Hermitian Gram Distributions over Finite Fields: Thresholds and Rank Structure

Let K be the quadratic extension of a finite field, equipped with its standard involution, and let T be a uniformly random rectangular matrix over K. We study the distribution of the Hermitian Gram matrix T-star-T and compare its pointwise probabilities with the uniform distribution on Hermitian matrices. Classical representation formulae determine the relevant fibres, but do not directly expose the normalized sign pattern or its sharp Witt-index threshold. We prove a sharp finite-field Witt-index threshold for the zero target, including a unique equality case and a quantitative supercritical lower bound. The zero-Gram theorem yields complete rank-by-rank sign classifications in both the full-support and underfilled rectangular regimes. In the full-support case, the sign sequence consists of a possibly empty positive prefix followed by an alternating tail. These classifications give exact total-variation formulae. We also prove, using a Parseval argument, that adding excess rows forces the Gram distribution toward the uniform Hermitian law, uniformly in the target dimension. The threshold is governed by maximal totally isotropic subspaces of Hermitian spaces. This record contains a preprint version of the manuscript.