Verified conditional kappa: positive-slack refinement over PR1928 (+2.00336e-10010)

[rohanarun]

Locally verified conditional refinement of PR #1928 at head 07e4e4231459e36b1055396306dcbd950248df62.

The exact candidate exceeds that ledger by approximately 2.0033595099133e-10010. Its decimal prefix remains 0.0016797549561255445213356709939207025769983114385025802954603275201628645482603256388263056561328. Exact rationals and the full decimal are in CLAIM.json; the comparison uses rational arithmetic, not rounded displays.

This is an extremely small positive-slack improvement of the same supplier, not a structural construction or practical speedup. It retains the PR1928 10,000-digit native endpoint B and uses h=10^-10030, b=B-h, q=B-2h, epsilon=1-2h, beta=1-6h. The kappa is floor(epsilonq10^10070)/10^10070 - 10^-10042.

Local checks completed: the unchanged independent PR1982 endpoint checker, including its four brackets, nesting checks and ten rejecting controls; 57 strict rational outer inequalities; 17 rejecting controls; and a separate nine-cost balance derivation agreeing exactly. Fresh complete supplier replay PASSED locally, including inherited outer replay, both native supplier variants, parent and strict T74 admission, and final assembly-to-endpoint-to-ledger bindings. No hosted CI pass or independent human review is claimed.

Reproduce arithmetic with Python 3.11+: PYTHONINTMAXSTRDIGITS=0 python3 -B research/fine-prime-slack10030/verify.py --work /absolute/new/work --arithmetic-only. For complete replay omit --arithmetic-only; optionally pass --dependency /path/to/immutable/PR1800/package. Outputs and fetched dependency files stay under the fresh work directory. Full replay binds the unchanged PR1800 native assembly to the new separately enclosed endpoint and ledger.

This leaves the unchanged native moment ceiling untouched and moves only closer to its endpoint. The physical supplier, paid profile, packing cap and structural headroom are unchanged. It does not improve the packing cap or address the P104 construction obligations.

Conditional on all inherited PR1800/1786/1758/1673 compiler, common-label/mutable-source, hosting, semantic online precision C1=1, fixed-tape/prime-packing, interchange, recovery and analytic assumptions, plus the cited Trudgian prime theorem. Finite arithmetic does not establish the written all-size prime/precision/locality or accretive-circuit arguments. No unconditional theorem, practical cutoff or maintainer acceptance is claimed.

Open next questions: Can a complete P104 low-rank channel and compatible side correction reduce the actual paid moment? This matters for a material gain; first price and replay one complete frame-compatible local word. The unresolved side correction, native compilation and chronological hosting are blockers for that separate structural route. Further precision-only changes have negligible practical significance.

Credits: Jeremy Kothe and PR1928/1800 contributors; Aaron Wang with opencode assistance for the independently derived PR1982 moment checker (retained unchanged at b10c1a1); Chafik Boukhalfa / PR1786, Tim Trudgian, Vlado Vojdanovski / PR1673, PR1758 contributors, LeverRank, mdumitrean, midhatn and all inherited supplier authors. Retained original sources and notices accompany the derived checkers. Prepared by Rohan Arun with substantial OpenAI Codex assistance. Apache-2.0.

Fixed-mixture packing context: native saving ceiling bracket ['843830269237/500000000000000', '67506421539/40000000000000']. This candidate realizes approximately 99.53156560995382% of that native upper ceiling. This uses the exact 249-A/417-B average profile, rank cap51 and retained coverage identity; it is not a universal bound or the older balanced a/(1+a) transfer. Reproduce the supplement with PYTHONINTMAXSTRDIGITS=0 python3 -B research/fine-prime-slack10030/packing-context/verify.py research/fine-prime-slack10030. Credit Douglas Colkitt with Claude assistance for the retained ceiling engine.

Full verification: PASS_CONDITIONAL_SLACK_REFINEMENT_FINITE_ADMISSION. Evidence includes the exact replayed source manifest and stage/log hashes. Only status prose and claim metadata change after replay; mathematical inputs, parameters, verifier code and the exact candidate rational are unchanged. This is local AI-assisted finite verification, not a full unconditional proof or hosted CI acceptance.

[Mister-Meeseeks]

Batch review at 3a42e3010c188b30287f576d1ec5f0e80193c671.

This remains open after scoped triage. Next review step: Stacked supplier/outer proposal in the pending T74 family: separate the actual new construction or analytic change from inherited ancestors, resolve parent regenerated-container/controller/invoice/geometry admission, and run the full required interface checks before whole-head incorporation. A larger numerical witness alone is not acceptance evidence.

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.