MMSE Identification of Finite Distributions and Structured Gaussian Mixtures: Finite Observations and Statistical Limits

This paper studies identification of a real probability law from the minimum mean-square error (MMSE) of a scalar Gaussian channel. It proves that equality of all right derivatives at zero determines a finite-support law within the finite-support class, up to translation and reflection. The result extends to finite atomic laws convolved with an unknown Gaussian, finite Gaussian mixtures with a unique maximal-variance component, and the closed class whose mean-variance component points are collinear. A local heat-zero argument also gives identification for symmetric laws with a square-exponential moment and a Lee-Yang reference class. The analytic proof recovers complete local contour amplitudes from Borel singularities and separates exponential and algebraic time scales. For every fixed component bound, polynomiality yields finite derivative cutoffs, finite projection designs, and compact Holder inverse estimates. In the closed classes these estimates include component collisions and zero weights. Quantitative compact-set results for the nonclosed unique-maximal-variance class are restricted to the stated compact families. Exact small-support calculations give derivative cutoffs five and seven in two subclasses, and a rational certificate constructs a three-atom collision through order seven. Arbitrarily long finite-jet agreement rules out an inverse power uniform in the number of components. Gaussian smoothing transfers finite-jet contacts to discrepancies small over the full nonnegative signal-to-noise axis. For Gaussian-noisy observations of MMSE values, uniform Gevrey-two derivative bounds and a shifted Chebyshev-Lobatto design convert a finite-jet inverse estimate into a constructive statistical upper bound. Adaptive lower bounds, combined with the author's previously established deterministic two-component stability estimates, give sharp polynomial minimax exponents in symmetric two-Gaussian classes. Logarithmic factors and general optimal derivative cutoffs remain open. The statistical observation model queries noisy MMSE values; it is distinct from observing independent samples of the input distribution. The deposit contains the manuscript PDF, LaTeX source, and a computation supplement with thirteen exact verification jobs. These regenerate posterior derivative formulas, replay rational collision certificates, check small-support cutoffs and arbitrary finite prefixes, and verify Gaussian-amplitude, Gaussian-addition, and double-zero Borel identities. The analytic identification and statistical theorems are proved in the manuscript.