A Sharp Dimension Bound for Pointwise Nilpotent but Nonformal Jordan Perturbation Families over F3

Abstract We study pointwise nilpotent but formally nonnilpotent matrix families over the finite field (\mathbb F_3), arising from linear perturbation spaces of a fixed unipotent Jordan block of size four. Let (J_4=I+E_{12}+E_{23}+E_{34}), and define (N(X)=J_4(I+X)-I) for (X\in\mathfrak{sl}_4(\mathbb F_3)). We investigate linear subspaces (K) for which (N(X)^4=0) at every (\mathbb F_3)-rational point, although the corresponding matrix polynomial in independent parameters is not identically zero. An exhaustive finite classification yields 3,685 admissible projective directions, 116,124 compatible pairs, and 19,354 distinct two-dimensional pointwise nilpotent subspaces, of which 1,053 are jointly nonformal. A complete affine-coset extension test finds no three-dimensional pointwise nilpotent extension of these nonformal planes. We combine this computational certificate with an algebraic restriction lemma for polynomials of total degree at most four to obtain a sharp dimension bound: every nonformal pointwise nilpotent linear subspace in the specified model has dimension at most two. An explicit two-dimensional example attains the bound. The result is a computer-assisted theorem whose finite classification component depends on the correctness and reproducibility of the accompanying exhaustive computations. We discuss the distinction between pointwise and formal polynomial identities over finite fields, the scope of the dimension bound, and related block constructions in dimension six. Research status and reproducibility This work is presented as a mathematical preprint supported by exhaustive finite computations and an algebraic proof argument. The computational classification has not been independently reproduced by an external reviewer, and the manuscript has not undergone formal peer review. The claimed dimension bound is conditional on the correctness of the accompanying computational certificate. No claim of definitive priority or global optimality outside the stated mathematical model is made. AI assistance disclosure OpenAI ChatGPT was used substantially in mathematical exploration, programming, certificate design, mathematical proof development, literature review, and manuscript drafting. The named author is responsible for reviewing the mathematics, checking the released files, verifying citations and originality claims, and deciding whether to submit and publish this work.