A sharp smoothing inequality in Gauss space and the exact threshold for local entropy concavity
Year: 2026
Type: preprint
Abstract: Unrefereed research candidate. In one sentence: near the Gaussian, the entropy of a noisy mixture of two copies of a symmetric signal is concave in the mixing weight exactly when the signal fraction is at most 4/7 (total signal-to-noise ratio 4/3), while with three or more copies concavity on the whole simplex of weights fails at every signal fraction. Take two independent copies of a symmetric random variable, mix them with weights √t and √(1−t), and add Gaussian noise; the signal fraction q is the share of the output variance that comes from the signal. Ball, Nayar and Tkocz conjectured that the entropy of the mixture is concave in t (without noise, for log-concave inputs). Luo (2026) disproved this and showed that symmetric near-Gaussian counterexamples exist for every q in (4/7, 1]. Schubr (2026) proved local concavity near the Gaussian for q < 1/2, and the Evidence Press release The critical signal fraction for local entropy concavity (doi:10.5281/zenodo.23910629) for q < 4/7, leaving q = 4/7 open. This paper settles the endpoint and extends that release: there is one radius δ* > 0, valid for every q ≤ 4/7 (δ* = 10−8 in the relative supremum norm is admissible), within which every admissible symmetric input has a concave entropy curve (Theorem C). With Luo's theorem, written out here with Luo's proof as Theorem D, the local-concavity set is exactly (0, 4/7], in the relative supremum norm and in every relative Cmb norm. The proof rests on a sharp smoothing inequality for bounded functions in Gauss space (Theorem A): for Ornstein–Uhlenbeck levels r < s < 1 it gives the largest exponent E(r,s) in ‖T√sg‖22 ≤ C‖g‖∞2−E‖T√rg‖2E; E ≥ 1 exactly when s ≤ √(2r(1+r)) − r, Luo's critical level, where the inequality holds with a logarithmic gain; Theorem B is a weighted form. For three or more copies (Theorem E), a fourth-cumulant obstruction of Madiman, Nayar and Tkocz (2020) is already present for three summands with noise: at each q in (0,1) there are smooth, strongly log-concave symmetric inputs arbitrarily close to the Gaussian whose entropy is not concave on the simplex, with the failure near a face; with Luo's theorem on an edge, no signal fraction in (0,1] gives local concavity on the whole simplex. Concavity on a fixed interior region of the simplex remains open. The effects are minute (the proved neighbourhood allows the input density to differ from the Gaussian density by a factor of at most 1 ± 10−8; the largest computed failure signal is about 6×10−8 nats); the interest is structural. The archive contains the paper (PDF and Markdown source), three figures, eleven Python check scripts plus comparison and figure scripts, the outputs of one complete run, a one-command replay (run_all.sh) that compares regenerated outputs with the shipped ones byte for byte, the software environment, the bibliography, a rating-free summary of the internal review, the research-gate record, CI and SHA-256 sums. Assurance: written analytic proofs; the computations are finite-precision diagnostics, not interval certificates and not part of any proof; internal AI-assisted review only (not peer review, independent reproduction or formal verification). Creator: Anonymous. Text, data and figures CC0-1.0; code MIT.
Author: Anonymous
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225790354