Lean statement

Colored-graph criterion for finite congruence lattices

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.FiniteCongruence.graph_criterion

Lean statement file: FiniteCongruenceGraph.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 206 link it.

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 graph_criterion (Lat : Type) [Fintype Lat] [Lattice Lat] [OrderBot Lat] :    Representable Lat ↔ HasGraphWitness Lat := 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.

structure OAI.FiniteCongruence.Algebra line 10

structure Algebra (Carrier : Type) where  numOps : ℕ  arity : Fin numOps → ℕ  op : (index : Fin numOps) → (Fin (arity index) → Carrier) → Carrier

definition OAI.FiniteCongruence.Compatible line 15

def Compatible {Carrier : Type} (alg : Algebra Carrier) (rel : Setoid Carrier) : Prop :=  ∀ (index : Fin alg.numOps) (left right : Fin (alg.arity index) → Carrier),    (∀ coord, rel (left coord) (right coord)) → rel (alg.op index left) (alg.op index right)

definition OAI.FiniteCongruence.Congruence line 19

def Congruence {Carrier : Type} (alg : Algebra Carrier) :=  {rel : Setoid Carrier // Compatible alg rel}

definition OAI.FiniteCongruence.Representable line 25

def Representable (Lat : Type) [PartialOrder Lat] : Prop :=  ∃ size : ℕ, 0 < size ∧ ∃ alg : Algebra (Fin size), Nonempty (Lat ≃o Congruence alg)

definition OAI.FiniteCongruence.Admissible line 28

def Admissible {Lat Carrier : Type} [LE Lat] (color : Carrier → Carrier → Lat)    (placement : Carrier → Carrier) : Prop :=  ∀ left right, color (placement left) (placement right) ≤ color left right

definition OAI.FiniteCongruence.Marked line 32

def Marked {Lat Carrier : Type} [LE Lat] (color : Carrier → Carrier → Lat)    (seeds : Finset (Carrier × Carrier)) (left right : Carrier) : Prop :=  ∃ placement : Carrier → Carrier, Admissible color placement ∧    ∃ edge ∈ seeds, placement edge.1 = left ∧ placement edge.2 = right

definition OAI.FiniteCongruence.Connected line 37

def Connected {Lat Carrier : Type} [LE Lat] (color : Carrier → Carrier → Lat)    (seeds : Finset (Carrier × Carrier)) (left right : Carrier) : Prop :=  Relation.EqvGen (Marked color seeds) left right

definition OAI.FiniteCongruence.IsGraphWitness line 41

def IsGraphWitness {Lat Carrier : Type} [Lattice Lat] [OrderBot Lat]    (color : Carrier → Carrier → Lat) : Prop :=  (∀ left right, color left right = color right left) ∧  (∀ left right, color left right = ⊥ ↔ left = right) ∧  (∀ left right middle, color left right ≤ color left middle ⊔ color middle right) ∧  (∀ lower upper : Lat, ¬ lower ≤ upper →    ∃ left right, color left right ≤ lower ∧ ¬ color left right ≤ upper) ∧  (∀ seeds : Finset (Carrier × Carrier), seeds.Nonempty → ∀ left right,    color left right ≤ seeds.sup (fun edge => color edge.1 edge.2) →      Connected color seeds left right)

definition OAI.FiniteCongruence.HasGraphWitness line 52

def HasGraphWitness (Lat : Type) [Lattice Lat] [OrderBot Lat] : Prop :=  ∃ size : ℕ, 0 < size ∧ ∃ color : Fin size → Fin size → Lat, IsGraphWitness color

Possible paper match, with no recorded review

No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.

Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: colored, criterion, finite, full, graph, interval, lattice, nonempty and three more.

Statements in Finite congruence lattices: characterization and undecidability come first because the release lists only that paper for this Lean statement.

Theorem 1.1 of Finite congruence lattices: characterization and undecidability

Theorem

The following statements hold.

(i)
A finite nonempty lattice is representable if and only if it satisfies the finite colored-graph criterion in f:criterion. For each candidate graph size the criterion is decidable by a finite test.
(ii)
There is no deterministic Turing machine that, given the order table of a finite nonempty lattice, always halts and decides \mathcal R(L).
(iii)
There is a nonrepresentable finite lattice L_0 of minimum cardinality, specified by the finite formulas in sec:minimum. Every finite lattice of smaller cardinality is representable, and \lvert L_0\rvert>7.
(iv)
Given the order table of a finite nonempty lattice L, it is undecidable whether L\cong[H,G] for some finite group G and subgroup H\le G, where [H,G] is the full subgroup interval.

