Lean statement

Integrability of both holomorphic tangent summands

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.SplitTangent.integrability_main

Lean statement file: SplitTangentIntegrability.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 052 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 integrability_main : IntegrabilityStatement.{u} := 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.SplitTangent.ComplexManifold line 15

structure ComplexManifold where  dim : ℕ  Carrier : Type u  [topology : TopologicalSpace Carrier]  [charts : ChartedSpace (Fin dim → ℂ) Carrier]  [holomorphicAtlas : IsManifold 𝓘(ℂ, Fin dim → ℂ) ω Carrier]  [hausdorff : T2Space Carrier]  [secondCountable : SecondCountableTopology Carrier]

abbreviation OAI.SplitTangent.Model line 28

abbrev Model (X : ComplexManifold.{u}) := Fin X.dim → ℂ

abbreviation OAI.SplitTangent.ModelWithCorners line 29

abbrev ModelWithCorners (X : ComplexManifold.{u}) := 𝓘(ℂ, Model X)

abbreviation OAI.SplitTangent.Tangent line 30

abbrev Tangent (X : ComplexManifold.{u}) (x : X.Carrier) :=  TangentSpace (ModelWithCorners X) x

abbreviation OAI.SplitTangent.ProjectiveSpace line 33

abbrev ProjectiveSpace (N : ℕ) := ℙ ℂ (Fin (N + 1) → ℂ)

definition OAI.SplitTangent.projectiveCoordinates line 37

def projectiveCoordinates {N : ℕ} (i : Fin (N + 1))    (p : ProjectiveSpace N) : Fin (N + 1) → ℂ :=  fun j => (p.rep i)⁻¹ * p.rep j

structure OAI.SplitTangent.ProjectiveEmbedding line 43

structure ProjectiveEmbedding (X : ComplexManifold.{u}) where  N : ℕ  map : X.Carrier → ProjectiveSpace N  injective : Function.Injective map  open_patch : ∀ i, IsOpen {x | (map x).rep i ≠ 0}  holomorphic : ∀ i, ContMDiffOn (ModelWithCorners X) 𝓘(ℂ, Fin (N + 1) → ℂ) ω    (fun x => projectiveCoordinates i (map x)) {x | (map x).rep i ≠ 0}  immersion : ∀ x i, (map x).rep i ≠ 0 → Function.Injective    (mfderiv (ModelWithCorners X) 𝓘(ℂ, Fin (N + 1) → ℂ)      (fun y => projectiveCoordinates i (map y)) x)

structure OAI.SplitTangent.RationalCurve line 55

structure RationalCurve (X : ComplexManifold.{u}) where  finite : ℂ → X.Carrier  infinity : ℂ → X.Carrier  finite_holomorphic : ContMDiff 𝓘(ℂ, ℂ) (ModelWithCorners X) ω finite  infinity_holomorphic : ContMDiff 𝓘(ℂ, ℂ) (ModelWithCorners X) ω infinity  overlap : ∀ z : ℂ, z ≠ 0 → finite z = infinity z⁻¹

definition OAI.SplitTangent.RationalCurve.value line 62

def RationalCurve.value {X : ComplexManifold.{u}} (g : RationalCurve X)    (z : OnePoint ℂ) : X.Carrier :=  z.elim (g.infinity 0) g.finite

abbreviation OAI.SplitTangent.PairPolynomial line 66

abbrev PairPolynomial (N : ℕ) := MvPolynomial (Fin 2 × Fin (N + 1)) ℂ

definition OAI.SplitTangent.Bihomogeneous line 69

def Bihomogeneous {N : ℕ} (p : PairPolynomial N) : Prop :=  ∃ a b : ℕ, ∀ m ∈ p.support,    (∑ i : Fin (N + 1), m (0, i)) = a ∧    (∑ i : Fin (N + 1), m (1, i)) = b

definition OAI.SplitTangent.pairEval line 74

def pairEval {X : ComplexManifold.{u}} (e : ProjectiveEmbedding X)    (p : PairPolynomial e.N) (xy : X.Carrier × X.Carrier) : ℂ :=  MvPolynomial.eval    (fun k : Fin 2 × Fin (e.N + 1) =>      if k.1 = 0 then (e.map xy.1).rep k.2 else (e.map xy.2).rep k.2) p

