Conditional κ = 0.00167975495612554452133567099392070257699831143850258029546032752016286454826032563882630565613284: recertified 1e-30000 supplier endpoint and 1e-30010 slacks (kappa +2.00336e-500 over PR1897): fine source-prime padding (…

[jeremykothe2]

Summary

Conditional κ = 0.0016797549561255445213356709939207025769…(198 digits)… (30046 decimals; the exact value is kappa in research/capped-fusion-t74-k554/outer/OUTER-CLAIM.json, SHA-256 of the decimal string c8ad5764d9f1eaf4f46cb506506d5c4396c8d61a2db59d6d945f16a2c8aa97a1).

Same construction as #1928/#1905 (PR1673 capped phase-cell fusion outer by Vlado Vojdanovski on the pivot-566+554 T74 supplier, with PR1786 fine source-prime padding, gauge-free interiors and the radix-distributed union schedule). New here: the supplier's complex endpoint is recertified on a 1e-30000 grid (accept at the endpoint, reject one grid step above, with the repaired even-term engines and an independent log oracle), with free strict slacks at 1e-30010. κ exceeds #1992 (rohanarun) by about 1e-10000, and #1928, #1897, #1876 and #1875 as well, all checked with exact Fraction comparisons in the ledger.

Verification

Fresh checkout of 322949a53488d64023fedb89a6774927707e57ba. Manifest: 236cff02c0d2358174226f69d3b5702ec151af6b5841501dd4be69963b5cd8f9. Full package-owned verify_release.py returned PASS_RELEASE in 3834.3 seconds, with fresh native replays, strict T74 admission and the outer ledger.

This package includes the #1956 carried bootstrap-cutoff repair (Josie Elliston), asserted exactly at proof time, with a control showing the uncarried cutoffs fail; and #1964's extended-label (Lemma G H1) stage (thakrarsagar), where the honest programs show 0 violations and the X/Y/Z adapter mutants are rejected. All 80 T74 residual wrappers were also checked exact with #2029's two kernels (512 columns, global phase).

See research/capped-fusion-t74-k554/README.md and research/capped-fusion-t74-k554/RELEASE.json for the full summary, proof description and reproduction commands.

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. 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.

[philippeVV]

Note for this head (322949a5): an independent exact broad-shift reconciliation is posted on PR1897 (#1897). The obligation passes for the shift the proof uses. Two things on this head need fixing:

• check_gaugefree.py:23 still carries the disabled or True line, without 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 corrected supplement.
• GAUGEFREE-INTERIORS.md :31/:84 state a linear leftover of ≤ 2 b0 θ B, which is false: there is an exact counterexample, and the true bound is 3 b0 θ B. Line :32 also says |s| "can exceed 1/2", which is false, since |s| ≤ 1/2.

Neither affects the conclusion, because §5 charges 4 b0 θ B. Package: https://github.com/philippeVV/integer-mult-bounds/tree/a3c40a6f/research/t74-broad-shift-reconcile

[philippeVV]

Scoped review of obligations O5 (from #1905) and O6 (from #1928) at head 322949a5 (prepared with Claude Code for philippeVV). No κ claim.

O5: adapters and native binding

• I rebuilt mutants X (exit straddle), Y (entry straddle) and Z (exit-block move) from their descriptions without using the PR generator. They come out byte-identical to the generator's output.
• On both fresh native directories, each mutant is rejected by the strict check, the permissive check and adapter_rule. Violations are 4/2/2 for X, 2/2/2 for Y and 113/113/114 for Z.
• The honest programs show 0 violations under every check.
• Three extra variants I added are also rejected: target touched after exit, source touched before entry, and a single exit adapter moved.
• Fresh native replays of new/{gauge44,extra}-word-566 match all 7 pinned artifact_semantic_sha256 entries in new/PLAN.json, plus the orientation-program hash.

O6: directed rounding and manifest

• Every rounding site in fine_mercator, fine_moment, check_ledger and the bootstrap/k-grid in assemble.py rounds in the conservative direction.
• I checked 24 engine enclosures against mpmath at 103,876 bits; all contain the true value.
• An independent recomputation of the ledger reproduces κ exactly.
• verify_manifest.py reports exact set equality over 383 files.

