Noncommutative Anisotropic Diffusion in Hilbert Space. II. Global Closure of the Logarithmic Gradient, Lower Bounds, and Nanosystem Applications

Abstract

This second part of the series develops the statistical and applied layer of the theory built in Part I [1]. Unlike current Hilbert diffusion models [2-5], we focus not only on well-posedness of the infinite-dimensional generative dynamics, but on the noncommutative $A$-geometry, explicit entropy constants, and checkable lower bounds. The analytic estimate of Part I reduces stability of the backward evolution to control of a validation error $\mathcal{E}_{val}$. We prove three results. First, we construct a cylindrically weak denoising label for the infinite-dimensional score field, consistent with the exact logarithmic gradient. Second, the local parametric closure is replaced by a global nonparametric score closure, its complexity controlled via the Dudley entropy integral and uniform empirical bounds in $L^2(ν_*;A)$. Third, for the trace-smoothed nonlinear class we construct minimax bounds by the Le Cam-Assouad method, showing the root statistical rate cannot be improved without additional quadratic structure. The final section applies the theory to anisotropic diffusion in a nanosystem model and checks the constants $c_A,C_A,C_{LSI}^A$ independently of any smallness condition, establishing explicit accuracy orders: the parametric trace-smoothed minimax score risk is $p/M$, while the uniform error of the risk functional is $\sqrt{p/M}$, showing the statistical plateau of order $p/M$ is not a proof artifact. The applied layer closes with an independent analytic benchmark for the isotropic case, a comparison with classical cell homogenization, and an approximation theorem for smooth logarithmic gradients by $A$-adapted spectral networks.