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.
