Peaks and Gaussian Limits for Powered Binomial-Sum Ratios

We determine the asymptotic maximum of the ratio between a truncated powered binomial sum and the corresponding full sum in a shorter row. This proves the maximum-value conjecture of Byun and Poznanović for every fixed positive integer power and positive weight, and extends it to all positive real powers. Every maximizing index lies within one half, up to an error of order n^-1, of an explicitly shifted linear function of the row length. We also determine the first lattice correction to the maximum. After normalization, the entire sequence satisfies a uniform local Gaussian limit, with convergence of all fixed absolute centered moments and an explicit constant correction to the mean. For the square power, we prove strict log-concavity for every positive weight and every row length. The asymptotic proof combines a geometric endpoint estimate, a Gaussian denominator estimate and global entropy localization.