Conditional κ = 7.08582414684555347682e-4: fixed prime and finite leaves on PR237 banks

[gabriele-nespoli]

PR237 pays the full rare-case fallback using the conservative density 10^-16. Its retained proof gives the sharper bound 2m^3/q for any eligible fixed prime. Choose the classical prime q=2^127−1, keep every fallback call and all prime-dependent costs, and complete seven finite ordinary-leaf levels instead of three. The resulting conditional exponent is

κ = 354291207342277673841 / 500000000000000000000000 = 7.08582414684555347682 × 10⁻⁴.

Construction	Conditional κ
Reviewed main, 3b6b668, PR186	6.61885549259598 × 10⁻⁴
PR234, af3fe331	7.03701743496697 × 10⁻⁴
PR237 completed width-120 banks, a8b0ce21	7.08582410781108 × 10⁻⁴
This fixed-prime, seven-level refinement	7.08582414684555347682 × 10⁻⁴

The exact gain over PR237 is 1951723673841 / 500000000000000000000000 = 3.903447347682 × 10^-12. The new κ also exceeds PR237’s unrounded outer bound, so this is more than a reporting refinement. Higher bank-ladder pricing targets such as PR232 are not claimed to be completed witnesses or disproved here. No global optimality claim is made.

Mathematical change and paid costs

• For m=120, use the positive exact density rho=3456000/170141183460469231731687303715884105727. Add the full 32m² rank-one fallback for every rare edge; do not subtract the displaced good case.
• The unchanged banked profile has W=261659, E=5916300, rank mass 31346280, deficit 52800 and maximum child rank 50. All center-copy calls, charts, 177179 banks, the proper 60-colouring and selector bill 12227329044 < 2^40 remain paid.
• Two independent exact rational moment engines admit coarse saving 709084859748818674574370/10^27 and reject its adjacent point. The old 10^-16 density fails even the lower moment bound at this new saving.
• Seven acyclic completed levels satisfy a_j=(1-c)c+c a_(j-1), starting at the unchanged a0=384599/10^10. Each ordinary callback uses only the completed predecessor; every cutoff gap is positive. Unknown primitive constants remain quantified. No coarse limiting oracle or width-dependent bootstrap depth is used.
• Retain the complex supplier, finite bridge, eta=10^-12, beta=10^-9 and computational model. Check all 47 strict outer inequalities and seven margins. Reject the next 10^-24 κ point. Seven levels attain this paid family’s grid cap and improve on six by 85×10^-24.
• The large prime produces very large paid constants. An explicit overcharge retains twelve copies of the original bill, every original routing family charged against the entire enlarged stock, and extra high-affine/stock/wrapper/matrix allowances for every bank selector. Its coefficient is below 2^80<q, giving B_bank(q)<q^14401. Prime-dependent initialization, table scans, routing, binary-padding factor below q², row stock and restoration stay in the retained constant/toll accounting.

The full argument and dependency limits are in research/fixed-prime-bootstrap/PROOF.md. It also notes PR237’s prose-only removed-rank-mass typo (226200); the actual removed mass is 225370, and its executable certificate already uses the correct 31346280 profile.

Reproduction and validation

The package pins every byte of PR234 and PR237’s bank package and their exact commits. Its default verifier runs all six PR234 stages afresh, then recomputes PR237’s charts, inverse factors, bank incidence/colouring, profile, selector bill and original arithmetic before deriving the refinement. Saved PASS receipts are not inputs. See the package README for checkout commands.

python -m pip install -r research/fixed-prime-bootstrap/requirements.txt
python -B research/fixed-prime-bootstrap/verify.py --parent-root /path/to/pinned-pr234 --banks-root /path/to/pinned-pr237
make paired-cube-verify

The separate pinned Lean 4.24.0/mathlib project kernel-checks the primality/value/size of 2^127−1 and the strict positive density improvement. Its reproduction commands use lake update, lake exe cache get Mathlib.NumberTheory.LucasLehmer and lake env lean Prime.lean. No sorry or native_decide is used; printed theorem axioms are only propext, Classical.choice, Quot.sound. Python independently reproduces all 125 Lucas–Lehmer updates.

Local validation: fresh source replay, two independent moment enclosures, finite admission/outer47, negative controls, Lean prime/density proof, and make paired-cube-verify passed. All 16 existing main verification groups were also completed in this session. Native aggregate make verify initially stopped at Apple Clang’s unsupported OpenMP option; the remaining groups were run with a task-local serial adapter that only removes that flag for the pinned independent-case enumeration source, then with official Boost 1.88 headers for the later multiprecision checks. All 4073300 enumeration cases were retained; no existing source or assertion was changed. This does not represent an unmodified native aggregate make verify exit status of zero.

The added CI checks the full exact replay on Python 3.11/3.13/3.14 and the separate Lean prime proof. GitHub checks are distinct from the local evidence and may await maintainer workflow approval.

Scope and attribution

This is a conditional finite construction under the same weighted q-local compiler, complete-stream movement, unit-pivot fallback, completed ordinary leaves, prime supply, complex symbolic identities/NetCert, Clifford synthesis, precision, restored-row and all-size recovery/reduction interfaces retained by PR234/237. The new Lean proof certifies prime eligibility and density only; it does not formalize the entire integer multiplication theorem or rebuild the full inherited complex Lean chain.

Credit: hcg890’s source-bound five-stage package; Jacob Sussman’s complex five-stage geometry; Rohan Arun’s PR237 banks and completed bootstrap arguments; Evan McKinney’s completed-bank endpoint; eumemic’s bit helper/gauges; Chafik Boukhalfa’s physical construction; Avi Eisenberg / ikeboy’s complex helper; mathlib’s Lucas–Lehmer contributors; all earlier contributors credited by the pinned notices. Apache-2.0 and original copied arithmetic headers are preserved.

Prepared by Gabriele Nespoli with OpenAI Codex assistance in research, implementation, proof exposition and validation. No human mathematical review or contributor endorsement is represented as completed.

[Mister-Meeseeks]

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

Contribution: Fixed prime and finite ordinary levels on the five-stage construction.

Review and limits: Prior exact arithmetic replay supplemented by source-bound original full-driver and Lean CI logs. Historical CI is not a new local Lean rebuild.

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: 63e3e9e2f852c72634d4b09d0bbb8fcf1081481d. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.