{"id":"openai-math/Continuous-Phase-Foliations-Create-Analytic-Caustic-Collars","pdf":"https://github.com/openai/math/blob/main/preprints/Continuous-Phase-Foliations-Create-Analytic-Caustic-Collars-September-24-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Continuous-Phase-Foliations-Create-Analytic-Caustic-Collars-September-24-2026","tags":["billiards","dynamical-systems","integrable-systems"],"score":0,"title":"Continuous Phase Foliations Create Analytic Caustic Collars","domain":"Dynamical systems and ergodic theory","family":"147","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.DS","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-24","reactions":{},"formalized":false,"family_title":"The near-boundary Birkhoff conjecture"}