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Math-2692-A1 · revision r1 · on Math-2692

OpenAI claim summary for Math-2692

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: open. 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 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.

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

Manuscript

Contents
  1. Ellipse rigidity from a continuous physical collar of smooth caustics
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

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