Lean statement

No conjugate points without a nonpositively curved metric

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.ThreeManifold.main_theorem

Lean statement file: ConjugatePoints.leanShow on the mapFamily 358. 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 MainStatement : Prop :=  ∃ (M : Type) (top : TopologicalSpace M),    letI := top    ∃ (_ : T2Space M) (_ : SecondCountableTopology M) (_ : CompactSpace M)      (_ : ConnectedSpace M) (charts : ChartedSpace E3 M),      letI := charts      ∃ (smooth : IsManifold I3 ∞ M),        letI := smooth        OrientedAtlas (M := M) ∧          (∃ g : SmoothMetric M, NoConjugatePoints g) ∧          ¬ ∃ g : SmoothMetric M, NonpositiveSectionalCurvature g⋮theorem main_theorem : MainStatement := 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.ThreeManifold.E3 line 11
```
abbrev E3 := Fin 3 → ℝ
```
abbreviation OAI.ThreeManifold.I3 line 12
```
abbrev I3 := 𝓘(ℝ, E3)
```

definition OAI.ThreeManifold.OrientedAtlas line 16

def OrientedAtlas : Prop :=  ∀ (c d : OpenPartialHomeomorph M E3), c ∈ atlas E3 M → d ∈ atlas E3 M →    ∀ x ∈ c.target, c.symm x ∈ d.source →      0 < LinearMap.det (fderiv ℝ (d ∘ c.symm) x).toLinearMap

abbreviation OAI.ThreeManifold.SmoothMetric line 23

abbrev SmoothMetric (M : Type) [TopologicalSpace M] [ChartedSpace E3 M]    [IsManifold I3 ∞ M] :=  ContMDiffRiemannianMetric I3 ∞ E3 (fun x : M => TangentSpace I3 x)

definition OAI.ThreeManifold.basisVector line 27

def basisVector (i : Fin 3) : E3 := Pi.single i 1

definition OAI.ThreeManifold.metricMatrix line 29

def metricMatrix (g : SmoothMetric M) (c : OpenPartialHomeomorph M E3) (x : E3) :    Matrix (Fin 3) (Fin 3) ℝ :=  fun i j => g.inner (c.symm x)    (mfderiv I3 I3 c.symm x (basisVector i))    (mfderiv I3 I3 c.symm x (basisVector j))

definition OAI.ThreeManifold.coordPartial line 35

def coordPartial (f : E3 → ℝ) (x : E3) (i : Fin 3) : ℝ :=  fderiv ℝ f x (basisVector i)

definition OAI.ThreeManifold.christoffel line 38

def christoffel (g : SmoothMetric M) (c : OpenPartialHomeomorph M E3)    (x : E3) (i j k : Fin 3) : ℝ :=  (1 / 2 : ℝ) * ∑ l : Fin 3, (metricMatrix g c x)⁻¹ i l *    (coordPartial (fun y => metricMatrix g c y l k) x j +     coordPartial (fun y => metricMatrix g c y l j) x k -     coordPartial (fun y => metricMatrix g c y j k) x l)

definition OAI.ThreeManifold.connectionTerm line 45

def connectionTerm (g : SmoothMetric M) (c : OpenPartialHomeomorph M E3)    (x u v : E3) : E3 :=  fun i => ∑ j : Fin 3, ∑ k : Fin 3, christoffel g c x i j k * u j * v k

definition OAI.ThreeManifold.curvatureCoefficient line 49

def curvatureCoefficient (g : SmoothMetric M) (c : OpenPartialHomeomorph M E3)    (x : E3) (l k i j : Fin 3) : ℝ :=  coordPartial (fun y => christoffel g c y l j k) x i -  coordPartial (fun y => christoffel g c y l i k) x j +    ∑ a : Fin 3, (christoffel g c x l i a * christoffel g c x a j k -      christoffel g c x l j a * christoffel g c x a i k)

definition OAI.ThreeManifold.curvatureTerm line 56

def curvatureTerm (g : SmoothMetric M) (c : OpenPartialHomeomorph M E3)    (x u v w : E3) : E3 :=  fun l => ∑ k : Fin 3, ∑ i : Fin 3, ∑ j : Fin 3,    curvatureCoefficient g c x l k i j * w k * u i * v j

definition OAI.ThreeManifold.NonpositiveSectionalCurvature line 61

def NonpositiveSectionalCurvature (g : SmoothMetric M) : Prop :=  ∀ (c : OpenPartialHomeomorph M E3), c ∈ atlas E3 M →    ∀ x ∈ c.target, ∀ u v : E3,      (∑ i : Fin 3, ∑ j : Fin 3,        metricMatrix g c x i j * curvatureTerm g c x u v v i * u j) ≤ 0

