[jeremykothe2]

Summary

Conditional κ = 0.001676855229071321886206456772274592619512448739510475262803365191051156655212009649515784691641384542992927543810394218063101696528820285122293539187175718645091286128304946879363305825544468149. PR1424's construction is unchanged. The certified complex endpoints (native and conservative) are refined from the inherited 1e-24 bracket to 1e-200: the retained engines still certify the 1e-24 bracket, and finer-constant copies of both rational engines certify accept at the endpoint and reject at endpoint+1e-200. The outer witness settings are deepened (bootstrap steps 80, eta = beta = 1e-205, backoff 1e-208, final kappa grid 1e-195) and the decimal fields are exact. kappa moves from 0.001676855229071321886206381217337893 to 0.001676855229071321886206456772274592619512448739510475262803365191051156655212009649515784691641384542992927543810394218063101696528820285122293539187175718645091286128304946879363305825544468149.

Verification

Fresh checkout of 968d7ea2be07c376ceaee24d659b4298628e9767. Manifest: 8f7c2d4c7696a3ee83eecc7b2d402ab7e29826eafafbe3b69fa34dd3198b6a44.
PASS package manifest: 159 files

The package's own unchanged verify.py (retained PR1297 provider and parent admission, owner reconstruction, parent lift, strict T74 admission with the 1e-24 outer kappa grid, geometry controls): PASS_T74_EXACT_FILL_COMPLETE_FINITE_ADMISSION 0.001676855229071321886206456772274592619512448739510475262803365191051156655212009649515784691641384542992927543810394218063101696528820285122293539187175718645091286128304946879363305825544468149; native reuse {'base': 'e8e6e78ff7422275c38324499e1f33fb2ecd30de9503aef2b8f1ee80d8c4df8e', 'extra': '7b0e9fbda32749aabc7e156d9fe09bceb0624a178628fe69e5dfea783462f905'}

Full package-owned verification passed in 285.08 seconds. The fresh receipt binds the exact package manifest and score.

See research/gmx-t74-ep200g195/README.md for reproduction commands and proof qualifications.

Credits

PR1424, PR1319 and PR1297 (rohanarun: outer witness refinement, 74-replica exact fill, four cold kernels 531/543/555/567); PR1377 finer kappa grid and PR1274 helper-donor kernel (jeremykothe2); PR1286, PR1193 (jeremydiviney), PR1148 (Thomas DiFiore), PR1195/1196 and the retained lineage named in those packages. Research direction, prioritisation and release strategy: jeremykothe2. Implementation assistance: oh-my-pi.

[philippeVV]

Small soundness note on research/gmx-t74-ep200g195/code/fine_mercator.py. No effect on this PR's reported κ.

for n in range(1, LOGN) with even LOGN = 830 stops on an odd term, so the partial sum s lies above log(1+z). It is nevertheless returned as the lower end of (s, s+t/LOGN). The retained engines/moment.py avoids this by summing to n = 224 (an even partial sum).

Exact check against an independent atanh enclosure: logarithm(x) returns a lower end above the true log for x = 68, 134, 444, even after the outward floor. Every input this PR's check_moment uses (73/r and 73) happens to survive the floor, with a least margin of about 0.0027 grid units. Rejections also require the sound fine_moment engine, so no decision changes, and the error is below one grid unit.

One-line fix: make LOGN odd (e.g. 831), keeping return s, s+t/LOGN. Reproduction (exact Fraction checker): https://github.com/philippeVV/integer-mult-bounds/blob/3106914f87c5cf190d2eebc2a11e715b24714eb8/research/fine-mercator-parity-audit/verify.py. Board note: #1166 (comment)

Posted by Philippe Van Velzen with Claude (Anthropic) assistance.

[Mister-Meeseeks]

Batch review at 968d7ea2be07c376ceaee24d659b4298628e9767.

This remains open after scoped triage. Next review step: Refines only the outer witness of the pending PR1319 family. Review the arithmetic separately, but whole-head admission requires resolving the parent regenerated-container binding and completing its controller/invoice/geometry replay; do not promote an unadmitted parent through a finer numerical grid.

This is a specific continuation point, not a rejection or a request to redo already recorded checks. Completed local evidence is retained separately from outstanding whole-head obligations.

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