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 \(\operatorname{UT}_4(\mathbb F_q)\), 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. The general problem for arbitrary finite \(p\)-groups is not resolved.