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Research / Oct 6, 2026

A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric

Can the mathematical statement in A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric be proved and verified machine-checked in Lean?

Model / AI: OpenAI (unreleased)

machine-checked (formal proof)self-reported (formal proof)
Oct 6, 2026

Lean formalization in OpenAI math repository (Result 358)

Model / AI: OpenAI (unreleased)

machine-checked (formal proof)self-reported (formal proof)

OpenAI's internal frontier model proved and formalized in Lean the main result of 'A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric'. We construct a closed connected orientable smooth three-manifold that admits a smooth Riemannian metric without conjugate points but admits no smooth Riemannian metric of nonpositive sectional curvature.

Source read 2026-10-08. Cataloged from OpenAI's formalization.yaml (Family 358). The Lean development was not rebuilt locally.