Section 1, Introduction Read it in the paper TeX source, line 60

Show the TeX

The following statements hold.
\begin{enumerate}[label=\textup{(\roman*)}]
\item A finite nonempty lattice is representable if and only if it satisfies
the finite colored-graph criterion in \Cref{f:criterion}.
For each candidate graph size the criterion is decidable by a finite test.
\item There is no deterministic Turing machine that, given the order table
of a finite nonempty lattice, always halts and decides \(\mathcal R(L)\).
\item There is a nonrepresentable finite lattice \(L_0\) of minimum
cardinality, specified by the finite formulas in \Cref{sec:minimum}.
Every finite lattice of smaller cardinality is representable, and
\(\lvert L_0\rvert>7\).
\item Given the order table of a finite nonempty lattice \(L\), it is
undecidable whether \(L\cong[H,G]\) for some finite group \(G\) and
subgroup \(H\le G\), where \([H,G]\) is the full subgroup interval.
\end{enumerate}

Other possible paper matches, closest wording first

Theorem 2.1 of Finite congruence lattices: characterization and undecidability. Words of the Lean statement, its names split into words, that also appear in the theorem: criterion, connected, graph, colored, congruence, edge and five more.
Corollary 1 of Finite congruence lattices: characterization and undecidability. Words of the Lean statement, its names split into words, that also appear in the theorem: witness, criterion, graph, integer, natural, representability and three more.
Proposition 43 of Finite congruence lattices: characterization and undecidability. Words of the Lean statement, its names split into words, that also appear in the theorem: congruence, size, algebra, order, representing, lattice and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: whose and intervals.
Theorem 15.1 of Finite congruence lattices: characterization and undecidability. Words of the Lean statement, its names split into words, that also appear in the theorem: carrier, congruence, size, algebra, representability, lattice and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: undecidability.
Proposition 16 of Finite congruence lattices: characterization and undecidability. Words of the Lean statement, its names split into words, that also appear in the theorem: partial, algebra, ordered and full. Found in the paper’s title, which its main theorem states: congruence, lattices and finite.
Proposition 4 of Finite congruence lattices: characterization and undecidability. Words of the Lean statement, its names split into words, that also appear in the theorem: algebra, representable, lattice and finite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: interval and subgroup.

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 31 s).
Reported 10 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 206’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 a supporting or a different result stated Their term: “supporting-only”.
The challenge statements state a lemma or auxiliary result, not the headline. Their record
Their note: graph_criterion (Lat) [Fintype Lat] [Lattice Lat] [OrderBot Lat] : Representable Lat ↔ HasGraphWitness Lat is only the paper's colored-graph characterization (docs: 'This selected theorem is the equivalence with the graph criterion; the paper's undecidability and subgroup-interval conclusions are outside it'). Nothing states that some finite lattice is NOT representable (the headline negative answer to the finite lattice representation problem) or that representability/subgroup-interval membership is undecidable.
Carlos Toledo’s cert-machine
Finds a supporting or a different result stated Their term: “DIFFERENT”.
The challenge statements state a different theorem from the headline. Their record
Their note: every clause of the headline: existence of a non-representable finite lattice, undecidability of representability, undecidability of subgroup-interval recognition. The Lean states instead a characterization (representable ⇔ colored-graph witness) that neither implies nor is implied by any headline clause.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization proves the paper's colored-graph characterization: a finite nonempty lattice is representable exactly when it admits the specified finite nonempty graph witness, whose edge colors encode the required congruence relations. This selected theorem is the equivalence with the graph criterion; the paper's undecidability and subgroup-interval conclusions are outside it.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states part of a paper’s theorem”. 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: 14 to its own declarations, 29 to Mathlib’s documentation. 10 declared names also link to the same declaration in OpenAI’s library. 1 identifier stays plain because it resolves to no single declaration: a variable Lean binds automatically, notation, or a name the scan of the pinned sources did not find. 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 graph_criterion 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 9 of the file’s 10 other declarations; with the theorem itself they span 36 lines. 2 of them are named in the statement itself.

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

structure Algebraline 10library
def Compatibleline 15library
def Congruenceline 19library
def Representableline 25in the statementlibrary
def Admissibleline 28library
def Markedline 32library
def Connectedline 37library
def IsGraphWitnessline 41library
def HasGraphWitnessline 52in the statementlibrary
theorem graph_criterionline 57selected theoremlibrary

Other Lean rules that may affect the statement 1

Lean can use instances, notation and attributes without naming them in the statement. These are rules for interpreting or finding mathematical objects. Atlas’s scan of names cannot follow all such uses; read the listed source as well.

