Lean statement

Crossing number of complete 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.Paper170.complete_graph_crossing_number

Lean statement file: CompleteCrossing.leanShow on the mapFamily 165. OpenAI lists this as a main result in its Lean catalogue.

See the closest candidates 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 complete_graph_crossing_number (order : ℕ) (at_least_three : 3 ≤ order) :    ordinaryCrossingNumber (completeGraph order) = hill order := 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.Paper170.Plane line 13
```
abbrev Plane := ℝ × ℝ
```

structure OAI.Paper170.Graph line 15

structure Graph (Vertex Edge : Type) where  src : Edge → Vertex  dst : Edge → Vertex  loopless : ∀ edge, src edge ≠ dst edge  simple : ∀ edge other, ((src edge = src other ∧ dst edge = dst other) ∨    (src edge = dst other ∧ dst edge = src other)) → edge = other

abbreviation OAI.Paper170.CompleteEdge line 22

abbrev CompleteEdge (order : ℕ) := {pair : Fin order × Fin order // pair.1 < pair.2}

definition OAI.Paper170.completeGraph line 24

def completeGraph (order : ℕ) : Graph (Fin order) (CompleteEdge order) where

Show the remaining 10 lines

  src edge := edge.val.1  dst edge := edge.val.2  loopless edge := ne_of_lt edge.property  simple edge other same := by    rcases same with same | reversed    · exact Subtype.ext (Prod.ext same.1 same.2)    · have increasing := edge.property      have other_increasing := other.property      rw [reversed.1, reversed.2] at increasing      exact (lt_asymm increasing other_increasing).elim

structure OAI.Paper170.ContinuousDrawing line 38

structure ContinuousDrawing (graph : Graph Vertex Edge) where  vertex : Vertex → Plane  vertex_injective : Function.Injective vertex  edge : ∀ edge : Edge, Path (vertex (graph.src edge)) (vertex (graph.dst edge))  edge_injective : ∀ selected, Function.Injective (edge selected)  interior_avoids_vertices : ∀ selected time, time ∈ Set.Ioo (0 : unitInterval) 1 →    ∀ other, edge selected time ≠ vertex other

definition OAI.Paper170.edgeInterior line 48

def edgeInterior (drawing : ContinuousDrawing graph) (edge : Edge) : Set Plane :=  drawing.edge edge '' Set.Ioo (0 : unitInterval) 1

structure OAI.Paper170.ProperCrossing line 51

structure ProperCrossing (first second : Set Plane) (point : Plane) : Prop where  chart_exists : ∃ chart : OpenPartialHomeomorph Plane Plane,    point ∈ chart.source ∧ chart point = (0, 0) ∧    chart '' (chart.source ∩ first) = chart.target ∩ {image | image.2 = 0} ∧    chart '' (chart.source ∩ second) = chart.target ∩ {image | image.1 = 0}

definition OAI.Paper170.crossingPoints line 57

def crossingPoints (drawing : ContinuousDrawing graph) : Set Plane :=  {point | ∃ edge other : Edge, edge ≠ other ∧    point ∈ edgeInterior drawing edge ∧ point ∈ edgeInterior drawing other}

structure OAI.Paper170.AdmissibleDrawing line 61

structure AdmissibleDrawing (graph : Graph Vertex Edge) extends ContinuousDrawing graph where  crossings_finite : (crossingPoints toContinuousDrawing).Finite  crossings_proper : ∀ edge other, edge ≠ other → ∀ point,    point ∈ edgeInterior toContinuousDrawing edge →    point ∈ edgeInterior toContinuousDrawing other →    ProperCrossing (edgeInterior toContinuousDrawing edge)      (edgeInterior toContinuousDrawing other) point  no_triples : ∀ first second third, first ≠ second → first ≠ third → second ≠ third →    edgeInterior toContinuousDrawing first ∩ edgeInterior toContinuousDrawing second ∩      edgeInterior toContinuousDrawing third = ∅

definition OAI.Paper170.crossingCount line 72

def crossingCount (drawing : AdmissibleDrawing graph) : ℕ :=  drawing.crossings_finite.toFinset.card

definition OAI.Paper170.ordinaryCrossingNumber line 75

def ordinaryCrossingNumber (graph : Graph Vertex Edge) : ℕ :=  sInf (Set.range (@crossingCount Vertex Edge graph))

definition OAI.Paper170.hill line 78

def hill (order : ℕ) : ℕ :=  (order / 2 * ((order - 1) / 2) * ((order - 2) / 2) * ((order - 3) / 2)) / 4

No confident match with a paper statement

No paper theorem is paired with this Lean statement.

Why none: no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.

Other possible paper matches, closest wording first

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: number, complete, graphs and crossing. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: 1/4 and n/2.
Theorem 1.1 of The crossing number of complete bipartite graphs. 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: number, complete, graphs and crossing.
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: least, drawing and crossings. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bound and lower.
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.
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, pairs and ordered.

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 64 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 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

“The formalized result proves Hill's proposed formula for the crossing number of the complete graph: for every $n\ge3$, $\mathrm{cr}(K_n)=\frac14\lfloor n/2\rfloor\lfloor(n-1)/2\rfloor\lfloor(n-2)/2\rfloor\lfloor(n-3)/2\rfloor$. It proves the lower bound and constructs a matching two-page drawing.”
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

89 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: 32 to its own declarations, 20 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 complete_graph_crossing_number 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 55 lines. 3 of them are named in the statement itself.

