P02 selection assessment

Selected title: Exact Dold repair for matrix powers and integral recurrences.

The main contribution is a complete spectral criterion for finite entrywise repair, together with the exact local prime exponents determined by Jordan blocks modulo each prime. The companion-module version gives the exact uniform multiplier for every admissible sampling exponent, and constructs one initialization attaining all its prime exponents at once. This replaces sufficient large-exponent bounds by a sharp classification and a finite invariant. The universal exponent and Lucas radical are consequences in the same paper.

The independent extension proof audit, current primary-source comparison, and root final transfer review document the evidence. The main proof was developed by the root, independently reconstructed by a separate AI agent, and reread in its complete final source by the root. No internal mathematical gap remains. Computational checks are supplementary; the proof controls all indices and all primes.

The sufficient Galois exponent mechanism was already in Luca–Ward and is credited. Minton's unsampled scalar trace classification is a distinct established result; the original AMS PDF was inaccessible, and its precise statement was checked through the published Byszewski–Graff–Ward survey. Rajs's current formal theorem was also compared. No matching complete simultaneous classification or exact local formula was found in that bounded source audit. That does not certify priority against all literature or unpublished work.

This is a substantial specialist candidate because it supplies a reusable exact invariant and both directions of a general theorem. The argument's ingredients are classical; their use is not itself claimed as a new technique. Significance rests on the sharp classification and its consequences, not on the number of corollaries or on correcting a printed multiplier. Neither an individual scalar power-subsequence classification nor nonnegative realizability is asserted.

The revised nine-page PDF compiled cleanly and was visually inspected. Complete Markdown and its conversion and math-syntax reports are included. All review and significance judgments are internal AI work; no external expert acceptance, assigned authorship, or citation impact is claimed.

Final artifact check

The root has now read the complete selected TeX, inspected every PDF page, and checked the matching complete Markdown conversion records. The source/PDF hashes and all resolved-reference, proof-body, and math-preservation checks are recorded in the collection manifest and per-paper reports. No internal mathematical finding remains open in this selected statement. Live GitHub visual rendering and human mathematical verification are separate from these local checks.