Problem
Research questions and friction points this paper is trying to address.
log-concave
Gaussian completely monotone conjecture
counterexamples
An Explicit Family of Log-Concave Counterexamples to the Gaussian Completely Monotone Conjecture
🤖 AI Summary
本文通过构造光滑且严格对数凹的反例，解决了高斯完全单调猜想问题，并采用解析方法在不同维度上验证了这些反例的有效性。
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
log-concave counterexamples
Gaussian completely monotone conjecture
entropy derivative
heat-flow formula
tensorization
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