Two remaining items

1. Self-authenticating parent receipt.
   • When --parent-receipt-cache and both *-native-from options are passed, verify.py:38-47 accepts a RECEIPT.json whose file hashes are checked only against files in that same unpinned cache directory.
   • In that case the retained PR1297 admission is skipped. The retained native dirs consumed at :55 and :108 are then bound only by FULL-WORD and a self-reported STEPS.json (:34-36).
   • Default runs without the cache are unaffected. Please pin the receipt digest, or have reviewers always run without the cache.
2. CRLF artifacts. outer/LEDGER.json (556 CRLF lines) and outer/INDEPENDENT.json are committed with CRLF line endings, but outer/run_outer.py:15 compares bytes. The check therefore fails on LF hosts, even though the regenerated files are identical apart from the CR characters. PR1897 worked around the same problem with an adapter.

Cross-ref: #1905, #1928.

[philippeVV]

Independent replay of the inherited T74/#1319 supplier admission (O1 from my earlier O3/O5/O6 notes). Evidence: o1-admission/ (fork commit 8b041d4b).

• verify.py passes all 11 stages (Python 3.14, natives bound by FULL-WORD). Supplier strict κ = 0.0016769381110224….
• Two checks that the selected T16 chain has and replay.py lacks now pass on this PR:
  • (a) T16-parity final binding. Role, call and chart equalities; controller and owner audit hashes inside the invoice; all strict_checks; 47 positive outer margins; provider content binding. Swapping the natives makes it fail. It also passes on PR1319's ancestor and on PR1625.
  • (b) 37-replica identity regression through the T74 code. It reproduces PR1297's κ 0.001676854978651125 and the pinned REGRESSION37 controls.
• Regenerated artifacts agree by semantic hash on Python 3.12 and 3.14: OWNERS, ORDER, DISPATCH, PROFILE, TEMPLATE, CONSTRUCTION and PLAN.

Requests:

1. Bind PLAN to the content hash (assignment_content_sha256) instead of container bytes. Patching gzip byte 9 fixes only the OS-byte cause seen on 3.12; another deflate build (Apple zlib, zlib-ng) would still break it.
2. Assert the controller mutation count (23 = 7 + 8×2) and correct the controller=15 label.
3. Record input hashes in GEOMETRY-CONTROLS.json.

Not covered: the outer layers behind 0.0016797550 (PR1673, PR1786, gauge-free interiors, union schedule).

Prepared with substantial Claude Code (Claude Opus 5.5) assistance; no independent human review claimed.

[philippeVV]

Outer-layer audit at 322949a5 (PR1673 fusion, radix union, gauge-free interiors, PR1786 fine primes and free ε). No κ claim. Scripts: outer-audit/ (fork commit e6bc8393).

Exact reproduction:

• κ: from new/SPEC.json I re-derive b = cl − 1e-30010, q = b − 1e-30010, ε = 1 − 2e-30010 and κ = ⌊εq·10^30046⌋/10^30046 − 10^-30022. This matches the OUTER-CLAIM and LEDGER rationals exactly.
• Moment enclosure: an mpmath interval enclosure at 110k bits gives M(b) < 1 and M(cl + 1e-30000) > 1.
• Interface binding: a fresh strict replay's ASSEMBLY matches complex_lower, profile and bit_lower.
• Ledger: regenerated LEDGER: 62/62 checks positive, 17 controls fire.
• Outer checks run directly: sched_test, sched_review, struct_check, check_gaugefree, check_fine_padding and replay_evidence (19/19) all pass.

Where the lift comes from: the supplier κ is b/(1+b), while the outer κ = εq tends to b. So the whole lift is b²/(1+b) ≈ 2.8168e-6. I ran an exact knockout over (ε, β, ξ), reverting one layer's row at a time:

• re-adding only PR1673's old full-scan row 1−ε gives back exactly the supplier κ, q/(1+q). The entire lift rests on PR1673's removal of the Vd per-axis scan.
• the other layers only allow β → 1.

layer removed (others kept)	loss in sup κ
fine primes	2.29e-6
fine primes, structured boundary kept (= PR1758's κ to 30 digits)	9.4e-9
gauge-free (linear guard)	2.55e-6
radix union	1.41e-6
per-visit tag	1.41e-6

Issues:

1. CRLF blocks the release path on LF hosts. outer/LEDGER.json (556 CR) and INDEPENDENT.json are committed with CRLF under -text. outer/run_outer.py:14-16 byte-compares them against the regenerated LF files, which differ only in CR characters, and exits. So verify_release.py never reaches any outer check on Linux. Please compare after stripping \r, or commit LF.
2. Stage E cannot fail. outer/check_gaugefree.py stage E (the circuit replay, around lines 119–134) has no assertion, so it only reports, yet the script prints "ALL STAGES PASS".
3. The native cache is self-authenticated. Like the parent receipt, a --native-cache entry is accepted on its own CACHE.json hashes (verify.py:70-75), and verify_release.py passes it whenever a cache is given.
4. Stale text:
   • verify_release.py:4 (45 checks / 5 controls) and :114 (ε = 1−2e-100);
   • outer/RETUNED-SLACKS.md:18-19 (grid and κ);
   • outer/UNION-SCHEDULE.md:107 (tau_eff);
   • README.md:5 ("unchanged formulas").

