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Feige's Conjecture

Probability & statistics · posed by Uriel Feige, 2004 · solved

1 attempt · 1 machine check

Statement

Let X1,…,XnX_1,\ldots,X_n be independent nonnegative random variables with EXi≤1\mathbb{E}X_i \le 1, and let SS be their sum. Is P(S<ES+1)≥1/e\mathbb{P}(S < \mathbb{E}S + 1) \ge 1/e? Feige proved the constant 1/131/13 and conjectured the sharp 1/e1/e. 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 δ≥1\delta \ge 1.

Context

Feige's 2004 bound, known across probability and TCS with two decades of partial results.

People

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Attempts

1 attempt

No person has examined this. There is 1 attempt here and 1 machine check recorded against it. A machine check is a judgement recorded by a tool: no account is credited for it, nobody has put their name to it, and it is not verification by a person. Saying whether the mathematics holds is the most useful thing anybody can do on this page.

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  • #1

    Attempt 1

    proof attemptChatGPT 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, Mark Stander ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    ChatGPT 5.6 Pro, GPT-5.6 Sol, Codex
    people
    Weibo Fu, Yanjun Han, Guanyang Wang, Jun Yan, Peng Zhang, Zhengqing Zhou, Zipei Nie, Jiaye Wei, Mark Stander

    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.

    Reviews

    1 machine check

    No person has reviewed this attempt. 1 machine check below — a machine check is not human verification.

    • Machine check · not human verification

      machine: correct

      Recorded from Lean ·

      scope Lean formalization of the result

      An end-to-end Lean formalization of the e−1e^{-1} 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.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

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