Conditional κ = 7.10346593569666e-4: fixed prime, seven finite levels and η = 10⁻²⁴ on #244's retimed source527 banks

[rohanarun]

Conditional κ = 355173296784832805399/(5·10^23) = 7.10346593569665610798e-4, +3.90e-12 over #244 (7.10346589668175e-4). Stacked on #244.

Change and mathematical scope

#244's retimed source527 word, banks and finite bill are 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 bisected on the 10⁻²⁷ grid; the old density fails at the new point.
• Seven finite ordinary levels instead of three, from the retained 384599/10^10, each with κ = 7.10346589668175e-4: retime source-pair additions to their delivery frames #244's cutoff rule at its fully charged q-power coefficient (< 2⁸⁰ < q).
• Outer-47 backoff η = 10⁻²⁴ instead of 10⁻¹², with β = 10⁻⁹ unchanged.

Hypotheses are exactly #244's; no construction changes. The large fixed prime makes initialization and table constants enormous but finite and paid, as argued in #235's PROOF.md.

Validation

• verify.py first runs κ = 7.10346589668175e-4: retime source-pair additions to their delivery frames #244's own immutable verifier from the same checkout (all eight mandatory stages and package integrity) and requires its κ.
• Coarse bisection bracketed by κ = 7.10346589668175e-4: retime source-pair additions to their delivery frames #244's engine and the independent atanh engine (base_two_moment.py).
• All 47 strict inequalities and 7 margins recomputed at η = 10⁻²⁴; the next 10⁻²⁴ κ grid point is rejected.
• Regression: with η = 10⁻¹² and three levels it reproduces κ = 7.10346589668175e-4: retime source-pair additions to their delivery frames #244's published κ exactly.
• Prime.lean kernel-checks q (workflow job prime-kernel). The paid-constant argument follows Conditional κ = 7.08582414684555347682e-4: fixed prime and finite leaves on PR237 banks #235's PROOF.md (proof argument, not a finite test).

python3 -m pip install -r research/five-stage-source527-retimed-fixed-prime/requirements.txt
python3 -B research/five-stage-source527-retimed-fixed-prime/verify.py --output /tmp/pr244-run

Full local replay: PASS (all eight #244 stages, package integrity, then PASS kappa=355173296784832805399/500000000000000000000000), about 11 minutes on 2 cores.

Credits

• eumemic (κ = 7.10346589668175e-4: retime source-pair additions to their delivery frames #244, κ = 7.10069340338651e-4: five-stage banks and F₂ movement fusion #210): the retimed source527 construction and verifier, 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).
• Gabriele Nespoli (Conditional κ = 7.08582414684555347682e-4: fixed prime and finite leaves on PR237 banks #235): the fixed prime, the seven levels and Prime.lean.
• hcg890 and Jacob Sussman (Conditional κ ≈ 7.037017435 × 10⁻⁴ > 2⁻¹¹: five-stage bit and complex suppliers #234); 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 seven-level transfer on PR244.

Review and limits: Source-bound complete Linux replay and arithmetic. Exact unchanged physical construction, paid rare-class fallback and strict assembly.

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: e56591cbf0b18b6d0f959f3742c05c7d8cc41f75. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.