A counterexample to Griffiths’ positivity conjecture
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Constructs ample rank-two bundles on \(\mathbb P^1\times\mathbb P^1\) with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths’ positivity conjecture already on the quadric surface. (OpenAI’s Lean note)
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Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
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.
Reported proof verification
One Lean statement in the release; three independent groups report that Comparator accepted runs for this file at the version shown here.
We give counterexamples to Griffiths’ conjecture in rank two on the quadric surface \(\mathbb P^1\times\mathbb P^1\). We construct an explicit bundle G whose coordinatewise power pullbacks, tensored with \(\mathcal O(1,1)\), are ample for every positive power, but admit no smooth strictly Griffiths-positive Hermitian metric for all sufficiently large powers.
One Lean statement in this family
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QuadricBundles.lean, the release lists it as a main result.
defSourceMainTheorem : 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.StrictlyGriffithsPositive⋮theoremmain_theorem : SourceMainTheorem := bysorry
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three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
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Independent comparisons and proof-verification reports
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MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
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
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method: 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.
Griffiths' positivity conjecture predicts that every ample holomorphic vector bundle on a smooth complex projective variety admits a smooth Hermitian metric with strictly Griffiths-positive curvature. 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.
The separate very-ampleness result for all admissible even exponents is not included.
AI-drafted comparisons, with no recorded human review
Ample bundles without Griffiths-positive metricsQuadricBundles
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: an explicit rank-two bundle on the quadric surface whose twisted power pullbacks are ample for every power but carry no Griffiths-positive metric for all large powers. The separate very-ampleness result is not included.
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Ample bundles without Griffiths-positive metricsQuadricBundles
Three reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“CERTIFIED”
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method
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.
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Their method
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…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
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.
Re-run through Comparator with Lean’s kernel: accepted (the check took 372 s; peak memory 6.6 GiB).
Date
11 October 2026
Version
at the release’s recorded version
Their method
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.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.