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Math-2326 · mathematics · Dynamical systems

Birkhoff Integrable-Billiards Conjecture

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
The only integrable planar inner billiards should be those inside ellipses.

Formulation references: [S1], [S2].

Short English statement

The only integrable planar inner billiards should be those inside ellipses.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2326-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2326

Manuscript 1 (family 147, 2026-09-24): Ellipse rigidity is proved in the C-infinity positive-curvature class with a complete continuous near-boundary foliation (phase or smooth physical caustics as stated). The broader problem-statement integrability sentence does not specify these regularity/foliation hypotheses, so no blanket equivalence is inferred.

Manuscript 2 (family 147, 2026-09-24): Ellipse rigidity is proved in the C-infinity positive-curvature class with a complete continuous near-boundary foliation (phase or smooth physical caustics as stated). The broader problem-statement integrability sentence does not specify these regularity/foliation hypotheses, so no blanket equivalence is inferred.

Birkhoff Integrable-Billiards Conjecture

OpenAI's claim

Manuscript 1 (family 147, 2026-09-24): A merely continuous invariant foliation of the full grazing phase annulus of a smooth positively curved convex billiard is claimed to create an analytic caustic collar and hence an elliptical table. Manuscript 2 (family 147, 2026-09-24): A smooth positively curved convex billiard table is claimed to be an ellipse whenever its near-boundary region has a continuous full foliation by smooth convex caustics, with no transverse differentiability required.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $\Omega\subset\mathbb R^2$ be bounded and strictly convex, with smooth regular simple closed boundary and positive curvature. In boundary arclength $s\in\mathbb R/L\mathbb Z$ and outgoing angle $\delta\in(0,\pi)$, let $T$ be the billiard reflection map on $M=(\mathbb R/L\mathbb Z)\times(0,\pi)$. Assume a $T$-invariant open annulus $U$ contains $(\mathbb R/L\mathbb Z)\times(0,\eta)$ for some $\eta>0$, and admits a homeomorphism $H:S^1\times(0,1)\to U$ for which each circle leaf is simple, closed, essential and individually $T$-invariant. OpenAI claims \[ \Omega=\{x\in\mathbb R^2:(x-c)^TQ(x-c)<1\} \] for some center $c$ and symmetric positive-definite real matrix $Q$. The assumed phase-space foliation near grazing is essential; circles are included.
Mathematical claim 2.
Let $\Omega$ be a bounded strictly convex planar domain with a smooth regular simple closed boundary of positive curvature. Assume a relatively open collar $A\subset\overline\Omega$ of the boundary and a homeomorphism $H_0:S^1\times[0,1)\to A$ whose level zero is $\partial\Omega$ and whose positive levels are smooth closed convex caustics. A caustic means a curve to which tangency of a billiard segment is preserved by reflection $v\mapsto v-2(v\cdot n)n$, with outward unit normal $n$. OpenAI claims \[ \Omega=\{x:(x-c)^TQ(x-c)<1\},\qquad Q=Q^T>0. \] The statement assumes a continuous collar of caustics and includes circles.

Outline of the argument

Manuscript 1 (family 147, 2026-09-24): Strict twist barriers and area recurrence rule out vertical leaf pairs and rational rotation plateaus, yielding unique periodic graph leaves. Stationary-mesh, Fourier, and buffered-division estimates then promote the continuous phase data to a physical analytic collar before the companion ellipse-rigidity theorem is applied.

Manuscript 2 (family 147, 2026-09-24): Periodic rational caustics give a Fourier action system whose near-identity derivative supports analytic Newton inversion. Values and parameter jets produce a joint glancing conjugacy; finite reflected-chain jet determination then controls complex continuation and rules out a first meromorphic wall, yielding global ellipse rigidity.

Formal verification

Family 147 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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