Finite MMSE Data for Two-Component Gaussian Mixtures: Sharp Stability and Identifiability Obstructions

This paper studies identification and inverse stability of real mixtures with at most two Gaussian components from derivatives at zero of their Gaussian-channel minimum mean-square error (MMSE). For centered symmetric mixtures, derivatives of orders zero, three, five, and six determine the law, and maximum order six is necessary. On bounded parameter sets that allow zero variances and vanishing weights, the optimal uniform Holder exponent is one fourteenth for both quadratic Wasserstein distance between laws and the Wasserstein distance between component measures in mean and standard-deviation coordinates. With positive variance bounds and a lower bound on nonzero weights, the sharp law exponent is two sevenths or one third according to an explicit weight threshold; the corresponding component exponents are one fourteenth and one twelfth. Without a weight floor, the sharp law exponent on a fixed positive variance band is the minimum of two sevenths and the upper variance divided by four times the variance-band width. The endpoint bound is attained, including at the transition where the upper variance is eight times the lower variance. The paper also proves identification through derivative order six for every two-component mixture with vanishing third cumulant and transfers symmetric identification to finite projections. For nonsymmetric mixtures, an exact rational certificate constructs two distinct distributions with identical derivatives through order six. Gaussian smoothing of this pair rules out uniform inverse exponents greater than one quarter even when the entire MMSE curve is observed. These results concern deterministic inverse continuity from exact derivatives; estimation of those derivatives from finite signal-to-noise samples is a separate problem. The deposit contains the manuscript PDF, LaTeX source, and a computation supplement. Seven verification programs regenerate derivative polynomials, symmetric and weight-threshold identities, a rational interval certificate for the sixth-order collision, independent cumulant formulas, and identities governing the transport exponents. The analytic identification and transport arguments are proved in the manuscript.