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.TalagrandDiscreteConvexity.talagrand_discrete_convexity
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.
defexceptional (k : ℕ) (D : Family α) : Family α := by classical exact Finset.univ.filter (fun S => ∀ f : Fin k → Finset α, (∀ j, f j ∈ D) → ¬ S ⊆ unionTuple f)
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: small, family and integer. Found in the paper’s title, which its main theorem states: talagrand, discrete and convexity.
Statements in Talagrand’s discrete-convexity conjecture come first because the release’s name for this Lean statement shares words with that paper’s title.
Theorem 1.1 of Talagrand’s discrete-convexity conjecture
Theorem
Set k=2^{75}. For every integer N\ge1, every p\in(0,1), and every family \mathcal D\subseteq2^{[N]},
\mu_p(\mathcal D)\ge1-\frac1k
\quad\Longrightarrow\quad
E_k(\mathcal D)\text{ is }p\text{-small}.
Set $k=2^{75}$. For every integer $N\ge1$, every $p\in(0,1)$, and every
family $\mathcal D\subseteq2^{[N]}$,
\[
\mu_p(\mathcal D)\ge1-\frac1k
\quad\Longrightarrow\quad
E_k(\mathcal D)\text{ is }p\text{-small}.
\]
Other possible paper matches, closest wording first
Theorem 2.1 of Graph Decompositions at the Integral Expectation Threshold. Words of the Lean statement, its names split into words, that also appear in the theorem: tuple, cost, discrete, cover, convexity, members and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: repeat and nonempty.
Corollary 1.2 of Talagrand’s discrete-convexity conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: density, unions, 1/2, small, family and integer.
Theorem 1.1 of Integral and fractional expectation thresholds are equivalent. Words of the Lean statement, its names split into words, that also appear in the theorem: independent, cover, members, finite, 1/2 and family. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nonempty.
Corollary 4.1 of Integral and fractional expectation thresholds are equivalent. Words of the Lean statement, its names split into words, that also appear in the theorem: monotonicity, finite, 1/2, small and family. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: assumption and nonempty.
Theorem 2.1 of Integral and fractional expectation thresholds are equivalent. Words of the Lean statement, its names split into words, that also appear in the theorem: probabilities, mass, finite, 1/2, small, family and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: least and nonempty.
Corollary 2.2 of Graph Decompositions at the Integral Expectation Threshold. Words of the Lean statement, its names split into words, that also appear in the theorem: contained, union, members and family.
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
three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. 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.
Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory; the run was still in progress at this snapshot (192 accepted, 224 pending).
AcceptedTheir term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 29 s; peak memory 2.1 GiB).
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 175’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: Two of the three claims are stated with explicit constants: talagrand_expectation_threshold_equivalence : qf F ≤ (25 * 512^4) * q F (increasing nonempty proper F) and talagrand_discrete_convexity with k = 2^75 (Small p (exceptional (2^75) D) from 1 - 1/2^75 ≤ familyMeasure p D). The summary's application, the Ascoli-He-Park-Talagrand graph-decomposition conjecture (also in the family title), has no challenge statement and the docs do not mention it.
OpenAI’s own note on what this Lean statement covers
“The formalized result proves Talagrand's discrete-convexity assertion with $k=2^{75}$. If a family of subsets of a finite nonempty ground set has Bernoulli-$p$ measure at least $1-1/k$, then the sets not contained in a union of $k$ members form a $p$-small family: they have a containment cover of total $p$-cost at most $1/2$. This holds for every $0<p<1$, with repeated members allowed in the union and no monotonicity assumption.”
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
57 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: 15 to its own declarations, 14 to Mathlib’s documentation. 9 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.
import Mathlib.Data.Fintype.Powersetimport Mathlib.Data.Finset.Unionimport Mathlib.Basic.Real.Basicimport Mathlib.Algebra.Order.BigOperators.Group.Finsetnamespace OAInamespace TalagrandDiscreteConvexityopenscoped BigOperatorsabbrevFamily (α : Type*) := Finset (Finset α)variable {α : Type*} [Fintype α] [DecidableEq α]noncomputablesection/-- Independent Bernoulli coordinates, including every absent coordinate. -/defbernoulliMass (p : ℝ) (s : Finset α) : ℝ := p ^ s.card * (1 - p) ^ (Finset.univ \ s).card/-- The probability of an arbitrary family, with no monotonicity condition. -/deffamilyMeasure (p : ℝ) (D : Family α) : ℝ := ∑ s ∈ D, bernoulliMass p s/-- Each distinct generator is charged once at the original density. -/defcost (p : ℝ) (G : Family α) : ℝ := ∑ I ∈ G, p ^ I.card/-- Every covered set contains a generator. -/defCovers (G A : Family α) : Prop := ∀ S ∈ A, ∃ I ∈ G, I ⊆ S/-- An actual finite containment cover with total cost at most one half. -/defSmall (p : ℝ) (A : Family α) : Prop := ∃ G : Family α, Covers G A ∧ cost p G ≤ 1 / 2/-- Exactly `k` entries, with no distinctness or disjointness restriction. -/defunionTuple {k : ℕ} (f : Fin k → Finset α) : Finset α :=Finset.univ.biUnion f/-- All subsets contained in no union of `k` members of the given family. -/defexceptional (k : ℕ) (D : Family α) : Family α := byclassicalexactFinset.univ.filter (fun S => ∀ f : Fin k → Finset α, (∀ j, f j ∈ D) → ¬ S ⊆ unionTuple f)/-- Smallness at the original density for every positive ground-set size. -/theoremtalagrand_discrete_convexity (N : ℕ) (_hN : 1 ≤ N) (p : ℝ) (hp0 : 0 < p) (hp1 : p < 1) (D : Family (Fin N)) (hD : 1 - 1 / (2 : ℝ) ^ 75 ≤ familyMeasure p D) :Small p (exceptional ((2 : ℕ) ^ 75) D) := bysorryendend TalagrandDiscreteConvexityend OAI
Every extracted statement of the accompanying papers
The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.
For every finite nonempty X and every nonempty proper increasing family \F\subseteq2^X,
q_f(\F)\le25\cdot512^4\,q(\F).
Both thresholds use the budget 1/2. The constant is independent of |X|, the sizes of the minimal members of \F, and the support of the fractional cover.
For every finite nonempty $X$ and every nonempty proper increasing
family $\F\subseteq2^X$,
\[
q_f(\F)\le25\cdot512^4\,q(\F).
\]
Both thresholds use the budget $1/2$. The constant is independent of
$|X|$, the sizes of the minimal members of $\F$, and the support of
the fractional cover.
Set $k=2^{75}$. For every integer $N\ge1$, every $p\in(0,1)$, and every
family $\mathcal D\subseteq2^{[N]}$,
\[
\mu_p(\mathcal D)\ge1-\frac1k
\quad\Longrightarrow\quad
E_k(\mathcal D)\text{ is }p\text{-small}.
\]
Corollary(Two unions at constant density loss)
There is a universal constant C\ge1 such that, for every integer N\ge1, every p\in(0,1), and every family \mathcal D\subseteq2^{[N]},
\mu_p(\mathcal D)>\frac12
\quad\Longrightarrow\quad
E_2(\mathcal D)\text{ is }(p/C)\text{-small}.
There is a universal constant $C\ge1$ such that, for every integer $N\ge1$,
every $p\in(0,1)$, and every family $\mathcal D\subseteq2^{[N]}$,
\[
\mu_p(\mathcal D)>\frac12
\quad\Longrightarrow\quad
E_2(\mathcal D)\text{ is }(p/C)\text{-small}.
\]
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.