definition OAI.ThreeManifold.alongDerivative line 67

def alongDerivative (g : SmoothMetric M) (c : OpenPartialHomeomorph M E3)    (x w : ℝ → E3) (s : ℝ) : E3 :=  deriv w s + connectionTerm g c (x s) (deriv x s) (w s)

definition OAI.ThreeManifold.IsGeodesicOn line 71

def IsGeodesicOn (g : SmoothMetric M) (γ : ℝ → M) (U : Set ℝ) : Prop :=  ContMDiffOn 𝓘(ℝ, ℝ) I3 ∞ γ U ∧    ∀ (c : OpenPartialHomeomorph M E3), c ∈ atlas E3 M →      ∀ s ∈ U, γ s ∈ c.source →        alongDerivative g c (c ∘ γ) (deriv (c ∘ γ)) s = 0

definition OAI.ThreeManifold.fieldCoordinates line 77

def fieldCoordinates (γ : ℝ → M) (J : ∀ s, TangentSpace I3 (γ s))    (c : OpenPartialHomeomorph M E3) (s : ℝ) : E3 :=  mfderiv I3 I3 c (γ s) (J s)

definition OAI.ThreeManifold.IsJacobiFieldOn line 81

def IsJacobiFieldOn (g : SmoothMetric M) (γ : ℝ → M)    (J : ∀ s, TangentSpace I3 (γ s)) (U : Set ℝ) : Prop :=  ∀ (c : OpenPartialHomeomorph M E3), c ∈ atlas E3 M →    ContDiffOn ℝ ∞ (fieldCoordinates γ J c) (U ∩ γ ⁻¹' c.source) ∧    ∀ s ∈ U, γ s ∈ c.source →      alongDerivative g c (c ∘ γ)        (alongDerivative g c (c ∘ γ) (fieldCoordinates γ J c)) s +      curvatureTerm g c (c (γ s)) (fieldCoordinates γ J c s)        (deriv (c ∘ γ) s) (deriv (c ∘ γ) s) = 0

definition OAI.ThreeManifold.NoConjugatePoints line 91

def NoConjugatePoints (g : SmoothMetric M) : Prop :=  ∀ (U : Set ℝ), IsOpen U → OrdConnected U →    ∀ (γ : ℝ → M), IsGeodesicOn g γ U →      ∀ (J : ∀ s, TangentSpace I3 (γ s)), IsJacobiFieldOn g γ J U →        ∀ a ∈ U, ∀ b ∈ U, a ≠ b → J a = 0 → J b = 0 → ∀ s ∈ U, J s = 0

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: conjugate, orientable, points, connected, riemannian, sectional, curvature, metric and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: without, everywhere, three, admits and closed.

Theorem 1.1 of A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric

Theorem

There exists a closed connected orientable smooth three-manifold M with a C^\infty Riemannian metric without conjugate points, such that M admits no C^\infty Riemannian metric whose sectional curvature is everywhere at most zero.

Section 1, Introduction Read it in the paper TeX source, line 27

Show the TeX

There exists a closed connected orientable smooth three-manifold $M$ with a $C^\infty$ Riemannian metric without conjugate points, such that $M$ admits no $C^\infty$ Riemannian metric whose sectional curvature is everywhere at most zero.

Other possible paper matches, closest wording first

Proposition 6.1 of A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric. Words of the Lean statement, its names split into words, that also appear in the theorem: real, nonpositive, point, field, connected, sectional and six more.
Proposition 2.3 of A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric. Words of the Lean statement, its names split into words, that also appear in the theorem: nonpositive, sectional, riemannian, curvature, metric, smooth and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: three, admits and closed.
Proposition 5.1 of A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric. Words of the Lean statement, its names split into words, that also appear in the theorem: single, field, geodesic, along and smooth. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constructed and closed.
Proposition 5.5 of A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric. Words of the Lean statement, its names split into words, that also appear in the theorem: field, connected, geodesic, along, smooth and manifold. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: resulting and closed.
Corollary 2.2 of A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric. Words of the Lean statement, its names split into words, that also appear in the theorem: topology, metric and manifold. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: cat and admits.
Theorem 2.1 of A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric. Words of the Lean statement, its names split into words, that also appear in the theorem: pi, space and manifold. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: cat, three, admits and closed.

