VibeMathedMath problems solved by AI
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Feige's Conjecture

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.

Result
Proved
Status
Resolved
AI contribution
AI-discovered
Method
Argument
Field
Probability
Posed by
Uriel Feige
Year posed
2004
Years open
22y
Solved
2026-07-27
Model
ChatGPT 5.6 Pro, GPT-5.6 Sol, Codex
Vendor
OpenAI
Collaborators
Weibo Fu, Yanjun Han, Guanyang Wang, Jun Yan, Peng Zhang, Zhengqing Zhou, Zipei Nie, Jiaye Wei, Mark Stander
Verification
Lean-verified
Publication
Preprint
Significance
35 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What the AI did

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.

Verification

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.

Sources

arXiv:2607.23980 - Sharp small-deviation inequalities for sums of independent nonnegative random variables

Changelog6 changes
  • HiddenPanther983changed More links from “Independent second proof (Nie and Wei), arXiv:2607.24528 | https://arxiv.org/abs/2607.2452…” to “Independent second proof (Nie and Wei), arXiv:2607.24528 | https://arxiv.org/abs/2607.2452…”
  • HiddenPanther983changed More links from “Independent second proof (Nie and Wei), arXiv:2607.24528 | https://arxiv.org/abs/2607.2452…” to “Independent second proof (Nie and Wei), arXiv:2607.24528 | https://arxiv.org/abs/2607.2452…”
  • HiddenPanther983changed Verification note from “An end-to-end Lean formalization of the $e^{-1}$ conjecture accompanies the primary paper,…” to “An end-to-end Lean formalization of the $e^{-1}$ conjecture accompanies the primary paper,…”
  • HiddenPanther983changed What the AI did from “The primary paper states plainly that the proof was found by ChatGPT 5.6 Pro, combining th…” to “The primary paper states plainly that the proof was found by ChatGPT 5.6 Pro, combining th…”
  • HiddenPanther983changed Collaborators from “Weibo Fu, Yanjun Han, Guanyang Wang, Jun Yan, Peng Zhang, Zhengqing Zhou, Zipei Nie, Jiaye…” to “Weibo Fu, Yanjun Han, Guanyang Wang, Jun Yan, Peng Zhang, Zhengqing Zhou, Zipei Nie, Jiaye…”
  • HiddenPanther983changed Statement from “Let $X_1,\ldots,X_n$ be independent nonnegative random variables with $\mathbb{E}X_i \le 1…” to “Let $X_1,\ldots,X_n$ be independent nonnegative random variables with $\mathbb{E}X_i \le 1…”

Discussion