Conditional κ = 7.09424410568126e-4: fixed prime, seven finite levels and η = 10⁻²⁴ on #240's five-stage banks

[rohanarun]

Conditional κ = 709424410568125629699/10^24 = 7.09424410568125629699e-4, +3.91e-12 over #240 (7.09424406659975e-4). Stacked on #240.

Change and mathematical scope

#240's banked 493-source five-stage word is unchanged. Only three fixed arithmetic parameters change:

• Fixed prime. Conditional κ = 7.08582414684555347682e-4: fixed prime and finite leaves on PR237 banks #235's q = 2^127−1. The proved rare-class density 2m³/q replaces the 10⁻¹⁶ envelope, and the full 32m² fallback is still paid. The coarse saving is 177482012714287685545103/(2.5·10^26) on the 10⁻²⁷ grid. The old density fails at this point.
• Seven finite ordinary levels instead of three, starting from the retained 384599/10^10, with PR234's cutoff checks.
• Outer-47 backoff η = 10⁻²⁴ instead of 10⁻¹², with β = 10⁻⁹ unchanged. This alone adds 2.1e-15.

The paid q-power constant is recomputed with #240's fresh finite-bill values: 62,150,081,742,953,667,360 < 2⁸⁰ < q.

Hypotheses are exactly #234's and #240's, and the construction does not change. The large fixed prime makes initialization and table constants enormous, but they stay finite and paid, as argued in #235's PROOF.md.

Validation

Finite exact checks, all run by verify.py:

• It first re-runs Conditional κ = 7.09424406659975e-4: five-stage completed banks with #210’s 493-source helper #240's own verifier from the same checkout (493 helper, telescope, five-stage lowering, geometry, primes, banks, 47 constraints): PASS.
• The coarse 10⁻²⁷ bisection is bracketed by PR234's engine and an independent atanh engine (base_two_moment.py).
• All 47 strict inequalities and the 7 margins of PR234's unchanged assembly are recomputed at η = 10⁻²⁴ and still hold. The next 10⁻²⁴ κ grid point is rejected.
• Regression: with η = 10⁻¹² and three levels, the verifier reproduces Conditional κ = 7.09424406659975e-4: five-stage completed banks with #210’s 493-source helper #240's published κ exactly.
• certificate.json stores the key values and the SHA-256 of the complete recomputed record. --dump PATH writes out the full record.

Formal check: Prime.lean kernel-checks the Lucas–Lehmer primality of q (the workflow includes #235's Lean prime-kernel job).

Proof argument, not a finite test: the paid finite constants follow #235's PROOF.md.

Reproduce (about 5 minutes; --full also runs PR234's fresh verifier):

git worktree add --detach ../pr210 13311491eb74a3a9a443ba4e318c68c263f064f0
git worktree add --detach ../pr234 af3fe331ca60c936229e060681ffccbc1c208678
python3 -m pip install sympy==1.14.0
python3 -B research/five-stage-493-fixed-prime/verify.py --pr210-root ../pr210 --pr234-root ../pr234 [--full]

Credits

• Gabriele Nespoli (Conditional κ = 7.08582414684555347682e-4: fixed prime and finite leaves on PR237 banks #235): the fixed prime, the seven levels, Prime.lean and base_two_moment.py, applied here to Conditional κ = 7.09424406659975e-4: five-stage completed banks with #210’s 493-source helper #240.
• hcg890 and Jacob Sussman (Conditional κ ≈ 7.037017435 × 10⁻⁴ > 2⁻¹¹: five-stage bit and complex suppliers #234).
• eumemic (κ = 7.10069340338651e-4: five-stage banks and F₂ movement fusion #210), with Dugongue (Conditional kappa 6.838643131376e-4: composed cascade frames, global cuts and four finite bootstraps #216) and sennemmi (κ = 6.94787513005285e-4: plateau retiming on PR210's 471-source word #230).
• Evan McKinney (Pack #187's bit residuals: conditional κ = 0.000676080316519385 #197).
• Chafik Boukhalfa (A stronger bit supplier: fixed-coordinate and face-diagonal bit local circuit on PR168 v4 (effective bit saving 6.7734e-4) #200).
• Avi Eisenberg (Pair donors to avoid parity erasure on the source-assisted v4 word: conditional κ = 6.647872e-4 #193).

Prepared by Rohan Arun with Anthropic Claude assistance.

🤖 Generated with Claude Code

https://claude.ai/code/session_01F5n4jqR72j3QDY71gGBVb1

[Mister-Meeseeks]

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

Contribution: Fixed prime and seven finite levels on PR240.

Review and limits: Source-bound Linux complete driver and prime-proof workflow. Parameter improvement on the same physical suppliers, with adjacent-grid controls and inherited interfaces.

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: ab482ea45dd73f0cb80796641528f0ab930c13b5. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.