Lean statement

Integral–fractional expectation-threshold comparison

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.TalagrandThreshold.talagrand_expectation_threshold_equivalence

Lean statement file: TalagrandExpectationThreshold.leanShow on the mapFamily 175. OpenAI lists this as a main result in its Lean catalogue.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem talagrand_expectation_threshold_equivalence [Nonempty α]    (F : Family α) (_hF : F.Nonempty) (hproper : F ≠ Finset.univ)    (hIncreasing : Increasing F) :    qf F ≤ ((25 : ℝ) * (512 : ℝ) ^ 4) * q F := by  sorry

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.

abbreviation OAI.TalagrandThreshold.Family line 14

abbrev Family (α : Type*) := Finset (Finset α)

definition OAI.TalagrandThreshold.cost line 20

def cost (p : ℝ) (G : Family α) : ℝ := ∑ S ∈ G, p ^ S.card

definition OAI.TalagrandThreshold.Covers line 22

def Covers (G F : Family α) : Prop := ∀ H ∈ F, ∃ S ∈ G, S ⊆ H

definition OAI.TalagrandThreshold.Small line 24

def Small (p : ℝ) (F : Family α) : Prop :=  ∃ G : Family α, Covers G F ∧ cost p G ≤ 1 / 2

definition OAI.TalagrandThreshold.fracCost line 27

def fracCost (p : ℝ) (g : Finset α → ℝ) : ℝ :=  ∑ S : Finset α, g S * p ^ S.card

definition OAI.TalagrandThreshold.FractionalCover line 30

def FractionalCover (F : Family α) (g : Finset α → ℝ) : Prop :=  (∀ S, 0 ≤ g S ∧ g S ≤ 1) ∧    ∀ H ∈ F, 1 ≤ ∑ S ∈ H.powerset, g S

definition OAI.TalagrandThreshold.FractionalSmall line 34

def FractionalSmall (p : ℝ) (F : Family α) : Prop :=  ∃ g : Finset α → ℝ, FractionalCover F g ∧ fracCost p g ≤ 1 / 2

definition OAI.TalagrandThreshold.Increasing line 37

def Increasing (F : Family α) : Prop :=  ∀ H ∈ F, ∀ K : Finset α, H ⊆ K → K ∈ F

definition OAI.TalagrandThreshold.integralFeasible line 40

def integralFeasible (F : Family α) : Set ℝ :=  {p | 0 ≤ p ∧ p ≤ 1 ∧ Small p F}

definition OAI.TalagrandThreshold.fractionalFeasible line 43

def fractionalFeasible (F : Family α) : Set ℝ :=  {p | 0 ≤ p ∧ p ≤ 1 ∧ FractionalSmall p F}

definition OAI.TalagrandThreshold.q line 46

def q (F : Family α) : ℝ := sSup (integralFeasible F)

definition OAI.TalagrandThreshold.qf line 48

def qf (F : Family α) : ℝ := sSup (fractionalFeasible 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: increasing, fractional, thresholds, cover, nonempty, 1/2 and family. Found in the paper’s title, which its main theorem states: integral and expectation. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: budget, proper and finite.

Statements in Integral and fractional expectation thresholds are equivalent come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of Integral and fractional expectation thresholds are equivalent

Theorem

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.

Section 1, Introduction Read it in the paper TeX source, line 52

Show the TeX

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.

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: cost, cover, nonempty, 1/2 and family. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: may and finite.
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: nonempty, 1/2, small and family. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: finite.
Theorem 1.1 of Graph Decompositions at the Integral Expectation Threshold. Found in the paper’s title, which its main theorem states: integral, threshold and expectation. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: empty and may.
Theorem 1.1 of Talagrand’s discrete-convexity conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: small and family. Found in the paper’s title, which its main theorem states: talagrand.
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: nonempty, 1/2, small and family. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: finite.
Proposition 3.1 of Talagrand’s discrete-convexity conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: small 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.

Carlos Toledo’s cert-machine

How they ran it, in their words

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…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

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.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

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).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 68 s; peak memory 2.1 GiB).
Reported 11 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

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.

Allowed axioms: propext, Quot.sound, Classical.choice

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.

MathVet
Finds part of it stated, or a narrower or weaker statement Their 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.
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their 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 Ascoli–He–Park–Talagrand graph-decomposition theorem (October-5 manuscript, not linked by the scope note).

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalized result proves $q_f(\mathcal F)\le25\cdot512^4 q(\mathcal F)$ for every nonempty proper increasing family on a finite nonempty ground set. Both thresholds use cover budget $1/2$, and fractional covers may assign weights to every subset, including the empty set.”
Quoted from OpenAI’s Lean note for this family

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

60 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: 23 to its own declarations, 15 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.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves talagrand_expectation_threshold_equivalence exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches all 12 other declarations of the file; with the theorem itself they span 25 lines. 4 of them are named in the statement itself.