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 288 s; peak memory 6.6 GiB).
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 358’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: MainStatement: there is a compact connected Hausdorff second-countable C^∞ 3-manifold with oriented atlas carrying a smooth metric with NoConjugatePoints (for every geodesic and Jacobi field on any open interval, J vanishing at two distinct parameters vanishes identically, constant geodesics included) while no smooth metric satisfies NonpositiveSectionalCurvature.
Definitions the group asks readers to compare with the paper: Hand-written chart-wise Christoffel symbols, curvature curvatureTerm, geodesics and Jacobi fields (IsGeodesicOn, IsJacobiFieldOn); sign conventions checked: ⟨R(u,v)v,u⟩ ≤ 0 is sectional curvature ≤ 0 and the Jacobi equation J'' + R(J,γ')γ' = 0 is the standard one
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 separates absence of conjugate points from nonpositive sectional curvature. It constructs a closed connected orientable smooth three-manifold with a smooth Riemannian metric having no conjugate points, while the same manifold admits no smooth metric of everywhere nonpositive sectional curvature. The no-conjugate-points assertion ranges over all geodesics and tangent-bundle Jacobi fields, including constant geodesics and unbounded time intervals. The separate no-focal-points and CAT(0) consequences are not included.”
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

116 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: 98 to its own declarations, 70 to Mathlib’s documentation. 18 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 main_theorem 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 18 other declarations of the file; with the theorem itself they span 79 lines. 1 of them is named in the statement itself.

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

abbrev E3line 11library
abbrev I3line 12library
def OrientedAtlasline 16library
abbrev SmoothMetricline 23library
def basisVectorline 27library
def metricMatrixline 29library
def coordPartialline 35library
def christoffelline 38library
def connectionTermline 45library
def curvatureCoefficientline 49library
def curvatureTermline 56library
def NonpositiveSectionalCurvatureline 61library
def alongDerivativeline 67library
def IsGeodesicOnline 71library
def fieldCoordinatesline 77library
def IsJacobiFieldOnline 81library
def NoConjugatePointsline 91library
def MainStatementline 97in the statementlibrary
theorem main_theoremline 109selected theorem

