Proved
Unit-slack bound proved by Nie and Wei ↗
Their separately authored proof reaches the same finite-dimensional constant.
Source dated Jul 27, 2026 · Observed Oct 13, 2026
Bound · version 1
The dimension-free unit-slack inequality
Statement
For every n ≥ 1 and independent nonnegative integrable X₁,…,Xₙ with E Xᵢ ≤ 1, , where .
Assumptions
- Positive finite number of independent variables.
- Nonnegative variables with each mean at most one; strict threshold.
Scope
The conventional unit-slack conjecture. The stronger finite-n theorem yields this bound and approaches 1/e as n grows.
Explanation
In Feige’s probability conjecture · publication v5
Bounds & computations · 1
Compare the quantity and hypotheses within each group. A finite check, conditional theorem and proposed improvement cover different scopes.
| Result | Value | Conditions |
|---|---|---|
| The dimension-free unit-slack inequality → Proved Selected source date: | ≥ 1/e | All positive finite dimensions; independent nonnegative variables with means at most one, strict threshold. |
In Feige’s probability conjecture · publication v7
Bounds & computations · 1
Compare the quantity and hypotheses within each group. A finite check, conditional theorem and proposed improvement cover different scopes.
| Result | Value | Conditions |
|---|---|---|
| The dimension-free unit-slack inequality → Proved Selected source date: | ≥ 1/e | All positive finite dimensions; independent nonnegative variables with means at most one, strict threshold. |
Evidence
Source reports
Proved
Sharp unit-slack bound proved by Fu and coauthors ↗
Theorem 1.1 at δ = 1, including finite-dimensional sharpness.
Source dated Jul 27, 2026 · Observed Oct 13, 2026
Connections
In this problem · publication v5 · Selected context
Feige’s probability conjecture →
Primary formulation · Proved
Related statements
The finite constants (n/(n+1))^n decrease to 1/e; each is strictly larger than 1/e.
Finite-n sharpness and the limiting universal bound are not identical statements.
Details & sources
The optimal bound for a fixed number of variables →
Proved · version 1
For every n ≥ 1, the optimal universal lower bound for is , over independent nonnegative integrable X₁,…,Xₙ with E Xᵢ ≤ 1.
- Fix positive finite n.
- Strict unit-slack event and individual means at most one.
Scope: Both July arguments support the same mathematical assertion. The pinned Lean declaration formalizes this fixed-dimensional optimality statement.
The dimension-free unit-slack inequality →
Proved · version 1
For every n ≥ 1 and independent nonnegative integrable X₁,…,Xₙ with E Xᵢ ≤ 1, , where .
- Positive finite number of independent variables.
- Nonnegative variables with each mean at most one; strict threshold.
Scope: The conventional unit-slack conjecture. The stronger finite-n theorem yields this bound and approaches 1/e as n grows.
- Sharp small-deviation inequalities for sums of independent nonnegative random variables →
- On Feige’s conjecture →
Relationship not independently reviewed.
Connection in Feige’s probability conjecture →Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v7
Feige’s probability conjecture →
Primary formulation · Proved
Related statements
The finite constants (n/(n+1))^n decrease to 1/e; each is strictly larger than 1/e.
Finite-n sharpness and the limiting universal bound are not identical statements.
Details & sources
The optimal bound for a fixed number of variables →
Proved · version 1
For every n ≥ 1, the optimal universal lower bound for is , over independent nonnegative integrable X₁,…,Xₙ with E Xᵢ ≤ 1.
- Fix positive finite n.
- Strict unit-slack event and individual means at most one.
Scope: Both July arguments support the same mathematical assertion. The pinned Lean declaration formalizes this fixed-dimensional optimality statement.
The dimension-free unit-slack inequality →
Proved · version 1
For every n ≥ 1 and independent nonnegative integrable X₁,…,Xₙ with E Xᵢ ≤ 1, , where .
- Positive finite number of independent variables.
- Nonnegative variables with each mean at most one; strict threshold.
Scope: The conventional unit-slack conjecture. The stronger finite-n theorem yields this bound and approaches 1/e as n grows.
- Sharp small-deviation inequalities for sums of independent nonnegative random variables →
- On Feige’s conjecture →
Relationship not independently reviewed.
Connection in Feige’s probability conjecture →Relationship not independently reviewed. This concerns its use in this problem.
Sources
On Feige’s conjecture →
- Theorem 1.3 and AI acknowledgement; arXiv submitted 27 July, HTML manuscript dated 24 August → · Original passage ↗
Recorded wording
Gives a separate argument for the unit-slack inequality and the bound δ(n/(n+δ))^n for 0 < δ ≤ 1 under unit means.
paraphrase · OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.
Zipei Nie · Jiaye Wei · Preprint · Publication date Jul 27, 2026
Revision & attribution
Source revision: 2026-07-27. Observed Oct 13, 2026.
Nie and Wei credit GPT-5.6 Sol for exploration and discovery, state that they checked the arguments and authored the text, and retain responsibility. Their argument uses the Vlassis–Thomas calibration theorem.
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.
Lean proof of the sharp unit-slack Feige bound →
- PalomarFeige.sharp_unit_slack_feige_complete; README and formalization metadata → · Original passage ↗
Recorded wording
The headline theorem asserts optimality of (n/(n+1))^n for every positive finite n in the strict unit-slack event.
paraphrase · OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.
Guanyang Wang · Peng Zhang · Project documentation
Public formal source; execution and statement correspondence are separate checks.
Revision & attribution
Source revision: 19c61d6cb556d1e967dbb0ada9f293c290837867. Observed Oct 13, 2026.
The project credits GPT-5.6 Sol Codex for formalization. This is a different contribution from the July discovery model. Authors report verification; OpenMath has not executed it.
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 · 2
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Sharp unit-slack bound proved by Fu and coauthors ↗
Theorem 1.1 at δ = 1, including finite-dimensional sharpness.
Source dated Jul 27, 2026 · Observed Oct 13, 2026 · Source passage →
Proved
Unit-slack bound proved by Nie and Wei ↗
Their separately authored proof reaches the same finite-dimensional constant.
Source dated Jul 27, 2026 · Observed Oct 13, 2026 · Source passage →
Record details
- Claim ID
- claim-feige-unit-slack
- Statement version
- 1
- SHA-256
- b955f33690953767f18856c5922298322b37125b939ac1a55a8c015ff26fb18e