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Lean statement

Ample bundles without Griffiths-positive metrics

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.QuadricCounterexample.main_theorem

Lean statement file: QuadricBundles.leanShow on the mapFamily 050. OpenAI lists this as a main result in its Lean catalogue.

See the candidate theoremWho ran the proof-verification programReport 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 SourceMainTheorem : Prop :=  ∃ G : AlgebraicRankTwoBundle, ∃ E : ℕ → AlgebraicRankTwoBundle,    (∀ m, 0 < m → IsPowerPullbackTwist G m (E m)) ∧    (∀ m, 0 < m → IsAmple (E m)) ∧    ∃ m₀ : ℕ, 0 < m₀ ∧ ∀ m, m₀ ≤ m →      ¬ ∃ h : SmoothHermitianMetric (E m), h.StrictlyGriffithsPositivetheorem main_theorem : SourceMainTheorem := 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.QuadricCounterexample.ProjectiveLine line 13

    abbrev ProjectiveLine := OnePoint ℂ

  • abbreviation OAI.QuadricCounterexample.QuadricSurface line 14

    abbrev QuadricSurface := ProjectiveLine × ProjectiveLine

  • definition OAI.QuadricCounterexample.reciprocalP1 line 16

    def reciprocalP1 (x : ProjectiveLine) : ProjectiveLine :=  x.elim (0 : ℂ) (fun z ↦ if z = 0 then ∞ else (z⁻¹ : ℂ))

  • definition OAI.QuadricCounterexample.affineP1 line 19

    def affineP1 (i : Fin 2) (z : ℂ) : ProjectiveLine :=  if i = 0 then reciprocalP1 z else (z : ProjectiveLine)

  • definition OAI.QuadricCounterexample.affineQuadric line 22

    def affineQuadric (i : Fin 2 × Fin 2) (z : ℂ × ℂ) : QuadricSurface :=  (affineP1 i.1 z.1, affineP1 i.2 z.2)

  • definition OAI.QuadricCounterexample.coordP1 line 25

    def coordP1 (i : Fin 2) (x : ProjectiveLine) : ℂ :=  (if i=0 then reciprocalP1 x else x).elim 0 id

  • definition OAI.QuadricCounterexample.coordQuadric line 27

    def coordQuadric (i : Fin 2 × Fin 2) (x : QuadricSurface) : ℂ × ℂ :=  (coordP1 i.1 x.1,coordP1 i.2 x.2)

  • definition OAI.QuadricCounterexample.powerP1 line 30

    def powerP1 (m : ℕ) (x : ProjectiveLine) : ProjectiveLine :=  x.elim ∞ (fun z ↦ ((z^m : ℂ) : ProjectiveLine))

  • definition OAI.QuadricCounterexample.powerQuadric line 32

    def powerQuadric (m : ℕ) (x : QuadricSurface) : QuadricSurface :=  (powerP1 m x.1,powerP1 m x.2)

  • abbreviation OAI.QuadricCounterexample.BaseCoordinates line 35

    abbrev BaseCoordinates := ℂ × ℂ

  • abbreviation OAI.QuadricCounterexample.FiberCoordinates line 36

    abbrev FiberCoordinates := Fin 2 → ℂ

  • abbreviation OAI.QuadricCounterexample.TwoMatrix line 37

    abbrev TwoMatrix := Matrix (Fin 2) (Fin 2) ℂ

  • abbreviation OAI.QuadricCounterexample.BasePolynomial line 38

    abbrev BasePolynomial := MvPolynomial (Fin 2) ℂ

  • definition OAI.QuadricCounterexample.baseEval line 39

    def baseEval (p : BasePolynomial) (z : BaseCoordinates) : ℂ :=  MvPolynomial.eval ![z.1,z.2] p

  • definition OAI.QuadricCounterexample.AlgebraicallyRegularOn line 42

    def AlgebraicallyRegularOn (U : Set BaseCoordinates) (f : BaseCoordinates → ℂ) : Prop :=  ∀ z ∈ U, ∃ p q : BasePolynomial, baseEval q z ≠ 0 ∧    ∀ w ∈ U, baseEval q w ≠ 0 → f w=baseEval p w/baseEval q w

  • definition OAI.QuadricCounterexample.principalChart line 46

    def principalChart (i : Fin 2 × Fin 2) (p : BasePolynomial) : Set QuadricSurface :=  {x | x ∈ range (affineQuadric i) ∧ baseEval p (coordQuadric i x) ≠ 0}

