Nonnegative-curvature Einstein classification and an L2 topological gap

What OpenAI claims

Classifies closed connected Einstein four-manifolds with positive Einstein constant and nonnegative sectional curvature: up to scaling, their universal Riemannian covers are the round S4, Fubini–Study ℂℙ², or a product of equal round two-spheres. A closed simply connected nonnegatively curved four-manifold is diffeomorphic to one of these whenever the scale-invariant L2 norm of its trace-free Ricci curvature lies below a universal positive constant. (Lean)

Papers (3)

1. Zero-Plane Rigidity for Einstein Four-Manifolds

   4 October 2026 · Preprint, not peer reviewed · PDF

   We prove that a closed Einstein four-manifold with positive Einstein constant, nonnegative sectional curvature, and a zero-curvature plane has universal Riemannian cover isometric to a product of two round two-spheres. Under the normalization Ric =3g, both spheres have radius 1/√3. The proof extends coupled estimates for the two Weyl curvature blocks to the boundary of the sectional-curvature cone and determines their equality case.

2. An L² Einstein Gap for Nonnegatively Curved Four-Manifolds

   5 October 2026 · Preprint, not peer reviewed · PDF

   We prove a universal, scale-invariant L2 gap for the trace-free Ricci tensor on simply connected closed four-manifolds with nonnegative sectional curvature. If the trace-free Ricci energy is sufficiently small, the manifold is diffeomorphic to S4, ℂℙ², or S²× S². The proof uses the classification of positive-Einstein, nonnegatively curved four-manifolds supplied by the companion zero-plane rigidity theorem, stated explicitly as the classification premise of our result. No auxiliary curvature, volume, diameter, injectivity-radius, or Sobolev bound is required. The conclusion concerns the smooth manifold, not an isometry of the original metric.

3. Positively curved Einstein four-manifolds

   23 September 2026 · Preprint, not peer reviewed · PDF

   We prove the classification conjecture for connected smooth closed Einstein four-manifolds with strictly positive sectional curvature. Up to positive scaling and isometry, every such manifold is the round four-sphere, the complex projective plane with its Fubini–Study metric, or real projective four-space with its round metric. No orientability assumption is needed.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.