What was already known
Villani conjectured this implication. Figalli, Gallouët and Rifford proved it for nonfocal manifolds; the case with conjugate cut points remained open.
AI involvement
OpenAI credits itself as manuscript author and states that its internal frontier model produced the results in this release. Individual human input is not detailed enough to establish task-only autonomy.
OpenAI is named as author on the manuscript, and its release attributes the results to an internal frontier model. The individual human input is not detailed enough to establish autonomy.
Importance
Resolves Villani’s injectivity-domain convexity conjecture in the compact setting without nonfocality.
Limitations
The conclusion is ordinary linear convexity; strict convexity and boundary regularity are not asserted. The paper’s arbitrary-potential local-injectivity claim lies outside the selected Lean scope.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned Lean note proves convexity of all open tangent injectivity domains under weak MTW without a nonfocality assumption.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- On every compact connected smooth Riemannian manifold of dimension at least two satisfying weak Ma–Trudinger–Wang curvature, each open tangent injectivity domain is convex.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. No named person is documented as explaining the central argument for this assessed contribution.
Correction and publication history
Primary manuscript, attribution, prior-work, and scoped checking review for non-notifying local inclusion.
Read this published revision