Integrate reviewed PR144: conditional kappa = 0.0004609169

.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/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:

CITATION.cff

@@ -1,6 +1,7 @@
 cff-version: 1.2.0
 message: >-
-  Please cite the final composition by Avi Eisenberg, eumemic and Alejandro Zarzuelo Urdiales, this project, the upstream
+  Please cite icekylinx’s paired-cube construction, an664’s completed-core sharing,
+  the constituent contributions, this project and the upstream
   manuscript and the constituent contributions used. Project-level authorship
   here identifies the maintainer; it does not assign sole authorship of the
   community result. See CONTRIBUTORS.md, NOTICE and source-specific manifests.
@@ -10,35 +11,32 @@ authors:
     given-names: Douglas
 license: Apache-2.0
 type: software
-version: "community-pr62-2026-10-08"
-date-released: 2026-10-08
+version: "community-pr144-2026-10-09"
+date-released: 2026-10-09
 repository-code: "https://github.com/CrocSwap/integer-mult-bounds"
 url: "https://github.com/CrocSwap/integer-mult-bounds"
 abstract: >-
   Community research on integer multiplication with conditional kappa =
-  25508460085039/500000000000000000 > 2^-15. Avi Eisenberg's interval strips
-  and core-aware pair assembly combine with eumemic's joint frame compiler
-  and Alejandro Zarzuelo Urdiales's exact parameter refinement. Includes
-  attributed alternative constructions, exact finite certificates, scoped
-  Lean checks and a conditional maintainer audit. The original OpenAI #109
-  framework remains assumed. No full formal verification, independent human
-  peer review or worldwide priority is claimed.
+  4609169/10000000000 = 0.0004609169. The selected construction combines
+  icekylinx's paired-cube motif, recursive compiler and integration with
+  an664's completed-core sharing, eumemic's producer, Zhihao Chen's physical
+  bit ledger and Swapnil Jain's underlying word and lifted frames. The
+  retained OpenAI #109 analytic and fixed-tape interfaces remain assumed.
+  Finite checks and written maintainer review do not constitute a formal
+  proof of multiplication or independent human peer review.
 preferred-citation:
   type: unpublished
-  title: "Interval strips, core-aware pair assembly and exact exponent refinement"
+  title: "Paired cubes and shared completed cores for conditional kappa = 4.609169e-4"
   authors:
-    - family-names: Eisenberg
-      given-names: Avi
-    - name: eumemic
-    - family-names: Zarzuelo Urdiales
-      given-names: Alejandro
+    - name: icekylinx
   year: 2026
-  url: "https://github.com/CrocSwap/integer-mult-bounds/pull/62"
+  url: "https://github.com/CrocSwap/integer-mult-bounds/pull/144"
   notes: >-
-    Cite PR62 together with eumemic's PR57 compiler and Alejandro Zarzuelo
-    Urdiales's PR61 refinement. Full constituent, parallel, maintainer and
-    upstream credits and assistance disclosures are retained in CONTRIBUTORS.md
-    and NOTICE; this entry does not replace those credits.
+    Cite the construction together with an664's PR128 sharing principle,
+    eumemic's PR117 producer, Zhihao Chen's PR97 ledger and Swapnil Jain's
+    underlying word. This entry identifies the submitted construction and
+    does not replace constituent attribution, earlier framework credit,
+    source licenses or assistance disclosures in CONTRIBUTORS.md and NOTICE.
 keywords:
   - integer multiplication
   - computational complexity

CONTRIBUTING.md

@@ -7,7 +7,9 @@ For a mathematical issue, identify the exact proposition, source label, or
 inequality; give a counterexample or the missing implication when possible;
 and distinguish numerical validation from an algorithmic proof obligation.
 For the current result, start with the
-[ternary review guide](docs/research/ternary-review.md).
+[current maintainer review](docs/research/community-round6-review.md),
+[selected certificate](certificates/paired-cube-network.json), and
+[reproduction guide](docs/paired-cube.md). Earlier notes are historical checkpoints.
 The [original audit](docs/audit.md) describes the retained upstream assumptions.
 
 For a parameter improvement, supply exact rational choices, the full dependency
@@ -20,9 +22,10 @@ Keep `upstream/` unchanged. Edit the generators under `scripts/`, then run:
 make verify
 ```
 
-Include regenerated certificates and patches in the same change. If the note
-changes, also run its corresponding PDF target (`make ternary-note` for the
-current result) and include the updated PDF. Review changes to
+Include regenerated certificates and patches in the same change. Run the
+selected incremental target, `make paired-cube-verify`, as well as the checks
+affected by your changes. The current proof is supplied as LaTeX source;
+historical PDF targets belong to their respective checkpoints. Review changes to
 claims in the README and note together. Finite tests should address a mathematical
 identity or a failure mode, rather than simply restating implementation details.
 

CONTRIBUTORS.md

@@ -1,13 +1,28 @@
 # Community contribution record
 
-**Latest reviewed batch: #50–#62.** The [round-two ledger](docs/research/community-round2-review.md)
-records the contribution and validation scope of every PR, including parallel
-and superseded numerical witnesses. The selected result combines Avi Eisenberg,
-eumemic and Alejandro Zarzuelo Urdiales. Rohan Gupta and Chafik Boukhalfa supplied
-the preceding reviewed compiler composition; Rohan Arun, RaD / hipotures and
-Rohan Garg supplied additional reviewed alternatives and tools.
-Earlier contributors retain all credit. Entries below describe their dated checkpoints.
-
+**Current reviewed construction: [PR #144](https://github.com/CrocSwap/integer-mult-bounds/pull/144), κ=0.0004609169.**
+The [round-six review](docs/research/community-round6-review.md) records the
+checked source heads, dependency arguments, finite controls and replay limits.
+
+- **icekylinx:** the principal construction and integration lineage: stopped
+  recursion (#104), partial gauges (#115), three-stage covers and weighted
+  local-ring compilation (#130), then paired cubes and the selected assembly
+  (#144), with the original assistance disclosures retained.
+- **an664:** completed-core workspace sharing (#128), adopted and explicitly
+  credited as a substantial conceptual dependency.
+- **eumemic:** the positive producer (#117) whose exact restrictions supply
+  the paired-cube query modules; original Claude disclosure retained.
+- **Zhihao Chen (jacklightChen):** the deferred physical bit word and reflected
+  ledger (#97), including fan order and endpoint accounting.
+- **Swapnil Jain:** the underlying deferred word, lifted frames and frozen
+  witness used by #97 and the selected bit subset.
+
+This is a cumulative community construction. Earlier semantic, routing,
+geometry, compiler and analytic contributions remain credited in `SOURCES.json`
+and `NOTICE`; acknowledging a dependency does not approve every separate claim
+in its author's other submissions. The [round-two ledger](docs/research/community-round2-review.md)
+retains the #50–#62 checkpoint, including parallel and superseded work.
+Entries below describe their dated checkpoints.
 
 Thank you to everyone contributing proofs, constructions, parameter improvements,
 independent implementations, checks, corrections and unsuccessful searches 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.