OpenAI claim summary for Math-2302
mathematicsTop Open Problemscontributionliterature_reportClaim: literature report
Summary of a claim published in openai/math; not a claim by this platform.
No conjugate points without nonpositive curvature: The closed orientable three-manifold has a smooth metric with no conjugate points, yet the underlying manifold carries no nonpositively curved metric. This negates the source implication even in its explicitly emphasized dimension-three residue.
The claimed counterexample lies within the question’s hypotheses and contradicts its universal conclusion.
Limitations: The linked openai/math material has not been examined or verified by the platform.
Manuscript
Contents
No conjugate points: flat-growth rigidity and the three-dimensional residue
OpenAI's claim
No conjugate points without nonpositive curvature: The manuscript claims a closed orientable three-manifold supporting a smooth metric without conjugate points but no smooth metric of everywhere nonpositive sectional curvature. The separation concerns two different existence properties on the same manifold.Mathematical statement of the claimed result
Mathematical claim 1.
A smooth orientable three-manifold $M$ is claimed to exist with all of the following properties: $M$ is closed and connected, one smooth metric has no conjugate points, and every smooth metric fails the condition $\sec\le0$ somewhere. Absence of conjugate points means there is no nonzero Jacobi field vanishing at two distinct times along a geodesic. The second assertion concerns every possible smooth metric on the same manifold.Outline of the argument
No conjugate points without nonpositive curvature: Topological gluing chooses a manifold with an obstruction to nonpositive curvature. Smooth product cores and a concave cometric are connected by controlled passage regions. Index-form estimates on successive geodesic passages and uniform bounds on returning geodesics exclude conjugate points for the constructed metric, while the topological obstruction survives the gluing.Formal verification
Family 358 describes Lean coverage for the closed three-manifold separation of no-conjugate-points and nonpositive-curvature existence.No conjugate points without nonpositive curvature: The manuscript links ConjugatePoints.lean (declaration OAI.ThreeManifold.main_theorem).
The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.
This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.
Status caveat
OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.Sources
- OpenAI, A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric, 2026-09-24; citation key
OAI:A-Three-Manifold-Without-Conjugate-Points-and-Without-a-Nonpositively-Curved-Metric-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - Family 358: Lean scope.
- Comparator ConjugatePoints.json.
- Comparator ConjugatePoints.lean.
- OpenAI catalogue entry.