[alejandrozu]

This refines PR #62's strongest stacked witness, interval strips and core-aware pair assembly combined with PR #57's joint frame compiler, at immutable source ad0f25ff7b23cff7f08ad237c2254e6ecf74257e.

The new conditional bound is T(n) = O(n (log n)^(1-kappa)), with
kappa = 25508460085039/500000000000000000 = 5.1016920170078e-5.
It exceeds #62's 5101691/10^11 = 5.101691e-5 by 1.0170078e-11, and exceeds the stronger old scoped limit 39859/781289859. Refined bit saving 102039046058023/2000000000000000000 and positive backoff h=1e-18 improve the parameters of the same physical construction. Eventual thresholds increase; the dyadic corollary remains 2^-15.

Fresh checks pass: exact characteristic upper envelope < 1, all 47 strict assembly inequalities, seven margins and eventual-bound arithmetic. 11 concrete Lean 4.31/Std proofs pass, computing the Padé rational expressions, upward-rounded weights, complete weighted moment envelope, positive slacks and strict κ comparisons. Four generic enclosure-interface proofs accompany the case. No omitted proofs, added project axioms or native decision shortcuts.

Construction checks are inherited from #62, as authorized for this refinement: its reported scalar/envelope/link validation, dirty-word/frame/profile/CRT/data checks are not rerun here. The new portable arithmetic checker pins the exact upstream source and arithmetic/assembly code hashes before import. This numerical certificate remains conditional on the analytic logarithm/exponential enclosures, source-profile linkage and all-size compiler/frame transfer, finite-alphabet tape layout, routing, prime-selection/recovery and eventual-threshold arguments. It is not a full formalization of integer multiplication or a measured runtime improvement.

The package also includes the author's separately scoped matrix/parity synthesis and exact matching search: sharp numerator-only modular late-division proofs; exact Gaussian shared-DAG interpreter; phase-child closure obstruction; all-cardinality feasibility/duality certificates; original matrix source and historical CI evidence. There are 171 standalone theorem endpoints. These background receipts retain their historical scope and do not constitute fresh #62 physical validation. Earlier #58/#60 refinements are preserved as historical evidence.

Reproduce via research/matrix-exponent-synthesis/candidate/README.md and lake build in the package. Source hashes and passing arithmetic/Lean receipts are included. The explanation is provided as LaTeX source; no new compiled PDF is required.

Alejandro Zarzuelo Urdiales is the human author/developer. The earlier Archivara exploration and his July matrix result were developed with multiple AI tools, followed by the October 7 revision at alejandrozu/openmath-2026-judging, commit 7203497dc990e47c2391bff1b9863408d817faeb. The current synthesis and proof/certificate tooling involved substantial OpenAI Codex assistance, most recently GPT-6.1 as recorded by the author. The new numerical gain is parameter-only; matrix arithmetic counts are not claimed as network ranks or as its source.

The current numerical construction retains Avi Eisenberg / ikeboy's #62 interval-strip and core-aware pair assembly, prepared with Anthropic Claude assistance, and eumemic's unchanged #57 joint frame compiler. Earlier #58/#60 editions retain Chafik Boukhalfa's composition/slot-priority credit and Rohan Gupta's dual-suffix credit. Known Dumas–Pernet–Sedoglavic, Rosowski, Gaussian three-product and matching-duality ingredients, all inherited licenses/notices, other community contributors and OpenAI retain their credit. No imported construction is claimed as a new invention.