The critical signal fraction for local entropy concavity

Unrefereed research candidate. In one sentence: near the Gaussian, Gaussian noise restores entropy concavity for symmetric inputs exactly when the noise variance exceeds three quarters of the signal variance (signal fraction q < 4/7, signal-to-noise ratio 4/3). Take two independent copies of a random signal, add Gaussian noise to each, and mix the noisy copies with weights √t and √(1−t). Ball, Nayar and Tkocz conjectured that the entropy of the mixture is concave in t for log-concave signals. Luo (2026) disproved this and showed that symmetric near-Gaussian counterexamples survive added noise whenever the signal is more than 4/7 of the total variance; Schubr (2026) proved local concavity below 1/2. This paper proves that for every q < 4/7 there is an explicit neighbourhood of the Gaussian (a uniform bound on f/φ − 1) in which every symmetric variance-one input has a strictly concave entropy curve (Theorem A), so with Luo's counterexamples the supremum of such signal fractions is exactly 4/7; whether q = 4/7 itself qualifies is open. For every q in (4/7, 1], one sequence of smooth strongly log-concave near-Gaussian inputs makes equal weighting a strict local entropy minimum, with exact curvature constant γ(q) = 21√5 q/(49q − 4) and a Gaussian-shaped local profile (Theorem B); the sign on (4/7, 2 − √2], Luo's Problem 10.3, was also proposed in an earlier public, unrefereed GitHub argument (1 November 2026), which is cited. The paper also proves that the Ornstein–Uhlenbeck smoothing inequality behind Theorem A is sharp, with 4/7 its fixed point and Luo's family the extremal Hermite test function (Proposition S); that Luo's family is strictly concave near equal weighting at q = 4/7 (Proposition E); and that the proof's radius shrinks like (4/7 − Q)9/4 (Remark R). The effects are minute (entropy dips of order 10−13 nats); the results are structural. The archive contains the manuscript (PDF, Markdown source and generated LaTeX), the figure and its script, the bibliography, five standard-library Python checkers with retained outputs in normal and optimized mode (exact identities, closed-form constants, the radius table, finite-n asymptotics and 16 deliberate mutants that must fail), a rating-free internal review summary, the research-gate record, the superseded first internal version, CI and SHA-256 sums. Assurance: written analytic proofs; finite-precision checks are not interval certificates; internal AI-assisted review only (not peer review, independent reproduction or formal verification). Creator: Anonymous. Text, data and figures CC0-1.0; code MIT.