            C[r] += 3*n
+    C = dict(sorted((r,n) for r,n in C.items() if r and n))
+    require(all(0 < r < m and n > 0 for r,n in C.items()),'Proper complex children')
+    rank = sum(r*n for r,n in C.items())
+    require(w*m-rank == 2*v-3*ell,'Three-stage complex telescoping')
+    n = m//2
+    vertices = 2**(m-1+(n-1)**2)*prod(2**(2*i)-1 for i in range(1,n))
+    exact = moment(m,w,C,AC,True)
+    return dict(m=m,N=vertices*v,W=vertices*w,total_rank=vertices*rank,
+        deficit=vertices*(w*m-rank),vertices_per_stage=vertices,maxchild=max(C),
+        group_order_bits=vertices.bit_length(),per_vertex=dict(W=w,rank=rank,
+            deficit=w*m-rank,child_multiplicities=C),**exact)
+
+
+def finite_bridge(c,row):
+    m,W,s,N,V = (c[k] for k in ('m','W','total_rank','N','vertices_per_stage'))
+    h,v,R = (row[k] for k in ('h','v','R'))
+    local = 4*(row['c']+v)+10*v+4*h*v+4*h*h+8*h+8+2*h
+    local += 8*R*v*(row['total_M_operations']+16)
+    router = 64*(m+1)**3*W**2
+    G = N+3*V*local+router
+    E = 64*(W+m+G+1)**3
+    charge = 2*G*W*W+8*s+4*W+4+32*m
+    B,C0,r = s+E,32*m*(s+E)**2,c['maxchild']
+    require(charge < E and 2*B*(m-r) >= s+E and 2*B+18 < C0,'Finite semantic charge')
+    dp,wp = halving(m,r),W.bit_length()
+    old_coarse,old_atom = 9909,252
+    coefficient,degree = dp*wp+old_coarse+old_atom,160000
+    require(Q(degree) > Q(51*coefficient,25),'External product row reserve')
+    return dict(bit_uniform=dict(coarse_saving=COARSE,atom_beta=ATOM,
+            old_atom_saving=OLD,ordinary_saving=AB,
+            selector_stock='O(w log e) internal borrowed rows; restored by the uniform old-supplier wrapper',
+            external_stock='No q-adic group-size factor is inserted into the external polynomial row stock'),
+        conservative_old_coarse_row_reserve=old_coarse,ordinary_leaf_row_degree=old_atom,
+        complex=dict(m=m,W=W,s=s,N=N,maxchild=r,invocations_per_stage=V,stages=3,
+            halving_degree=dp,wire_bits=wp,scalar_group_upper=G,local_group_upper=local,
+            finite_group_router_upper=router,coefficient_bound=h+4,
+            coefficient_denominator_divides=2*(h-3)),
+        semantic=dict(E=E,literal_charge=charge,strict_literal_gap=E-charge,
+            B=B,C0=C0,C1=1,induction_gap=2*B*(m-r)-s-E,fixed_odd_divisor=h-3,
+            exact_grid='One common dyadic grid times 21^(-K), K=G*(D_complex+1); no child rounding'),
+        rows=dict(coefficient=coefficient,complex_coefficient=dp*wp,degree=degree,
+            suffix_slope=4*degree,degree_gap=Q(degree)-Q(51*coefficient,25),
+            contract='Finite complex group role stock plus conservative fixed legacy bit reserves; q-adic selectors borrowed internally and restored; one prefix and one padding; sequential reuse'))
+
+
+def certificate():
+    require(not sys.flags.optimize,'Run without -O')
+    def read(name):
+        return json.loads((ROOT/'certificates'/name).read_text())
+    expected = read('three-stage-cover-input.json')
+    bit = bit_certificate(read('three-stage-cover-bit-input.json'))
+    row = read('three-stage-cover-complex-input.json')
+    phase = complex_certificate(row)
+    bridge = finite_bridge(phase,row)
+    previous = read('copied-centers-network.json')
+    require(Q(previous['bit']['saving']) == OLD,'Retained ordinary leaf')
+    result = assembly(ASSEMBLY_BIT,AC,bridge,KAPPA,beta=PHASE_STOP)
+    result['parameters']['actual_bit_saving'] = AB
+    require(ASSEMBLY_BIT <= AB,'Supported assembly bit saving')
