Fixed-prime eight-level gen5 descent: conditional κ = 744728654228590183370616/10^27

[jacklightChen]

The actual PR291 supplier admits the slightly larger conditional bound

κ = 744728654228590183370616/10^27 = 0.000744728654228590183370616.

This is 4006913183370616/10^27 ≈ 4.0069×10⁻¹² above #291 (744728650221677/10^18). It transfers the existing fixed-prime and eight-finite-level methods to the current collective/descent costs. It is a small arithmetic refinement, not a new coding mechanism or a crossing of 2⁻¹⁰. The bit side still limits the bound. No experimental structural gain is included.

One self-contained package, research/gen5-descent-fixed-prime/, added on current main; no existing project file changes. All139 original Git blobs at PR291 77d92c521e442c802b6fb4717bd8d066971a4b82, source proofs, notices and licenses are preserved.

Complete paid transfer

• Actual60 physical replicas; stock1,244,515;545charts and4,765,800bank assignments. Normalization byfive does not change the copy count.
• Literal28,510,800children/rank149,077,800; normalized m120/W248903/rank29,815,560/deficit52,800/maxchild50. Both cleanup and arbitrary dirty old-value return remain paid.
• Normalizer815 and selector932,937,188,400<2^40; full current 90505949107835401 coefficient/q^14401 plus inherited primitive, table, wrapper and preceding-level constants.
• q=2^127−1, rho3456000/q, the entire32m² rank-one fallback, two exact interval moment engines, eight finite acyclic ordinary levels, eta10⁻²⁴/beta10⁻⁹,47strict constraints/seven margins. Adjacent coarse/final10⁻²⁷ points and wrong stock/copy/selector substitutions reject.

Actual local validation

The PR287 parent supplier was locally validated (368.141574s); PR290's new450-entry kernel (14.874811s), global geometry/complete bank and mathematical continuation (84.053883s) were then validated. For PR291, the exact accepted PR290 word equals its fixed input hash. The new89-frame descent, full physical/source/COPY/annihilator paths, new Gram witnesses, every five-stage instruction-to-bank binding and full new fee/finite assembly passed in48.899878s. Unchanged scalar/COPY projection, signed decoder, complex dependencies and bank geometry/assignments were reused by exact input identity; not rerun. Current-cost prime/eight-level assembly passed in0.820591s.

An initial local metadata comparison failed before new descent mathematics because in-memory integer keys differed from JSON string keys; normalization fixed it. Original source/logs are preserved locally. No accepted mathematical result was invalidated or successful supplier replayed for that fix.

The package hash/syntax preflight passes. The default verifier below runs the complete original supplier, recomputes all125Lucas–Lehmer steps and compares the complete exact mathematical certificate. This standalone fresh driver is supplied for reproduction; local claims above are the actual incremental runs, not a claim that the full driver was redundantly run.

python3 -m pip install -r research/gen5-descent-fixed-prime/source/requirements.txt
python3 -B research/gen5-descent-fixed-prime/verify.py --output /tmp/gen5-descent-prime-fresh

Python3.11+, SymPy1.14.0, assertions on, a new output directory, no network after dependencies. --check-package checks only hashes/syntax.

All inherited all-size compiler/Clifford/tensor, weighted common-chart, restored-row, local-ring fallback, routing, prime, precision/recovery, complex symbolic and analytic/fixed-tape interfaces remain assumptions. No new Lean/native/fullglobalcomplex or dense all-size phase proof; no unconditional theorem or practical runtime claim.

Credits: Rohan Arun (#291/#287/#243), Chafik Boukhalfa (#290/#284/#282), DreamingOfClouds (#285/#276), eumemic, Dugongue, hcg890, Jacob Sussman, Evan McKinney and the retained supplier lineage; Gabriele Nespoli (#235), Rohan Arun (#243) and sennemmi (#253) for fixed-prime/finite-level methods. Zhihao Chen (jacklightChen), with OpenAI Codex assistance, contributed exact delta admission, current-cost transfer, negative controls, source closure and reproducible packaging. Original assistance disclosures, all NOTICE/LICENSE files and historical acknowledgments remain. No exclusive priority claim for the arithmetic method.

[Mister-Meeseeks]

Thank you @jacklightChen. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Fixed-prime eight-level transfer on PR291.

Review and limits: Fresh complete supplier and fixed-prime eight-level replay passed in 576s, including both interval engines, prime admissibility, finite invoice, strict margins and rejected adjacent grid points. This includes the parent replay; no new Lean build claimed.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: e65a690f87bdade17b361f9c66554861bb85a5c1. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.