definition OAI.SplitTangent.IsZariskiOpenPair line 82

def IsZariskiOpenPair {X : ComplexManifold.{u}} (e : ProjectiveEmbedding X)    (U : Set (X.Carrier × X.Carrier)) : Prop :=  ∃ equations : Set (PairPolynomial e.N),    (∀ p ∈ equations, Bihomogeneous p) ∧    U = {xy | ∃ p ∈ equations, pairEval e p xy ≠ 0}

definition OAI.SplitTangent.RationallyConnected line 90

def RationallyConnected {X : ComplexManifold.{u}} (e : ProjectiveEmbedding X) : Prop :=  ∃ U : Set (X.Carrier × X.Carrier), IsZariskiOpenPair e U ∧ U.Nonempty ∧ Dense U ∧    ∀ xy ∈ U, ∃ g : RationalCurve X, ∃ a b : OnePoint ℂ,      g.value a = xy.1 ∧ g.value b = xy.2

structure OAI.SplitTangent.TangentSplitting line 97

structure TangentSplitting (X : ComplexManifold.{u}) where  projector : ∀ x : X.Carrier, Tangent X x →L[ℂ] Tangent X x  idempotent : ∀ x v, projector x (projector x v) = projector x v  holomorphic : ContMDiff    ((ModelWithCorners X).prod 𝓘(ℂ, Model X))    ((ModelWithCorners X).prod 𝓘(ℂ, Model X)) ω    (fun v : TangentBundle (ModelWithCorners X) X.Carrier =>      (⟨v.proj, projector v.proj v.2⟩ : TangentBundle (ModelWithCorners X) X.Carrier))  first_positive : ∀ x, 0 < Module.finrank ℂ (LinearMap.range (projector x).toLinearMap)  second_positive : ∀ x, 0 < Module.finrank ℂ (LinearMap.ker (projector x).toLinearMap)

definition OAI.SplitTangent.TangentSplitting.first line 108

def TangentSplitting.first {X : ComplexManifold.{u}} (S : TangentSplitting X) (x : X.Carrier) :    Submodule ℂ (Tangent X x) := LinearMap.range (S.projector x).toLinearMap

definition OAI.SplitTangent.TangentSplitting.second line 111

def TangentSplitting.second {X : ComplexManifold.{u}} (S : TangentSplitting X) (x : X.Carrier) :    Submodule ℂ (Tangent X x) := LinearMap.ker (S.projector x).toLinearMap

definition OAI.SplitTangent.HolomorphicSectionOn line 116

def HolomorphicSectionOn (X : ComplexManifold.{u})    (V : ∀ x : X.Carrier, Tangent X x) (U : Set X.Carrier) : Prop :=  ContMDiffOn (ModelWithCorners X) ((ModelWithCorners X).prod 𝓘(ℂ, Model X)) ω    (fun x => (⟨x, V x⟩ : TangentBundle (ModelWithCorners X) X.Carrier)) U

definition OAI.SplitTangent.Integrable line 122

def Integrable (X : ComplexManifold.{u}) (D : ∀ x : X.Carrier, Submodule ℂ (Tangent X x)) : Prop :=  ∀ U : Set X.Carrier, IsOpen U → ∀ V W : ∀ x : X.Carrier, Tangent X x,    HolomorphicSectionOn X V U → HolomorphicSectionOn X W U →    (∀ x ∈ U, V x ∈ D x) → (∀ x ∈ U, W x ∈ D x) →    ∀ x ∈ U, VectorField.mlieBracketWithin (ModelWithCorners X) V W U x ∈ D x

definition OAI.SplitTangent.IntegrabilityStatement line 130

def IntegrabilityStatement : Prop :=  ∀ (X : ComplexManifold.{u}), ConnectedSpace X.Carrier → CompactSpace X.Carrier →    2 ≤ X.dim → ∀ e : ProjectiveEmbedding X, RationallyConnected e →    ∀ S : TangentSplitting X, Integrable X S.first ∧ Integrable X S.second

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: points, positive, integrable, projective, complex, manifold, connected and holomorphic. Found in the paper’s title, which its main theorem states: rationally, bundles and tangent. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: two, least, rank and dimension.