import Mathlibnamespace OAInoncomputable sectionopen Set Manifold Bundleopen scoped ContDiffnamespace ThreeManifoldabbrev E3 := Fin 3 → ℝabbrev I3 := 𝓘(ℝ, E3)variable {M : Type} [TopologicalSpace M] [ChartedSpace E3 M]def OrientedAtlas : Prop :=  ∀ (c d : OpenPartialHomeomorph M E3), c ∈ atlas E3 M → d ∈ atlas E3 M →    ∀ x ∈ c.target, c.symm x ∈ d.source →      0 < LinearMap.det (fderiv ℝ (d ∘ c.symm) x).toLinearMapvariable [IsManifold I3 ∞ M]abbrev SmoothMetric (M : Type) [TopologicalSpace M] [ChartedSpace E3 M]    [IsManifold I3 ∞ M] :=  ContMDiffRiemannianMetric I3 ∞ E3 (fun x : M => TangentSpace I3 x)def basisVector (i : Fin 3) : E3 := Pi.single i 1def metricMatrix (g : SmoothMetric M) (c : OpenPartialHomeomorph M E3) (x : E3) :    Matrix (Fin 3) (Fin 3) ℝ :=  fun i j => g.inner (c.symm x)    (mfderiv I3 I3 c.symm x (basisVector i))    (mfderiv I3 I3 c.symm x (basisVector j))def coordPartial (f : E3 → ℝ) (x : E3) (i : Fin 3) : ℝ :=  fderiv ℝ f x (basisVector i)def christoffel (g : SmoothMetric M) (c : OpenPartialHomeomorph M E3)    (x : E3) (i j k : Fin 3) : ℝ :=  (1 / 2 : ℝ) * ∑ l : Fin 3, (metricMatrix g c x)⁻¹ i l *    (coordPartial (fun y => metricMatrix g c y l k) x j +     coordPartial (fun y => metricMatrix g c y l j) x k -     coordPartial (fun y => metricMatrix g c y j k) x l)def connectionTerm (g : SmoothMetric M) (c : OpenPartialHomeomorph M E3)    (x u v : E3) : E3 :=  fun i => ∑ j : Fin 3, ∑ k : Fin 3, christoffel g c x i j k * u j * v kdef curvatureCoefficient (g : SmoothMetric M) (c : OpenPartialHomeomorph M E3)    (x : E3) (l k i j : Fin 3) : ℝ :=  coordPartial (fun y => christoffel g c y l j k) x i -  coordPartial (fun y => christoffel g c y l i k) x j +    ∑ a : Fin 3, (christoffel g c x l i a * christoffel g c x a j k -      christoffel g c x l j a * christoffel g c x a i k)def curvatureTerm (g : SmoothMetric M) (c : OpenPartialHomeomorph M E3)    (x u v w : E3) : E3 :=  fun l => ∑ k : Fin 3, ∑ i : Fin 3, ∑ j : Fin 3,    curvatureCoefficient g c x l k i j * w k * u i * v jdef NonpositiveSectionalCurvature (g : SmoothMetric M) : Prop :=  ∀ (c : OpenPartialHomeomorph M E3), c ∈ atlas E3 M →    ∀ x ∈ c.target, ∀ u v : E3,      (∑ i : Fin 3, ∑ j : Fin 3,        metricMatrix g c x i j * curvatureTerm g c x u v v i * u j) ≤ 0def alongDerivative (g : SmoothMetric M) (c : OpenPartialHomeomorph M E3)    (x w : ℝ → E3) (s : ℝ) : E3 :=  deriv w s + connectionTerm g c (x s) (deriv x s) (w s)def IsGeodesicOn (g : SmoothMetric M) (γ : ℝ → M) (U : Set ℝ) : Prop :=  ContMDiffOn 𝓘(ℝ, ℝ) I3 ∞ γ U ∧    ∀ (c : OpenPartialHomeomorph M E3), c ∈ atlas E3 M →      ∀ s ∈ U, γ s ∈ c.source →        alongDerivative g c (c ∘ γ) (deriv (c ∘ γ)) s = 0def fieldCoordinates (γ : ℝ → M) (J : ∀ s, TangentSpace I3 (γ s))    (c : OpenPartialHomeomorph M E3) (s : ℝ) : E3 :=  mfderiv I3 I3 c (γ s) (J s)def IsJacobiFieldOn (g : SmoothMetric M) (γ : ℝ → M)    (J : ∀ s, TangentSpace I3 (γ s)) (U : Set ℝ) : Prop :=  ∀ (c : OpenPartialHomeomorph M E3), c ∈ atlas E3 M →    ContDiffOn ℝ ∞ (fieldCoordinates γ J c) (U ∩ γ ⁻¹' c.source) ∧    ∀ s ∈ U, γ s ∈ c.source →      alongDerivative g c (c ∘ γ)        (alongDerivative g c (c ∘ γ) (fieldCoordinates γ J c)) s +      curvatureTerm g c (c (γ s)) (fieldCoordinates γ J c s)        (deriv (c ∘ γ) s) (deriv (c ∘ γ) s) = 0def NoConjugatePoints (g : SmoothMetric M) : Prop :=  ∀ (U : Set ℝ), IsOpen U → OrdConnected U →    ∀ (γ : ℝ → M), IsGeodesicOn g γ U →      ∀ (J : ∀ s, TangentSpace I3 (γ s)), IsJacobiFieldOn g γ J U →        ∀ a ∈ U, ∀ b ∈ U, a ≠ b → J a = 0 → J b = 0 → ∀ s ∈ U, J s = 0def MainStatement : Prop :=  ∃ (M : Type) (top : TopologicalSpace M),    letI := top    ∃ (_ : T2Space M) (_ : SecondCountableTopology M) (_ : CompactSpace M)      (_ : ConnectedSpace M) (charts : ChartedSpace E3 M),      letI := charts      ∃ (smooth : IsManifold I3 ∞ M),        letI := smooth        OrientedAtlas (M := M) ∧          (∃ g : SmoothMetric M, NoConjugatePoints g) ∧          ¬ ∃ g : SmoothMetric M, NonpositiveSectionalCurvature gtheorem main_theorem : MainStatement := by  sorryend ThreeManifoldendend 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.

A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric

Theorem

Section 1, Introduction Read it in the paper TeX source, line 27

Show the TeX

There exists a closed connected orientable smooth three-manifold $M$ with a $C^\infty$ Riemannian metric without conjugate points, such that $M$ admits no $C^\infty$ Riemannian metric whose sectional curvature is everywhere at most zero.

Configuration and file facts

File: lean/ComparatorChallenges/ConjugatePoints.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Geometry/ConjugatePoints/ThreeManifold.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
Imports from outside Mathlib: None
Local vocabulary: 18 local vocabulary declarations
Lines of Lean, without comments or blank lines: 79
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