Not covered:

• PR1673's main theorem;
• PR1758's Theorem 4.1;
• PR1786's setup/online precision and locality lemmas;
• the all-size interfaces.

The finite replays use PR1673's regime (halo 3, core 8), not fine θ with halo p^48 and core p^128.

Prepared with substantial Claude Code (Claude Opus 5.5) assistance; no independent human review claimed.

[philippeVV]

Follow-up: scoped audit of PR1673's fusion step as #2044 uses it. Per the knockout in my previous comment, the whole lift comes from removing the per-axis Vd row. No κ claim. Files: outer-audit/pr1673-fusion/ (fork commit bc4ed55b).

Question. Does dropping that row hide an uncharged per-axis pass at #2044's point? The point is ε = 1−2e-30010, β = 1−6e-30010, θ ~ p⁻², halo p⁴⁸ and core p¹²⁸.

Cost census. I took every cost and side term written in phase_fusion_note.tex and in its PHASE-LAYOUT, TENSOR-COMPILER, SUPPLIER-SUPPORT, HALO and CAPPED notes, and evaluated each exactly at #2044's point. All are paid except two, and this PR already replaces both (its controls fire on the old rows):

• PR1673's per-axis scan of U (note:250-252), with exponent ε+β+½−ξ ≈ 1.59;
• its precision term g·a·B (note:276-283), with exponent β+ξ ≈ 1.91.

No per-axis full-volume pass found. Per group there are polylog full scans, H calls and router exposures. Per-axis work touches only the fractions μ = W/B and gA/L of the volume.

Exact simulations:

• Product-diagonal scan on a tape model: at most 5 head moves and 2 multiplications per record, for any g ≤ 18. I re-ran this and it reproduces exactly.
• Fine-θ q-embedding: the membership predicate holds exhaustively, and the group embedding takes one pass at (1+T/S) moves per record, independent of g.
• Five-factor group in the dense-union regime: matches ⊗Qᵢ exactly, with 3 dense global calls for every g.

Re-runs. The vendored PR1673 verify.py passes all 21 control replays. check_capped_outer.py and check_matrix_corollary.py regenerate byte-identical output.

Minor issues:

• HALO-THEOREM.md:30 assumes K0² = O(B). That fails here, since K0 ≈ g·log s ≈ p^(1−4e-30010). Its fallback metadata term is paid with a large margin.
• Note:254 says "a fixed number of full scans" per group, but the tensor compiler uses polylog many. The 1−ε+β margin absorbs this.

Not checked: the support-local global H call costing V·g^(1−q) (SUPPLIER-SUPPORT-AUDIT §6-8). The fused-Gaussian row rests entirely on it, at a margin of about 2e-30010, and its side conditions hold only with 2–6e-30010 exponent slack. Also not checked: PR1758 Theorem 4.1, PR1786's lemmas, and the all-size interfaces.

Prepared with substantial Claude Code (Claude Opus 5.5) assistance; no independent human review claimed.

[philippeVV]

Follow-up: the support-local global H call (vendored PR1673 SUPPLIER-SUPPORT-AUDIT §6-8, note:122-136) at #2044's point. My previous comment listed this as the one unchecked lemma under the fused-Gaussian row. No κ claim. Files: h-call-support/ (fork commit 76af2562).

Exponent. q = b − 1e-30010 is limited by two strict ledger checks: q_below_internal (a_eff − q = 5e-30011, where a_eff sits below the selected bit rate a* = 0.0017334 from PR589) and q_below_leaf.

No cost that grows with d.

• Exposure moves at most 2gK positions.
• Routing costs O(V K (kG)^(1−a)), independent of the gap between fields.
• Descriptors are paid by a log²p-bit suffix.
• Deferral replaces the original d^(1−c) reservation term (compact-control-layout.tex:65-72). It gives R(h) ≤ 3h/128 + 3 ≤ h/16 for h ≥ 48, and Σ h_j^(1−q) ≤ 1.067·g^(1−q).
• The proof needs only the finite guard K ≥ G + 4⌈log₂p⌉ + 10 (compact-control-movement.tex:149-154).

