Mixing-weight entropy concavity near the Gaussian below one-half signal

Scott Schubr

Abstract

We establish a Gaussian neighborhood of mixing-weight entropy concavity for symmetric variance-one inputs below one-half retained signal. For each prescribed upper signal fraction less than one half, an explicit uniform relative-density radius guarantees concavity, strictly for non-Gaussian inputs, for every mixing weight and every smaller positive signal fraction. This gives a negative answer, on that signal range, to Congyi Luo's question about near-Gaussian counterexample sequences, including sequences with strong log-concavity and any prescribed finite Gaussian moment matching. The proof controls arbitrary bounded even perturbations through an infinite-mode Hermite estimate that absorbs nonlinear curvature errors relative to the quadratic energy. Complementary finite-jet estimates yield a common all-weight restoration threshold for families with a common absolute-moment envelope and a positive finite Hermite discrepancy, as well as eventual concavity for each fixed symmetric non-Gaussian input with all moments finite. They also force every fixed Gaussian-moment discrepancy of a family with nonnegative small-signal curvature to decay faster than every fixed algebraic power of the signal fraction. A uniform log-concave-class endpoint estimate and exactly moment-matched exponential-tail examples distinguish the resulting restoration regimes.

Mixing-weight entropy concavity near the Gaussian below one-half signal

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