  • structure OAI.QuadricCounterexample.AlgebraicRankTwoBundle line 49

    structure AlgebraicRankTwoBundle where
    Show the remaining 17 lines
      numberOfCharts : ℕ  chart : ℕ → Fin 2 × Fin 2  equation : ℕ → BasePolynomial  finiteCover : ∀ i, numberOfCharts ≤ i → equation i=0  covers : ∀ x : QuadricSurface, ∃ i, x ∈ principalChart (chart i) (equation i)  transition : ℕ → ℕ → QuadricSurface → TwoMatrix  transition_self : ∀ i x, x ∈ principalChart (chart i) (equation i) →    transition i i x=1  transition_mul : ∀ i j k x,    x ∈ principalChart (chart i) (equation i) →    x ∈ principalChart (chart j) (equation j) →    x ∈ principalChart (chart k) (equation k) →    transition i j x * transition j k x=transition i k x  transition_regular : ∀ i j a b, AlgebraicallyRegularOn    {z | baseEval (equation i) z ≠ 0 ∧      affineQuadric (chart i) z ∈ principalChart (chart j) (equation j)}    (fun z ↦ transition i j (affineQuadric (chart i) z) a b)

  • definition OAI.QuadricCounterexample.AlgebraicRankTwoBundle.domain line 70

    def domain (V : AlgebraicRankTwoBundle) (i : ℕ) : Set QuadricSurface :=  principalChart (V.chart i) (V.equation i)

  • definition OAI.QuadricCounterexample.AlgebraicRankTwoBundle.coordinateDomain line 72

    def coordinateDomain (V : AlgebraicRankTwoBundle) (i : ℕ) : Set BaseCoordinates :=  {z | baseEval (V.equation i) z ≠ 0}

  • definition OAI.QuadricCounterexample.AlgebraicRankTwoBundle.dualTransition line 75

    def dualTransition (V : AlgebraicRankTwoBundle) (i j : ℕ) (x : QuadricSurface) : TwoMatrix :=  (V.transition j i x).transpose

  • definition OAI.QuadricCounterexample.hyperplaneTransition line 80

    def hyperplaneTransition (i j : Fin 2) (x : ProjectiveLine) : ℂ :=  if i=j then 1 else coordP1 i x

  • definition OAI.QuadricCounterexample.polarizationTransition line 82

    def polarizationTransition (i j : Fin 2 × Fin 2) (x : QuadricSurface) : ℂ :=  hyperplaneTransition i.1 j.1 x.1 * hyperplaneTransition i.2 j.2 x.2

  • definition OAI.QuadricCounterexample.IsPowerPullbackTwist line 85

    def IsPowerPullbackTwist (G : AlgebraicRankTwoBundle) (m : ℕ)    (E : AlgebraicRankTwoBundle) : Prop :=  E.numberOfCharts=G.numberOfCharts ∧ E.chart=G.chart ∧  (∀ i, E.equation i = (MvPolynomial.bind₁ fun j ↦ MvPolynomial.X j ^ m) (G.equation i)) ∧  ∀ i j x, x ∈ E.domain i → x ∈ E.domain j →    E.transition i j x = polarizationTransition (G.chart i) (G.chart j) x •      G.transition i j (powerQuadric m x)

  • structure OAI.QuadricCounterexample.SymmetricSection line 93

    structure SymmetricSection (V : AlgebraicRankTwoBundle) (n : ℕ) where  coefficient : ℕ → Fin (n+1) → BaseCoordinates → ℂ  regular : ∀ i k, AlgebraicallyRegularOn (V.coordinateDomain i) (coefficient i k)  compatible : ∀ i j x, x ∈ V.domain i → x ∈ V.domain j → ∀ v : FiberCoordinates,    (∑ k : Fin (n+1), coefficient i k (coordQuadric (V.chart i) x) *      ((V.dualTransition i j x).mulVec v 0)^k.val *      ((V.dualTransition i j x).mulVec v 1)^(n-k.val)) =    ∑ k : Fin (n+1), coefficient j k (coordQuadric (V.chart j) x) * v 0^k.val * v 1^(n-k.val)

9 more definitions are reached the same way; the full file lists them.

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: smooth, strictly, algebraic, two, metric, ample, griffiths, hermitian and five more. Found in the paper’s title, which its main theorem states: surface and quadric. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: admits.

