# Verification report

Complete self-audited proof of sharp simultaneous coefficient-ratio bounds for every dimension d and integer dilation n>=d+1, attained by the unimodular simplex. Explicit finite formulas characterize every admissible multiplier/threshold pair in the Beck-Stapledon question, including the root/ratio-only variant and the literal one-dimensional edge case of the original theorem. One whole frozen source record and one source question are resolved by a finite characterization; no efficient or short closed form is claimed.

Written proof: positive log-concave convolution kernels, monotone shifted coefficient ratios and weighted averages; previously established real-rootedness; a symmetric decomposition for the alternating chain; the simplex obstruction; Eulerian asymptotics from above; and explicit Cauchy bounds reducing each infinite integer tail to finitely many rational comparisons. Prior root bounds, chain bounds and sufficiency of the Eulerian multiplier are credited. Novel contributions are sharp simplex ratio extremality, multiplier necessity, and the full finite tradeoff.

Exact table verification: sixteen pairs in dimensions one through eight. Direct integer polynomial products and translation produce a positive constant and nonnegative higher coefficients on each full half-line. Evaluations independently match the finite-difference Ehrhart formula. Failure at the immediately smaller threshold, or the dimension-one chain obstruction, proves minimality. Exact simplex ratios prove the other minimum. No numerical root finder or floating-point arithmetic is used.
Table verifier SHA-256: d610961f2745bf056d6cc19c467cb75a1a0ea15f6df73f60c9c4d59a19a047ab.
Table certificate SHA-256: 86815aae1b65a892c3494cd9ef1a8cce4c1d6a3df976a86d5013d5994b1c3a3e.

Supplementary finite checks: 4090 basis-ratio checks, 7200 nonnegative numerators, 25200 first-chain comparisons and 4116 conditional full-chain examples. These support but do not replace the written all-dimension proof.
Kernel verifier SHA-256: e156ec88c3eede10ba1de132a19d3c1a5751ee2497faaf85f0d2ec0329ab5ef9.
Kernel certificate SHA-256: 93f043c82d71d49864958eb4af35457adb2086521e4389b26a61dc1a5cc04e60.

Known mathematical gaps in the stated theorems: none found in the author's detailed self-audit. Independent review and proof-assistant verification: not performed. Absolute priority: not certified by the bounded primary-source search. The seven-page PDF compiled in the built-in editor and Tectonic; all final pages were visually inspected. Author name is prominent, affiliation and contacts are a footnote, and AI assistance is disclosed in prose.