∑math20

Dynamical systems and ergodic theory · Result 147

The near-boundary Birkhoff conjecture

2 papers

Resolves the near-boundary Birkhoff conjecture for smooth strictly convex planar billiards of positive curvature. Such a billiard is an ellipse whenever a full grazing annulus is continuously foliated by individually invariant essential curves. A continuous physical collar of smooth closed convex caustics also suffices.

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▶Introduction: the problem and why it mattersin preparation
▶The presented solutionin preparation

Papers 2

First page of Continuous Phase Foliations Create Analytic Caustic Collars

Continuous Phase Foliations Create Analytic Caustic Collars

OpenAI · September 24, 2026

For a smooth strictly convex planar billiard of positive curvature, a continuous foliation of a full grazing annulus by individually invariant essential curves forces an analytic boundary and a jointly analytic collar of smooth strictly convex caustics. The physical-collar rigidity theorem proved in the companion article then implies that the table is an ellipse.

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@misc{OAI:Continuous-Phase-Foliations-Create-Analytic-Caustic-Collars-September-24-2026,
  author = {{OpenAI}},
  title = {{Continuous Phase Foliations Create Analytic Caustic Collars}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Continuous-Phase-Foliations-Create-Analytic-Caustic-Collars-September-24-2026/paper.pdf}{OAI:Continuous-Phase-Foliations-Create-Analytic-Caustic-Collars-September-24-2026}},
  year = {2026}
}
First page of Rigidity of Smooth Billiards with a Continuous Caustic Collar

Rigidity of Smooth Billiards with a Continuous Caustic Collar

OpenAI · September 24, 2026

We prove that a smooth strictly convex planar billiard with positive curvature is an ellipse whenever a full neighborhood of its boundary is continuously foliated by smooth closed convex caustics. No differentiability across the leaves is assumed. This resolves the near-boundary Birkhoff conjecture in the stated smooth class.

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Cite
@misc{OAI:Rigidity-of-Smooth-Billiards-with-a-Continuous-Caustic-Collar-September-24-2026,
  author = {{OpenAI}},
  title = {{Rigidity of Smooth Billiards with a Continuous Caustic Collar}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Rigidity-of-Smooth-Billiards-with-a-Continuous-Caustic-Collar-September-24-2026/paper.pdf}{OAI:Rigidity-of-Smooth-Billiards-with-a-Continuous-Caustic-Collar-September-24-2026}},
  year = {2026}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.