Theorem 1.1 of Ample rank-two bundles on the quadric surface without Griffiths-positive metrics

Theorem

There is an algebraic vector bundle G of rank two on S such that, for every positive integer m, the bundle

E_m=f_m^*G\otimes P

is ample. There is an integer m_0 such that E_m admits no smooth Hermitian metric with strictly Griffiths-positive Chern curvature for every m\ge m_0.

Section 1, Introduction Read it in the paper TeX source, line 23
Show the TeX
There is an algebraic vector bundle $G$ of rank two on $S$ such that,
for every positive integer $m$, the bundle
\[
 E_m=f_m^*G\otimes P
\]
is ample. There is an integer $m_0$ such that $E_m$ admits no smooth
Hermitian metric with strictly Griffiths-positive Chern curvature
for every $m\ge m_0$.

Other possible paper matches, closest wording first

  • Proposition 4.1 of Ample rank-two bundles on the quadric surface without Griffiths-positive metrics. Words of the Lean statement, its names split into words, that also appear in the theorem: base, holomorphic, point, finite, smooth, strictly and eight more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: times and admits.
  • Proposition 2.3 of Ample rank-two bundles on the quadric surface without Griffiths-positive metrics. Words of the Lean statement, its names split into words, that also appear in the theorem: dual, algebraic, two, fiber, rank, vector and one more.
  • Proposition 3.1 of Ample rank-two bundles on the quadric surface without Griffiths-positive metrics. Words of the Lean statement, its names split into words, that also appear in the theorem: ample, positive, integer and bundle.
  • Proposition 5.4 of Ample rank-two bundles on the quadric surface without Griffiths-positive metrics. Words of the Lean statement, its names split into words, that also appear in the theorem: finite, fiber, hermitian and bundle.

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 372 s; peak memory 6.6 GiB).

    Reported 11 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 050’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: Headline stated; the 'very-ampleness for all admissible even exponents' result is not (docs). All notions (algebraic bundle on P1xP1, ampleness, Hermitian metric, Griffiths positivity) are bespoke, with no bridge to Mathlib objects.

    Definitions the group asks readers to compare with the paper: Bespoke AlgebraicRankTwoBundle (4 charts by polynomial principal opens + regular transition matrices), IsAmple V := ∃ n > 0, ∃ N, ∃ s : Fin N → SymmetricSection V n, GivesClosedProjectiveEmbedding s (a 4-clause custom 'closed embedding' predicate), SmoothHermitianMetric on the dual bundle (dualMatrix), StrictlyGriffithsPositive via metricMixedHessian/holomorphicMetricDerivative; a mistake in any of these would silently change what is proved, and the statement is vacuous-prone since it ends in ¬ ∃ h

  • 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 counterexample starts with a rank-two algebraic bundle $G$ on $\mathbb P^1\times\mathbb P^1$. Its coordinatewise power pullbacks, tensored with $\mathcal O(1,1)$, are ample for every positive power but admit no such metric for all sufficiently large powers.”

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

181 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: 135 to its own declarations, 66 to Mathlib’s documentation. 35 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.

