The near-boundary Birkhoff conjecture: billiards integrable near the boundary are ellipses
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.
Exact FrontierDelta
Scope and record
Occurred: Sep 24, 2026
Delta type: SOURCE CLAIM
Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.
Canonical aliases: The near-boundary Birkhoff conjecture: billiards integrable near the boundary are ellipses · Near-boundary Birkhoff conjecture
Confidence: Not scored
Registry verification: unreviewed · announcement · partial
Attribution
VibeMathed
registry · event recorded by
Unreleased internal OpenAI model
model · ai model contributor · OpenAI
Lineage and corrections
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. parent of this event
This event attributed to Unreleased internal OpenAI model
The near-boundary Birkhoff conjecture: billiards integrable near the boundary are ellipses parent of this event
The near-boundary Birkhoff conjecture: billiards integrable near the boundary are ellipses evidence for this event
Companion: Rigidity of Smooth Billiards with a Continuous Caustic Collar evidence for this event
VibeMathed record: The near-boundary Birkhoff conjecture: billiards integrable near the boundary are ellipses evidence for this event
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