OpenAI · 186-01 · 2026-10-05 · Combinatorics

A uniform influence bound for hypergraph properties

preprints/A-uniform-influence-bound-for-hypergraph-properties-October-5-2026

Family 186. The Friedgut–Kalai graph and hypergraph threshold conjectures.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

Proves the Friedgut–Kalai threshold-width conjectures for graphs and fixed-uniformity hypergraphs. For fixed $0 < ε < 1/2$ , every nontrivial increasing relabeling-invariant property crosses from probability $ε$ to $1 - ε$ within width $O ((lo g n)^{- 2})$ for graphs and $O_{r} ((lo g n)^{- r / (r - 1)})$ for $r$ -uniform hypergraphs, $r \geq 3$ . The hypergraph influence bound also applies to nonmonotone properties.

For every fixed integer r ≥ 3, we prove that every relabeling-invariant Boolean property of simple r-uniform hypergraphs on n vertices satisfies $Var_{p} (f) \leq C_{r} I_{p} (f) / (lo g n)^{r / (r - 1)}$ . The constant depends only on r, and the bound holds uniformly for all $0 < p < 1$ without a monotonicity assumption. For increasing properties, it gives the corresponding threshold-width bound with exponent $r / (r - 1)$ , proving the hypergraph threshold-width conjecture of Friedgut and Kalai.

Preprint source and Lean scope

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A Lean scope note is stored for family 186, but it does not name this preprint. It names 186-02. This page still has the full preprint source. No compilation is asserted. O.1 stays NOT_CLOSED.