Exact Word-Map Distributions on Four-by-Four Unitriangular Groups

We give the exact probability distribution of every group word, with no constants and any number of variables, on the full unitriangular group UT4(Fq), for every finite field. A nonzero exponent sum modulo the characteristic makes the word map uniform. Otherwise the distribution is determined by the rank of its degree-two noncommutative coefficient matrix and an exact bilinear zero count on the matrix kernel. The formula includes characteristics two and three and proves the Amit-Ashurst lower bound for every nonempty word fiber on this family. It also classifies the word image as the whole group, its derived subgroup, its center or the identity. A separate elementary split-extension argument gives an identity-fiber bound on full unitriangular groups in every dimension. This is a complete scoped theorem, not a solution of the full Amit conjecture or of AMR-011-0049 for arbitrary finite p-groups. The bilinear zero count is not claimed to have a polynomial-time algorithm, and the elementary split-extension observation is not claimed as new. The source archive includes the English manuscript, detailed self-audit, two exact finite implementations and their outputs. The finite computations check the proof but do not establish its arbitrary-word or arbitrary-field quantifiers. This AI-assisted, self-audited preprint is unrefereed. The related primary literature is explicitly credited. Novelty and absolute priority remain undetermined; no independent specialist review or proof-assistant verification is claimed.