import Mathlibnamespace OAInoncomputable sectionnamespace QuadricCounterexampleopen Set Filter Topology Metric OnePointopen scoped Matrix.Norms.Elementwise ComplexOrderabbrev ProjectiveLine := OnePoint ℂabbrev QuadricSurface := ProjectiveLine × ProjectiveLinedef reciprocalP1 (x : ProjectiveLine) : ProjectiveLine :=  x.elim (0 : ℂ) (fun z ↦ if z = 0 then ∞ else (z⁻¹ : ℂ))def affineP1 (i : Fin 2) (z : ℂ) : ProjectiveLine :=  if i = 0 then reciprocalP1 z else (z : ProjectiveLine)def affineQuadric (i : Fin 2 × Fin 2) (z : ℂ × ℂ) : QuadricSurface :=  (affineP1 i.1 z.1, affineP1 i.2 z.2)def coordP1 (i : Fin 2) (x : ProjectiveLine) : ℂ :=  (if i=0 then reciprocalP1 x else x).elim 0 iddef coordQuadric (i : Fin 2 × Fin 2) (x : QuadricSurface) : ℂ × ℂ :=  (coordP1 i.1 x.1,coordP1 i.2 x.2)def powerP1 (m : ℕ) (x : ProjectiveLine) : ProjectiveLine :=  x.elim ∞ (fun z ↦ ((z^m : ℂ) : ProjectiveLine))def powerQuadric (m : ℕ) (x : QuadricSurface) : QuadricSurface :=  (powerP1 m x.1,powerP1 m x.2)abbrev BaseCoordinates := ℂ × ℂabbrev FiberCoordinates := Fin 2 → ℂabbrev TwoMatrix := Matrix (Fin 2) (Fin 2) ℂabbrev BasePolynomial := MvPolynomial (Fin 2) ℂdef baseEval (p : BasePolynomial) (z : BaseCoordinates) : ℂ :=  MvPolynomial.eval ![z.1,z.2] pdef AlgebraicallyRegularOn (U : Set BaseCoordinates) (f : BaseCoordinates → ℂ) : Prop :=  ∀ z ∈ U, ∃ p q : BasePolynomial, baseEval q z ≠ 0 ∧    ∀ w ∈ U, baseEval q w ≠ 0 → f w=baseEval p w/baseEval q wdef principalChart (i : Fin 2 × Fin 2) (p : BasePolynomial) : Set QuadricSurface :=  {x | x ∈ range (affineQuadric i) ∧ baseEval p (coordQuadric i x) ≠ 0}structure AlgebraicRankTwoBundle where  numberOfCharts : ℕ  chart : ℕ → Fin 2 × Fin 2  equation : ℕ → BasePolynomial  finiteCover : ∀ i, numberOfCharts ≤ i → equation i=0  covers : ∀ x : QuadricSurface, ∃ i, x ∈ principalChart (chart i) (equation i)  transition : ℕ → ℕ → QuadricSurface → TwoMatrix  transition_self : ∀ i x, x ∈ principalChart (chart i) (equation i) →    transition i i x=1  transition_mul : ∀ i j k x,    x ∈ principalChart (chart i) (equation i) →    x ∈ principalChart (chart j) (equation j) →    x ∈ principalChart (chart k) (equation k) →    transition i j x * transition j k x=transition i k x  transition_regular : ∀ i j a b, AlgebraicallyRegularOn    {z | baseEval (equation i) z ≠ 0 ∧      affineQuadric (chart i) z ∈ principalChart (chart j) (equation j)}    (fun z ↦ transition i j (affineQuadric (chart i) z) a b)namespace AlgebraicRankTwoBundledef domain (V : AlgebraicRankTwoBundle) (i : ℕ) : Set QuadricSurface :=  principalChart (V.chart i) (V.equation i)def coordinateDomain (V : AlgebraicRankTwoBundle) (i : ℕ) : Set BaseCoordinates :=  {z | baseEval (V.equation i) z ≠ 0}def dualTransition (V : AlgebraicRankTwoBundle) (i j : ℕ) (x : QuadricSurface) : TwoMatrix :=  (V.transition j i x).transposeend AlgebraicRankTwoBundledef hyperplaneTransition (i j : Fin 2) (x : ProjectiveLine) : ℂ :=  if i=j then 1 else coordP1 i xdef polarizationTransition (i j : Fin 2 × Fin 2) (x : QuadricSurface) : ℂ :=  hyperplaneTransition i.1 j.1 x.1 * hyperplaneTransition i.2 j.2 x.2def IsPowerPullbackTwist (G : AlgebraicRankTwoBundle) (m : ℕ)    (E : AlgebraicRankTwoBundle) : Prop :=  E.numberOfCharts=G.numberOfCharts ∧ E.chart=G.chart ∧  (∀ i, E.equation i = (MvPolynomial.bind₁ fun j ↦ MvPolynomial.X j ^ m) (G.equation i)) ∧  ∀ i j x, x ∈ E.domain i → x ∈ E.domain j →    E.transition i j x = polarizationTransition (G.chart i) (G.chart j) x •      G.transition i j (powerQuadric m x)structure SymmetricSection (V : AlgebraicRankTwoBundle) (n : ℕ) where  coefficient : ℕ → Fin (n+1) → BaseCoordinates → ℂ  regular : ∀ i k, AlgebraicallyRegularOn (V.coordinateDomain i) (coefficient i k)  compatible : ∀ i j x, x ∈ V.domain i → x ∈ V.domain j → ∀ v : FiberCoordinates,    (∑ k : Fin (n+1), coefficient i k (coordQuadric (V.chart i) x) *      ((V.dualTransition i j x).mulVec v 0)^k.val *      ((V.dualTransition i j x).mulVec v 1)^(n-k.val)) =    ∑ k : Fin (n+1), coefficient j k (coordQuadric (V.chart j) x) * v 0^k.val * v 1^(n-k.val)def SymmetricSection.value {V : AlgebraicRankTwoBundle} {n : ℕ}    (s : SymmetricSection V n) (i : ℕ) (x : QuadricSurface) (v : FiberCoordinates) : ℂ :=  ∑ k : Fin (n+1), s.coefficient i k (coordQuadric (V.chart i) x) * v 0^k.val * v 1^(n-k.val)def ProjectiveAlgebraicCone {N : ℕ} (C : Set (Fin N → ℂ)) : Prop :=  ∃ equations : Set (MvPolynomial (Fin N) ℂ),    (∀ p ∈ equations, ∃ d : ℕ, p.IsHomogeneous d) ∧    C = {y | ∀ p ∈ equations, MvPolynomial.eval y p=0}def sectionVector {V : AlgebraicRankTwoBundle} {n N : ℕ}    (s : Fin N → SymmetricSection V n) (i : ℕ) (x : QuadricSurface)    (v : FiberCoordinates) : Fin N → ℂ := fun j ↦ (s j).value i x vdef GivesClosedProjectiveEmbedding {V : AlgebraicRankTwoBundle} {n N : ℕ}    (s : Fin N → SymmetricSection V n) : Prop :=  (∀ i x, x ∈ V.domain i → ∀ v : FiberCoordinates, v ≠ 0 → sectionVector s i x v ≠ 0) ∧  (∀ i j x y, x ∈ V.domain i → y ∈ V.domain j → ∀ v w : FiberCoordinates,    v ≠ 0 → w ≠ 0 → (∃ c : ℂ, c ≠ 0 ∧ sectionVector s i x v = c • sectionVector s j y w) →    x=y ∧ ∃ c : ℂ, c ≠ 0 ∧ v=c • ((V.dualTransition i j x).mulVec w)) ∧  ProjectiveAlgebraicCone ({0} ∪ {a | ∃ i x, x ∈ V.domain i ∧    ∃ v : FiberCoordinates, v ≠ 0 ∧ ∃ c : ℂ, c ≠ 0 ∧ a=c • sectionVector s i x v}) ∧  ∀ i x, x ∈ V.domain i → ∀ v : FiberCoordinates, v ≠ 0 →    ∃ j : Fin N, ∃ b : Fin 2, v b ≠ 0 ∧ sectionVector s i x v j ≠ 0 ∧    ∃ p q : Fin 3 → MvPolynomial (Fin N) ℂ,      (∀ k, MvPolynomial.eval (fun l ↦ sectionVector s i x v l / sectionVector s i x v j) (q k) ≠ 0) ∧      ∀ i' x', x' ∈ V.domain i' → ∀ v' : FiberCoordinates, v' ≠ 0 →        sectionVector s i' x' v' j ≠ 0 →        (∀ k, MvPolynomial.eval (fun l ↦ sectionVector s i' x' v' l / sectionVector s i' x' v' j) (q k) ≠ 0) →        x' ∈ V.domain i ∧ ((V.dualTransition i i' x').mulVec v') b ≠ 0 ∧        (∀ k : Fin 3,          (![ (coordQuadric (V.chart i) x').1, (coordQuadric (V.chart i) x').2,             ((V.dualTransition i i' x').mulVec v') (1-b) /               ((V.dualTransition i i' x').mulVec v') b ] : Fin 3 → ℂ) k =          MvPolynomial.eval (fun l ↦ sectionVector s i' x' v' l / sectionVector s i' x' v' j) (p k) /          MvPolynomial.eval (fun l ↦ sectionVector s i' x' v' l / sectionVector s i' x' v' j) (q k))def IsAmple (V : AlgebraicRankTwoBundle) : Prop :=  ∃ n : ℕ, 0 < n ∧ ∃ N : ℕ, ∃ s : Fin N → SymmetricSection V n,    GivesClosedProjectiveEmbedding sdef holomorphicMetricDerivative (H : BaseCoordinates → TwoMatrix)    (z u : BaseCoordinates) : TwoMatrix :=  (1/2 : ℂ) • (fderiv ℝ H z u - Complex.I • fderiv ℝ H z (Complex.I • u))def metricMixedHessian (H : BaseCoordinates → TwoMatrix)    (z u : BaseCoordinates) : TwoMatrix :=  (1/4 : ℂ) • ((fderiv ℝ (fun y ↦ fderiv ℝ H y) z u) u +    (fderiv ℝ (fun y ↦ fderiv ℝ H y) z (Complex.I • u)) (Complex.I • u))structure SmoothHermitianMetric (V : AlgebraicRankTwoBundle) where  dualMatrix : ℕ → BaseCoordinates → TwoMatrix  smooth : ∀ i, ContDiffOn ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (dualMatrix i) (V.coordinateDomain i)  positive : ∀ i z, z ∈ V.coordinateDomain i → (dualMatrix i z).PosDef  compatible : ∀ i j x, x ∈ V.domain i → x ∈ V.domain j →    dualMatrix j (coordQuadric (V.chart j) x) =      (V.dualTransition i j x).conjTranspose * dualMatrix i (coordQuadric (V.chart i) x) *        V.dualTransition i j xdef SmoothHermitianMetric.StrictlyGriffithsPositive {V : AlgebraicRankTwoBundle}    (h : SmoothHermitianMetric V) : Prop :=  ∀ i z, z ∈ V.coordinateDomain i → ∀ u : BaseCoordinates, u ≠ 0 →    (metricMixedHessian (h.dualMatrix i) z u -      (holomorphicMetricDerivative (h.dualMatrix i) z u).conjTranspose *        (h.dualMatrix i z)⁻¹ * holomorphicMetricDerivative (h.dualMatrix i) z u).PosDefdef SourceMainTheorem : Prop :=  ∃ G : AlgebraicRankTwoBundle, ∃ E : ℕ → AlgebraicRankTwoBundle,    (∀ m, 0 < m → IsPowerPullbackTwist G m (E m)) ∧    (∀ m, 0 < m → IsAmple (E m)) ∧    ∃ m₀ : ℕ, 0 < m₀ ∧ ∀ m, m₀ ≤ m →      ¬ ∃ h : SmoothHermitianMetric (E m), h.StrictlyGriffithsPositivetheorem main_theorem : SourceMainTheorem := by  sorryend QuadricCounterexampleendend 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.

