A Counterexample to the Gaussian Completely Monotone Conjecture

2026-05-12

AI Summary

This work disproves the Gaussian Complete Monotonicity (GCM) conjecture, which posits that all time derivatives of entropy along the heat flow alternate in sign. By constructing an explicit probability measure on the real line, we provide the first counterexample to GCM: the fifth-order time derivative of entropy is shown to be positive at a certain time. This counterexample simultaneously refutes both the Gaussian optimality conjecture and the entropy power conjecture, and demonstrates that even log-concave measures can violate GCM. Combining heat flow analysis, explicit computation of entropy derivatives, and a carefully designed construction of the probability measure—partially aided by GPT-5.5 Pro—this study delivers a pivotal counterexample that clarifies longstanding relationships among central conjectures in information theory and probabilistic inequalities.

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.

Problem

• Gaussian completely monotone conjecture
• entropy
• heat flow
• log-concave measure
• entropy power conjecture

Innovation

• Gaussian completely monotone conjecture
• entropy along heat flow
• counterexample
• log-concave measure
• entropy power conjecture