Nonuniformity of Free-Gradient Heat Semigroups under Finite Fisher Information

Let X_1,...,X_m be a nonempty finite bounded selfadjoint tuple with finite joint nonmicrostates free Fisher information, and let Delta be the generator of the closed polynomial free-gradient form. We prove that its heat semigroup cannot converge uniformly to the identity in L2 on the operator-norm unit ball of the generated von Neumann algebra. For each tuple we obtain a strictly positive lower bound valid at every positive time, with no restriction excluding one or two variables. The witnesses are coordinate exponentials. A direct proof of the classical marginal L3-density implication, the conjugate-variable adjoint formula, and a one-sided Fejer/Riesz estimate control their generator norm, while their energy grows linearly. Resolvent duality yields the nonuniformity bound. Existing rigidity results then imply non-L2-rigidity and, for at least two variables, primeness. The result addresses the 2006 AIM Free Analysis uniformity question. This preprint is unrefereed and makes no absolute priority claim. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: AIM-PROBABILITY-0111. Paper page: https://eulersolve.org/papers/aim-probability-0111/