AI 中文总结
研究费格猜想，借助GPT - 5.6 Sol并基于前人突破给出简短证明，证明对于期望为\(1\)的\(n\)个独立非负随机变量，其和小于\(n + 1\)的概率至少为\((\frac{n}{n + 1})^n \geq \frac{1}{e}\)，还讨论了后续工作影响。
AI 中文摘要
我们给出了费格猜想的一个简短证明：对于\(n\)个期望为\(1\)的独立非负随机变量，它们的和小于\(n + 1\)的概率至少为\((\frac{n}{n + 1})^n \geq \frac{1}{e}\)。该证明借助GPT - 5.6 Sol完成，基于弗拉西斯和托马斯在无分布\(p\)值的有限样本有效性方面建立加夫克猜想的近期突破。我们还讨论了明、拉姆达斯、沈、王和沃迪 - 史密斯后续工作的影响。
英文摘要
We present a short proof of Feige's conjecture: for $n$ independent nonnegative random variables with expectation one, the probability that their sum is less than $n+1$ is at least $\left(\frac{n}{n+1}\right)^n\ge \frac{1}{e}$. The proof was obtained with the assistance of GPT-5.6 Sol and builds on the recent breakthrough of Vlassis and Thomas establishing Gaffke's conjecture on the finite-sample validity of a distribution-free $p$-value. We also discuss the implications of the subsequent work of Ming, Ramdas, Shen, Wang, and Waudby-Smith.
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