Side conditions hold only from galactic sizes. At ε = 1−2e-30010 and β = 1−6e-30010 they start holding at log₂ p ≈ 1.7–5e30014 (thresholds.py). Router reach is not load-bearing, because the full-span fallback gives the same bound.

Exact bookkeeping model (sim_deferred.py). It covers support-limited exposure, deferred phases, row padding and role split, with an adversarial dirty-control supplier. The result equals H^⊗g exactly in 12/12 configurations, and all 4 negative controls fail. I re-ran it and it reproduces. It checks bookkeeping only; the supplier's internal cost is still analytic.

Nits (none changes the bound):

• SUPPLIER-SUPPORT-AUDIT.md:109 calls the guard O(g·a·d + g polylog p) "O(p)". That is false at both points: g·d ≈ p² at Conditional κ = 0.00167975495612554452133567099392070257699831143850258029546032752016286454826032563882630565613284: recertified 1e-30000 supplier endpoint and 1e-30010 slacks (kappa +2.00336e-500 over PR1897): fine source-prime padding (… #2044's point. TENSOR-COMPILER.md:23-27 has the same claim; both are superseded by the gauge-free guard.
• K = 64G in SUPPORT-ROUTER.md:25, but 128G in SUPPLIER-SUPPORT-AUDIT.md:45 and note:124.
• SUPPORT-ADVERSARIAL-AUDIT.md and check_supplier_capacity.py cover only g = p^0.093.
• check_supplier_capacity.py:28 tests K + 0.907·log p rather than K + log₂ L.

Not checked:

• the retained equal-width arbitrary-gap interchange (a hypothesis at note:63);
• whether T74's mixed-width profile satisfies the compact-layout contract (whole-call row hosting, common frame, stopped recurrence);
• the all-size interfaces.

Prepared with substantial Claude Code (Claude Opus 5.5) assistance; no independent human review claimed.

[philippeVV]

Follow-up: does T74's mixed-width profile satisfy the compact-layout contract behind the H call (V·g^(1−q), τ = 1−a_eff)? No κ claim. Script: layout-contract/ (fork commit d1e01310).

1. Recurrence: needs the mixed weighted-tree version, and the vendored text cites the uniform one.

• The vendored PR1673 text cites "the original m,W,s recurrence" (radical_reduction_ultra/README.md:35, through compact-control-layout.tex:11,79-83, with children of e/m axes). Read literally, that does not cover T74's rank-48 children: the tree depth is about 1.653·log₂ h, not log₇₃ h.
• Expanding to rank one makes the uniform recurrence apply, but it only gives 1 − log₇₃(mass/W) = 0.0011430885. That value is the same for T16 and T74, since deficit/W = 5/14 in both.
• The ledger's b ≈ 0.0016797 needs the mixed weighted-tree version, which is already in the repo: notes/endpoint-gauge-complex.tex:186-225 ("These arguments do not assume equal leaf depth") and notes/batched-complex-rows.tex.
• T74 meets that version's hypotheses: 73² = 5329 > 2·48² = 4608, the deficit is 23680, and the native moment holds at b. Request: please cite that section in README.md/OUTER-AUDIT.md.

2. Stop. STOP = 1e-30016 gives (1−stop)b − a ≈ 5.0e-30011 and (1−stop)b − q ≈ 1.0e-30010. PR1673's vendored value of 1e-9 would fail at this point (−1.68e-12), but RETUNED-SLACKS.md:15 already records the change. The mixed guard e ≥ 2m holds only from log₂ h ≥ 7.19e30016.

3. Row stock.

• physical − live = 490, the same as T16. The stock is 2644 bits.
• Row coefficient: 2·2644 + 9909 + 252 + 2·2048 = 19545 < 20161. T16's is 16469, so the margin here is much smaller.
• In the support-local call this gives q₀ ≤ 39 K-chunks, so R(h) ≤ 3h/128 + 41 ≤ h/16 for h ≥ 1050.
• Correction to my previous comment: its "3h/128 + 3, h ≥ 48" assumed a one-chunk row stock. The conclusion is unchanged; the geometric factor 1.067 still holds.

4. Ledger inputs. check_ledger.py uses only (m,W) = (73,66304), the T74 profile, and b from SPEC. I found nothing in it that is proven only for T16.

Not discharged (same as for T16): the all-size common-frame/mutable-source and whole-call hosting hypotheses.

Prepared with substantial Claude Code (Claude Opus 5.5) assistance; no independent human review claimed.