Statements in Integrability of split tangent bundles on rationally connected manifolds come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of Integrability of split tangent bundles on rationally connected manifolds

Theorem (Automatic integrability)

Let X be a smooth connected projective complex manifold of dimension at least two. Suppose that two general points of X lie on the image of a morphism \PP^1\to X. For every specified holomorphic decomposition

T_X=E_1\oplus E_2

into subbundles of positive rank, both E_1 and E_2 are integrable.

Section 1, Introduction Read it in the paper TeX source, line 21

Show the TeX

\label[theorem]{thm:main}
Let \(X\) be a smooth connected projective complex manifold of dimension
at least two. Suppose that two general points of \(X\) lie on the image
of a morphism \(\PP^1\to X\). For every specified holomorphic decomposition
\[
T_X=E_1\oplus E_2
\]
into subbundles of positive rank, both \(E_1\) and \(E_2\) are integrable.

Other possible paper matches, closest wording first

Corollary 5.1 of Integrability of split tangent bundles on rationally connected manifolds. Words of the Lean statement, its names split into words, that also appear in the theorem: curve, one, rationally, compact, point, rational and six more.
Corollary 1.2 of Universal-cover splitting for compact Kähler manifolds. Words of the Lean statement, its names split into words, that also appear in the theorem: splitting, positive, integrable, manifolds, projective, complex and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: summands and ranks.
Theorem 1.1 of Universal-cover splitting for compact Kähler manifolds. Words of the Lean statement, its names split into words, that also appear in the theorem: dim, compact, positive, integrable, manifolds, complex and three more. Found in the paper’s title, which its main theorem states: splitting. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: dimension and ranks.
Corollary 1.3 of Universal-cover splitting for compact Kähler manifolds. Words of the Lean statement, its names split into words, that also appear in the theorem: bundle, splitting, positive, integrable, projective, complex and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: rank and dimension.
Proposition 2.1 of Integrability of split tangent bundles on rationally connected manifolds. Words of the Lean statement, its names split into words, that also appear in the theorem: integrable, tangent, map and holomorphic. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: summand.
Theorem C.1 of Universal-cover splitting for compact Kähler manifolds. Words of the Lean statement, its names split into words, that also appear in the theorem: nonempty, charts, rational, projective, connected, map and one more.

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 471 s; peak memory 6.5 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 052’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 disagree 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: Both headline theorems are stated; the 'corresponding product decomposition' for rationally connected manifolds (composition of the two, needing a Kahler metric from the embedding) is not a single statement. Complex manifold, Kahler metric, integrability and rational connectedness are all bespoke encodings.
Definitions the group asks readers to compare with the paper: Bespoke ComplexManifold/KahlerMetric (closedness of the fundamental form via a cyclic derivative sum in charts), Integrable (Frobenius condition through chartwise Lie brackets of holomorphic local sections), OrdinaryUniversalCover, CompatibleProduct, ProjectiveEmbedding (injective holomorphic immersion into P^N) and RationallyConnected (dense Zariski-open set of pairs joined by a RationalCurve); plausible but unchecked against Mathlib/standard notions
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — DECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note says so. Their record
Their note: The product decomposition for rationally connected projective manifolds (the last clause) is not stated; neither are the Kahler paper's further corollaries.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization proves integrability of both summands in a holomorphic splitting of the tangent bundle of a rationally connected projective manifold. More precisely, for a compact connected complex manifold of dimension at least two with a projective embedding witnessing rational connectedness, each positive-rank summand of a holomorphic tangent-bundle splitting is integrable.”
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

142 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: 69 to its own declarations, 55 to Mathlib’s documentation. 21 declared names also link to the same declaration in OpenAI’s library. 7 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names 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 integrability_main 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 20 other declarations of the file; with the theorem itself they span 83 lines. 1 of them is named in the statement itself.