Names in the namespace OAI.Paper170 are listed without it.

abbrev Planeline 13library
structure Graphline 15library
abbrev CompleteEdgeline 22library
def completeGraphline 24in the statementlibrary
structure ContinuousDrawingline 38library
def edgeInteriorline 48library
structure ProperCrossingline 51library
def crossingPointsline 57library
structure AdmissibleDrawingline 61library
def crossingCountline 72library
def ordinaryCrossingNumberline 75in the statementlibrary
def hillline 78in the statementlibrary
theorem complete_graph_crossing_numberline 81selected theoremlibrary

import Mathlib.Data.Set.Cardimport Mathlib.Order.Lattice.Natimport Mathlib.Topology.Instances.Real.Lemmasimport Mathlib.Topology.OpenPartialHomeomorph.Defsimport Mathlib.Topology.Pathnamespace OAInoncomputable sectionnamespace Paper170abbrev Plane := ℝ × ℝstructure Graph (Vertex Edge : Type) where  src : Edge → Vertex  dst : Edge → Vertex  loopless : ∀ edge, src edge ≠ dst edge  simple : ∀ edge other, ((src edge = src other ∧ dst edge = dst other) ∨    (src edge = dst other ∧ dst edge = src other)) → edge = otherabbrev CompleteEdge (order : ℕ) := {pair : Fin order × Fin order // pair.1 < pair.2}def completeGraph (order : ℕ) : Graph (Fin order) (CompleteEdge order) where  src edge := edge.val.1  dst edge := edge.val.2  loopless edge := ne_of_lt edge.property  simple edge other same := by    rcases same with same | reversed    · exact Subtype.ext (Prod.ext same.1 same.2)    · have increasing := edge.property      have other_increasing := other.property      rw [reversed.1, reversed.2] at increasing      exact (lt_asymm increasing other_increasing).elimvariable {Vertex Edge : Type}structure ContinuousDrawing (graph : Graph Vertex Edge) where  vertex : Vertex → Plane  vertex_injective : Function.Injective vertex  edge : ∀ edge : Edge, Path (vertex (graph.src edge)) (vertex (graph.dst edge))  edge_injective : ∀ selected, Function.Injective (edge selected)  interior_avoids_vertices : ∀ selected time, time ∈ Set.Ioo (0 : unitInterval) 1 →    ∀ other, edge selected time ≠ vertex othervariable {graph : Graph Vertex Edge}def edgeInterior (drawing : ContinuousDrawing graph) (edge : Edge) : Set Plane :=  drawing.edge edge '' Set.Ioo (0 : unitInterval) 1structure ProperCrossing (first second : Set Plane) (point : Plane) : Prop where  chart_exists : ∃ chart : OpenPartialHomeomorph Plane Plane,    point ∈ chart.source ∧ chart point = (0, 0) ∧    chart '' (chart.source ∩ first) = chart.target ∩ {image | image.2 = 0} ∧    chart '' (chart.source ∩ second) = chart.target ∩ {image | image.1 = 0}def crossingPoints (drawing : ContinuousDrawing graph) : Set Plane :=  {point | ∃ edge other : Edge, edge ≠ other ∧    point ∈ edgeInterior drawing edge ∧ point ∈ edgeInterior drawing other}structure AdmissibleDrawing (graph : Graph Vertex Edge) extends ContinuousDrawing graph where  crossings_finite : (crossingPoints toContinuousDrawing).Finite  crossings_proper : ∀ edge other, edge ≠ other → ∀ point,    point ∈ edgeInterior toContinuousDrawing edge →    point ∈ edgeInterior toContinuousDrawing other →    ProperCrossing (edgeInterior toContinuousDrawing edge)      (edgeInterior toContinuousDrawing other) point  no_triples : ∀ first second third, first ≠ second → first ≠ third → second ≠ third →    edgeInterior toContinuousDrawing first ∩ edgeInterior toContinuousDrawing second ∩      edgeInterior toContinuousDrawing third = ∅def crossingCount (drawing : AdmissibleDrawing graph) : ℕ :=  drawing.crossings_finite.toFinset.carddef ordinaryCrossingNumber (graph : Graph Vertex Edge) : ℕ :=  sInf (Set.range (@crossingCount Vertex Edge graph))def hill (order : ℕ) : ℕ :=  (order / 2 * ((order - 1) / 2) * ((order - 2) / 2) * ((order - 3) / 2)) / 4theorem complete_graph_crossing_number (order : ℕ) (at_least_three : 3 ≤ order) :    ordinaryCrossingNumber (completeGraph order) = hill order := by  sorryend Paper170endend 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/CompleteCrossing.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Combinatorics/CompleteCrossing/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.Set.Card, Mathlib.Order.Lattice.Nat, Mathlib.Topology.Instances.Real.Lemmas, Mathlib.Topology.OpenPartialHomeomorph.Defs, Mathlib.Topology.Path
Imports from outside Mathlib: None
Local vocabulary: 12 local vocabulary declarations
Lines of Lean, without comments or blank lines: 55
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