Nonuniformity of Free-Gradient Heat Semigroups under Finite Fisher Information

Let a nonempty finite bounded selfadjoint tuple have finite joint nonmicrostates free Fisher information, and let \(A\) be the positive selfadjoint operator associated with the closure of its polynomial free-gradient form. We prove that \(e^{-tA}\) does not converge uniformly to the identity in \(L^2\) on the operator-norm unit ball as \(t\downarrow0\) . The conclusion holds for every positive number of variables, including one and two. For each coordinate, oscillatory unitaries \(e^{isX_i}\) have energy asymptotic to a positive constant times \(s\) , while \(\|Ae^{isX_i}\|_2=O(s)\) . The latter bound follows from the joint conjugate-variable formula and an \(L^3\) estimate for a one-sided Fejér operator. Resolvent duality, or independently a spectral-moment argument, yields an explicit positive defect at every positive time. We include a direct proof of the marginal \(L^3\) -density estimate and verify the exponential generator domains. The resulting deformation is admissible for Peterson's \(L^2\) -rigidity; for at least two variables, established factoriality and nonamenability results then give primeness. No compact-resolvent or nonamenability-set hypothesis is used in the semigroup theorem. No absolute priority claim is made.