Research audit: tighter Gaussian correction-power endpoint bound (no new kappa)

[bg-grillo-alan]

Change and mathematical scope

Research audit / analytic refinement; no new kappa. In eumemic's Gaussian correction-power lemma, two correlated endpoint squares are bounded separately. Combining them first replaces the exponential prefactor

exp(pi alpha^2 (1/(4 theta) + 1/2))

with

exp(pi alpha^2 floor(s/2)^2 / (s^2 theta)).

The Gaussian decay term is unchanged. A written all-path proof and rationally certified Neumann residual counts are included. For p=128, alpha=4, s=15, t=16, the old endpoint bound priced with the same decay needs 6 terms; the refined bound needs 5. The original simplified corollary needs 18, but only the 6-to-5 comparison isolates this contribution. Across the declared finite count screen, 1,876 of 3,296 cases improve in that comparison.

A concrete Gaussian row shows why the prefactor cannot be discarded entirely. A separate scalar control distinguishes a power residual from absolute inverse-tail error; the latter retains an extra residual bit and the explicit ||E||<1/2 premise.

Only research/gaussian-endpoint-potential-audit/ is added. No rounded implementation, tape-cost improvement, optimality claim, new supplier or multiplication exponent is proposed. Existing generators, the selected frontier and upstream/ are unchanged.

Validation

python -B research/gaussian-endpoint-potential-audit/verify.py
python -B -m unittest discover -s research/gaussian-endpoint-potential-audit -p test_verify.py -v

Expected marker: PASS_GAUSSIAN_ENDPOINT_POTENTIAL.

• Exact checks: 646 coprime pairs, 112,672 single steps, 9,460 path cases, 3,296 count cases. A separate integer-scaled path calculation agrees with the rational interpreter; no predecessor checker is imported.
• Eight test groups pass normally and under -O, including formal polynomial matrix squares retaining wrapped nonzero steps, incorrect arithmetic, source drift/missing inputs, receipt corruption/types, domain controls and output preservation.
• The complete receipt matches under optimized Python and in an isolated minimal copy in a path containing spaces. Four inputs have SHA256 and exact base Git-blob pins. Constants use rational Taylor enclosures.
• Selected-record verification and its five existing tests pass. Staged whitespace checks pass.
• Executed on Windows / CPython 3.14.5. No Linux, hosted CI, full make verify, cold supplier replay or Lean result is claimed. Finite checks are separate from the written universal argument.

Pins, overlap and attribution

Base: f567884aeb8851b6e794af1184119a3bfbe1e631. OS-random assignment seed 62, adversarial mathematical audit. The local claim records three candidates, the finite domain and stopping rule. No pinned sprint coordination issue existed. Final PR metadata inspection covered arrivals through #901 and targeted all-state Gaussian/potential searches; no duplicate endpoint refinement was found in that bounded inspection.

Credit eumemic's PR5 (d3d370c34937503bd3fa63c83d505a05e8e50893) for the operator and potential, icekylinx for the retained chirp enclosure, and Harvey–van der Hoeven for Gaussian resampling. Effy Elden's PR684 audits domain/windows and explicitly excludes Neumann evaluation; no pending-PR code is imported. PR5's partial review and PR371's unresolved rounding work are not promoted to full approval.

Prepared for Alan Grillo (@bg-grillo-alan) with substantial OpenAI Codex assistance in derivation, implementation, testing and prose. Apache-2.0; predecessor notices retained. No independent human review is claimed. Selected conditional kappa remains 1661634653669567/10^18, with its inherited realization, hosting, precision/tape and analytic obligations.

[Mister-Meeseeks]

100-PR batch review at e17fd726b23ff009518c2fc7fa0c423e3c25a46d.

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

Reviewed scope: Gaussian endpoint-potential audit with exact source comparisons and scoped numerical bounds; inherited analytic premises retained.

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.