Feige's conjecture on sums of nonnegative random variables settled
For independent nonnegative random variables with mean at most 1 and sum S, the probability that S is below its mean plus one is at least 1/e, the sharp constant Feige conjectured in 2004; three independent proofs appeared within days.
- Lab
- Independent
- Model
- ChatGPT 5.6 Pro
- Field
- Mathematics
- Date
- 2026-07-27
- Human collaborators
- Weibo Fu, Yanjun Han, Guanyang Wang, Jun Yan, Peng Zhang, Zhengqing Zhou
- Problem posed
- 2004 · open 22 yrs
What was found
Feige proved in 2004 that for independent nonnegative random variables X_1..X_n with E[X_i] <= 1 and S their sum, P(S < E[S] + 1) is bounded below by a positive constant, established 1/13, and conjectured the sharp value 1/e. The primary paper states plainly in its abstract that the proof was found by ChatGPT 5.6 Pro, which combined the Vlassis-Thomas Dirichlet calibration theorem with Grunbaum-type convex geometry; the authors then checked, revised and rewrote the argument. The sharper form determines the optimal small-deviation bound. An end-to-end Lean formalization accompanies the paper, covering the Vlassis-Thomas theorem, Grunbaum's centroid theorem and the combining argument, developed with Codex.
Novelty check
Feige's conjecture is documented as open from his 2004 paper on sums of independent random variables, where he proved the weaker constant 1/13. The sharp 1/e constant is recorded as an open problem in the subsequent small-deviation literature and had stood for 22 years. No prior proof of the sharp constant appears; the near-simultaneous arrival of three independent proofs within days confirms it was live and open, not settled and forgotten.
Caveats and known objections
Not peer-reviewed; all three proofs are preprints. The Lean formalization accompanies the primary paper and is author-produced. Autonomy is graded ai-led rather than autonomous because the human authors posed the problem, then checked, revised and rewrote the argument for publication. The three independent proofs are genuine corroboration of the result, though two of the three were themselves AI-assisted, so they are not fully independent of the method.
Independent checks
Zipei Nie and Jiaye Wei (independent second proof): independently proved the same sharp 1/e bound, obtained with assistance from GPT-5.6 Sol · link ↗
Mark Stander (independent third proof): further independent proof of the conjecture · link ↗
Disagree with these grades?
Bring a citation: a grade moves on evidence, not on argument.
Or on GitHub: submit a check challenge the grade send a correction or send a pull request
Flag this for triage
Signals order the review queue and nothing else. They are never published, and they never move a grade: that takes a citation.
Entry history (1 event)
- AddedEntered the registry graded Formally verified and AI-led.
Entries are never deleted. A grade that does not hold up is downgraded on the record, with the reason beside it.
Graded formally verified for verification and ai-led for autonomy. What these mean.
Cite this entry
whataifound.org. (2026). Feige's conjecture on sums of nonnegative random variables settled. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-07-27-feige-conjectureBibTeX
@misc{whataifound-independent-2026-conjecture,
title = {Feige's conjecture on sums of nonnegative random variables settled},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Formally verified. Autonomy: AI-led.},
url = {https://whataifound.org/finding/2026-07-27-feige-conjecture}
}