Lean statement

Crossing number of complete bipartite graphs

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.Zarankiewicz.mainTarget_proof

Lean statement file: BipartiteCrossing.leanShow on the mapFamily 165. 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.

def MainTarget : Prop :=  ∀ m n : ℕ, 0 < m → 0 < n →    (∃ D : AdmissibleDrawing m n, crossingCount D = axisPairs m * axisPairs n) ∧    (∀ D : AdmissibleDrawing m n, axisPairs m * axisPairs n ≤ crossingCount D)⋮theorem mainTarget_proof : MainTarget := 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.

definition OAI.Zarankiewicz.axisPairs line 17

def axisPairs (r : ℕ) : ℕ := (r/2) * ((r-1)/2)

abbreviation OAI.Zarankiewicz.Plane line 20
```
abbrev Plane := ℝ × ℝ
```

abbreviation OAI.Zarankiewicz.BipartiteVertex line 21

abbrev BipartiteVertex (m n : ℕ) := Fin m ⊕ Fin n

abbreviation OAI.Zarankiewicz.BipartiteEdge line 22

abbrev BipartiteEdge (m n : ℕ) := Fin m × Fin n

structure OAI.Zarankiewicz.ContinuousDrawing line 24

structure ContinuousDrawing (m n : ℕ) where  vertex : BipartiteVertex m n → Plane  vertex_injective : Function.Injective vertex  edge : ∀ e : BipartiteEdge m n, Path (vertex (.inl e.1)) (vertex (.inr e.2))  edge_injective : ∀ e, Function.Injective (edge e)  interior_avoids_vertices : ∀ e t, t ∈ Set.Ioo (0 : unitInterval) 1 →    ∀ v, edge e t ≠ vertex v

definition OAI.Zarankiewicz.edgeInterior line 32

def edgeInterior {m n : ℕ} (D : ContinuousDrawing m n) (e : BipartiteEdge m n) : Set Plane :=  D.edge e '' Set.Ioo (0 : unitInterval) 1

definition OAI.Zarankiewicz.ProperCrossing line 35

def ProperCrossing (A B : Set Plane) (x : Plane) : Prop :=  ∃ chart : OpenPartialHomeomorph Plane Plane,    x ∈ chart.source ∧ chart x = (0,0) ∧    chart '' (chart.source ∩ A) = chart.target ∩ {y | y.2 = 0} ∧    chart '' (chart.source ∩ B) = chart.target ∩ {y | y.1 = 0}

definition OAI.Zarankiewicz.crossingPoints line 41

def crossingPoints {m n : ℕ} (D : ContinuousDrawing m n) : Set Plane :=  {x | ∃ e f : BipartiteEdge m n, e ≠ f ∧ x ∈ edgeInterior D e ∧ x ∈ edgeInterior D f}

structure OAI.Zarankiewicz.AdmissibleDrawing line 44

structure AdmissibleDrawing (m n : ℕ) extends ContinuousDrawing m n where  crossings_finite : (crossingPoints toContinuousDrawing).Finite  crossings_proper : ∀ e f, e ≠ f → ∀ x,    x ∈ edgeInterior toContinuousDrawing e → x ∈ edgeInterior toContinuousDrawing f →    ProperCrossing (edgeInterior toContinuousDrawing e) (edgeInterior toContinuousDrawing f) x  no_triples : ∀ e f g, e ≠ f → e ≠ g → f ≠ g →    edgeInterior toContinuousDrawing e ∩ edgeInterior toContinuousDrawing f ∩      edgeInterior toContinuousDrawing g = ∅

definition OAI.Zarankiewicz.crossingCount line 53

def crossingCount {m n : ℕ} (D : AdmissibleDrawing m n) : ℕ := D.crossings_finite.toFinset.card

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: integers. Found in the paper’s title, which its main theorem states: bipartite and crossing. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: positive.

Statements in The crossing number of complete bipartite graphs come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of The crossing number of complete bipartite graphs

Theorem

For all positive integers m,n,

\crn(K_{m,n})=d_m d_n.

