The near-boundary Birkhoff conjecture: billiards integrable near the boundary are ellipses
In a planar billiard in a strictly convex domain , a caustic is a curve such that a trajectory tangent to it stays tangent after every reflection. Ellipses are integrable: confocal ellipses foliate a neighbourhood of the boundary by caustics. The Birkhoff conjecture, traditionally attributed to Birkhoff and first printed by Poritsky (1950), asserts that every integrable strictly convex billiard is an ellipse. Lazutkin found a Cantor family of caustics near the boundary for every smooth positively curved table; Bialy proved rigidity under integrability of the whole phase space; Avila-De Simoi-Kaloshin and Kaloshin-Sorrentino proved local versions near ellipses. Is a smooth, strictly convex, positively curved billiard table an ellipse whenever it is integrable near the boundary?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Dynamical systems; convex billiards and integrability
- Posed by
- Traditionally attributed to G. D. Birkhoff (Dynamical Systems, 1927); first printed formulation by Hillel Poritsky, Ann. of Math. 51 (1950), as the manuscripts cite
- Year posed
- 1950
- Years open
- 76y
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 45 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1 (phase foliations): if a -invariant open annulus containing in the billiard phase space is homeomorphically foliated by individually invariant essential curves, then is an ellipse. The companion proves the same conclusion from a continuous collar of smooth closed convex caustics, via analytic regularity of the boundary and complex continuation. Only continuity across leaves is assumed. Not covered: integrability only on a Cantor set, rational integrability, or integrability away from the boundary; the full Birkhoff conjecture (global integrability as typically defined) is reduced to but not identical with this case.
What the AI did
The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 of each September 24 manuscript was read against the Birkhoff conjecture as the papers state it. One proves the ellipse conclusion when a full neighbourhood of the boundary is continuously foliated by smooth closed convex caustics; the other derives such a collar from a continuous foliation of a grazing phase annulus by invariant essential curves, and invokes the first. The proofs were not refereed. Scope stated by the papers: boundary with positive curvature, a full collar (a Cantor family of caustics does not suffice), no assumption on dynamics away from the boundary. No Lean formalization exists.