Sharp threshold width for monotone graph properties
Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.
Theorem selected for verification:OAI.Problem313.sharp_threshold_width
The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.
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
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
Definitions needed to read it
Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.
defgraphQuantile {n : ℕ} (a : ℝ) (f : GraphConfig n → Bool) : ℝ := sInf {p : ℝ | p ∈ Set.Icc (0 : ℝ) 1 ∧ graphMean p f ≥ a}
Possible paper match, with no recorded review
Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.
Words of the Lean statement, its names split into words, that also appear in the theorem: sharp, threshold, graphs, 1/2, vertex, nontrivial, increasing, invariant and three more. Found in the paper’s title, which its main theorem states: properties. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: varepsilon.
Statements in A Sharp Threshold Bound for Monotone Graph Properties come first because the release lists only that paper for this Lean statement.
Theorem 1.1 of A Sharp Threshold Bound for Monotone Graph Properties
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 integer $n\ge2$, every nontrivial increasing family $\mathcal P$ of graphs on $n$ vertices invariant under all vertex permutations, and every $0<\varepsilon<1/2$,
\begin{equation}
p_{1-\varepsilon}(\mathcal P)-p_\varepsilon(\mathcal P)
\le \frac{2^{19}}{(\log n)^2}\log\frac1{2\varepsilon}.
\end{equation}
Other possible paper matches, closest wording first
Theorem 1.2 of A Sharp Threshold Bound for Monotone Graph Properties. Words of the Lean statement, its names split into words, that also appear in the theorem: vertex, invariant, permutations, integer and log.
Corollary 4.1 of A uniform influence bound for hypergraph properties. Words of the Lean statement, its names split into words, that also appear in the theorem: 1/2, nontrivial, increasing, property and log. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: varepsilon and uniform.
Theorem 1.1 of A uniform influence bound for hypergraph properties. Words of the Lean statement, its names split into words, that also appear in the theorem: property, integer and log. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constant.
Proposition 3.2 of A Sharp Threshold Bound for Monotone Graph Properties. Words of the Lean statement, its names split into words, that also appear in the theorem: integer and log. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: uniform.
How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.
Reported Comparator runs
two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.
For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.
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 (.github/workflows/openai-math-kernel.yml), one j…
AcceptedTheir term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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.
AcceptedTheir term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.
This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.
Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.
Do family 186’s Lean statements express the advertised result?
These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
Finds part of it stated, or a narrower or weaker statementTheir term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
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).
Finds part of it stated, or a narrower or weaker statementTheir term: “NARROWER — UNLINKED”.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link. Their record
Their note: 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.
OpenAI’s own note on what this Lean statement covers
“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 $\varepsilon$.”
An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.
Supporting material
The complete Lean statement file
63 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.
Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.
Links in the file: 27 to its own declarations, 31 to Mathlib’s documentation. 13 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.
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 integer $n\ge2$, every nontrivial increasing family $\mathcal P$ of graphs on $n$ vertices invariant under all vertex permutations, and every $0<\varepsilon<1/2$,
\begin{equation}
p_{1-\varepsilon}(\mathcal P)-p_\varepsilon(\mathcal P)
\le \frac{2^{19}}{(\log n)^2}\log\frac1{2\varepsilon}.
\end{equation}
Theorem(Variance and influence)
For every integer n\ge2, every Boolean function f on E_n invariant under vertex permutations, and every 0<p<1,
For every integer $n\ge2$, every Boolean function $f$ on $E_n$ invariant under vertex permutations, and every $0<p<1$,
\begin{equation}
\Var_p(f)\le\frac{2^{17}}{(\log n)^2}I_p(f).
\end{equation}
No monotonicity assumption is needed.
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,
A Sharp Threshold Bound for Monotone Graph Properties
TeX label thm:threshold, build/sections/introduction.tex, line 12
The Lean statement says
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
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
Recorded comparisons with the paper
Atlas records no contributed comparison of this statement with the paper.
Contributed Comparator run reports
Atlas records no contributed report of running Comparator for this statement file.
Questions to resolve when comparing this statement with the paper
Both independent readers find this family's challenges state less than its headline, while the atlas's AI draft classes this file as stating a paper's theorem as the paper does; check which part of the headline it covers.