Proved
Conditional mathematical implication in the cited source ↗
Observed Oct 13, 2026
Theorem · version 1
The universal constant is a limit across dimensions
Statement
If the sharp fixed-dimensional unit-slack constant is (n/(n+1))^n for every n ≥ 1, then the infimum of P(S < E S + 1) over all positive finite n and admissible distributions is 1/e. No fixed finite n attains this dimension-free infimum.
Assumptions
- Independent nonnegative integrable variables, each mean at most one.
- Allow the positive finite number of variables to vary.
Scope
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.
Evidence
Source reports
Connections
In this problem · publication v6 · Selected context
Feige’s probability conjecture →
Conditional implication · Proved
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v7
Feige’s probability conjecture →
Conditional implication · Proved
Relationship not independently reviewed. This concerns its use in this problem.
Sources
Sharp small-deviation inequalities for sums of independent nonnegative random variables →
- Theorem 1.1, Corollary 1.2 and AI disclosure → · Original passage ↗
Recorded wording
Proves two parameter-range lower bounds, with the finite-n bound sharp when δ ≥ 1.
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
Conditional mathematical implication in the cited source ↗
Observed Oct 13, 2026 · Source passage →
Record details
- Claim ID
- claim-feige-conjecture-consequence
- Statement version
- 1
- SHA-256
- c00ec15684cbe9b89e93415be5856bfed3c524b9311c56c5b0421f416a103cdf