Sharp Coefficient Ratios and Optimal Constants for Dilated Ehrhart Series

For every dimension d and integer dilation n >= d+1, the ratios of the coefficients of a dilated Ehrhart h-star polynomial to its last coefficient are maximized simultaneously by the unimodular simplex. The same sharp inequality holds for every nonnegative numerator of degree at most d under the Veronese operator. Combined with known real-rootedness and alternating-inequality results, this gives explicit finite rational-polynomial formulas for every admissible pair of coefficient multiplier and dilation threshold in the Beck-Stapledon theorem and its Ehrhart corollary. The smallest full-chain threshold is d+2 and the smallest eventual integer multiplier is one more than the largest Eulerian number of order d; these two minima need not occur together. Exact polynomial certificates give the least thresholds at that multiplier through dimension eight. The general-numerator root/ratio-only version and the one-dimensional edge case of the original Oberwolfach formulation are treated separately. Previously established root and alternating-inequality bounds, and the prior sufficiency of the Eulerian multiplier, are explicitly credited. The main contributions are the sharp ratio extremality, necessity of the multiplier, and finite optimization of their full tradeoff. AI-assisted and self-audited preprint; no independent review, formal verification, or absolute priority certification is claimed.