Uniform influence and sharp thresholds for graph and hypergraph properties
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Proves the Friedgut–Kalai threshold-width conjectures for graphs and fixed-uniformity hypergraphs. For fixed \(0\lt \varepsilon\lt 1/2\), every nontrivial increasing relabeling-invariant property crosses from probability ε to \(1-\varepsilon\) within width \(O((\log n)^{-2})\) for graphs and \(O_r((\log n)^{-r/(r-1)})\) for r-uniform hypergraphs, r ≥ 3. The hypergraph influence bound also applies to nonmonotone properties. (OpenAI’s Lean note)
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Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
The other possible matches are Theorem 1.1 in A uniform influence bound for hypergraph properties () and Theorem 1.2(Variance and influence) in the same paper ().
For every integer n\ge2, every nontrivial increasing family \mathcal P of graphs on n vertices invariant under all vertex permutations, and every 0<\varepsilon<1/2,
For every fixed integer r ≥ 3, we prove that every relabeling-invariant Boolean property of simple r-uniform hypergraphs on n vertices satisfies \(\mathop{\mathrm{Var}}\nolimits _p(f)\le C_r I_p(f)/(\log n)^{r/(r-1)}\). The constant depends only on r, and the bound holds uniformly for all \(0\lt p\lt 1\) 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.
We prove the Friedgut–Kalai sharp-threshold conjecture. For every integer n ≥ 2, every nontrivial increasing family of graphs on n vertices invariant under all vertex permutations, and every \(0\lt \varepsilon\lt 1/2\), the edge probabilities at which its probability equals ε and \(1-\varepsilon\) differ by at most \(C\log(1/(2\varepsilon))/(\log n)^2\), for a universal constant C.
One Lean statement in this family
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For every integer n\ge2, every nontrivial increasing family \mathcal P of graphs on n vertices invariant under all vertex permutations, and every 0<\varepsilon<1/2,
SharpThreshold.lean, the release lists it as a supporting statement.
theoremsharp_threshold_width : ∀ (n : ℕ) (f : GraphConfig n → Bool) (ε : ℝ),2 ≤ n → VertexInvariant f → IncreasingGraphProperty f → NontrivialGraphProperty f → 0 < ε → ε < (1 / 2 : ℝ) → graphQuantile (1 - ε) f - graphQuantile ε f ≤ ((2 : ℝ) ^ 19 / (Real.log (n : ℝ)) ^ 2) * Real.log (1 / (2 * ε)) := bysorry
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Independent comparisons and proof-verification reports
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MathVet finds that the Lean statements express part of the advertised result (their term: “partial”).
Their note:Only the graph case is stated: sharp_threshold_width for vertex-invariant increasing nontrivial f on GraphConfig n, graphQuantile (1-ε) f - graphQuantile ε f ≤ (2^19/(log n)^2) * log(1/(2ε)). The r-uniform hypergraph width O_r((log n)^{-r/(r-1)}) for r >= 3 and the nonmonotone influence bound named in the summary have no statement (docs: graph case only).
“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.
Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNLINKED”).
The gap they name:The whole hypergraph half (threshold width O_r((log n)^{-r/(r−1)}) for r ≥ 3, and the Var ≤ C_r·I/(log n)^{r/(r−1)} influence bound for nonmonotone properties) — the October 5 manuscript, which the scope note does not link.
“NARROWER — UNLINKED”, in their words: The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
The Friedgut–Kalai sharp-threshold conjecture concerns how quickly a nontrivial increasing graph property appears in the independent-edge random graph. For every $n\ge2$, every such property invariant under vertex relabeling, and $0<\varepsilon<1/2$, the formalization proves that the edge-probability interval between probabilities $\varepsilon$ and $1-\varepsilon$ has width at most $2^{19}\log(1/(2\varepsilon))/(\log n)^2$. The constant is uniform over the graph property, $n$, and $\varepsilon$.
AI-drafted comparisons, with no recorded human review
Sharp threshold width for monotone graph propertiesSharpThreshold
Unreviewed AI draft: states a paper’s theorem as the paper does. States the paper’s Friedgut–Kalai theorem with its constant: threshold width at most 2^19 log(1/(2ε))/(log n)² for symmetric increasing graph properties. The variance–influence theorem is not part of the statement.
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Sharp threshold width for monotone graph propertiesSharpThreshold
Two reports, all at the release’s recorded version.
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Family 186The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Compare the proposed paper and Lean matches
Family 186Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing.Open the form
Why this passage is shown
Theorem 1.1
In A uniform influence bound for hypergraph properties, 5 October 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Words they share
hypergraph, property
Review
Atlas records no review confirming that this is the theorem OpenAI’s description refers to.
Only a special case is stated, or a co-equal headline claim has no challenge statement.
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.