Conditional κ = 0.001676855229071321: finer certified endpoints (1e-200) and deeper outer witness on PR1424 (82 bootstrap steps, eta=beta=1e-210, kappa grid 1e-199); construction unchanged

[jeremykothe2]

Summary

Conditional κ = **0.0016768552290713218862064567722745926195124487395104752628033651910511566552120096495157846916413845429929275438103942180631016965288202851222935391871757186450912861283049468793633058255444681490343
**. Repairs the fine Mercator engine of #1479/#1510/#1513 (odd term count made its lower log end an upper bound; finding PR1519 philippeVV), adds an independent exact atanh oracle stage, recomputes every endpoint certificate with the repaired engine, and keeps PR1424's construction 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 82, eta = beta = 1e-210, backoff 1e-213, final kappa grid 1e-199) and the decimal fields are exact. kappa moves from 0.001676855229071321886206381217337893 to 0.0016768552290713218862064567722745926195124487395104752628033651910511566552120096495157846916413845429929275438103942180631016965288202851222935391871757186450912861283049468793633058255444681490343.

Verification

Fresh checkout of 527d7623009f295b7311e98ae36f5e2fbdbbc3e5. Manifest: d4d4737daa86ae45560bd8a8eb50c3f3307c32a7806855d71c9bea9afb540229.
PASS package manifest: 160 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 0.0016768552290713218862064567722745926195124487395104752628033651910511566552120096495157846916413845429929275438103942180631016965288202851222935391871757186450912861283049468793633058255444681490343; native reuse {'base': 'e8e6e78ff7422275c38324499e1f33fb2ecd30de9503aef2b8f1ee80d8c4df8e', 'extra': '7b0e9fbda32749aabc7e156d9fe09bceb0624a178628fe69e5dfea783462f905'}

Full package-owned verification passed in 298.74 seconds. The fresh receipt binds the exact package manifest and score.

See research/gmx-t74-ep200r/README.md for reproduction commands and proof qualifications.

Credits

Finding of the fine-Mercator lower-bound bug: PR1519 (philippeVV). 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.

[Mister-Meeseeks]

Batch review at 527d7623009f295b7311e98ae36f5e2fbdbbc3e5.

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.