Conditional κ = 0.0016797549561255445213356709939207025769983114385025802954603275201628645482603256388263056561328: recertified 1e-10000 supplier endpoint and 1e-10010 slacks (kappa +2.00336e-500 over PR1897): fine source-prime padding (PR1786) at the free epsilon -> 1 point, with gauge-free interiors and the radix-distributed union schedule on PR1673's capped phase-cell fusion outer, pivot-566+554 T74 supplier

[jeremykothe2]

Summary

Conditional κ = 0.0016797549561255445213356709939207025769…(10046 digits). Capped phase-cell fusion outer (PR1673, Vlado Vojdanovski) on this repository's T74 supplier (pivot-566 and pivot-554 helper-donor kernels, certified complex endpoint 0.00167975495612554452133567...), free strict slacks at 1e-10010 or below, with fine source-prime padding (PR1786, chafreaky / Chafik Boukhalfa, via Tim Trudgian's prime theorem; vendored) which makes the grouped chirp guard beta + 2 xi - 2 < 1, the fractional-shift gauge-free interiors with accretive shallow circuits (PR1758, outer/GAUGEFREE-INTERIORS.md), the radix-distributed union schedule with per-visit tag work, and the explicit Strassen boundary. New here: epsilon is a free parameter of the PR1786 ledger (every row holds for every 1-epsilon > 1e-10010), so the explicit point is epsilon = 1 - 2e-10010, beta = 1 - 6e-10010, giving kappa = 0.00167975495612554452133567099392070257..., 3.4e-10013 below the supplier's complex saving q, hence the ceiling of any outer of this form (outer/FINE-PADDING-COMPOSITION.md). 62 strict inequalities, 17 rejecting controls, an independent balance derivation with an epsilon sweep, PR1786's finite prime/underflow controls and the gauge-free identity checks are regenerated; the supplier's full native admission is rerun. Conditional, with all inherited all-size hypotheses and the cited prime theorem; the fine-padding lemmas (PR1786 is a draft) and the composition need independent review; asymptotic margins are 2e-10010. Endpoint recertification: native and conservative complex endpoints refined to 1e-10000 (accept at endpoint, reject at endpoint + 1e-10000) with the repaired fine engines evaluated in directed-rounding integer fixed point; kappa exceeds PR1897 by 2.00336e-500, PR1876 by 8.11857e-401, PR1875 by 6.06686e-301 and PR1839 by 6.56133e-91 (62 strict inequalities, 17 rejecting controls).

Verification

Fresh checkout of follow-up head 07e4e4231459e36b1055396306dcbd950248df62 (follow-up to the original head efb276be7206b824c23c7042046be54acc1cac4b). Manifest: 7a15370f528a86703b9ffb0391c160e5b3a0eccde0e48b06fb61cad3376ae1de.
PASS package manifest: 383 files

PASS_RELEASE in 539.88 seconds (full verify_release.py on a fresh copy of the committed package, native cache reused; the receipt binds this manifest and the unchanged kappa).

Follow-up changes (kappa byte-identical to the original head; chain exponents, CLAIM.json, new/SPEC.json unchanged):

• Carried bootstrap cutoffs (repair of Audit: selected bootstrap cutoffs omit the carried constant (κ unchanged) #1956, Josie Elliston). The earlier cutoff formula omitted the constant carried from the previous stage. Stage j now uses the least power of two L_j with L_jgap_j^2 >= 36 and L_jgap_j >= 2(4+ell_j), where ell_0 = 512 and ell_{j+1} = max(512, ell_j + ceil(L_j*(a_{j+1}-a_j))) + 1, both asserted exactly in integers at proof time; a control asserts that the old uncarried cutoffs fail the carry rule at stages 2-7.
• Lemma G label/adapter stage (Research audit: the selected #1094 local programs satisfy Lemma G's extended-label H1 (HPE-aware check, 0 violations); the selected chain runs no tag check (published checker needs a cold-HPE patch; the frame-only replay accepts three adapter mutants) (no new kappa) #1964, thakrarsagar; Lemma G by DreamingOfClouds, Lemma G of #484: clean proof and Lean 4 formalisation of its core #509/Lemma A of #484: the realization of paid wrappers and spectators, proved in Lean 4 #516). New lemmag-labels stage beside the frame-only local replay: label_check (strict and permissive) and adapter_rule report 0 violations on each provider's LOCAL-PROGRAMS in both orientations, and the X (exit straddle), Y (entry straddle) and Z (exit adapters moved before a phase-B add) adapter mutants are REJECTED.

Supplier: the package's own verify.py (retained PR1297 provider and parent admission, owner reconstruction, parent lift, strict T74 admission, geometry controls, now with the lemmag-labels stage) passed inside the release run.

Credits

PR1786 (chafreaky / Chafik Boukhalfa): fine source-prime padding, its proof and checks, vendored under vendor/pr1786; Tim Trudgian: effective prime number theorem (arXiv:1401.2689). PR1673 (Vlado Vojdanovski, with AI assistance): capped phase-cell fusion outer construction, its proof and exact evidence, vendored under vendor/pr1673. PR1758 (IdeaOuter and contributors): fractional-shift gauge-free interiors, accretive shallow-circuit lemma, per-visit tag accounting. Jeremy Kothe and contributors (PR1731 ledger lineage, radix-union schedule), LeverRank (independent schedule simulation), Gohberg, Semencul, Kailath, Kung, Morf, Bitmead, Anderson, Pan (structured Toeplitz-like algorithms of the retained alternative), Francois Le Gall et al. and Chafik Boukhalfa (PR1699). Supplier: PR1625 (mdumitrean), PR1661 (midhatn), pivot-554 kernel (PR1672 lineage) and the PR1319/PR1297/PR1274/PR1286 lineage as in earlier packages. PR1897 (M6LI), PR1876, PR1875 and PR1828 (rohanarun): the endpoint and slack refinements that this package exceeds; rohanarun also PR1319 and PR1297; chafreaky PR1786; Vlado Vojdanovski PR1673; mdumitrean PR1625; midhatn; philippeVV PR1519 (odd-term Mercator finding); Kothe and the existing lineage.

[Mister-Meeseeks]

Review follow-up for frozen head 07e4e4231459e36b1055396306dcbd950248df62.

This remains open for directed-rounding/scaling, full manifest and native-reuse review, including the carried-cutoff and label/adapter corrections associated with #1956 and #1964. The shared all-size fine-prime, gauge-free and fusion obligations also remain open. The finite release and tiny fixed slacks do not by themselves certify those implications.

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.