Ample rank-two bundles on the quadric surface without Griffiths-positive metrics

Theorem

There is an algebraic vector bundle G of rank two on S such that, for every positive integer m, the bundle

E_m=f_m^*G\otimes P

is ample. There is an integer m_0 such that E_m admits no smooth Hermitian metric with strictly Griffiths-positive Chern curvature for every m\ge m_0.

Section 1, Introduction Read it in the paper TeX source, line 23
Show the TeX
There is an algebraic vector bundle $G$ of rank two on $S$ such that,
for every positive integer $m$, the bundle
\[
 E_m=f_m^*G\otimes P
\]
is ample. There is an integer $m_0$ such that $E_m$ admits no smooth
Hermitian metric with strictly Griffiths-positive Chern curvature
for every $m\ge m_0$.
Configuration and file facts
OpenAI’s Lean catalogue
OpenAI lists this as a main result in its Lean catalogue.
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
34 local vocabulary declarations
Lines of Lean, without comments or blank lines
134
Second proof-verification program (nanoda)
Off in OpenAI's configuration.

Statement comparison

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Plectis Atlas

Ample bundles without Griffiths-positive metrics

Family 050: A counterexample to Griffiths’ positivity conjecture

Candidate pairing

Theorem

There is an algebraic vector bundle G of rank two on S such that, for every positive integer m, the bundle

E_m=f_m^*G\otimes P

is ample. There is an integer m_0 such that E_m admits no smooth Hermitian metric with strictly Griffiths-positive Chern curvature for every m\ge m_0.

Ample rank-two bundles on the quadric surface without Griffiths-positive metrics

TeX label thm:main, build/sections/01-introduction.tex, line 23

The Lean statement says

def SourceMainTheorem : Prop :=  ∃ G : AlgebraicRankTwoBundle, ∃ E : ℕ → AlgebraicRankTwoBundle,    (∀ m, 0 < m → IsPowerPullbackTwist G m (E m)) ∧    (∀ m, 0 < m → IsAmple (E m)) ∧    ∃ m₀ : ℕ, 0 < m₀ ∧ ∀ m, m₀ ≤ m →      ¬ ∃ h : SmoothHermitianMetric (E m), h.StrictlyGriffithsPositivetheorem main_theorem : SourceMainTheorem := 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.

Atlas selected this possible match because the two statements share words. No review of whether they mean the same thing is recorded.three independent groups report that Comparator accepted runs for this file at the release version shown here.Statement review: none recorded

34 local vocabulary declarationsAllowed axioms: propext, Quot.sound, Classical.choiceopenai/math at fd4aeebwcook04.github.io/plectis-atlas/statements/QuadricBundles.html

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