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arXiv · 2605.11656

A Counterexample to the Gaussian Completely Monotone Conjecture

Yuzhou Gu·Mark Sellke
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Abstract

We provide an explicit probability measure on $\mathbb{R}$ for which the fifth time derivative of the entropy along the heat flow is positive at some time. This disproves the Gaussian completely monotone (GCM) conjecture (Cheng-Geng '15) and therefore also the Gaussian optimality conjecture (McKean '66) and the entropy power conjecture (Toscani '15). Our proof also implies the existence of a log-concave probability measure on $\mathbb{R}$ for which the GCM conjecture fails at some order. The explicit counterexample was found by GPT-5.5 Pro.

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Yuzhou Gu, Mark Sellke. 2026-05-12. A Counterexample to the Gaussian Completely Monotone Conjecture. https://arxiv.org/abs/2605.11656

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