A Counterexample to the Gaussian Completely Monotone Conjecture
Probability
2026-05-13 v1 Information Theory
math.IT
Abstract
We provide an explicit probability measure on 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 for which the GCM conjecture fails at some order. The explicit counterexample was found by GPT-5.5 Pro.
Cite
@article{arxiv.2605.11656,
title = {A Counterexample to the Gaussian Completely Monotone Conjecture},
author = {Yuzhou Gu and Mark Sellke},
journal= {arXiv preprint arXiv:2605.11656},
year = {2026}
}