Scoped negative result: Gaussian E8 copy floor 40 excludes three stages

[thakrarsagar]

Result

Research note and scoped negative result. No new kappa or physical supplier.

PR620 leaves the copy cost of asymmetric Gaussian E8 decoders open. This contribution proves a lower bound of 40 Gaussian rank units / 80 rational rank units, for any finite number of scalar binary totals and the pairing-rank charge of #446. It does not assume symmetric decoding or full-rank totals.

Under the separate #595/#620 deficit accounting, D/T <= 120(s-1) - 80s = 40s-120. Thus the three-stage Gaussian format cannot have positive deficit, even before helper, source-access, frame, restoration and recursive expenses. Five or more stages remain unresolved. The actual abstract decoder optimum remains in [40,60].

The proof extends @Mister-Meeseeks' concurrent #703 averaged clique method to the distinct 60-line Gaussian geometry. All 640 maximum cliques receive weight 1/192. Every support pair is covered by closure to geometric flats. The exact census checks 2,579,856 nonempty unordered flat pairs, and an explicit rank-two/rank-three pair with pairing rank one shows this uniform weighting is sharp. This does not prove the actual decoder optimum is 40.

Reproduce

python3 -B research/gaussian-clique-copy-bound/verify.py

Expected:

PASS_GAUSSIAN_CLIQUE_COPY_BOUND: cost >= 40 Gaussian / 80 rational; 2579856 flat pairs; 6 negative controls; kappa unchanged

Python 3.10+ and standard library only. The package contains the proof, portable independent geometry, exact receipt, source pins, local sprint claim, attribution and validation record.

Validation on macOS arm64:

• Fresh exact replay on Python 3.14.8 and 3.11.15, with all six built-in rejection controls and identity/asymmetric positive controls.
• Relocated minimal-input replay from an unrelated working directory.
• Modified source, corrupted certificate and optimized Python are rejected.
• Independent Fraction elimination checks the sharp witness and arithmetic cases. Separate development review compared 1,000 seed-46 integer matrices and the original E8 root construction.
• Selected-record integrity, local documentation links and staged whitespace pass.

The unchanged repository-wide suite and selected construction's expensive cold replay were not rerun. No Linux CI pass or proof-assistant verification of the multiplication theorem is claimed.

Provenance and limits

Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. Contribution: 9671c5d.
Structural decoder track, OS assignment seed 46. No pinned coordination issue existed, so the claim and result update are local. Final overlap refresh covered 551 open PRs through #732; main was unchanged. #717's schedule erratum was checked and does not change the deficit premise used here.

Credit Joel Pulikkan / @GamingPuzzled's #620 for the geometry, prior searches and open question; @DreamingOfClouds' #446/#595 for pairing-rank and conditional accounting; and Douglas R. Colkitt / @Mister-Meeseeks' #703 at 358383e2f1b0f292d4d6eb897c06fd8558a4f5e6 for the precursor weighted-clique method. The relevant #620 source files still match its current head e9cf03bbdf7094b5a83f6dbd1d74bfd77a05fb87. No predecessor executable code is imported.

Scope is whole Gaussian-line supports, scalar F2 totals and this fitting pattern. Sub-line supports, vector-valued totals, other geometries and changed stage accounting are outside scope. The inherited realization, compiler, hosting, precision, tape and analytic assumptions remain. upstream/, the selected frontier and all existing files are unchanged.

Prepared by Sagar Thakrar (@thakrarsagar) with substantial OpenAI Codex assistance in proof development, implementation, verification and writing. Apache-2.0. No independent human review is claimed.

[Mister-Meeseeks]

100-PR batch review at 9671c5d72f3119ba26f3774f40aa7130c67bcf77.

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

Reviewed scope: Gaussian scalar-total pairing-rank floor 40; closure-to-flats and clique certificate cover arbitrary supports in the declared model, optimum remains in [40,60].

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.