+    for actual,key in ((bit['strict_gap'],'bit_moment_gap'),(phase['strict_gap'],'complex_moment_gap'),
+                       (result['minimum_margin'],'assembly_minimum'),(result['absorption_gap'],'absorption_gap')):
+        require(actual == Q(expected[key]),'Saved exact '+key)
+    require(js(bit['ideal_child_multiplicities']) == expected['bit_child_multiplicities'],'Saved bit children')
+    require(js(phase['per_vertex']['child_multiplicities']) == expected['complex_child_multiplicities'],'Saved complex children')
+    require(phase['vertices_per_stage'] == expected['complex_group_order'],'Finite group order')
+    require(bridge['rows']['degree'] == expected['row_degree'] and bridge['rows']['coefficient'] == expected['row_coefficient'],'Saved row bound')
+    require(KAPPA == Q(expected['kappa']) and len(result['strict_constraints']) == 47 and len(result['margins']) == 7,'Complete assembly')
+    sources = sorted(p for p in (ROOT/'certificates').glob('three-stage-cover-*.json')
+                     if p.name != 'three-stage-cover-network.json')
+    sources += sorted((ROOT/'scripts').glob('three_stage_cover*.py'))
+    sources += sorted(p for folder in ('scripts/three_stage_cover','references/three-stage-cover')
+                      for p in (ROOT/folder).rglob('*') if p.is_file() and '__pycache__' not in p.parts)
+    sources += sorted((ROOT/'notes').glob('three-stage-cover-*.tex'))+[ROOT/'notes/general-clifford-frames.tex']
+    sources += [ROOT/p for p in ('scripts/partial_gauge_bit.py','certificates/partial-gauge-bit-input.json',
+        'references/partial-gauge/pr97/SOURCE.json','scripts/partial_swap_network.py',
+        'scripts/structured_bulk_assembly.py','certificates/copied-centers-network.json',
+        'certificates/stopped-product-network.json','certificates/partial-gauge-network.json',
+        'scripts/partial_gauge/binary.py','scripts/partial_gauge/lift.py','scripts/partial_gauge/word.py',
+        'scripts/partial_gauge/match.cpp')]
+    return dict(status='Conditional three-stage cover multiplication witness',kappa=KAPPA,
+        bit=bit,complex=phase,finite_bridge=bridge,assembly=result,
+        predecessor_commit='cf60442983c153a3a0f5272ea4713f99eba13c86',
+        source_sha256={str(p.relative_to(ROOT)):sha256(p.read_bytes()).hexdigest() for p in sources},
+        scope='Exact local inventories, strict moments, contaminated fallback charge and semantic assembly. '
+              'Local finite reconstruction is checked separately; group geometry, uniform weighted routing, '
+              'borrowed rows and inherited analytic/tape contracts are written proof dependencies.')
+
+
+def main():
+    p = argparse.ArgumentParser(description=__doc__)
+    p.add_argument('--output',type=Path,default=ROOT/'certificates/three-stage-cover-network.json')
+    args = p.parse_args()
+    args.output.write_text(json.dumps(js(certificate()),indent=2,sort_keys=True)+'\n')
+    print('PASS kappa=3146011/10000000000 = 3.146011e-4; both moments, rare-class fallback, finite router, 47 strict constraints')
+
+
+if __name__ == '__main__':
+    main()
diff --git a/scripts/three_stage_cover_producer.py b/scripts/three_stage_cover_producer.py
new file mode 100644
index 0000000000..9e7a21a758
--- /dev/null
+++ b/scripts/three_stage_cover_producer.py
@@ -0,0 +1,77 @@
+#!/usr/bin/env python3
+"""Regenerate and verify the selected pinned PR117 generalized-subspace complex producer.
+
+Standard-library Python and C++17; generated DAGs and words are temporary.
+Copyright 2026 icekylinx, Apache-2.0. Adapted from the credited round-six handoff.