Section 1, Introduction Read it in the paper TeX source, line 29

Show the TeX

For all positive integers $m,n$,
\[
 \crn(K_{m,n})=d_m d_n.
\]

Other possible paper matches, closest wording first

Corollary 2.3 of The crossing number of complete bipartite graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: real and pairs. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: m/2.
Proposition 5.1 of The crossing number of complete graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: drawing and crossings.
Theorem 1.1 of The crossing number of complete graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: integer. Found in the paper’s title, which its main theorem states: crossing. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: n/2.
Proposition 4.3 of The crossing number of complete graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: drawing and crossings.

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.

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

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 165’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.

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: complete_graph_crossing_number : 3 ≤ n → ordinaryCrossingNumber (completeGraph n) = hill n (Hill's formula; the ℕ-division by 4 is exact because the product of the four floors is divisible by 4) and MainTarget : ∀ m n > 0, (∃ admissible drawing with Z(m)Z(n) crossings) ∧ (∀ admissible drawings, Z(m)Z(n) ≤ crossings) state both formulas for all sizes, for drawings by arbitrary continuous simple arcs with crossings counted as points. The minimum ranges only over the custom AdmissibleDrawing class (finitely many crossing points, each a topologically transversal crossing, no triple points), which is the usual crossing-number class.
Definitions the group asks readers to compare with the paper: structure AdmissibleDrawing ... extends ContinuousDrawing with crossings_finite, crossings_proper : ProperCrossing ... (local chart straightening the two arcs to the axes) and no_triples; ordinaryCrossingNumber := sInf (Set.range crossingCount) (sInf ∅ = 0, but a nonempty class is forced for n ≥ 5 since hill n > 0); the two files carry separate copies of the definitions (Zarankiewicz vs Paper170 namespaces)
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“Zarankiewicz's crossing-number conjecture predicts $\mathrm{cr}(K_{m,n})=\lfloor m/2\rfloor\lfloor(m-1)/2\rfloor\lfloor n/2\rfloor\lfloor(n-1)/2\rfloor$. The formalized result proves this equality for every positive $m,n$ and constructs a drawing attaining it.”
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

67 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: 35 to its own declarations, 15 to Mathlib’s documentation. 12 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 mainTarget_proof 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 11 other declarations of the file; with the theorem itself they span 35 lines. 1 of them is named in the statement itself.

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

def axisPairsline 17library
abbrev Planeline 20library
abbrev BipartiteVertexline 21library
abbrev BipartiteEdgeline 22library
structure ContinuousDrawingline 24library
def edgeInteriorline 32library
def ProperCrossingline 35library
def crossingPointsline 41library
structure AdmissibleDrawingline 44library
def crossingCountline 53library
def MainTargetline 55in the statementlibrary
theorem mainTarget_proofline 60selected theoremlibrary

