# Verification report

## Accepted theorem and limitations

Over a prime field F_p, let S be a nonempty set of d distinct shifts and let h be nonconstant with h(s) nonzero for every s in S. The manuscript classifies the kernel K_h of reduction from the subgroup generated by X-s modulo h to that generated modulo rad(h). It determines its exact order, invariant factors, and integer relation lattice for arbitrary degrees and multiplicities of the irreducible factors. The nontrivial cyclic factors have at most two adjacent p-exponents.

This does not determine the reduced subgroup, solve the squarefree or irreducible growth problem, or close the whole source record AIM-COMPUTATION-0090. General finite fields cannot be substituted for prime fields in the formula. The literal linear-modulus obstruction already in the research record is not claimed as a new counterexample.

## Proof audit

The originating researcher's detailed self-audit checks the primary projection as a subgroup-preserving power map, simultaneous relations across all local factors, injectivity of splitting-field base change, the logarithmic numerator isomorphism, the necessity-only derivative step before descent, exact Frobenius valuation scaling, and the final sufficient derivative criterion only at multiplicities at most p. Smith form of the final lattice then gives the invariant factors. No unresolved mathematical gap was identified within the stated theorem. This is not independent expert review or proof-assistant certification.

## Exact regression evidence

The first standard-library Python checker constructs the finite quotient rings and enumerates the generated groups directly. It verifies 7,484 instances, 379,582 generated-group elements, and 8,359 torsion counts. This includes 850 repeated-factor, 5,418 nonsplit-factor, and 1,846 mixed-factor instances. The largest quotient ring in this check has 3,125 elements.

The second checker compares the explicit relation lattice with direct quotient-ring powering in ten configurations, including high multiplicities and prime 101. It checks 7,730 exponent vectors and 7,730 signed-period translates. Small configurations exhaust the principal periods; larger configurations use a fixed random seed and a specified critical-layer grid. It shares the first checker's basic polynomial arithmetic, but the actual relation test uses ring powers, not the claimed derivative criterion.

Normal and optimized runs of each checker agree byte for byte. Explicit runtime checks, not optimizable assertions, enforce correctness. The source-package preparation also replays both scripts in both modes after extraction and compares their receipts. These are finite tests supporting a written general proof, not a formal verification of the theorem.

## Source and prior-work comparison

The official AIM question and remarks were inspected, including printed pages 24-25 of the hash-verified primary PDF. The published 2004 Voloch paper was read in full, and its printed page 236 was visually checked; the older author preprint was not substituted for the journal version. Bernstein's published introduction and congruence-counting results were also compared. Voloch studies the same generated groups and records the ambient principal-unit order, but no exact generated-kernel classification was located in the inspected sources. This bounded comparison does not establish global novelty or priority.

## Package and presentation

The English manuscript has five pages. The built-in compiler and Tectonic both succeeded. All five rendered pages were visually inspected; no clipping, overfull boxes, undefined references, or missing characters were found. The preparation script records a portable rebuild with identical page content and extracted text before acceptance. SHA-256 bindings, replay receipts, and the visual audit are retained locally. Third-party full texts are not included in the upload archive.

The paper is an unrefereed preprint. This local report does not itself imply any DOI, deposit, site release, or Hugging Face notice has been completed.