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Math-2308-A1 · revision r1 · on Math-2308

OpenAI claim summary for Math-2308

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: partially solved. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Manuscript 1 (family 147, 2026-09-24): The theorem answers the boundary-collar rigidity question with a complete continuous collar, a C-infinity boundary and positive curvature; existence of caustics at selected rotation numbers or a nonspecified closed convex curve alone is not shown sufficient.

Manuscript 2 (family 147, 2026-09-24): The theorem answers the boundary-collar rigidity question with a complete continuous collar, a C-infinity boundary and positive curvature; existence of caustics at selected rotation numbers or a nonspecified closed convex curve alone is not shown sufficient.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Boundary integrability and the explicit confocal-caustic dictionary
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Mathematical claim 2.
  6. Outline of the argument
  7. Formal verification
  8. Status caveat
  9. Sources

Boundary integrability and the explicit confocal-caustic dictionary

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