Certify precision-40 κ above PR #3133

[Gonzalo-Mier]

Change and mathematical scope

This precision-only follow-up refines the certified endpoint for PR #3133's unchanged rank-7 coupled words, literal T74 allocation and paid profile. It certifies:

• C40 = 0.0017297760894051203342384678198935738683
• κ = C40 - 10^-86 - 10^-62

The finite candidate exceeds PR #3133's public claim by about 4.6782e-25. Both rational moment engines certify the endpoint, and the exact comparator proves the increase. This is a conditional finite candidate, not an all-size theorem or selected-frontier update. The complete source-export, native, replica-admission and outer-release replay passed; adjacent-grid and negative controls reject, and the precision-30 candidate was not released. The construction and selected frontier are unchanged; selected κ remains 0.001672660197076918.

Task and overlap

Task complex-new-helper-inventory, priority revision 6; scope is limited to refining the certified endpoint of the existing T74 candidate. Package base commit: 997e6a9ca7b4f95d50e366f61412fd640d72a88a. Exact construction dependency: PR #3133 at 51eb9c4ee9e4722010d308aa3309e039d3940cc2. PR #3136 independently audited the parent words but does not contain this endpoint refinement. The full predecessor lineage is in CREDITS.md; no construction from concurrent packages is imported.

Validation

Registered environment: CPython 3.12.15, gmpy2 2.3.2, mpmath 1.3.0, NumPy 1.26.4, SciPy 1.15.3 and SymPy 1.14.0. Reproduce from a full clone containing the pinned source Git objects:

python3.12 -B scripts/research_package.py --package research/rank7-coupled-586-588-carrier-fold-precision40
python3.12 -B research/rank7-coupled-586-588-carrier-fold-precision40/reproduce.py \
  --repo /path/to/full-repository-with-source-objects \
  --work /tmp/rank7-precision40-replay --jobs 2

The complete source-export, native, replica-admission and outer-release replay passed, with receipts binding the exact words, allocation, claim and manifest. The work reused only the authenticated, hash-checked retained bootstrap/cache. No all-size theorem or full multiplication theorem is claimed.

Credit

The complete predecessor lineage remains in CREDITS.md. Precision-40 refinement by Gonzalo Mier with substantial OpenAI Codex assistance; Apache-2.0. No independent human review is claimed.

Open next questions

1. Test the next endpoint grid against the independent interval enclosure.
2. Close the inherited 1.4053e-9 integer-rounding gap with a literal T740 allocation.
3. Test whether a chronological B_WARP change can unlock helpers 584 and 590 without returning the savings in restoration.