Names in the namespace OAI.TalagrandThreshold are listed without it.

abbrev Familyline 14in the statementlibrary
def costline 20library
def Coversline 22library
def Smallline 24library
def fracCostline 27library
def FractionalCoverline 30library
def FractionalSmallline 34library
def Increasingline 37in the statementlibrary
def integralFeasibleline 40library
def fractionalFeasibleline 43library
def qline 46in the statementlibrary
def qfline 48in the statementlibrary
theorem talagrand_expectation_threshold_equivalenceline 52selected theoremlibrary

import Mathlib.Data.Fintype.Powersetimport Mathlib.Data.Finset.Unionimport Mathlib.Basic.Real.Basicimport Mathlib.Algebra.Order.Archimedean.Real.Basicimport Mathlib.Algebra.Order.BigOperators.Group.Finsetimport Mathlib.Algebra.BigOperators.Group.Finset.Piecewisenamespace OAInamespace TalagrandThresholdopen scoped BigOperatorsabbrev Family (α : Type*) := Finset (Finset α)variable {α : Type*} [Fintype α] [DecidableEq α]noncomputable sectiondef cost (p : ℝ) (G : Family α) : ℝ := ∑ S ∈ G, p ^ S.carddef Covers (G F : Family α) : Prop := ∀ H ∈ F, ∃ S ∈ G, S ⊆ Hdef Small (p : ℝ) (F : Family α) : Prop :=  ∃ G : Family α, Covers G F ∧ cost p G ≤ 1 / 2def fracCost (p : ℝ) (g : Finset α → ℝ) : ℝ :=  ∑ S : Finset α, g S * p ^ S.carddef FractionalCover (F : Family α) (g : Finset α → ℝ) : Prop :=  (∀ S, 0 ≤ g S ∧ g S ≤ 1) ∧    ∀ H ∈ F, 1 ≤ ∑ S ∈ H.powerset, g Sdef FractionalSmall (p : ℝ) (F : Family α) : Prop :=  ∃ g : Finset α → ℝ, FractionalCover F g ∧ fracCost p g ≤ 1 / 2def Increasing (F : Family α) : Prop :=  ∀ H ∈ F, ∀ K : Finset α, H ⊆ K → K ∈ Fdef integralFeasible (F : Family α) : Set ℝ :=  {p | 0 ≤ p ∧ p ≤ 1 ∧ Small p F}def fractionalFeasible (F : Family α) : Set ℝ :=  {p | 0 ≤ p ∧ p ≤ 1 ∧ FractionalSmall p F}def q (F : Family α) : ℝ := sSup (integralFeasible F)def qf (F : Family α) : ℝ := sSup (fractionalFeasible F)endtheorem talagrand_expectation_threshold_equivalence [Nonempty α]    (F : Family α) (_hF : F.Nonempty) (hproper : F ≠ Finset.univ)    (hIncreasing : Increasing F) :    qf F ≤ ((25 : ℝ) * (512 : ℝ) ^ 4) * q F := by  sorryend TalagrandThresholdend 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.

Integral and fractional expectation thresholds are equivalent

Theorem

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.

Section 1, Introduction Read it in the paper TeX source, line 52

Show the TeX

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.

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}.

Section 1, Introduction Read it in the paper TeX source, line 43

Show the TeX

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}.

Section 1, Introduction Read it in the paper TeX source, line 109

Show the TeX

There is a universal constant $C\ge1$ such that, for every integer $N\ge1$,
every $p\in(0,1)$, and every family $\mathcal D\subseteq2^{[N]}$,
\[
\mu_p(\mathcal D)>\frac12
\quad\Longrightarrow\quad
E_2(\mathcal D)\text{ is }(p/C)\text{-small}.
\]

Configuration and file facts

File: lean/ComparatorChallenges/TalagrandExpectationThreshold.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Combinatorics/ExpectationThreshold/Main.lean
Allowed axioms: propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib.Data.Fintype.Powerset, Mathlib.Data.Finset.Union, Mathlib.Basic.Real.Basic, Mathlib.Algebra.Order.Archimedean.Real.Basic, Mathlib.Algebra.Order.BigOperators.Group.Finset, Mathlib.Algebra.BigOperators.Group.Finset.Piecewise
Imports from outside Mathlib: None
Local vocabulary: 12 local vocabulary declarations
Lines of Lean, without comments or blank lines: 25
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

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