instance {Carrier : Type} (alg : Algebra Carrier) : PartialOrder (Cong… line 22

import Mathlib.Data.Finset.Lattice.Foldimport Mathlib.Data.Fintype.Defsimport Mathlib.Data.Setoid.Basicimport Mathlib.Order.Hom.Basicnamespace OAInamespace FiniteCongruencestructure Algebra (Carrier : Type) where  numOps : ℕ  arity : Fin numOps → ℕ  op : (index : Fin numOps) → (Fin (arity index) → Carrier) → Carrierdef Compatible {Carrier : Type} (alg : Algebra Carrier) (rel : Setoid Carrier) : Prop :=  ∀ (index : Fin alg.numOps) (left right : Fin (alg.arity index) → Carrier),    (∀ coord, rel (left coord) (right coord)) → rel (alg.op index left) (alg.op index right)def Congruence {Carrier : Type} (alg : Algebra Carrier) :=  {rel : Setoid Carrier // Compatible alg rel}instance {Carrier : Type} (alg : Algebra Carrier) : PartialOrder (Congruence alg) :=  inferInstanceAs (PartialOrder {rel : Setoid Carrier // Compatible alg rel})def Representable (Lat : Type) [PartialOrder Lat] : Prop :=  ∃ size : ℕ, 0 < size ∧ ∃ alg : Algebra (Fin size), Nonempty (Lat ≃o Congruence alg)def Admissible {Lat Carrier : Type} [LE Lat] (color : Carrier → Carrier → Lat)    (placement : Carrier → Carrier) : Prop :=  ∀ left right, color (placement left) (placement right) ≤ color left rightdef Marked {Lat Carrier : Type} [LE Lat] (color : Carrier → Carrier → Lat)    (seeds : Finset (Carrier × Carrier)) (left right : Carrier) : Prop :=  ∃ placement : Carrier → Carrier, Admissible color placement ∧    ∃ edge ∈ seeds, placement edge.1 = left ∧ placement edge.2 = rightdef Connected {Lat Carrier : Type} [LE Lat] (color : Carrier → Carrier → Lat)    (seeds : Finset (Carrier × Carrier)) (left right : Carrier) : Prop :=  Relation.EqvGen (Marked color seeds) left rightdef IsGraphWitness {Lat Carrier : Type} [Lattice Lat] [OrderBot Lat]    (color : Carrier → Carrier → Lat) : Prop :=  (∀ left right, color left right = color right left) ∧  (∀ left right, color left right = ⊥ ↔ left = right) ∧  (∀ left right middle, color left right ≤ color left middle ⊔ color middle right) ∧  (∀ lower upper : Lat, ¬ lower ≤ upper →    ∃ left right, color left right ≤ lower ∧ ¬ color left right ≤ upper) ∧  (∀ seeds : Finset (Carrier × Carrier), seeds.Nonempty → ∀ left right,    color left right ≤ seeds.sup (fun edge => color edge.1 edge.2) →      Connected color seeds left right)def HasGraphWitness (Lat : Type) [Lattice Lat] [OrderBot Lat] : Prop :=  ∃ size : ℕ, 0 < size ∧ ∃ color : Fin size → Fin size → Lat, IsGraphWitness color/-- A finite lattice is representable as the full congruence lattice of a nonempty finitealgebra exactly when it has a finite nonempty colored-graph witness. -/theorem graph_criterion (Lat : Type) [Fintype Lat] [Lattice Lat] [OrderBot Lat] :    Representable Lat ↔ HasGraphWitness Lat := by  sorryend FiniteCongruenceend 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.

Finite congruence lattices: characterization and undecidability

Theorem

The following statements hold.

(i)
A finite nonempty lattice is representable if and only if it satisfies the finite colored-graph criterion in f:criterion. For each candidate graph size the criterion is decidable by a finite test.
(ii)
There is no deterministic Turing machine that, given the order table of a finite nonempty lattice, always halts and decides \mathcal R(L).
(iii)
There is a nonrepresentable finite lattice L_0 of minimum cardinality, specified by the finite formulas in sec:minimum. Every finite lattice of smaller cardinality is representable, and \lvert L_0\rvert>7.
(iv)
Given the order table of a finite nonempty lattice L, it is undecidable whether L\cong[H,G] for some finite group G and subgroup H\le G, where [H,G] is the full subgroup interval.

Section 1, Introduction Read it in the paper TeX source, line 60

Show the TeX

The following statements hold.
\begin{enumerate}[label=\textup{(\roman*)}]
\item A finite nonempty lattice is representable if and only if it satisfies
the finite colored-graph criterion in \Cref{f:criterion}.
For each candidate graph size the criterion is decidable by a finite test.
\item There is no deterministic Turing machine that, given the order table
of a finite nonempty lattice, always halts and decides \(\mathcal R(L)\).
\item There is a nonrepresentable finite lattice \(L_0\) of minimum
cardinality, specified by the finite formulas in \Cref{sec:minimum}.
Every finite lattice of smaller cardinality is representable, and
\(\lvert L_0\rvert>7\).
\item Given the order table of a finite nonempty lattice \(L\), it is
undecidable whether \(L\cong[H,G]\) for some finite group \(G\) and
subgroup \(H\le G\), where \([H,G]\) is the full subgroup interval.
\end{enumerate}

Configuration and file facts

File: lean/ComparatorChallenges/FiniteCongruenceGraph.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 206 link it.
Solution module: OAI.Algebra.Universal.GraphCriterion
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.Finset.Lattice.Fold, Mathlib.Data.Fintype.Defs, Mathlib.Data.Setoid.Basic, Mathlib.Order.Hom.Basic
Imports from outside Mathlib: None
Local vocabulary: 10 local vocabulary declarations
Lines of Lean, without comments or blank lines: 38
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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