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

structure ComplexManifoldline 15library
abbrev Modelline 28library
abbrev ModelWithCornersline 29library
abbrev Tangentline 30library
abbrev ProjectiveSpaceline 33library
def projectiveCoordinatesline 37library
structure ProjectiveEmbeddingline 43library
structure RationalCurveline 55library
def RationalCurve.valueline 62library
abbrev PairPolynomialline 66library
def Bihomogeneousline 69library
def pairEvalline 74library
def IsZariskiOpenPairline 82library
def RationallyConnectedline 90library
structure TangentSplittingline 97library
def TangentSplitting.firstline 108library
def TangentSplitting.secondline 111library
def HolomorphicSectionOnline 116library
def Integrableline 122library
def IntegrabilityStatementline 130in the statementlibrary
theorem integrability_mainline 135selected 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.

attribute [instance] ComplexManifold.topology ComplexManifold.charts line 24

import Mathlibnamespace OAI noncomputable sectionopen scoped Manifold ContDiff LinearAlgebra.Projectivizationopen Set Bundleuniverse unamespace SplitTangent structure ComplexManifold where  dim : ℕ  Carrier : Type u  [topology : TopologicalSpace Carrier]  [charts : ChartedSpace (Fin dim → ℂ) Carrier]  [holomorphicAtlas : IsManifold 𝓘(ℂ, Fin dim → ℂ) ω Carrier]  [hausdorff : T2Space Carrier]  [secondCountable : SecondCountableTopology Carrier]attribute [instance] ComplexManifold.topology ComplexManifold.charts  ComplexManifold.holomorphicAtlas ComplexManifold.hausdorff  ComplexManifold.secondCountableabbrev Model (X : ComplexManifold.{u}) := Fin X.dim → ℂabbrev ModelWithCorners (X : ComplexManifold.{u}) := 𝓘(ℂ, Model X)abbrev Tangent (X : ComplexManifold.{u}) (x : X.Carrier) :=  TangentSpace (ModelWithCorners X) xabbrev ProjectiveSpace (N : ℕ) := ℙ ℂ (Fin (N + 1) → ℂ) def projectiveCoordinates {N : ℕ} (i : Fin (N + 1))    (p : ProjectiveSpace N) : Fin (N + 1) → ℂ :=  fun j => (p.rep i)⁻¹ * p.rep j structure ProjectiveEmbedding (X : ComplexManifold.{u}) where  N : ℕ  map : X.Carrier → ProjectiveSpace N  injective : Function.Injective map  open_patch : ∀ i, IsOpen {x | (map x).rep i ≠ 0}  holomorphic : ∀ i, ContMDiffOn (ModelWithCorners X) 𝓘(ℂ, Fin (N + 1) → ℂ) ω    (fun x => projectiveCoordinates i (map x)) {x | (map x).rep i ≠ 0}  immersion : ∀ x i, (map x).rep i ≠ 0 → Function.Injective    (mfderiv (ModelWithCorners X) 𝓘(ℂ, Fin (N + 1) → ℂ)      (fun y => projectiveCoordinates i (map y)) x) structure RationalCurve (X : ComplexManifold.{u}) where  finite : ℂ → X.Carrier  infinity : ℂ → X.Carrier  finite_holomorphic : ContMDiff 𝓘(ℂ, ℂ) (ModelWithCorners X) ω finite  infinity_holomorphic : ContMDiff 𝓘(ℂ, ℂ) (ModelWithCorners X) ω infinity  overlap : ∀ z : ℂ, z ≠ 0 → finite z = infinity z⁻¹def RationalCurve.value {X : ComplexManifold.{u}} (g : RationalCurve X)    (z : OnePoint ℂ) : X.Carrier :=  z.elim (g.infinity 0) g.finiteabbrev PairPolynomial (N : ℕ) := MvPolynomial (Fin 2 × Fin (N + 1)) ℂ def Bihomogeneous {N : ℕ} (p : PairPolynomial N) : Prop :=  ∃ a b : ℕ, ∀ m ∈ p.support,    (∑ i : Fin (N + 1), m (0, i)) = a ∧    (∑ i : Fin (N + 1), m (1, i)) = bdef pairEval {X : ComplexManifold.