Dimension-Free Gaussian-Reference Stability for Scale and Covariance Mixtures

This preprint compares the entropy increase under normalized independent self-addition with the entropy-power deficit against an optimally chosen Gaussian reference. It proves a linear comparison with an absolute dimension-independent constant for arbitrary Gaussian shared-scale mixtures with an integrable absolute logarithmic scale, and for common-mean two-component Gaussian covariance mixtures. The constant is uniform in the mixing law, component weights, and covariance contrast; an optimal numerical value is not supplied. For a vanishing component weight and diverging covariance separation, the optimized ratio tends to one half without a restriction on their relative rates. For general centered isotropic inputs with finite Gaussian divergence, the paper gives a sharp optimization envelope in terms of the divergence lost under standard Gaussian smoothing. It also characterizes Gaussian factors of stationary references and localizes global minimizers. For arbitrary nearby covariance mixtures, a Frobenius-norm condition gives dimension-independent local comparison bounds, with limiting ratio seven twelfths for each fixed nontrivial mixing law. Further results include total-variation Bernstein stability for shared scales, nonlinear latent-information bounds, attainment and localization of reference scales, fixed-law information-dimension corrections, coexistence of distinct global reference optima, and singular covariance asymptotics described by nested subspace medians. A covariance-cube example explains why small entropy deficit need not concentrate the latent covariances. A global comparison for unrestricted mixtures of three or more covariance components remains outside the results proved here; the singular geometric asymptotics keep the configuration fixed. The deposit contains the manuscript PDF, LaTeX source, and a computation supplement. Five Python programs reproduce 191 exact algebraic identities, rational inequalities, sign certificates, and finite examples, with complete SHA-256 manifest verification and exact agreement with archived JSON reports. These finite checks complement the analytic proofs and do not replace their all-dimensional estimates and limiting arguments.