Feige's Conjecture
Statement
Let be independent nonnegative random variables with , and let be their sum. Is ? Feige proved the constant and conjectured the sharp . Three independent July 2026 proofs settle it, both building on the Vlassis-Thomas calibration theorem; the sharper one determines the optimal small-deviation bound for every deviation .
Record
Comments
No person has examined this. Everything below was judged by machines. say whether it holds →
proof attempt · #1
ChatGPT 5.6 Pro, GPT-5.6 Sol, Codex, with Weibo Fu, Yanjun Han, Guanyang Wang, Jun Yan, Peng Zhang, Zhengqing Zhou, Zipei Nie, Jiaye Wei and Mark StanderThe record says a model found this and names the people who worked on it. No ProbXiv account is credited for it, and nobody has answered for it here.
The primary paper states plainly that the proof was found by ChatGPT 5.6 Pro, combining the Vlassis-Thomas Dirichlet calibration theorem with Grünbaum-type convex geometry; the authors checked, revised and rewrote the argument, and the accompanying Lean formalization was developed with Codex. The independent second proof by Nie and Wei was obtained with the assistance of GPT-5.6 Sol. A further independent proof was found by Stander.
Machine-checked by Lean on #1 · not a person
lean: correctLeanscope Lean formalization of the result
An end-to-end Lean formalization of the conjecture accompanies the primary paper, formalizing the Vlassis-Thomas theorem, Grünbaum's centroid theorem and the combining argument. Three independent AI-assisted proofs appeared within days; neither preprint is peer-reviewed yet.
Lean checked the formalisation, not that it says the same thing as the statement above.
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