Conditional kappa refinement: 500-digit enclosures and complete native-proof reuse

[M6LI]

Exact conditional saving:

kappa = 0.001679754956125544521335670993920702576998311438502580295460327520162864548260325638826305656132844632274451316908586220972070448120165412142771300636692956715734962527390622927714845418099553756337112475558813976784140553038674557881956561879800689254171219767487483694675693056982788922757319255565606685566411725238449650999339827075715283714270157705762309226038611771955655377501414652136546310881185746917580802513640545967852810801680966856133978576162943792322682482152988532923811998938861379689883167389728443314312230815059559906903773756886872476136837071871203030423643505345473735135533

At submission (2026-10-14 13:30:03 UTC), this exceeded source-pinned PR1875 (d31827d128197345be2d8d5649f5103abb2759ba) by approximately 6.066855666120612e-301, and PR1876's400-digit point by approximately 8.118574691758080e-401. PR1876 now reports a full finite replay at 6a442373e6127b7e0985d277726bcfd9a16a1f36, with that same exact point. PR1904 appeared afterward (13:38:26 UTC, head 091ba6e64169230d13285d12563390ea76f44208) reporting a higher1000-digit point by approximately 2.003359509912251e-500, with native reuse; this PR therefore makes no current-leading claim. These are extremely small arithmetic refinements of the same paid construction, with no practical speedup claimed. Exact rational comparisons use primary source files.

The package preserves PR1800 (46d865ab92e58098538b8c5348b6a131c99abf45) byte-for-byte under parent/. Its matching-profile600-digit endpoint operand comes from PR1789 (2f5f9dbb8b8ce9f9a5f90b234e4b03ca3f4ba87b). That operand is only a nomination: two separately implemented directed rational/fixed-point series enclosures prove the accepted point on the actually admitted native profile. Both reject an above-root point and a one-count rank48 mutation.

The endpoint gaps, r0 and layout slack are 10^-500, epsilon is 1-2*10^-500, beta is 1-6*10^-500, stop is 10^-510, and the final rational uses a 10^600 floor and 10^-550 backoff. All slacks remain positive; epsilon=1 is rejected. The main row still limits this family to the unchanged supplier's moment root.

Reproduce with Python3.12+, NumPy2.3.5, SymPy1.14.0 and mpmath1.3.0:

python -B research/fine-prime-slack500/verify_refinement.py --main "$PWD" --work /absolute/absent/work --jobs 2 --timeout-seconds 1800

The portability adapter preserves every frozen parent byte and runs the unchanged parent checker code in an authenticated execution copy. Only CRLF/LF bytes in the two generated parent JSON ledgers and their two manifest digest entries may change. Full-byte comparisons retain every other byte, including mathematical values, JSON types, ordering and whitespace. Both original and execution inventories are checked before and after. The refined ledgers use the same strict newline-only comparison. Format/mathematical-payload mutation controls run with normal and optimized Python. Runtime portability receipts and both regenerated ledgers are retained as CI artifacts.

Default verification is cold and does not reuse native results. The dedicated workflow runs this command on Python3.12 and3.13, requires all committed sources unchanged, and preserves proof receipts and logs. Hosted CI must be checked at the final published head; local PASS is not a hosted-CI claim.

Local evidence was obtained in two source-bound stages. First, the complete fresh native parent and an earlier105-slack point passed in 1181.734seconds on Windows/CPython3.12.14, with all397 inputs unchanged, four complete complex native word chains, owner reconstruction, invoice,47 supplier inequalities and all required control chains. A separate AI-agent audit authenticated that actual execution. The updated500-point verification then passed in 8.282seconds, including new57 strict inequalities,17 reason-matched controls, both new actual-profile enclosures and the exact broad-shift guard. It reused that finite native evidence only after authenticating every one of397 baseline source files and1199 execution files, before and after the new checks, under the hard-pinned reviewed seal. This reuse is explicit in the receipt; it is not described as a second native replay. The fresh run had an8GiB aggregate Job Object limit/four-CPU affinity; the new stage had2GiB/two CPUs and used about69.5MB. All400 final package files remained unchanged.

The independent Gaussian guard repairs an inherited vacuous or True row: the shifted constant is d^2-d^2/L, not the inherited plus sign. Three exact identities and four mathematical mutation controls pass. Parent bytes and notices remain frozen. The bit supplier is PR589 cited unchanged and repriced, not a fresh rebuild of its bit circuit.

The conclusion remains conditional on the inherited compiler, common-label/mutable-source realization, whole-call hosting, semantic precision, fixed-tape/prime-packing, interchange, recovery, analytic transfer, bit H1-H5, fine-prime padding, gauge-free accretive circuit and union-schedule arguments. Finite arithmetic does not prove those uniform statements, a full Lean multiplication theorem, linear time or an unconditional record. The maintainer-selected result is unchanged. Open next questions and exact reuse invalidation are documented in the package README.

Credits: Jeremy Kothe and PR1800/1839 contributors; Rohan Arun and the concurrent PR1828/1875/1876 precision work; midhatn/PR1789's matching-profile operand; Chafik Boukhalfa/PR1786 and Tim Trudgian's prime theorem; Vlado Vojdanovski/PR1673; PR1758 and the complete retained supplier lineage. New code, independent AI-agent enclosures and reviews used substantial OpenAI Codex assistance. No independent human review is claimed. Apache-2.0; original notices are retained.

The source-bound local verification summary is committed at certificates/fine-prime-slack500-local.json; the separately reviewed actual500 closure has SHA256 0dd6face3587680ec64527c533f5a039f883af4961a4b6c4ceea1f73ab0349b1.