{u}} (e : ProjectiveEmbedding X)    (p : PairPolynomial e.N) (xy : X.Carrier × X.Carrier) : ℂ :=  MvPolynomial.eval    (fun k : Fin 2 × Fin (e.N + 1) =>      if k.1 = 0 then (e.map xy.1).rep k.2 else (e.map xy.2).rep k.2) p def IsZariskiOpenPair {X : ComplexManifold.{u}} (e : ProjectiveEmbedding X)    (U : Set (X.Carrier × X.Carrier)) : Prop :=  ∃ equations : Set (PairPolynomial e.N),    (∀ p ∈ equations, Bihomogeneous p) ∧    U = {xy | ∃ p ∈ equations, pairEval e p xy ≠ 0} def RationallyConnected {X : ComplexManifold.{u}} (e : ProjectiveEmbedding X) : Prop :=  ∃ U : Set (X.Carrier × X.Carrier), IsZariskiOpenPair e U ∧ U.Nonempty ∧ Dense U ∧    ∀ xy ∈ U, ∃ g : RationalCurve X, ∃ a b : OnePoint ℂ,      g.value a = xy.1 ∧ g.value b = xy.2 structure TangentSplitting (X : ComplexManifold.{u}) where  projector : ∀ x : X.Carrier, Tangent X x →L[ℂ] Tangent X x  idempotent : ∀ x v, projector x (projector x v) = projector x v  holomorphic : ContMDiff    ((ModelWithCorners X).prod 𝓘(ℂ, Model X))    ((ModelWithCorners X).prod 𝓘(ℂ, Model X)) ω    (fun v : TangentBundle (ModelWithCorners X) X.Carrier =>      (⟨v.proj, projector v.proj v.2⟩ : TangentBundle (ModelWithCorners X) X.Carrier))  first_positive : ∀ x, 0 < Module.finrank ℂ (LinearMap.range (projector x).toLinearMap)  second_positive : ∀ x, 0 < Module.finrank ℂ (LinearMap.ker (projector x).toLinearMap)def TangentSplitting.first {X : ComplexManifold.{u}} (S : TangentSplitting X) (x : X.Carrier) :    Submodule ℂ (Tangent X x) := LinearMap.range (S.projector x).toLinearMapdef TangentSplitting.second {X : ComplexManifold.{u}} (S : TangentSplitting X) (x : X.Carrier) :    Submodule ℂ (Tangent X x) := LinearMap.ker (S.projector x).toLinearMap def HolomorphicSectionOn (X : ComplexManifold.{u})    (V : ∀ x : X.Carrier, Tangent X x) (U : Set X.Carrier) : Prop :=  ContMDiffOn (ModelWithCorners X) ((ModelWithCorners X).prod 𝓘(ℂ, Model X)) ω    (fun x => (⟨x, V x⟩ : TangentBundle (ModelWithCorners X) X.Carrier)) U def Integrable (X : ComplexManifold.{u}) (D : ∀ x : X.Carrier, Submodule ℂ (Tangent X x)) : Prop :=  ∀ U : Set X.Carrier, IsOpen U → ∀ V W : ∀ x : X.Carrier, Tangent X x,    HolomorphicSectionOn X V U → HolomorphicSectionOn X W U →    (∀ x ∈ U, V x ∈ D x) → (∀ x ∈ U, W x ∈ D x) →    ∀ x ∈ U, VectorField.mlieBracketWithin (ModelWithCorners X) V W U x ∈ D x def IntegrabilityStatement : Prop :=  ∀ (X : ComplexManifold.{u}), ConnectedSpace X.Carrier → CompactSpace X.Carrier →    2 ≤ X.dim → ∀ e : ProjectiveEmbedding X, RationallyConnected e →    ∀ S : TangentSplitting X, Integrable X S.first ∧ Integrable X S.secondtheorem integrability_main : IntegrabilityStatement.{u} := by  sorryend SplitTangentendend 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.

Universal-cover splitting for compact Kähler manifolds

Theorem

Let X be a compact connected Kähler manifold of complex dimension n\geq2, and let

T_X=E_1\oplus E_2

be a specified decomposition into integrable holomorphic subbundles of positive ranks r_1,r_2. If \pi:\widetilde X\to X is the ordinary universal covering map, there are connected simply connected complex manifolds Y_1,Y_2 and a biholomorphism \Phi:\widetilde X\to Y_1\times Y_2 such that

\dd\Phi(\pi^*E_i)=\pr_i^*T_{Y_i}\qquad(i=1,2),

where \dd\pi identifies T_{\widetilde X} with \pi^*T_X. In particular, \dim_\C Y_i=r_i.

