Conditional κ = 0.000710433690953069816941696: fixed-prime refinement on PR251

[sennemmi]

Draft refinement stacked on PR #251's unmerged physical construction at 2b030c06a811473ae8ed49dce07bc7ea50b72e9e. The 37 transported entrances, physical word, frames, roles, lifetimes, routes, and charged child count remain unchanged.

Exact result

The refinement certifies

kappa = 5550263210570857944857 / 7812500000000000000000000
      = 710433690953069816941696 / 10^27

This exceeds PR #251's exact 710433687047883/10^18 by exactly 3905186816941696/10^27. PR #252's exact value is lower. The bit supplier remains binding; this is an exponent improvement through a sharper conditional arithmetic bound on the existing physical witness, not a new DAG cost reduction.

Mathematical mechanism

• Fix q = 2^127 - 1; for PR κ = 7.10433687047883e-4: expand transported entrances with shared frame intersections #251's literal m = 120 profile, replace the old rare-class envelope by the proved density 2m^3/q = 3456000/q < 10^-16.
• Keep the full 32m^2 fallback charge for every rare child. The exact finite coefficient remains 37443766921470961 < 2^80 < q.
• Recompute the coarse bound with the public moment engine and an independent rational atanh engine; both reject the adjacent 10^-27 grid point.
• Extend the finite ordinary recurrence to eight levels and set only the outer backoff to eta = 10^-24 (beta = 10^-9 unchanged). Recheck all 47 strict assembly inequalities and seven margins; the next final grid point fails.

The complete numbers, intervals, cutoffs, and comparison are pinned in research/five-stage-source527-37-entrances-fixed-prime/expected.json and recomputed by the verifier.

Physical witness and reproduction

The verifier replays PR #251's eight mandatory source stages from its pinned commit, checks all 20,107 formal F2 columns in both directions, and confirms dirty restoration, reflected frame ledgers, entrance transport, replicas, and bank assignments before applying the arithmetic refinement.

python3 -m pip install -r research/five-stage-source527-37-entrances-fixed-prime/requirements.txt
python3 -B research/five-stage-source527-37-entrances-fixed-prime/verify.py --output /tmp/source527-pr251-fixed-prime

The output path must be a fresh directory outside the checkout. The custom workflow also kernel-checks the prime and density facts in Prime.lean; that Lean check does not prove the multiplication theorem.

CI coverage

The existing verify.yml workflow runs all 16 make verify-* groups, including the test group, across Python 3.11, 3.13, and 3.14, plus the certificate, immutable-offline, ranked-pair-finite, Gaussian, historical, and matrix formal checks. The new path-scoped workflow adds the full PR #251 replay and fixed-prime certificate check plus the Lean prime-kernel check. This PR remains Draft until these cloud checks finish and the public frontier is rechecked.

Conditional scope and attribution

This retains PR #251 / #244 / #234's all-size weighted compiler, common weighted chart, restored-row bound, selector and bank interfaces, exact-tape routing, prime-supply assumptions, precision/recovery interface, complex symbolic correctness, and analytic reduction. It is not an unconditional multiplication theorem and does not establish practical runtime.

Apache-2.0. The 37-entrance physical construction remains eumemic's PR #251 work. The fixed-prime and finite-level method originates in Gabriele Nespoli's PR #235 and was adapted to PR #249 in Rohan Arun's PR #252. Original notices and AI disclosures remain in the inherited source. This PR's adaptation and certificate were prepared by sennemmi with substantial OpenAI Codex assistance; no authorship of the inherited physical construction is claimed.

[sennemmi]

Archiving this Draft as part of the research closeout. It contributed an exact fixed-prime refinement on PR #251 with conditional κ 710433690953069816941696/10^27. The full research-artifact, refinement, terminal and recycled-bit workflows passed at head 7eff1c4971882c0d4b67e9c4f5eb3e642adc44ad; PR #251 remains open. The branch and certificate are retained. Later comparable work, including #261, superseded this value, so no further maintenance is planned.