高斯完全单调猜想的显式对数凹反例族
An Explicit Family of Log-Concave Counterexamples to the Gaussian Completely Monotone Conjecture
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中文总结 AI 辅助
该研究构造了各维均光滑严格对数凹的高斯完全单调猜想反例，一维显式族$f_m$的符号$m$阶熵导数为负，通过张量积得高维示例，证明由GPT-5.6 Sol Pro完成。
中文摘要 AI 辅助
我们在每一维中构造了光滑、严格对数凹的高斯完全单调猜想反例。在一维情况下，它们形成显式族$ f_m $，其在时间零处的符号$ m $阶熵导数对所有足够大的$ m $均为负，且该不等式对所有足够小的正时间持续成立。通过与宽高斯因子进行张量积可得到高维示例。该论证是解析且自包含的，它将符号问题简化为圆上的双频熵计算，并通过高斯窗傅里叶模式的精确热流公式将所得渐近结果转移到实轴上。该证明由GPT-5.6 Sol Pro在作者指导下完成。
英文摘要
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.
发表机构
- Stanford University（斯坦福大学）
- Shanghai Jiao Tong University（上海交通大学）
机构由 AI 辅助整理，请以论文原文为准。