Section 1, Introduction Read it in the paper TeX source, line 100

Show the TeX

Let $X$ be a compact connected K\"ahler manifold of complex dimension
$n\geq2$, and let
\[
T_X=E_1\oplus E_2
\]
be a specified decomposition into integrable holomorphic subbundles of
positive ranks $r_1,r_2$. If $\pi:\widetilde X\to X$ is the ordinary
universal covering map, there are connected simply connected complex
manifolds $Y_1,Y_2$ and a biholomorphism
$\Phi:\widetilde X\to Y_1\times Y_2$ such that
\[
\dd\Phi(\pi^*E_i)=\pr_i^*T_{Y_i}\qquad(i=1,2),
\]
where $\dd\pi$ identifies $T_{\widetilde X}$ with $\pi^*T_X$.
In particular, $\dim_\C Y_i=r_i$.

Corollary (Non-uniruled projective manifolds)

Let X be a smooth connected non-uniruled projective complex manifold, and let T_X=E_1\oplus E_2 be a specified holomorphic splitting into subbundles of positive ranks. Then both summands are integrable and the ordinary universal cover has a compatible product decomposition as in Theorem thm:main.

Section 1, Introduction Read it in the paper TeX source, line 126

Show the TeX

Let $X$ be a smooth connected non-uniruled projective complex manifold,
and let $T_X=E_1\oplus E_2$ be a specified holomorphic splitting into
subbundles of positive ranks. Then both summands are integrable and the
ordinary universal cover has a compatible product decomposition as in
Theorem~\ref{thm:main}.

Corollary

Let X be a smooth connected projective complex manifold of dimension n\geq2 with nef and big canonical bundle K_X. Every specified holomorphic splitting T_X=E_1\oplus E_2 into positive-rank subbundles is integrable and induces a compatible product decomposition of the ordinary universal cover as in Theorem thm:main.

Section 1, Introduction Read it in the paper TeX source, line 156

Show the TeX

Let $X$ be a smooth connected projective complex manifold of dimension
$n\geq2$ with nef and big canonical bundle $K_X$. Every specified
holomorphic splitting $T_X=E_1\oplus E_2$ into positive-rank subbundles
is integrable and induces a compatible product decomposition of the
ordinary universal cover as in Theorem~\ref{thm:main}.

Integrability of split tangent bundles on rationally connected manifolds

Theorem (Automatic integrability)

T_X=E_1\oplus E_2

into subbundles of positive rank, both E_1 and E_2 are integrable.

Section 1, Introduction Read it in the paper TeX source, line 21

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\label[theorem]{thm:main}
Let \(X\) be a smooth connected projective complex manifold of dimension
at least two. Suppose that two general points of \(X\) lie on the image
of a morphism \(\PP^1\to X\). For every specified holomorphic decomposition
\[
T_X=E_1\oplus E_2
\]
into subbundles of positive rank, both \(E_1\) and \(E_2\) are integrable.

Corollary (Compatible product)

Under the hypotheses of thm:main, there are smooth projective manifolds X_1,X_2 and an isomorphism X\simeq X_1\times X_2 under which

E_i=\pr_i^*T_{X_i}\qquad (i=1,2).

Section 1, Introduction Read it in the paper TeX source, line 37

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\label[corollary]{cor:product}
Under the hypotheses of \cref{thm:main}, there are smooth projective
manifolds \(X_1,X_2\) and an isomorphism \(X\simeq X_1\times X_2\)
under which
\[
E_i=\pr_i^*T_{X_i}\qquad (i=1,2).
\]

Configuration and file facts

File: lean/ComparatorChallenges/SplitTangentIntegrability.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 052 link it.
Solution module: OAI.Geometry.SplitTangent.RationalVariation
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: 20 local vocabulary declarations
Lines of Lean, without comments or blank lines: 86
Second proof-verification program (nanoda): Off in OpenAI's configuration.

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