Conditional κ = 0.001696395766736081: exact pivot-518 and 506 helper-donor kernel(s) (PR1625 shape, KernelsOnBorrow) on the borrowed words, exact frame refit (WildIdeas), 1e-3000 certified supplier endpoint, fixed-grid-maximum kappa C - 1/…

[jeremykothe2]

Summary

Conditional κ = 0.001696395766736081340151201755361527911765706899936336210617… (3046 decimals; the exact value is kappa in research/gmx-t74-borrow43-k518-506/outer/OUTER-CLAIM.json, SHA-256 of the decimal string 4921e9d2f65f912a2ecde54842170f357c1fd452bfdcd402e12f48b2c0ef39ac). Applies PR2226's anticipatory dirty borrowing (Cinna Davis / 1337isnot1337) to our T74 local words: 43 helpers per call are removed by borrowing dirty space from other registers (480 to 437 helpers, local mass 5590 to 5385), with frames re-solved, fresh natives, PLAN rebound and SPEC refrozen, under the PR1673 capped phase-cell fusion outer; re-solves the frames of the borrowed gauge44 word exactly (Jeremy Kothe (jeremykothe2)); and, new in this package, applies two exact helper-donor kernels, at pivots 518 and 506, (the PR1625 mdumitrean shape; search and exact pricing by Jeremy Kothe (jeremykothe2)) to both borrowed words with all frames re-solved (local profile {1:2341, 2:505, 3:420, 4:125, 5:42, 8:8}, mass 5385 unchanged; audited forward and dual by new/borrow/audit_local.py, with tamper controls on the kernel reads; native chains, rebind, discovery and the strict assembly rerun). The supplier native and conservative complex endpoints are certified to 1e-3000 (accept at the endpoint, reject at endpoint + 1e-3000), grid_digits 2995, margin_digits 3005, bootstrap_steps 1116, and the PR1956 carried bootstrap cutoffs and PR1964 Lemma G label stage are retained. Conditional final kappa = 0.0016963957667360813401512017553615279117…(3046 digits) (supplier strict-assembly kappa 0.0016935228816887139163412051031782853465…(2995 digits)). Retuned to the fixed-grid maximum of PR2180 / PR2161 (dataisfire): kappa = C - 1/F - B exactly (floor grid 1e-3046, backoff 1e-3022, C the certified endpoint; outer/PROOF-GRID-MAX.md). It exceeds PR2288 (the same package without the kernel) by 8.48163E-8, PR2269 by 1.48628E-7, PR2265 by 1.29832E-5 (63 strict inequalities: the 62 retained ones plus the explicit main-binding row gaussian_above_main, 17 rejecting controls). The single 3000-digit rejection moment of the strict assembly uses PR2168's exact-integer coarse exponential (code/exact_coarse.py, EXACT-ARITHMETIC-DERIVATION.json; intervals identical to the retained engine).

Verification

Fresh checkout of 9f67406e9972be9a97139f68ca29ea27a7545655 (text-only follow-up to baebf01eb36a461510e93098da5a25f78deaa985: public credit wording and summary rewording; kappa unchanged; then text-only README follow-up correcting stated comparison figures per #2485; kappa unchanged). Manifest: 7cfea1abc46338778aba1cdaefcdbc3e0a5c7a8171cd6557888e835aa8dc69fd.
PASS package manifest: 429 files

Supplier: the package's own unchanged verify.py (retained PR1297 provider and parent admission, owner reconstruction, parent lift, strict T74 admission, geometry controls): PASS_T74_EXACT_FILL_COMPLETE_FINITE_ADMISSION, native reuse {'base': 'e8e6e78ff7422275c38324499e1f33fb2ecd30de9503aef2b8f1ee80d8c4df8e', 'extra': '7b0e9fbda32749aabc7e156d9fe09bceb0624a178628fe69e5dfea783462f905'}; its complex endpoint 169639576673608134015120175536... brackets the ledger's b = 1696395766736081340151201755361527911765…(3078 digits)/1000000000000000000000000000000000000000…(3081 digits). Outer (PR1673 capped phase-cell fusion): regenerated exact ledger (63 strict inequalities: the 62 retained rows plus the explicit main-binding row gaussian_above_main, the fixed-grid maximum kappa = C - 1/F - B asserted, 17 rejecting controls incl. the old g-factor union row, the old linear and old-theta chirp guards, the old 2*beta tag test, epsilon at 1 or at the r0 slack and beta outside its window, larger endpoint rejected by the native moment), independent balance derivation, the exact union-schedule simulation, the (retained, not used for kappa) structured-boundary displacement-rank check, the gauge-free identity/symbol/circuit checks, PR1786's finite prime/underflow controls and 19 exact evidence replays all pass. Conditional kappa 0.0016963957667360813401512017553615279117…(3046 digits) (union schedule outer/UNION-SCHEDULE.md; explicit Strassen boundary tau=45/16 (the structured boundary of outer/STRUCTURED-BOUNDARY.md is not used); gauge-free interiors outer/GAUGEFREE-INTERIORS.md; fine source-prime padding PR1786 at the free point epsilon=1-(1025/1024)e-3080, outer/FINE-PADDING-COMPOSITION.md; fixed-grid maximum of PR2180/PR2161 (dataisfire), outer/PROOF-GRID-MAX.md; exact-integer coarse exponential PR2168/PR2184 (dataisfire) at the one 3000-digit rejection call).

Full package-owned verification passed in 138 seconds on the new head (PASS_RELEASE, kappa identical to the previous head). The fresh receipt binds the exact package manifest and score.

See research/gmx-t74-borrow43-k518-506/README.md for reproduction commands and proof qualifications.

Credits

PR2226 (Cinna Davis / 1337isnot1337 with OpenAI Codex): anticipatory dirty borrowing, the alias rule, K_G handoffs and the exact free-operator replay (vendored under new/code/borrow/vendor). Jeremy Kothe (jeremykothe2): exact re-solve of the frames of the borrowed gauge44 word (FusionPairs exact-frame-toolkit frames.optimal_frames), new work in this package (new/borrow/inputs/refit_43base.json). PR1625 (mdumitrean): the pivot-566 helper-donor kernel shape reused here with pivots 518 and 506. Jeremy Kothe (jeremykothe2): exact kernel search, pricing and frame re-solve on the borrowed words, new work in this package (new/borrow/inputs/kernel_43base.json, kernel_43extra.json). 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. PR1956 (Josie Elliston: the bootstrap-cutoff carry finding and repair), 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. PR1964 (thakrarsagar / Sagar Thakrar): Lemma G label and adapter-rule checks and the X/Y/Z adapter mutants (code/lemmag_checks.py), Lemma G by DreamingOfClouds (PR509 / PR516). PR2161 / PR2180 (dataisfire, OpenAI Codex-assisted): the fixed-grid maximum point and its proof, re-derived at this endpoint; PR2044 (Kothe) free-epsilon refinement they stack on; PR1786 (chafreaky / Chafik Boukhalfa) and PR1673 (Vlado Vojdanovski) ledger formulas. PR2168 / PR2184 (dataisfire): exact-integer coarse exponential and its one-site derivation. PR2269 and PR2265 are the records exceeded. PR2286 (thakrarsagar) side-condition audit and philippeVV (PR1897 / PR2044 comments): the text defects (minimum-margin claim, xi aside, HALO K0^2 branch, p^10 umbrella, vacuous or-True guards, stale xi digits) corrected in outer/FINE-PADDING-COMPOSITION.md section 4, outer/check_ledger.py, outer/struct_check.py and outer/check_gaugefree.py.