Research: Gaussian nonlinear message copy floor 40b under explicit width pricing

[ToikY]

Research note / scoped negative result; no new κ or physical supplier. Fixed-support, deterministic, zero-error nonlinear block messages on the 60 Gaussian E8 lines satisfy

C >= max(ceil(2 sum_t b_t / 3), max_clique_Q sum_(t in Q) b_t).

For uniform b-bit demand, this gives 40b Gaussian / 80b rational pairing-rank units. Direct messages give upper bound 60b; the nonlinear optimum remains unresolved. Demands confined to any nonorthogonal clique attain the concentration bound.

The essential premise is explicit: each q-bit message pays q times its Gaussian source/reader pairing rank, additively. This does not establish physical width pricing for arbitrary nonlinear circuits. With the additional inherited Dfc stage accounting, even the optimistic budget is D/T <= 40b(s-3): three stages cannot give positive deficit while granting every other cost zero. Five or more stages and changed models remain open.

The proof composes Sagar Thakrar's #738 Gaussian scalar geometric certificate, Alexander Templeton's #785 conditional-transcript argument, and Michael Fischler's #814 unequal-demand/concentration composition. It adds the Gaussian nonlinear/block-message theorem and concrete controls of its assumptions. No method-priority claim. #770/#868 concern Gaussian clique-certificate ceilings and have distinct scope.

Reproduce offline with Python 3.10+ and the standard library:

python3 -I -B research/gaussian-nonlinear-copy-floor/verify.py
python3 -I -B research/gaussian-nonlinear-copy-floor/test_package.py

Expected main marker:

PASS_GAUSSIAN_NONLINEAR_COPY_FLOOR lines=60 cliques=640 flat_pairs=2579856 floor=40b no_new_kappa

Validation on CPython 3.12.14: the complete unchanged #738 calculation covers 2,579,856 nonempty flat pairs and its six rejection controls. New checks cover 640 maximum cliques, every port's 128 incidences, all 8,704 maximal cliques for demand bounds, 49,152 correct nonlinear receiver decodes, and 2,621,440 conditional projection assignments. A separately written truth-table checker exhibits actual omitted-inversion/dropped-bit failures and correct-code counterexamples to weakened pricing or side-information premises. Three focused package tests pass, including receipt/source corruption, omission, invalid demands and output preservation. Both a minimal relocated copy and the exact staged Git-blob export pass the complete isolated replay with a byte-identical receipt. Selected-record integrity and whitespace checks pass. Optimized Python is deliberately rejected because the retained geometry uses assertions.

Only research/gaussian-nonlinear-copy-floor/ is added. Base f567884; immutable predecessor heads:

• Scoped negative result: Gaussian E8 copy floor 40 excludes three stages #738: 9671c5d72f3119ba26f3774f40aa7130c67bcf77
• Scoped negative result: Hurwitz copy floor for fixed-support nonlinear messages #785: 83e78085ae9c110a34bff87e71387292d85e47c0
• Research: Eisenstein nonlinear message copy floor 22b under explicit width pricing #814: d91c5152b314391406b0ec792b6765a5cf0af2f1

Six #738 files and two predecessor proofs are retained verbatim with SHA256/Git-blob pins and original attribution. Further lineage and adjacent work are credited in NOTICE.md. OS-random seed 44, structural decoder track; three candidates and the result/link are in the local claim because no sprint coordination issue exists. Final overlap refresh through #887 found no duplicate; all inspected predecessor heads stayed unchanged.

Fixed supports/widths, independent demands, orthogonal free side information and additive per-bit pricing are essential. All-size/compiler, realization, hosting, precision/tape/exact recovery and analytic obligations remain inherited. Full repository/cold supplier suites, hosted CI and formal verification of the multiplication theorem are not claimed. Existing files, selected frontier and upstream/ remain unchanged; accepted conditional κ remains 1661634653669567/10^18.

Prepared for @ToikY with substantial OpenAI Codex assistance, including separate AI model/proof review and a separately implemented finite checker. No independent human peer review. Apache-2.0.

[Mister-Meeseeks]

100-PR batch review at 8c9964b7ab4419409d20762c8d78591701008d04.

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

Reviewed scope: Gaussian nonlinear block-message copy floor under explicit per-bit pairing prices and side-information assumptions.

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.

[ywz65]

CLAIM / adaptive Gaussian follow-up: extend PR888 fixed-support nonlinear transcript bound to a paid public prefix-free header choosing supports and prefix-free variable-length payloads. The proposed source-bound tax is E[C] >= (2/3)sum b_t + 4E[length M] - (10/3)H(M), hence at least (2/3)(sum b_t + E[length M]), with a separate clique concentration bound. Header tariff4 is the explicit full Gaussian-space source/reader contract, not a claimed native cost theorem. PR738 geometry and PR888 transcript argument and Jerry Chen Hurwitz adaptive-routing entropy accounting are retained and credited; PR2607 adaptive affine Eisenstein result is distinct. Main0fdd7d82, priority4; picker822a27d6015294b31b90467ba9f189a5. P104 and complex-ceiling first steps passed; scope changes to this excluded adaptive helper-observation premise. Overlap804openPRs16:13UTC/1239Discussionrecords15:22UTC with bounded freshness. Target proof, exact actual-geometry adaptive code and pricing/prefix/side-information controls within1hour; currently local-only. ywz65 with Codex assistance.

[ywz65]

DONE: source-bound proof and complete controls submitted as PR2641, #2641 at c63fbb8. Expected Gaussian header penalty and conditional prefix-free payload extension passed registered3.12 and Git-free optimized3.14 replay; separate proof review clear. Scope remains explicit full-space public header/additive pairing tariff, no native supplier. After main revision5 update, narrowed to this result and stopped dependent construction expansion. Prior geometry/transcript/paid-header methods retain credit.