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An Explicit Family of Log-Concave Counterexamples to the Gaussian Completely Monotone Conjecture
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 fm whose signed mth 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.
Related
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BibTeX
@misc{zou2026explicit,
title = {{An Explicit Family of Log-Concave Counterexamples to the Gaussian Completely Monotone Conjecture}},
author = {Jiayang Zou and Luyao Fan and Jiayang Gao and Jia Wang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30275},
eprint = {2608.30275},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30275},
}
RIS
TY - UNPB
TI - An Explicit Family of Log-Concave Counterexamples to the Gaussian Completely Monotone Conjecture
AU - Jiayang Zou
AU - Luyao Fan
AU - Jiayang Gao
AU - Jia Wang
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30275
UR - https://arxiv.org/abs/2608.30275
AB - 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 mth 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.
ER -