Feige’s probability conjecture

4 selected milestones · Source coverage checked 2026-10-13 · Historical annotation and mathematical review remain separate.

1. 2006 publication

   Foundation

   Feige’s probability question

   The July papers cite Uriel Feige’s journal formulation of a lower-tail bound using independence and first moments.

   Scope & review: The conference version dates to 2004; this milestone uses the journal year.

   Statements & sources

   • The dimension-free unit-slack inequality →

   • Sharp small-deviation inequalities for sums of independent nonnegative random variables ↗Theorem 1.1, Corollary 1.2 and AI disclosure
   • On Feige’s conjecture ↗Theorem 1.3 and AI acknowledgement; arXiv submitted 27 July, HTML manuscript dated 24 August

   Historical annotation review: pending. The date refers to the publication, with the precision supported by the source.

2. Jul 27, 2026 preprint

   Bound established

   Separately authored proofs of the unit-slack bound

   Fu and coauthors and Nie–Wei post arguments reaching the sharp finite-n unit-slack constant.

   Scope & review: Authorship, AI contributions and mathematical input credits remain separate.

   Statements & sources

   • The dimension-free unit-slack inequality →
   • The optimal bound for a fixed number of variables →
   • The July bounds in two deviation ranges →

   • Sharp small-deviation inequalities for sums of independent nonnegative random variables ↗Theorem 1.1, Corollary 1.2 and AI disclosure
   • On Feige’s conjecture ↗Theorem 1.3 and AI acknowledgement; arXiv submitted 27 July, HTML manuscript dated 24 August

   Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.

3. Aug 18, 2026 preprint

   Proposal

   A broader Samuels proof is proposed

   Ling’s later paper claims a formula for arbitrary mean vectors and all positive deviations.

   Scope & review: The general formula has its own assessment and is not covered by the unit-slack Lean theorem.

   Statements & sources

   • The later proposal for every positive deviation →

   • On Samuels’ Conjecture ↗Theorem 1.1 and Corollary 1.3; arXiv submitted 18 August, manuscript dated 19 August

   Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.

4. Date unavailable formalization

   Formalization

   Unit-slack optimality has a public Lean proof

   Wang and Zhang’s project publishes the exact fixed-dimensional optimality declaration.

   Scope & review: The pinned source is inspected, without an OpenMath proof execution or an inferred first-publication date.

   Statements & sources

   • The optimal bound for a fixed number of variables →

   • Lean proof of the sharp unit-slack Feige bound ↗PalomarFeige.sharp_unit_slack_feige_complete; README and formalization metadata

   Historical annotation review: pending. The date refers to the formalization, with the precision supported by the source.