Conditional κ = 7.10392190350878e-4: fixed prime, seven finite levels and η = 10⁻²⁴ on #249's source527 banks

[rohanarun]

Conditional κ = 142078438070175647403/(2·10^23) = 7.10392190350878237015e-4, +3.90e-12 over #249 (7.10392186446856e-4). Stacked on #249.

Change and mathematical scope

#249's source527 word (with its 21 transported entrances), 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 = 7.10392186446856e-4: transport dirty compensations to 21 entrance frames #249'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 #249'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 = 7.10392186446856e-4: transport dirty compensations to 21 entrance frames #249's own immutable verifier from the same checkout (all eight mandatory stages and package integrity) and requires its κ.
• Coarse bisection bracketed by κ = 7.10392186446856e-4: transport dirty compensations to 21 entrance frames #249'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 = 7.10392186446856e-4: transport dirty compensations to 21 entrance frames #249'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-entrances-fixed-prime/requirements.txt
python3 -B research/five-stage-source527-entrances-fixed-prime/verify.py --output /tmp/pr249-run

Full local replay: PASS (all eight #249 stages, package integrity, then PASS kappa=142078438070175647403/200000000000000000000000), about 11 minutes on 2 cores.

Credits

• eumemic (κ = 7.10392186446856e-4: transport dirty compensations to 21 entrance frames #249, 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 source527 construction, entrance transport, retiming 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

[chntnm]

Independent replay: PASS at 45121f7

I replayed this PR's documented command on a separate machine. It reproduces the stated κ. This is a finite reproduction report, not a review of the proof argument.

Result

PASS kappa=142078438070175647403/200000000000000000000000 = 0.000710392190350878237015

• The parent κ = 7.10392186446856e-4: transport dirty compensations to 21 entrance frames #249 verifier ran fresh from the same checkout. All eight stages passed (raw, bit, scalar, primes, banks, complex, math, finite), with inputs_unchanged: true and κ = 88799023305857/125000000000000000.
• Parent manifest SHA-256: 86e8bc5182079c00877077c8e4c7b2690ba2486bec385c0857f1794cd2e6bedb (131 files).
• The recomputed summary equals the committed certificate.json (no --write).
• git status showed zero changed tracked files after the run.

Conditions

Head	45121f754db92f91d1e105061125e6b1fa4c2dc0
OS	ubuntu:24.04 container (Docker on a Windows host)
Python	3.12.3, assertions enabled, sympy==1.14.0
Network	disabled (--network none), so the run is offline as described
Command	python3 -B research/five-stage-source527-entrances-fixed-prime/verify.py --output /tmp/pr252-run
Wall time	5 min 54 s total, 347 s of it in the #249 stages

What this run does not cover

1. The rare-class density. verify.py sets rho = 2m^3/q and uses it in the paid moment. Its only tie to the proof is lines 86–87, which require the strings 2m^3/q and 32m^2 to appear in three-stage-cover-bit.tex and INTEGRATED_PROOF.md. So this pass confirms the arithmetic downstream of that density. The density bound itself, and the claim that the constants for q = 2^127 − 1 are finite and paid, rest on Conditional κ = 7.08582414684555347682e-4: fixed prime and finite leaves on PR237 banks #235's written argument and still need a human reader. The PR description already says this. I note it so a reader does not take the pass as covering that step.
2. Prime.lean. I did not run the Lean job. The Python Lucas–Lehmer check of 2^127 − 1 (125 iterations) did run and pass.
3. Inherited hypotheses. I did not review the κ = 7.10392186446856e-4: transport dirty compensations to 21 entrance frames #249 / 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 construction or the Weighted dual-suffix carriers: conditional κ = 7.9961717358e-5 #109 framework assumptions.

One observation on size

The gain over #249 is 3.90e-12 absolute, about 5.5e-9 relative. It comes from three arithmetic parameters and no change to the construction, as the description states.

---

Replay and report prepared with Anthropic Claude (Claude Code) assistance.

[Mister-Meeseeks]

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

Contribution: Fixed-prime transfer on PR249's transported entrances.

Review and limits: Source-bound full Linux replay; seven finite ordinary levels and strict parameter backoffs, not an asymptotic self-bootstrap.

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: 45121f754db92f91d1e105061125e6b1fa4c2dc0. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.