Terminal elimination on #144's shared bit cores: conditional κ = 4.646633e-4

.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

.github/workflows/bit-elim-144.yml

@@ -0,0 +1,29 @@
+name: Verify terminal elimination on the paired-cube bit word
+
+on:
+  push:
+  pull_request:
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  certificate:
+    runs-on: ubuntu-24.04
+    timeout-minutes: 30
+    strategy:
+      fail-fast: false
+      matrix:
+        python-version: ['3.11', '3.14']
+    steps:
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1 # v7
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97 # v7
+        with:
+          python-version: ${{ matrix.python-version }}
+      - name: Reproduce PR144, regenerate the elimination, replay and reprice
+        run: python3 research/bit-elim-144/verify.py
+      - name: Require committed files to remain unchanged
+        run: git diff --exit-code

.github/workflows/verify.yml

@@ -16,7 +16,7 @@ jobs:
       fail-fast: false
       matrix:
         python-version: ['3.11', '3.13', '3.14']
-        group: [community, producers, certificates, ternary, research, tests, strips, clones, positive, joint, pair]
+        group: [community, producers, certificates, ternary, research, tests, strips, clones, positive, joint, pair, partial-gauge, three-stage-cover, paired-cube]
     steps:
       - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1 # v7
         with:

Makefile

@@ -15,6 +15,9 @@ community-followup-check:
 verify:
 	$(MAKE) verify-community
 	$(MAKE) verify-producers
+	$(MAKE) verify-partial-gauge
+	$(MAKE) verify-three-stage-cover
+	$(MAKE) verify-paired-cube
 	$(MAKE) verify-certificates
 	$(MAKE) verify-ternary
 	$(MAKE) verify-research
@@ -30,6 +33,7 @@ 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 +432,52 @@ 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
+
+.PHONY: paired-cube-producer paired-cube-bit paired-cube-certificate paired-cube-verify verify-paired-cube
+paired-cube-producer:
+	python3 scripts/paired_cube_producer.py
+
+paired-cube-bit:
+	python3 scripts/paired_cube_bit.py
+
+paired-cube-certificate:
+	python3 scripts/paired_cube_network.py
+
+paired-cube-verify: paired-cube-producer paired-cube-bit paired-cube-certificate
+
+verify-paired-cube: paired-cube-verify

