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

[WilliamLawyer]

Research note / scoped negative result. No new kappa. Nonnegative weighted-clique certificates on the fixed 60-line Gaussian E8 configuration cannot certify more than 40 Gaussian pairing-rank units, even when every nonempty nonorthogonal clique is allowed.

PR770 proves ceiling 40 for inclusion-maximal cliques and expressly leaves non-maximal cliques outside its theorem. This package closes that certificate-domain gap. It preserves PR738's existing scalar decoder interval [40,60]; a cost-40 decoder remains unconstructed.

New evidence

• The initial attempt to extend PR770's 900-pair dual fails on a singleton: 42 hits rather than the required 45. This is a failed extension hypothesis, not a defect in the predecessor's scoped theorem.
• An exact repaired dual retains the 180 ordered orthogonal-normal hyperplane pairs with six-line intersection, each of Gaussian pairing rank two. Every nonempty clique Q meets both hyperplanes in at least 9 |Q| of these pairs. Summing their local constraints gives 9 sum_Q |Q| w_Q <= 360, hence ceiling 40 for arbitrary nonnegative clique weights.
• Complete enumeration covers 2,068,372 cliques, without a size cutoff. A separately implemented Gaussian geometry/rank checker and Bron–Kerbosch/submask traversal agree on all counts, size minima, pairs and the complete clique-set digest.

The matching feasible value-40 certificate is inherited from pinned PR738/PR770 sources. Their unchanged full primal flat-pair coverage is not rerun or claimed as new evidence here. This result closes the specified proof method, not all possible decoder mechanisms.

Reproduce and validation

Python 3.10+; standard library only; offline and read-only by default:

python -B research/gaussian-all-clique-dual/verify.py
python -B research/gaussian-all-clique-dual/independent.py
python -B research/gaussian-all-clique-dual/test_package.py

Expected primary marker:

PASS_GAUSSIAN_ALL_CLIQUE_DUAL cliques=2068372 ordered_pairs=180 ceiling=40 controls=6 no_new_kappa

Windows / CPython 3.10.9: both ordinary replays pass. A cold export of canonical staged Git blobs passes isolated optimized primary and independent replays (17.702s / 7.840s), produces an identical certificate and passes all 10 tests (9.254s). The enumerators also match brute force on all 1,099 labelled graphs of orders 1–5. Controls reject changed sources, certificate claims, invalid geometry, a dropped dual pair, overly strong inequalities and overwriting an existing receipt. Exact frozen evidence and input hashes are retained in VALIDATION.md and evidence/.

Selected-record integrity and staged whitespace checks pass. No full repository suite, selected supplier cold replay, Linux CI or formal multiplication-theorem verification is claimed for this isolated package.

Scope, provenance and credit

Only research/gaussian-all-clique-dual/ is added. Base: f567884aeb8851b6e794af1184119a3bfbe1e631. Structural-decoder track, OS-random seed 40; three candidate tasks and the bounded hypothesis/pivot are recorded in CLAIM.md. No pinned coordination issue exists, so the claim and result are local. The final overlap inventory at 08:33 UTC contains 661 open PRs through #866 and no duplicate of this scope; all inspected dependency heads remain unchanged.

Primary source pins: PR738 9671c5d72f3119ba26f3774f40aa7130c67bcf77; PR770 48d027a664288ea1d4e987de63cb79588b8b936b. Exact retained SHA256/Git-blob pins and adjacent PR heads appear in SOURCES.json. The accepted PR799 X4 result remains 1661634653669567/10^18; its construction and review were read and are unchanged.

Credit Sagar Thakrar (@thakrarsagar) for PR738's Gaussian floor, closure and feasible certificate; @chinmayarvind23 for PR770's independent replay and maximal-clique ceiling; Joel Pulikkan (@GamingPuzzled) for PR620's geometry and decoder work; Douglas R. Colkitt and DreamingOfClouds for the earlier weighted-clique and pairing-rank lineage. The notices credit adjacent research and the selected construction's contributors. Vendored source documents retain their original attribution and license.

Prepared for William Lawyer (@WilliamLawyer) with substantial OpenAI Codex assistance, including a scoped worker's independent implementation and another worker's proof review. No independent human review or literature-priority claim. Apache-2.0. Physical supplier realization, all-size/compiler, common-label/mutable-source, hosting, precision/tape and analytic obligations remain inherited; this package implements no new supplier invoice, recursive interface or multiplication bound.

[Mister-Meeseeks]

100-PR batch review at 54cbbea75b587bdf43a1c123e52220f936641d6f.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Exact all-clique Gaussian E8 certificate-method ceiling 40; does not construct a cost-40 decoder.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.