The current402-file portability revision passed a further source-bound integration in 16.281seconds, with the complete prior native-proof reuse explicit and all402 source hashes unchanged. Both original hosted Python3.12/3.13 runs passed their12 native stages but stopped at the inherited ledger CRLF/LF comparison; those failures are retained. The corrected dedicated cold CI run38576700891 passed on Python3.12 and3.13 at exact head fdf717a838bb3d4c78396474131bd1ab5f18165d. Both jobs rebuilt four distinct native word chains, passed all12 native stages, the500-point arithmetic and independent enclosures,57 strict inequalities,17 rejecting controls, broad-shift checks and normal/optimized format controls, and required committed sources unchanged. Both complete uploaded proof-artifact subsets were independently audited; bit evidence remains cited/repriced and all-size hypotheses remain. The committed local certificates precede this CI result; the linked run provides the current hosted evidence. Independent portability and actual-integration reviews have SHA256 ae1ddc2a8ca4a4a7b54bb42ca3abfb7e2e4a411363439bb70a7daf06f101915b and ba70010efffc911fab899a14a53908e9ea3bec4f20e309cc24a1022feed9edeb; source-bound summary: certificates/fine-prime-slack500-portability.json.

Clarification of finite cutoff and proof scope — 2026-10-14

The parent's recorded bootstrap and conservative_bootstrap cutoff_log2 values at PAYLOAD=2048 are fixed-allowance arithmetic samples, not certified thresholds for the actual algorithms including carried constants. Our source-bound local AI-agent reviews find that 58 of the 60 samples fail the written minimal-carry test. This does not change the numerical κ calculation or disprove the conditional asymptotic bound. The finding follows Rohan Arun's original PR387 proof mirror, pinned at 0b1cccf8 and Josie Elliston's PR1956 audit.

Under the inherited uniform all-size, completed-algorithm, stopped-construction, stock-fit and paid compiler premises, stage-dependent dominating constants and sufficiently large finite cutoffs can instead be chosen sequentially: fix the previous algorithm, then the stopped procedure and its uniform thresholds, then the current cutoff, then the next completed time constant. This preserves the same fixed-depth exponent recurrence; it does not certify the stored samples, assert actual 2^2048 bounds, or supply a numerical cutoff certificate. Josie Elliston's PR1965 discusses a separate stock/time absorption repair.

PR1964-based extended-label checks pass for both providers and orientations and reject scalar-neutral X/Y/Z adapter controls; H2 and all-size obligations remain inherited. These are AI-agent reviews, not independent human peer review. This clarification supplies no new theorem, frontier result, source change or native replay.

[Mister-Meeseeks]

Review follow-up for frozen head fdf717a838bb3d4c78396474131bd1ab5f18165d.

This remains open for a review of whole-head provenance, native-reuse binding, directed enclosures and the outer proof chain. In particular, reconcile the broad-shift guard correction with the inherited outer argument and check parent/source agreement. No independent full replay of this release was completed in this tranche.

Recorded scope, attribution and execution evidence. We are leaving this open with the outstanding work recorded; previously completed checks remain credited. Thank you for the contribution.

[philippeVV]

This is independent evidence for the review item "reconcile the broad-shift guard correction with the inherited outer argument" (6174289936). The check is exact (sympy 1.13.3 plus Fractions, no imports from this PR) and was run against the latest head carrying this line, PR2044 322949a5. Package: research/t74-broad-shift-reconcile/. It prints PASS_BROAD_SHIFT_RECONCILE in about 3.5 minutes.

• The disabled or True in check_gaugefree.py:23 is in a branch the proof does not use.
  • It tests s = −δ/λ, while the proof uses s = −δ(1−θ), whose assertions are live.
  • Corrected, the inherited expression's residual is −2bδ(2θy − δ(1+θ)²)/(1+θ), which matches the Conditional kappa: verified 400-digit refinement over PR1875 (+6.06686e-301) #1876/Conditional kappa refinement: 500-digit enclosures and complete native-proof reuse #1897 correction.
• The inherited |δ| ≤ 1/2 holds, and more strongly: δ = β′_m/ρ, so |δ| ≤ 1/(2ρ). With the proof's shift, 1 − 2|s| ≥ 2θ/(1+θ) ≥ θ.
• Exact grid:
  • Coverage: θ = p/q with q ≤ 40, c ∈ (1/12)ℤ ∩ [−2,2), every phase cell, every dyadic block and the centre anchor. That is 287,256 blocks and 7,568,034 exact entries.
  • The §5 bound θB²/4 + 4θB holds everywhere. The largest ratio is 1120/1681.
• Two wording errors in GAUGEFREE-INTERIORS.md. Neither affects the conclusion.
  • :32 says |s| "can exceed 1/2 by θ/2". In fact |s| = |β′_m| ≤ 1/2, with equality attained at β′ = −1/2 (φ(π) = 0), so the shrink is still needed.
  • :31 says the "linear leftover … ≤ 2 b0 θ B (total)", and :84 has the analogous 2 b0 θ B'. This is false. Counterexample: θ = 1/40, c = −19/12, B = 8, m = −48, δ = 203/492 gives 1421/3362·b0 ≈ 0.423·b0 > 0.4·b0. The stated coefficient bounds only give 3 b0 θ B. §5 charges 4 b0 θ B, so the proof is unaffected.
• Side check: enabling the second disabled assertion, struct_check.py:130 (far blocks zero), in a local copy passes exactly.

The other obligations (supplier admission, outer schedule, all-size, adapters, rounding, native reuse) are not addressed here. No κ claim.