NOTICE

@@ -302,3 +302,86 @@ inherited PR #48 sources. The stacked witness in research/pair-assembly/frame
 uses eumemic's PR #57 joint frame compiler unchanged; its driver and verifier
 are adaptations of PR #57's skip_frame_compiler.py, skip_frame_compose.py and
 verify_skip_frame.py. Original notices are retained.
+
+Stopped product-ring extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The opposite-bank generic projector factorization, componentwise atom-ring
+adapters, stopped recurrence and outer ordinary-interchange conversion,
+aligned pair-star producer with rational centers, exact denominator-21
+grid and selected (23,23)/(24,24) composition are contributed by icekylinx
+with substantial OpenAI GPT-6 Astra and Codex assistance. This extension
+builds on merged PR #36, 11817ccacb564bb7f98789c20dc11d3fece207e3, and is
+integrated on upstream main 56b66d58297deca1d7dd130247d720e960f77a37.
+
+The retained ordinary leaf uses the preceding projector, corner and
+copied-center proofs. The selected two-stage topology retains Aurel Prosz
+(Paureel), Swapnil Jain and Zhihao Chen's credits; the semantic assembly
+retains PR #21/#23 and RaD's attribution. Paired producers, carrier matching,
+original source notices and each imported license remain unchanged.
+
+Partial-gauge extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The pair-first cube assembly, compatible backward frame enlargement,
+partial source-gauge schedule and stopped whole-projector composition are
+contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex
+assistance, extending PR #104 at 948ce1510df750f4c18b96bdaef436a86f8bf834.
+
+The physical deferred bit word, reflected event accounting, fan-order
+clarification and signed two-stage endpoint are adopted from Zhihao Chen
+(jacklightChen), PR #97, e15350b66e4bf796153a68c240a6cc1b788abe3e, with
+Codex assistance. Its deferred readouts, V leaves, lifted frames and frozen
+witness originate in Swapnil Jain's 741e7aa078392553815df7926ee17ac5e25a8c38,
+with the original Claude disclosure retained. Cyclic interval and core-aware
+pair primitives build on Avi Eisenberg (ikeboy), PR #62,
+ad0f25ff7b23cff7f08ad237c2254e6ecf74257e, with Anthropic Claude assistance.
+Prior two-stage, copied-center, ordinary-leaf, semantic/bulk, routing and
+precision credits remain in force. Imported source notices and licenses
+are preserved under references/partial-gauge/; see SOURCES.json.
+
+Three-stage cover extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The arbitrary-subspace Clifford frame theorem, three-stage Cayley geometry,
+weighted q-adic GL cover, unit-lower weighted compiler, charged local-ring
+fallback, uniform batching, internal row borrowing and exact composition
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra assistance. This
+repository integration uses Codex assistance and extends
+PR #115, cf60442983c153a3a0f5272ea4713f99eba13c86.
+
+The selected local complex DAG and support/label replay are adopted from
+Copyright 2026 eumemic, PR #117, cbb05ce504d571546d9b7794c186a613c659c3bf,
+with its original Apache-2.0 terms and Claude assistance disclosure.
+The unchanged bit physical word retains Zhihao Chen (jacklightChen),
+PR #97, and Swapnil Jain's underlying witness and assistance notices.
+The prior opposite-bank, stopped ordinary-leaf, copied-center, topology,
+semantic/bulk, precision and analytic credits remain applicable.
+Original source notices and license bytes are preserved in
+references/three-stage-cover/pr117/ and references/partial-gauge/.
+
+Paired-cube and shared-core extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The paired-coordinate triple-cube identity, signed overlap channels,
+original-source involution, coordinate-star centers, concrete frame/carrier
+compiler, chronological source gauges, PR97 gauge-subset selection,
+orthogonal cross-stage routing and exact supplier/assembly integration
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra assistance.
+Repository integration uses Codex assistance and extends PR #130 at
+6a9970a530119174507904e23592fd59ede19a5d.
+
+The substantial completed-core workspace-sharing principle is credited to
+an664, PR #128, 530588a019b4a74f09180680c9e3961bf649ec89, with its reported
+OpenAI Codex assistance. No source code, partition or serialized witness
+from that PR is imported. The new routing and paired-motif integration
+are described separately and claim no exclusive priority for broad sharing,
+paired-coordinate, Fourier or carrier-matching ideas.
+
+Selected coarse queries restrict the retained eumemic PR #117 positive DAG;
+its original Apache-2.0 and Claude assistance notices remain unchanged.
+The bit word and original frames retain Zhihao Chen's PR #97 and Swapnil
+Jain's frozen witness, notices and historical assistance disclosures.
+Earlier generalized Clifford, three-stage, uniform weighted recursion,
+ordinary-leaf, semantic/bulk, precision and analytic credits remain in force.
+Source-specific license bytes are preserved under references/.

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.

README.md

@@ -3,6 +3,113 @@
 **Community research maintained by Douglas Colkitt — conditional on the original
 OpenAI #109 framework.**
 
+## Paired-cube and shared-core extension
+
+The construction contributed by **icekylinx**, extending
+[PR #130](https://github.com/CrocSwap/integer-mult-bounds/pull/130), gives
+
+$$
+T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad
+\kappa=\frac{4609169}{10000000000}=4.609169\times10^{-4}.
+$$
+
+A signed paired-cube producer and coordinate-star centers complete the
+identity using the original source registers. Completed dirty cores reuse
+one auxiliary bank across three orthogonal blocks, adopting an664's PR #128
+sharing principle. The bit branch selects certified gauges from the retained
+PR #97 / Swapnil word. All local transitions, copied centers, complement
+calls, finite routers and rare-class fallback remain charged. The analytic,
+uniform-recursion and fixed-tape hypotheses are retained.
+
+[Proof source](notes/paired-cube-note.tex) ·
+[Exact certificate](certificates/paired-cube-network.json) ·
+[Incremental reproduction](docs/paired-cube.md)
+
+```sh
+make paired-cube-verify
+```
+
+## 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
 
 $$