Proved
Earlier lower bound 1/13 ↗
Only the exact statement and passage cited here; not an independent OpenMath assessment.
Observed Oct 21, 2026
Bound · version 1
Feige’s first universal lower bound: 1/13
Statement
For every n ≥ 1 and independent nonnegative integrable X₁,…,Xₙ with E Xᵢ ≤ 1, P(S < E S + 1) ≥ 1/13, where S = Σᵢ Xᵢ.
Assumptions
- Independent nonnegative integrable variables with means at most one.
- Reported in the 2026 paper’s account of earlier work.
Scope
An earlier universal lower bound, superseded by the sharp constant 1/e.
Evidence
Source reports
Connections
In this problem · publication v7 · Selected context
Feige’s probability conjecture →
Related result · Proved
An earlier lower bound on the unit-slack probability, before the sharp 1/e.
Relationship not independently reviewed. This concerns its use in this problem.
Sources
Sharp small-deviation inequalities for sums of independent nonnegative random variables →
- §1.1, Background and related work; references [Fei04], [HZZ10], [GHLL20] → · Original passage ↗
Recorded wording
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).
paraphrase · OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.
Weibo Fu · Yanjun Han · Guanyang Wang · Jun Yan · Peng Zhang · Zhengqing Zhou · Preprint · Publication date Jul 27, 2026
Revision & attribution
Source revision: 2026-07-27. Observed Oct 13, 2026.
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.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Earlier lower bound 1/13 ↗
Only the exact statement and passage cited here; not an independent OpenMath assessment.
Observed Oct 21, 2026 · Source passage →
Record details
- Claim ID
- claim-feige-bound-one-thirteenth
- Statement version
- 1
- SHA-256
- 3ca1f122df7023887ebdfee9a9a335282fb7e27c649d73998a02808bf08dc972