Lean statement

Taming implies compatibility

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.TamingCompatibility.taming_implies_compatibility

Lean statement file: TamingCompatibility.leanShow on the mapFamily 342. 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 taming_implies_compatibility    [T2Space X] [SecondCountableTopology X] [CompactSpace X] [ConnectedSpace X]    (J : AlmostComplexStructure X)    (h : ∃ α : TwoForm X, IsSymplectic α ∧ Tames α J) :    ∃ η : TwoForm X, IsSymplectic η ∧ Compatible η J := 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.TamingCompatibility.Space line 10
```
abbrev Space := EuclideanSpace ℝ (Fin 4)
```
abbreviation OAI.TamingCompatibility.Model line 11
```
abbrev Model := 𝓘(ℝ, Space)
```

abbreviation OAI.TamingCompatibility.TwoForm line 16

abbrev TwoForm (X : Type*) [TopologicalSpace X] [ChartedSpace Space X] :=  (x : X) → (TangentSpace Model x) [⋀^Fin 2]→L[ℝ] ℝ

structure OAI.TamingCompatibility.AlmostComplexStructure line 19

structure AlmostComplexStructure (X : Type*) [TopologicalSpace X]    [ChartedSpace Space X] [IsManifold Model ∞ X] where  endomorphism : (x : X) → TangentSpace Model x →L[ℝ] TangentSpace Model x  square : ∀ x v, endomorphism x (endomorphism x v) = -v  smooth : ContMDiff Model.tangent Model.tangent ∞    (fun p : TangentBundle Model X =>      (⟨p.proj, endomorphism p.proj p.2⟩ : TangentBundle Model X))

definition OAI.TamingCompatibility.eval line 27

def eval (α : TwoForm X) (x : X) (u v : TangentSpace Model x) : ℝ :=  α x ![u, v]

definition OAI.TamingCompatibility.pullback line 30

noncomputable def pullback (α : TwoForm X) (f : Space → X) :    Space → Space [⋀^Fin 2]→L[ℝ] ℝ :=  fun y => (α (f y)).compContinuousLinearMap (mfderiv Model Model f y)

definition OAI.TamingCompatibility.IsSmooth line 34

def IsSmooth (α : TwoForm X) : Prop :=  ∀ (f : Space → X) (U : Set Space), IsOpen U →    ContMDiffOn Model Model ∞ f U → ContDiffOn ℝ ∞ (pullback α f) U

definition OAI.TamingCompatibility.IsClosed line 38

def IsClosed (α : TwoForm X) : Prop :=  ∀ (f : Space → X) (U : Set Space), IsOpen U →    ContMDiffOn Model Model ∞ f U → ∀ y ∈ U,      extDerivWithin (pullback α f) U y = 0

definition OAI.TamingCompatibility.IsNondegenerate line 43

def IsNondegenerate (α : TwoForm X) : Prop :=  ∀ x v, (∀ w, eval α x v w = 0) → v = 0

definition OAI.TamingCompatibility.IsSymplectic line 46

def IsSymplectic (α : TwoForm X) : Prop :=  IsSmooth α ∧ IsClosed α ∧ IsNondegenerate α

definition OAI.TamingCompatibility.Tames line 49

def Tames (α : TwoForm X) (J : AlmostComplexStructure X) : Prop :=  ∀ x v, v ≠ 0 → 0 < eval α x v (J.endomorphism x v)

definition OAI.TamingCompatibility.IsInvariant line 52

def IsInvariant (α : TwoForm X) (J : AlmostComplexStructure X) : Prop :=  ∀ x u v, eval α x (J.endomorphism x u) (J.endomorphism x v) = eval α x u v

definition OAI.TamingCompatibility.Compatible line 55

def Compatible (α : TwoForm X) (J : AlmostComplexStructure X) : Prop :=  Tames α J ∧ IsInvariant α J

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: connected, real, almost, complex, manifold, tamed, compatible, symplectic and three more. Found in the paper’s title, which its main theorem states: implies and compatibility. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: four.

Theorem 1.1 of Taming implies compatibility on four-manifolds

Theorem

Suppose that a smooth almost complex structure J on a closed connected smooth real four-manifold X is tamed by a symplectic form. Then there is a smooth symplectic form compatible with J.

Section 1, Introduction Read it in the paper TeX source, line 23

Show the TeX

Suppose that a smooth almost complex structure \(J\) on a closed
connected smooth real four-manifold \(X\) is tamed by a symplectic
form. Then there is a smooth symplectic form compatible with \(J\).

