Research: Gaussian E8 weighted maximal-clique ceiling 40 and independent replay

[chinmayarvind23]

Change and mathematical scope

The Gaussian E8 copy floor is 40, but can changing clique weights improve that certificate? This research note and independent verification proves that 40 is the exact optimum for all nonnegative weighted inclusion-maximal-clique certificates, including nonuniform weights and mixtures of maximal cliques of different sizes.

Concurrent #738 already establishes the decoder lower bound 40, closure reduction and uniform maximum-clique sharpness. It was discovered during the final overlap refresh, after this package's proof and implementations were developed. This submission credits that result and adds the stronger method ceiling: #738's inspected proof gives a general ceiling of 48; here 40 suffices for the specified maximal-clique class. Certificates using non-maximal cliques remain outside this extension. No new kappa or physical supplier is claimed.

The short ceiling argument sums the local certificate constraints over the 900 ordered orthogonal-root hyperplane pairs. Each pair has Gaussian pairing rank two. A maximal clique of size k meets both hyperplanes in exactly 900-30k cases, which is at least 45k because k<=12. Therefore every such weighted certificate has value at most 1800/45=40. Uniform weights 1/192 on the 640 maximum cliques attain it.

Two separately implemented exact replays independently confirm the lower bound on all 2,582,128 unordered pairs of 2,272 flats. The primary uses eight rational coordinates and exact pairing ranks; the second uses four Gaussian-integer coordinates, cofactor-based flat generation and nonzero-minor certificates. It also enumerates all 8,704 inclusion-maximal cliques. Adding empty-support pairs explains the count difference from #738 and is not claimed as a stronger mathematical domain. The abstract decoder optimum remains in [40,60].

Validation

From the repository root, Python 3.11+ and its standard library:

python -I -B -c "import subprocess, sys; base='research/gaussian-e8-copy-bound/'; [subprocess.run([sys.executable, '-I', '-B', base + name], check=True) for name in ('verify.py', 'independent_check.py')]"

Expected:

PASS_GAUSSIAN_E8_COPY_BOUND lower=40 upper=60 flats=2272 pairs=2582128 controls=8
PASS_INDEPENDENT_GAUSSIAN_COPY_BOUND lower=40 pairs=2582128 maximal_cliques=8704 controls=10

The package includes the proof, deterministic receipts, source hashes, local sprint claim, attribution and a validation record. Both checkers import no predecessor implementation and retain their correctness checks under optimized Python.

Validation on CPython 3.12.15 / Linux in a digest-pinned, network-disabled container: both complete replays pass normally and with -O; all 18 internal rejection controls pass. All 21 external QA cases have the expected outcomes, including relocated minimal-input runs, byte-identical receipt regeneration, and rejection of altered source bytes, altered default receipts, omitted sources, escaping manifest paths and existing output files. A packaging check caught Windows manifest line-ending normalization; the manifest and its receipt metadata were normalized before publication, followed by targeted replay and receipt checks on the exact staged Git blobs. Verifier code and all seven pinned inputs were unchanged. Selected-record/source-inventory integrity, local documentation links and staged whitespace pass. The package's VALIDATION.md records the commands and timings. Full make verify, the selected supplier's cold replay, proof-assistant verification and remote CI were not run or claimed.

Scope and provenance

Only research/gaussian-e8-copy-bound/ is added. Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. The selected construction and upstream/ remain unchanged. The theorem concerns whole Gaussian-line supports, binary scalar totals, the fixed fitting pattern and source/reader pairing-rank charge. It does not cover vector totals, rational sub-line supports or altered supplier models. Neither finite replay is a formal verification of the multiplication theorem.

Credit Sagar Thakrar / @thakrarsagar's #738 (9671c5d72f3119ba26f3774f40aa7130c67bcf77), Joel Pulikkan / @GamingPuzzled's #620 geometry and open question, Douglas R. Colkitt / @Mister-Meeseeks' #703 weighted-clique method (358383e2f1b0f292d4d6eb897c06fd8558a4f5e6), and @DreamingOfClouds' #446/#595 pairing model and conditional accounting. No #738 code is imported. The original #703 proof and notice are retained unchanged.

OS-random assignment seed 6, bit-supplier track. No pinned coordination issue was found, so the claim is local. Final overlap refresh covered Gaussian search results and new PR metadata through #763, followed by the newly arrived #767's distinct planar/incidence decoder subclass obstructions. Main and the inspected #703/#738 heads remained unchanged. Prepared for @chinmayarvind23 with substantial OpenAI Codex assistance in proof development, implementation, verification and writing. No independent human review is claimed. Apache-2.0.

[Mister-Meeseeks]

100-PR batch review at 48d027a664288ea1d4e987de63cb79588b8b936b.

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

Reviewed scope: Independent Gaussian floor 40 and exact ceiling 40 for nonnegative weights on maximal cliques; does not cover the larger nonmaximal-clique method.

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.