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Math-2692 · mathematics · Dynamical systems

Ellipse rigidity from a continuous physical collar of smooth caustics

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let a bounded planar billiard table have a C-infinity strictly convex boundary with everywhere positive curvature. If a whole inner boundary collar is continuously foliated by individually C-infinity strictly convex closed caustics, with no differentiability of the transverse foliation assumed, must the boundary be an ellipse?

Short English statement

Let a bounded planar billiard table have a C-infinity strictly convex boundary with everywhere positive curvature. If a whole inner boundary collar is continuously foliated by individually C-infinity strictly convex closed caustics, with no differentiability of the transverse foliation assumed, must the boundary be an ellipse?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2692-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2692

Manuscript 1 (family 147, 2026-09-24): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Ellipse rigidity from a continuous physical collar of smooth caustics

OpenAI's claim

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. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
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): 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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