Other possible paper matches, closest wording first

Theorem 1.2 of Taming implies compatibility on four-manifolds. Words of the Lean statement, its names split into words, that also appear in the theorem: real, two, taming, compatible, symplectic, invariant and three more.
Proposition 2.2 of Taming implies compatibility on four-manifolds. Words of the Lean statement, its names split into words, that also appear in the theorem: two, compatible, symplectic, invariant, smooth, form and one more.
Corollary 1.3 of Taming implies compatibility on four-manifolds. Words of the Lean statement, its names split into words, that also appear in the theorem: taming, compatible, symplectic, smooth and form.
Theorem 5.1 of Taming implies compatibility on four-manifolds. Words of the Lean statement, its names split into words, that also appear in the theorem: almost, manifold, complex, invariant, smooth and closed. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: four.
Proposition 8.4 of Taming implies compatibility on four-manifolds. Words of the Lean statement, its names split into words, that also appear in the theorem: invariant, forms 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

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 342’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: taming_implies_compatibility: for a closed connected smooth 4-manifold X and a smooth almost complex structure J, if some symplectic form tames J then some symplectic form η satisfies Compatible η J (tames and J-invariant), with J fixed and no integrability or cohomology-class condition.
Definitions the group asks readers to compare with the paper: Differential forms are hand-built: TwoForm X := ∀ x, (T_x X)[⋀²]→L[ℝ] ℝ with IsSmooth (pullbacks along every smooth chart-like map f : ℝ⁴ → X are C^∞) and IsClosed (extDerivWithin of the pullback vanishes); AlmostComplexStructure with smoothness as a map of tangent bundles. Reads as the standard notions.
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 establishes this for every closed connected smooth four-manifold, keeping the almost-complex structure fixed. No integrability assumption or prescribed cohomology class for the compatible form is required.”
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

68 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: 57 to its own declarations, 32 to Mathlib’s documentation. 14 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 taming_implies_compatibility 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 13 other declarations of the file; with the theorem itself they span 39 lines. 5 of them are named in the statement itself.

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

abbrev Spaceline 10library
abbrev Modelline 11library
abbrev TwoFormline 16in the statementlibrary
structure AlmostComplexStructureline 19in the statementlibrary
def evalline 27library
def pullbackline 30library
def IsSmoothline 34library
def IsClosedline 38library
def IsNondegenerateline 43library
def IsSymplecticline 46in the statementlibrary
def Tamesline 49in the statementlibrary
def IsInvariantline 52library
def Compatibleline 55in the statementlibrary
theorem taming_implies_compatibilityline 58selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen scoped Manifold ContDiffnamespace TamingCompatibilityabbrev Space := EuclideanSpace ℝ (Fin 4)abbrev Model := 𝓘(ℝ, Space)variable {X : Type*} [TopologicalSpace X] [ChartedSpace Space X]  [IsManifold Model ∞ X]abbrev TwoForm (X : Type*) [TopologicalSpace X] [ChartedSpace Space X] :=  (x : X) → (TangentSpace Model x) [⋀^Fin 2]→L[ℝ] ℝstructure AlmostComplexStructure (X : Type*) [TopologicalSpace X]    [ChartedSpace Space X] [IsManifold Model ∞ X] where  endomorphism : (x : X) → TangentSpace Model x →L[ℝ] TangentSpace Model x  square : ∀ x v, endomorphism x (endomorphism x v) = -v  smooth : ContMDiff Model.tangent Model.tangent ∞    (fun p : TangentBundle Model X =>      (⟨p.proj, endomorphism p.proj p.2⟩ : TangentBundle Model X))def eval (α : TwoForm X) (x : X) (u v : TangentSpace Model x) : ℝ :=  α x ![u, v]noncomputable def pullback (α : TwoForm X) (f : Space → X) :    Space → Space [⋀^Fin 2]→L[ℝ] ℝ :=  fun y => (α (f y)).compContinuousLinearMap (mfderiv Model Model f y)def IsSmooth (α : TwoForm X) : Prop :=  ∀ (f : Space → X) (U : Set Space), IsOpen U →    ContMDiffOn Model Model ∞ f U → ContDiffOn ℝ ∞ (pullback α f) Udef IsClosed (α : TwoForm X) : Prop :=  ∀ (f : Space → X) (U : Set Space), IsOpen U →    ContMDiffOn Model Model ∞ f U → ∀ y ∈ U,      extDerivWithin (pullback α f) U y = 0def IsNondegenerate (α : TwoForm X) : Prop :=  ∀ x v, (∀ w, eval α x v w = 0) → v = 0def IsSymplectic (α : TwoForm X) : Prop :=  IsSmooth α ∧ IsClosed α ∧ IsNondegenerate αdef Tames (α : TwoForm X) (J : AlmostComplexStructure X) : Prop :=  ∀ x v, v ≠ 0 → 0 < eval α x v (J.endomorphism x v)def IsInvariant (α : TwoForm X) (J : AlmostComplexStructure X) : Prop :=  ∀ x u v, eval α x (J.endomorphism x u) (J.endomorphism x v) = eval α x u vdef Compatible (α : TwoForm X) (J : AlmostComplexStructure X) : Prop :=  Tames α J ∧ IsInvariant α Jtheorem taming_implies_compatibility    [T2Space X] [SecondCountableTopology X] [CompactSpace X] [ConnectedSpace X]    (J : AlmostComplexStructure X)    (h : ∃ α : TwoForm X, IsSymplectic α ∧ Tames α J) :    ∃ η : TwoForm X, IsSymplectic η ∧ Compatible η J := by  sorryend TamingCompatibilityendend 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.