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Math-2415-A1 · revision r1 · on Math-2415

OpenAI claim summary for Math-2415

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: 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 150, 2026-09-25): For triangles (three edges), rational angle triples form a Lebesgue-null set and the theorem gives ergodicity of all remaining triangles; polygons with arbitrary other prescribed edge counts are not handled.

Manuscript 2 (family 150, 2026-10-05): For triangles (three edges), rational angle triples form a Lebesgue-null set and the theorem gives ergodicity of all remaining triangles; polygons with arbitrary other prescribed edge counts are not handled.

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

Manuscript

Contents
  1. Full-Measure Families of Ergodic Irrational Polygons
  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

Full-Measure Families of Ergodic Irrational Polygons

OpenAI's claim

Manuscript 1 (family 150, 2026-09-25): The full unit-speed billiard flow of each nondegenerate triangle with an angle irrational relative to $\pi$ is claimed to be ergodic, without a Diophantine or genericity condition. Manuscript 2 (family 150, 2026-10-05): An irrational angle divided by $\pi$ is claimed to force weak mixing for the full billiard flow of any nondegenerate Euclidean triangle, using normalized Liouville measure.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $Q$ be the interior of any Euclidean triangle that is nondegenerate and has an angle whose ratio to $\pi$ is irrational. For the unit-speed specular billiard flow $\Phi_t$ on $Q\times S^1$, discarding trajectories that hit vertices in either time direction, OpenAI claims ergodicity with respect to \[ d\mu=\frac{dA\,d\theta}{2\pi\operatorname{Area}(Q)}. \] Precisely, if a measurable $A$ satisfies $\mu(\Phi_t^{-1}A\mathbin{\triangle} A)=0$ for every real $t$, then $\mu(A)$ is $0$ or $1$. This ranges over every such triangle, rather than almost every parameter or a fixed direction.
Mathematical claim 2.
For every nondegenerate Euclidean triangle $Q$ having at least one angle irrational relative to $\pi$, OpenAI claims that its unit-speed specular billiard flow $\Phi_t$ is weakly mixing under normalized Liouville measure \[ d\mu=\frac{dA\,d\theta}{2\pi\operatorname{Area}(Q)}. \] Equivalently, the simultaneous product flow $(x,y)\mapsto(\Phi_tx,\Phi_ty)$ is ergodic for $\mu\otimes\mu$. Vertex-hitting trajectories are discarded. Weak mixing is the claim, without an assertion of ordinary mixing.

Outline of the argument

Manuscript 1 (family 150, 2026-09-25): On the flat double, positional smoothing and long cone-avoiding segments bound endpoint gradients of angular Fourier tails. Two successive weak-limit passages show that all coefficient gradients vanish without a Sobolev assumption; irrational cone rotation removes every nonzero angular mode, leaving a constant invariant function.

Manuscript 2 (family 150, 2026-10-05): Local transport estimates force a possible eigenfunction to be a plane wave with only direction-dependent amplitude. Reflections through the irrational vertex make that amplitude constant, and reflection at the third side excludes any nonzero frequency; the spectral criterion then gives weak mixing.

Formal verification

For family 150, the scope page documents the following selections. The selection establishes ergodicity and the almost-everywhere flow law for an irrational-angle triangle, including the exceptional-trajectory null set. The weak-mixing companion is not selected. The documented statement covers the main claim of manuscript 1. Its selected declaration also covers the clause used for this problem. Manuscript 2 is not among the papers selected by that page, so the family link does not formalize its main conclusion. It does not establish formal coverage of the complete problem clause. Comparator files: IrrationalTriangleBilliard.lean. Declarations: OAI.TriangularBilliards.irrational_triangle_billiard.

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