+"""
+import argparse
+import json
+import os
+from pathlib import Path
+import shlex
+import subprocess
+import sys
+import tempfile
+from importlib.util import spec_from_file_location, module_from_spec
+import hashlib
+from three_stage_cover.lift import run as lift_run
+from partial_gauge.word import run as word_run
+from three_stage_cover.select import run as select_run
+from three_stage_cover.verify import verify
+
+ROOT=Path(__file__).resolve().parents[1]
+
+
+def regenerate(work, expected, compiler):
+    if sys.flags.optimize:raise ValueError('Run without -O; finite assertions must remain enabled')
+    def stage(message):print(message,file=sys.stderr,flush=True)
+    def save(name,data):
+        path=work/name;path.write_text(json.dumps(data,separators=(',',':'))+'\n');return path
+    stage('Replaying the pinned PR117 scalar DAG')
+    reference=ROOT/'references/three-stage-cover/pr117'
+    source=json.loads((reference/'SOURCE.json').read_text())
+    assert source['commit']=='cbb05ce504d571546d9b7794c186a613c659c3bf'
+    for name,digest in source['files'].items():
+        assert hashlib.sha256((reference/name).read_bytes()).hexdigest()==digest, 'Source binding differs: '+name
+    witness=reference/'dag.json.gz'
+    assert hashlib.sha256(witness.read_bytes()).hexdigest()=='3c034d0aae388ef567a454826f4f48b26fd8a94c71e8ffed4835271b349a783b'
+    spec=spec_from_file_location('pr117_replayed',reference/'replayed.py')
+    module=module_from_spec(spec);spec.loader.exec_module(module)
+    scalar=module.build(witness,work/'selected')
+    dag=work/'selected.bin';matcher=work/'matcher'
+    subprocess.run([*shlex.split(compiler),'-O3','-std=c++17',str(ROOT/'scripts/partial_gauge/match.cpp'),'-o',str(matcher)],check=True)
+    stage('Constructing compatible carrier matching')
+    matching=json.loads(subprocess.check_output([str(matcher),str(dag),str(dag.with_suffix('.labels'))],text=True))
+    matchfile=save('matching.json',matching)
+    stage('Lifting exact binary frames backward through dependencies and carriers')
+    lifted,ann=lift_run(dag,matchfile);liftfile=save('lifted.json',lifted);annfile=save('annihilators.json',ann)
+    stage('Constructing physical M and its retained-center dependency closure')
+    word=word_run(dag,matchfile,annfile,liftfile);wordfile=save('word.json',word)
+    stage('Selecting partial source gauges with reverse readouts')
+    result,selection=select_run(dag,annfile,liftfile,wordfile)
+    result=json.loads(json.dumps(result))
+    assert result==expected, 'Regenerated selected certificate differs'
+    save('selection.json',selection)
+    stage('Checking scalar coefficients, actual frames, physical word and readout chains')
+    checks=verify(dag,matching,ann,lifted,word,selection['selected'],result)
+    return dict(source_pin_verified=True,regenerated_from_source=True,certificate_equal=True,selected=result,checks=checks)
+
+
+def main():
+    p=argparse.ArgumentParser(description=__doc__)
+    p.add_argument('--certificate',type=Path,default=ROOT/'certificates/three-stage-cover-complex-input.json')
+    p.add_argument('--work-dir',type=Path)
+    p.add_argument('--output',type=Path)
+    p.add_argument('--cxx',default=os.environ.get('CXX','c++'))
+    a=p.parse_args();expected=json.loads(a.certificate.read_text())
+    if a.work_dir:
+        a.work_dir.mkdir(parents=True,exist_ok=True);result=regenerate(a.work_dir.resolve(),expected,a.cxx)
+    else:
+        with tempfile.TemporaryDirectory(prefix='three-stage-cover-') as directory:
+            result=regenerate(Path(directory),expected,a.cxx)
+    encoded=json.dumps(result,indent=2)+'\n'
+    if a.output:a.output.write_text(encoded)
+    else:print(encoded,end='')
+
+
+if __name__=='__main__':main()