An Explicit Family of Log-Concave Counterexamples to the Gaussian Completely Monotone Conjecture
2026-08-31 • Information Theory
Information Theory
AI summaryⓘ
The authors found smooth examples that go against a mathematical idea called the Gaussian completely monotone conjecture. In simple terms, they showed that a certain property expected to always be positive can actually become negative, starting with functions in one dimension and then expanding to higher dimensions. Their proof uses detailed analysis involving entropy and heat flow, and they leveraged GPT-5.6 Sol Pro to help develop the argument. This work challenges previously held assumptions about Gaussian functions.
Gaussian functionslog-concavecompletely monotoneentropy derivativesheat flowFourier modestensorizationanalytic proofsigned derivativeentropy calculation
Authors
Jiayang Zou, Luyao Fan, Jiayang Gao, Jia Wang
Abstract
We construct smooth, strictly log-concave counterexamples to the Gaussian completely monotone conjecture in every dimension. In one dimension, they form an explicit family $f_m$ whose signed $m$th entropy derivative at time zero is negative for every sufficiently large $m$; the inequality persists for all sufficiently small positive times. Tensorization with a broad Gaussian factor gives the higher-dimensional examples. The argument is analytic and self-contained. It reduces the sign to a two-frequency entropy calculation on the circle and transfers the resulting asymptotic to the real line through an exact heat-flow formula for Gaussian-windowed Fourier modes. The proof was developed by GPT-5.6 Sol Pro under the authors' guidance.