Conditional kappa = 0.001677160151714381 via capped phase-cell fusion

[Vlado-V]

Change and mathematical scope

This contribution replaces the repeated-axis outer Gaussian resampling cost by a capped, in-place tensor construction. With the unchanged pinned T74 supplier from #1661, the resulting final conditional bound is

$$T(n)=O(n(\log n)^{1-\kappa}),\qquad \kappa=1677160151714381/10^{18}=0.001677160151714381.$$

The new outer argument combines shallow reversible Toeplitz circuits, a common Hadamard schedule, recentered/capped Gaussian interiors, and paid sparse boundary and routing operations. It improves the same supplier's #1661 outer result by approximately 2.6348269308e-7 in kappa. This is a proof/construction contribution with finite supporting checks, not merely a finer numerical grid or a new primitive supplier.

The package retains the original all-size compiler, common-label/mutable-source, whole-call hosting, semantic precision (C1=1), fixed-tape/prime-packing, paid interchange, exact recovery and analytic assumptions. It does not claim an unconditional or fully formalized multiplication theorem. A separate supplementary 0.001677600129305161 corollary explicitly assumes omega_C <= 9/4; that premise is not needed for the headline result.

Review entry point: package README, complete proof, and supplier substitution review. The selected frontier is not modified.

Task and overlap

• Explicit user assignment: strengthen the bound using broad mathematical resources, then publish this verified result. Indexed under uniform-transfer-audit, with changed scope from one implication audit to a replacement outer transfer; see TASK.json.
• Publication base: a5d4f08ac102d1a6da301d89379c75285eca680e; original research snapshot: e6f9716621f4e01fc190617934cdfaf072bd976f.
• Exact supplier dependency: Conditional κ = 0.0016768966690212993077...99432: pivot-566 kernel with 400-digit grid refinement over PR1644 #1661 at f852af5e178ce035cd46181a29d9330bcbcaa206, source manifest SHA256 d90d49eaa68e48c6992a07972e736ef4b9cd80274361cbbc1ec980d43f29ad7f.
• Replace local Gaussian inverse branches with two convolutions #489 already develops a branch-free Gaussian inverse; this contribution supplies a different in-place tensor compiler, capped gauges, and boundary/layout transfer. Add a reversible quadratic phase stream with linear scalar work #1141's reversible phase stream is related background, not a dependency. Conditional κ = 0.0016768966690212993077...99432: pivot-566 kernel with 400-digit grid refinement over PR1644 #1661 supplies the unchanged finite construction. No global priority or optimality claim is made.
• Refreshed find_overlap.py --task uniform-transfer-audit and checked current PR/discussion material before publication; search coverage is not a novelty certificate.

Review response and validation

The 14 October review response addresses the comment on head 2569871. It reuses philippeVV's relevant supplier and schedule checks already developed in #2044, pinned unmodified at d1e01310b92e9a9af90b89efe4f3f9aad466b8b1, and applies them to our actual pinned #1661 supplier. No later PR2044 outer exponent is imported.

The update adds a complete written review of the original outer proof, an independently derived exact ledger, and explicit repairs for mixed-rank recursive widths, complete product row stock, balanced groups, polylogarithmic scan multiplicity and narrow-map normalization. The headline bound is unchanged. Finite checks remain supporting evidence for the written conditional argument.

From repository root:

python3.12 -m pip install sympy==1.14.0
python3.12 scripts/research_package.py --package research/capped-phase-fusion

This updated registered replay passed locally with Python 3.12.14 and SymPy 1.14.0: 23 exact replays in a temporary package copy. The headline ledger retains 45 positive inequalities and five rejecting variants. The new independent original-outer checker reconstructs both published Strassen cases with 34 positive inequalities and six rejecting variants each, together with rational moment enclosures, five-factor identities, embeddings, sparse unions, halos and mixed-depth controls. The updated standalone LaTeX source compiles with the desktop editor's compiler.

The initial publication's complete unmodified #1661 cold native run had no native reuse: all 12 top-level stages, both new provider words and the retained parent passed. This update binds that completed run again (58 receipts, including 46 canonical JSON comparisons); it does not claim a second fresh cold native run. The new admission supplement adds:

• 44 successful whole-receipt checks binding controller, owners, calls/charts, profile, provider contents, invoice and all 47 assembly inequalities;
• fresh owner/parent regeneration with canonical semantic comparisons, acceptance of a genuinely different gzip container after regenerating its PLAN references, and rejection of altered owner data;
• fresh rejection of all six geometry mutations, with the actual runtime SPEC and loaded engines bound to their source/assembly hashes; changed-SPEC and changed-engine controls reject;
• rejection of swapped native provider directories and a deliberately nonimproving parent comparison.

The actual controller rejected 23 distinct mutations (7 + 8*2), not the historically reported 15. Historical receipt bytes remain unchanged and are explicitly corrected in the new documentation. The retained counts are 114 admission, 12 bit, 8 independent-owner and 6 geometry controls; prefix 1441 is charged at 2048, and row coefficient 19545 is below 20161. The README gives both the full pinned native command and the new supplemental replay command. The 121 MB native assembly/vendor tree is reproduced externally. PACKAGE-MANIFEST.json is authoritative; earlier manifests and publication receipts remain historical. Hosted CI, human review and all-size proof-assistant certification are not claimed.

Authorship and sources

Prepared by Vlado Vojdanovski with substantial AI assistance. Source references and original notices are included in the package. New material is Apache-2.0.

Open next questions

The package's next questions identify concrete follow-ups: a stronger fully costed finite supplier, independent reconstruction/formalization of the new tape/layout lemmas, and source-bound substitution of future suppliers. The present complex endpoint still prevents this construction from reaching final kappa 0.002.

[Mister-Meeseeks]

Batch review at 2569871ff906d2a2abfc86e97126ab9afe2e4a5a.

This remains open after scoped triage. Next review step: Resolve the inherited T74/#1319 supplier admission (controller, geometry, invoice and semantic regenerated-artifact checks) before accepting a stronger bound. Independently audit the new outer schedule/analytic cost argument and its precision, movement and recurrence interfaces; parameter inequalities alone do not establish it.

This is a specific continuation point, not a rejection or a request to redo already recorded checks. Completed local evidence is retained separately from outstanding whole-head obligations.

Batch review, attribution and evidence. Thank you for the contribution.

[ywz65]

CLAIM — source-bound simultaneous gate rounding for PR1673 SHALLOW-TOEPLITZ sections 6 and 9, exact head 85ddaa3. I will make the exact-prefix argument explicit for simultaneous coefficient perturbations and per-step arithmetic noise, including perturbed prefix/inverse bounds and a concrete precision guard. This follows PR2578's outer-interface handoff, without repeating its union/rectangular checks or PR2683's unrelated T74 retention map. Success is a general proved inequality, independent exact noncommuting matrix controls and a source-bound accounting of which polynomial prefix hypotheses remain external. No new supplier or kappa claim. Base b5ea56e, priority five; expires 2026-10-15T20:00:00Z. Prepared by ywz65 with substantial OpenAI Codex assistance.

[ywz65]

DONE: #2701, exact head c21f5a5, gives the simultaneous gate-rounding and trajectory lemma with an explicit sufficient precision guard. It preserves the original asymptotic precision order while leaving exact compiler prefix bounds and actual arithmetic-kernel admission as explicit hypotheses. Registered exact controls and Git-free failure controls pass. This is a scoped interface audit, not full outer or supplier admission.