Conditional κ=710465204542247142217249/10^27: eight-level refinement of #254

[sennemmi]

Result

This is an arithmetic refinement of the 132-pair physical construction in #254. The word, frames, banks, routes, roles, replicas, and physical invoice are unchanged.

The candidate is the exact rational

κ = 710465204542247142217249 / 10^27

It improves #254 by 3909417142217249/10^27. The latest public comparable value is #255, which reports 710465204542247142217/10^24; this candidate is strictly higher by 249/10^27 (about 2.49e-22). This is a small arithmetic endpoint improvement, not a new physical DAG saving.

Mathematical change

For the actual m = 120 profile, fix q = 2^127 - 1 and replace the inherited rare-class envelope with the exact density rho = 2m^3/q = 3456000/q, while retaining the full 32m^2 fallback for every rare child. Extend the ordinary finite recurrence to eight completed levels and use eta = 10^-24, beta = 10^-9. The verifier checks both independent rational interval engines, all eight finite cutoff inequalities, all 47 strict outer constraints and seven margins, and rejection of the adjacent 10^-27 endpoint.

The counted finite coefficient remains 62447396383377201 < 2^80 < q. Selectors, charts, routes, copies, replicas, bridges, and restoration remain charged.

Reproduction and CI

From the repository root:

python3 -m pip install -r research/cohort-kernel-entrances132-fixed-prime/requirements.txt
python3 -B research/cohort-kernel-entrances132-fixed-prime/verify.py --output /tmp/cohort-pr254-fixed-prime

The wrapper regenerates and checks PR254's source witness with all seven native checkers, then recomputes this exact certificate; it does not accept stored PASS receipts as proof. The added Actions workflow repeats that command and kernel-checks the fixed-prime Lean file. The repository verification workflow also runs its full Python-version matrix, finite/certificate groups, and formal jobs. Those Actions runs are in progress; this Draft makes no claim that they have passed yet.

Conditions and limits

The result inherits PR249/PR254's conditional interfaces: the all-size weighted compiler, common weighted chart, restored-row bound, selectors and routing, prime supply, precision/recovery, complex symbolic correctness, and analytic reduction. The physical witness and arithmetic are finitely replayed, but this is not an unconditional multiplication theorem or a practical-runtime claim. Prime.lean certifies the prime and density facts only.

Attribution

The physical witness and its packaging are Dugongue's #254 contribution. The fixed-prime density and finite-level method originate in Gabriele Nespoli's #235 work; the related #252 adaptation is Rohan Arun's work. hvaliconAI's #255 is credited as the parallel seven-level refinement; this candidate extends the exact endpoint to eight levels. The retained complex saving comes from the source527 math_check.py; its full PR234/PR193, source527 contributor, Apache-2.0, and Claude-assistance notices remain in research/source527-five-stage-crossover/references/source527/NOTICE.md. This extension's exact profile recomputation, eight-level certificate, and CI integration were prepared by sennemmi with substantial OpenAI Codex assistance. The package is Apache-2.0 and retains the inherited notices and source hashes. Credits do not imply endorsement.

[sennemmi]

Archiving this Draft as part of the research closeout. It contributed an eight-level fixed-prime refinement on PR #254 with conditional κ 710465204542247142217249/10^27. The dedicated refinement, terminal and recycled-bit workflows passed at head 83a421d10da96af35bfad95310b55ce1291d5109; the root research-artifact workflow was cancelled. PR #254 remains open. The branch and certificates are retained. Later comparable work, including #261, superseded this value; no further maintenance is planned.