Proves two parameter-range lower bounds, with the finite-n bound sharp when δ ≥ 1.
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The exact passages cited in OpenMath. Each passage identifies how its wording was recorded.
Feige proved the first universal lower bound, 1/13 (2004); it was later improved to 1/8 by He, Zhang and Zhang (2010), to approximately 0.14 by Garnett (2020), and to 0.1798 by Guo, He, Ling and Liu (2020).
Passage ID: feige-conjecture:entry:fu-history:v1
Source item: §1.1, Background and related work; references [Fei04], [HZZ10], [GHLL20]
Source record: feige-conjecture:source:fu:v1
Proves two parameter-range lower bounds, with the finite-n bound sharp when δ ≥ 1.
Passage ID: feige-conjecture:entry:fu:v1
Source item: fu
Source record: feige-conjecture:source:fu:v1
Explicit citations in versioned problem publications. A citation records a use; it does not confer proof or independent verification.
Problem publication v7
Source assessment
Proved
Earlier lower bound 1/8
Only the exact statement and passage cited here; not an independent OpenMath assessment.
Observed Oct 21, 2026
Source assessment
Proved
Earlier lower bound 0.1798
Only the exact statement and passage cited here; not an independent OpenMath assessment.
Observed Oct 21, 2026
Source assessment
Proved
Earlier lower bound 1/13
Only the exact statement and passage cited here; not an independent OpenMath assessment.
Observed Oct 21, 2026
Source assessment
Proved
Conditional mathematical implication in the cited source
Finite-dimensional equality examples attain their own larger constants. The limiting value 1/e is approached as n grows, rather than attained in one finite dimension.
Observed Oct 13, 2026
Source assessment
Proved
Sharp unit-slack bound proved by Fu and coauthors
Theorem 1.1 at δ = 1, including finite-dimensional sharpness.
Source date Jul 27, 2026 · Observed Oct 13, 2026
Source assessment
Proved
Sharp unit-slack bound proved by Fu and coauthors
Theorem 1.1 at δ = 1, including finite-dimensional sharpness.
Source date Jul 27, 2026 · Observed Oct 13, 2026
Construction
A finite-dimensional extremizer
Proof explanation
Fu–Han–Wang–Yan–Zhang–Zhou account
Source assessment
Proved
Two parameter-range bounds proved in the cited paper
Lower bound δ(n/(n+δ))^n for 0 < δ < 1, and (1−1/(n+δ))^n for δ ≥ 1. Sharpness is asserted only in the second range.
Source date Jul 27, 2026 · Observed Oct 13, 2026
Question
Feige’s probability conjecture
Connection
The finite constants (n/(n+1))^n decrease to 1/e; each is strictly larger than 1/e.
Definition
Strict small-deviation event
History
Separately authored proofs of the unit-slack bound
Further reading
Read the mathematical argument; inspect its Claim for assessment and scope.
Recorded authorship
Weibo Fu · Yanjun Han · Guanyang Wang · Jun Yan · Peng Zhang · Zhengqing Zhou
Weibo Fu, Yanjun Han, Guanyang Wang, Jun Yan, Peng Zhang, Zhengqing Zhou
The authors report GPT-5.6 Pro assistance in proof discovery; the calibration theorem of Vlassis and Thomas and Grünbaum-type inequalities are mathematical inputs, not authors of this paper.
OpenMath source records preserve exact citations. Observation dates order these records; they do not establish the latest upstream paper or proof.
Sharp small-deviation inequalities for sums of independent nonnegative random variables
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