Research: Gaussian E8 all-clique certificate ceiling is 40 (no new kappa)

research/gaussian-all-clique-dual/CLAIM.md

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+# Local sprint claim
+
+Contributor: William Lawyer, with substantial OpenAI Codex assistance.
+Start: 2026-10-13 07:54:33 UTC. Expected finish: by 09:54:33 UTC.
+Track: structural decoder research. Assignment seed: **40**, one operating-system
+draw using Python `secrets.randbelow(100)`.
+Base: `f567884aeb8851b6e794af1184119a3bfbe1e631`.
+
+Three narrow candidates were considered:
+
+1. Test a mixed partial-hyperplane rectangle decoder on the Gaussian E8 unit.
+   PR767 leaves partial supports open, but a positive supplier would require
+   a new complete cost and realization argument.
+2. Test PR770's explicit hyperplane-pair dual on **all** nonorthogonal cliques,
+   including non-maximal cliques. Its inspected proof expressly excludes them.
+3. Extend the Gaussian scalar floor to fixed-support nonlinear messages.
+   PR785 supplies this mechanism on Hurwitz geometry; that transfer offers
+   less separation from existing work than the all-clique certificate gap.
+
+Selected candidate: **2**. Seed 40 selects the second candidate by `seed % 3`;
+its useful diversification is the previously excluded certificate domain.
+Object: the 60 Gaussian E8 lines and PR770's 900 ordered orthogonal-normal
+hyperplane pairs. Hypothesis: each nonempty nonorthogonal clique Q meets both
+hyperplanes in at least `45 |Q|` ordered pairs. Test by complete integer
+clique enumeration, with a separately constructed geometry and traversal.
+Deliverable: a solver-free exact dual certificate closing this method's gap,
+or an explicit failed-dual witness; no new kappa claim.
+
+The live inventory retrieved at 07:58:21 UTC contains 654 open PRs through
+#857 and no open issues or pinned coordination issue. Scope overlaps were
+checked in PR738, PR767, PR770, PR785, PR798 and PR801; their exact inspected
+heads will be retained with the result. The ledger marks PR620 partially
+reviewed; pending PR claims are not accepted supplier results.
+No external reservation issue is opened. This local claim is the reservation.
+
+Research will stop by 09:24:33 UTC, leaving the final 30 minutes for verification
+and handoff. One evidence-driven pivot is allowed if the proposed dual fails.
+
+## Result update
+
+Result posted at 2026-10-13 08:41:32 UTC:
+[PR868: Gaussian E8 all-clique certificate ceiling is 40](https://github.com/CrocSwap/integer-mult-bounds/pull/868).
+Research note / scoped negative result, **no new kappa**. The first result
+commit is `7cdd3ac12254290c55bac602d5e293dea4d981ae`.
+
+The original 900-pair extension fails concretely on singleton `{0}`:
+42 hits rather than 45. One evidence-driven pivot retains the 180 ordered
+orthogonal-normal hyperplane pairs with six-line intersection. Each has
+Gaussian pairing charge two. Every nonempty clique has at least nine hits
+per vertex, bounding every nonnegative weighted-clique certificate by 40.
+This closes PR770's expressly excluded non-maximal-clique domain; the
+predecessor's feasible value 40 and scalar decoder interval [40,60] are
+unchanged. A cost-40 decoder and a stronger multiplication bound are not
+constructed.
+
+Domain and stopping rule: every nonempty clique in the fixed 60-line Gaussian
+E8 nonorthogonality graph, extending until no vertex can be added; no sample
+or size cutoff. Completed coverage: **2,068,372 cliques**, sizes 1–12, with
+all counts and minima in `CERTIFICATE.json`. Seed 40 selected the task; the
+coverage itself is deterministic and exhaustive. The increasing-order
+traversal and an independent Gaussian-coordinate, Bron–Kerbosch/submask
+implementation agree on the complete domain digest
+`e07f4d5b163728561aa881fd76ba32092d93a2914c2718019964f8dd013e8efd`.
+
+The code and mathematical inputs pass ordinary replays and an isolated,
+optimized cold export from staged Git blobs. All 10 package tests pass,
+including source/certificate corruption controls and exhaustive small-graph
+checks. `VALIDATION.md` records the measured checks and exclusions. A separate
+scoped AI worker reviewed the proof; no independent human review is claimed.
+
+The refreshed overlap inventory at 08:33 UTC contains 661 open PRs through
+#866, no coordination issue, no duplicate of this scope, and unchanged heads
+for all eight inspected source PRs. Exact pins and predecessor credits are
+retained. No external claim requires an update; this is the single local
+claim result update. Research stopped after the complete certificate and
+independent replay, within the 120-minute budget, leaving ample verification
+and handoff time. Only the new package changes; no selected frontier or
+`upstream/` files are edited.

research/gaussian-all-clique-dual/LICENSE

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research/gaussian-all-clique-dual/NOTICE.md

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+# Attribution and licenses
+
+Prepared for **William Lawyer (@WilliamLawyer)** with substantial **OpenAI
+Codex assistance** in research selection, mathematics, proof review,
+implementation, verification and writing. Separate Codex workers reviewed
+the dual deduction and built an independent replay within the claimed
+scope. These are independent implementations and AI reviews; no independent
+human review is claimed.
+
+Credit **Sagar Thakrar (@thakrarsagar), PR738** for the Gaussian scalar copy
+floor of 40, closure to geometric flats, the uniform 640-clique certificate,
+and its exact whole-support pairing-rank replay. His unchanged proof is
+retained at `vendor/pr738-PROOF.md`.
+
+Credit **@chinmayarvind23, PR770** for independent replay of
+that floor and the ceiling 40 for all inclusion-maximal-clique weights. The
+explicit limitation to maximal cliques is the starting point of this note.
+Its unchanged proof and notice are retained under `vendor/pr770-*`.
+The 180-pair dual and complete non-maximal-clique coverage are this note's
+additional evidence. No predecessor executable code is imported.
+
+Credit **Joel Pulikkan (@GamingPuzzled), PR620** for the extension-ring E8
+geometry, decoder question, earlier searches and symmetric cost-60 upper
+bound; its retained report is `vendor/pr620-extension-REPORT.md`.
+Credit **Douglas R. Colkitt (@Mister-Meeseeks), PR703** for the earlier
+weighted-clique method on Hurwitz geometry, and **DreamingOfClouds,
+PR446/PR595** for the conditional source/reader pairing-rank and format
+accounting used by that line of work.
+
+**Charles Bethin's PR767**, **Alexander Templeton's PR785**, **Jonathan Lam's
+PR798**, **Rohan Guliani's PR801**, and **PR814's Eisenstein nonlinear-message
+extension** were checked as overlapping or adjacent research. Their geometry,
+decoder subclasses or nonlinear-message domains differ. No claims from
+their PR titles are treated as the selected supplier result.
+
+The selected PR799 construction and its Jeremy Kothe / Chafik Boukhalfa /
+DreamingOfClouds / eumemic and ancestral realization lineage retain the
+credit in the base review and notices. This contribution changes none of
+their source artifacts or selected result.
+
+[SOURCES.json](SOURCES.json) binds exact inspected heads and retained file
+bytes. The Apache-2.0 license applies to new files and the retained source
+documents; their original notices and attributions are preserved.
+Relative references inside unchanged vendored documents refer to their
+original source packages at the pinned commits, rather than this package.
+Standard
+E8 coordinates, basic Hermitian rank identities and graph algorithms are
+not claimed as novel. No global literature-priority claim is made.