import Mathlib.Topology.Pathimport Mathlib.Topology.OpenPartialHomeomorph.Defsimport Mathlib.Topology.Instances.Real.Lemmasimport Mathlib.Data.Set.Finite.Basicimport Mathlib.Data.Finset.Attrimport Mathlib.Tactic.Commonimport Mathlib.Tactic.Finiteness.Attrimport Mathlib.Tactic.SetLikeimport Mathlib.Util.CompileInductivenamespace OAInoncomputable sectionnamespace Zarankiewiczdef axisPairs (r : ℕ) : ℕ := (r/2) * ((r-1)/2)abbrev Plane := ℝ × ℝabbrev BipartiteVertex (m n : ℕ) := Fin m ⊕ Fin nabbrev BipartiteEdge (m n : ℕ) := Fin m × Fin nstructure ContinuousDrawing (m n : ℕ) where  vertex : BipartiteVertex m n → Plane  vertex_injective : Function.Injective vertex  edge : ∀ e : BipartiteEdge m n, Path (vertex (.inl e.1)) (vertex (.inr e.2))  edge_injective : ∀ e, Function.Injective (edge e)  interior_avoids_vertices : ∀ e t, t ∈ Set.Ioo (0 : unitInterval) 1 →    ∀ v, edge e t ≠ vertex vdef edgeInterior {m n : ℕ} (D : ContinuousDrawing m n) (e : BipartiteEdge m n) : Set Plane :=  D.edge e '' Set.Ioo (0 : unitInterval) 1def ProperCrossing (A B : Set Plane) (x : Plane) : Prop :=  ∃ chart : OpenPartialHomeomorph Plane Plane,    x ∈ chart.source ∧ chart x = (0,0) ∧    chart '' (chart.source ∩ A) = chart.target ∩ {y | y.2 = 0} ∧    chart '' (chart.source ∩ B) = chart.target ∩ {y | y.1 = 0}def crossingPoints {m n : ℕ} (D : ContinuousDrawing m n) : Set Plane :=  {x | ∃ e f : BipartiteEdge m n, e ≠ f ∧ x ∈ edgeInterior D e ∧ x ∈ edgeInterior D f}structure AdmissibleDrawing (m n : ℕ) extends ContinuousDrawing m n where  crossings_finite : (crossingPoints toContinuousDrawing).Finite  crossings_proper : ∀ e f, e ≠ f → ∀ x,    x ∈ edgeInterior toContinuousDrawing e → x ∈ edgeInterior toContinuousDrawing f →    ProperCrossing (edgeInterior toContinuousDrawing e) (edgeInterior toContinuousDrawing f) x  no_triples : ∀ e f g, e ≠ f → e ≠ g → f ≠ g →    edgeInterior toContinuousDrawing e ∩ edgeInterior toContinuousDrawing f ∩      edgeInterior toContinuousDrawing g = ∅def crossingCount {m n : ℕ} (D : AdmissibleDrawing m n) : ℕ := D.crossings_finite.toFinset.carddef MainTarget : Prop :=  ∀ m n : ℕ, 0 < m → 0 < n →    (∃ D : AdmissibleDrawing m n, crossingCount D = axisPairs m * axisPairs n) ∧    (∀ D : AdmissibleDrawing m n, axisPairs m * axisPairs n ≤ crossingCount D)theorem mainTarget_proof : MainTarget := by  sorryend Zarankiewiczendend 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.

The crossing number of complete graphs

Theorem

For every integer n\ge3,

\crn(K_n)=Z(n):= \frac14\left\lfloor\frac n2\right\rfloor \left\lfloor\frac{n-1}2\right\rfloor \left\lfloor\frac{n-2}2\right\rfloor \left\lfloor\frac{n-3}2\right\rfloor.

Section 1, Introduction Read it in the paper TeX source, line 117

Show the TeX

For every integer \(n\ge3\),
\[
\crn(K_n)=Z(n):=
\frac14\left\lfloor\frac n2\right\rfloor
\left\lfloor\frac{n-1}2\right\rfloor
\left\lfloor\frac{n-2}2\right\rfloor
\left\lfloor\frac{n-3}2\right\rfloor.
\]

The crossing number of complete bipartite graphs

Theorem

For all positive integers m,n,

\crn(K_{m,n})=d_m d_n.

Section 1, Introduction Read it in the paper TeX source, line 29

Show the TeX

For all positive integers $m,n$,
\[
 \crn(K_{m,n})=d_m d_n.
\]

Configuration and file facts

File: lean/ComparatorChallenges/BipartiteCrossing.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Combinatorics/Crossing/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.Topology.Path, Mathlib.Topology.OpenPartialHomeomorph.Defs, Mathlib.Topology.Instances.Real.Lemmas, Mathlib.Data.Set.Finite.Basic, Mathlib.Data.Finset.Attr, Mathlib.Tactic.Common, Mathlib.Tactic.Finiteness.Attr, Mathlib.Tactic.SetLike, Mathlib.Util.CompileInductive
Imports from outside Mathlib: None
Local vocabulary: 11 local vocabulary declarations
Lines of Lean, without comments or blank lines: 35
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.

Plectis Atlas