Conditional κ = 7.10346593569665e-4: fixed-prime refinement on PR244

[sennemmi]

Conditional result

This stacks on PR244, commit a568d94f941929232ab393c7d33dc5a30e017892, and refines its retimed 5,891,100-call source527 profile. The exact candidate is

• PR244 base: 710346589668175/10^18
• Fixed prime with eta = 10^-12: 710346593567535/10^18
• Final with eta = 10^-24: 710346593569665/10^18
• Exact gain over PR244: 3901490/10^18

The gain splits into 3899360/10^18 from replacing the inherited rare-class density envelope with 2m^3/q, and 2130/10^18 from rechecking PR243's smaller outer backoff on PR244's profile. κ is still a conditional finite result, not an unconditional multiplication theorem.

Mathematical and physical certificate

The inherited profile has m=120, normalized stock W=260987, rank mass 31265640, deficit 52800, largest child 50, and 60 literal physical replicas with stock 1304935. It retains every bank selector call (525064856400) and the complete 32m^2 rare-case fallback; this refinement changes only their justified frequency bound, not the physical word or fallback inventory.

The fixed prime is q=2^127-1=170141183460469231731687303715884105727, chosen before the input or atom width. Its per-edge rare-class density is rho=3456000/q < 10^-16. The certificate independently recomputes two rational moment bounds, a five-level finite ordinary recurrence, all 47 strict assembly inequalities and seven margins, and rejection of the adjacent 10^-18 κ grid point. The finite coefficient is 93574191906738001 < 2^80.

The pinned parent verifier reconstructs PR244's 364 frame changes and independent complete path census, then checks the F₂ helper, literal inverse, restored sources and dirty registers, prime witnesses, all 4,976,100 bank assignments, routes, complex supplier, and finite bill. The original PR244 package reports two identical full eight-stage replays.

Reproduction and CI

python3 -m pip install -r research/five-stage-source527-fixed-prime/requirements.txt
python3 -B research/five-stage-source527-fixed-prime/verify.py --output /tmp/source527-fixed-prime-run

The new source527-fixed-prime workflow runs that full pinned replay and exact refinement, and separately kernel-checks Prime.lean using pinned Lean 4.24.0/mathlib. The repository's standard PR workflow also covers all 16 make verify targets across Python 3.11, 3.13, and 3.14, plus its formal, ranked-pair, certificate, and immutable-offline jobs. The upstream pull_request Actions are gated with action_required; GitHub's workflow-run approval API returned 403 because this account lacks repository-admin rights. I have not treated that gate as a pass. To complete validation, the exact head commit is running in the contributor fork: the full source replay, exact refinement, and Lean prime proof and the repository-wide verification matrix are queued. I will add their final results here.

Local preflight checked the exact rational certificate, both moment engines, the backoff decomposition, finite cutoffs, all 47 constraints and seven margins, package and source hashes, Python syntax, and git diff --check. The expensive eight-stage parent replay is delegated to the Draft CI rather than repeated locally.

Attribution, licence, and limits

Apache-2.0. The physical construction is stacked from eumemic's PR244 retiming and PR210 source527/bank work. The fixed-prime method and byte-pinned Prime.lean come from Gabriele Nespoli's PR235. Rohan Arun's PR243 introduced the eta = 10^-24 backoff; its Anthropic Claude disclosure is retained. Existing notices for earlier suppliers and contributors remain in the pinned parent package. This transfer and its verification package were prepared by OpenAI Codex with substantial AI assistance; these credits do not imply endorsement by upstream contributors.

The inherited all-size compiler, weighted/common-ancestor chart, restored-row, selector, routing, exact-tape, prime-supply, precision/recovery, complex symbolic-correctness, ordinary-leaf, and analytic-transfer interfaces remain assumptions. CI checks the finite construction and certificate; it does not discharge those theorem interfaces or establish an unconditional bound.

[sennemmi]

Archiving this Draft as part of the research closeout. It contributed an exact fixed-prime refinement on PR #244, moving the conditional value from 710346589668175/10^18 to 710346593567535/10^18. The full research-artifact, fixed-prime, terminal and recycled-bit workflows passed at head 6e97365e5e2952eb8d9d5cb15b7004cd3b18583f; PR #244 remains open. Its method and certificate stay preserved here. Later comparable work, including #261, superseded this value; no further maintenance is planned.