OpenAI claim summary for Math-295

Griffiths Positivity Conjecture for Ample Vector Bundles

OpenAI's claim

On the quadric surface, a family of rank-two ample bundles is claimed to admit no smooth strictly Griffiths-positive metric for all sufficiently large coordinatewise power pullbacks. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.

On $S=\mathbb P^1\times\mathbb P^1$ let $P=\mathcal O_S(1,1)$ and let $f_m$ raise homogeneous coordinates in both factors to the $m$th power. OpenAI claims a rank-two algebraic bundle $G$ such that \[ E_m=f_m^*G\otimes P\text{ is ample for every }m\ge1, \] while for every sufficiently large $m$, no smooth Hermitian metric on $E_m$ has strictly Griffiths-positive Chern curvature. Strict positivity means positive curvature in each nonzero tangent and fiber direction. The single fixed $G$ produces all members of the sequence.

Outline of the argument

Manuscript 1 (family 050, 2026-09-24): Compatible graph identifications use finite-order transformations for gluing but an expanding product for the obstruction. Explicit symmetric-power sections certify ampleness; a limiting positive-metric argument would produce a measure invariant under the graph transformations, contradicting the expanding dynamics.

Formal verification

For family 050, the scope page documents the following selections. The selected power-pullback family is ample in every degree but lacks a smooth strictly positive Griffiths metric beyond a threshold. The additional even-exponent very-ampleness certificate is not selected. The documented statement covers the main claim of manuscript 1. Its selected declaration also covers the clause used for this problem. Comparator files: QuadricBundles.lean. Declarations: OAI.QuadricCounterexample.main_theorem.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

• OpenAI, Ample rank-two bundles on the quadric surface without Griffiths-positive metrics, 2026-09-24; citation key OAI:ample-rank-two-bundles-on-the-quadric-surface-without-griffiths-positive-metrics-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• Family 050: Lean scope.
• Comparator QuadricBundles.json.
• Comparator QuadricBundles.lean.
• OpenAI catalogue entry.