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Math-2683 · mathematics · Dynamical systems

Weak mixing of every triangular billiard with an irrational angle

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every Euclidean triangle having at least one angle whose ratio to pi is irrational, is its billiard flow weakly mixing for normalized Liouville measure, after deleting trajectories that hit a vertex?

Short English statement

For every Euclidean triangle having at least one angle whose ratio to pi is irrational, is its billiard flow weakly mixing for normalized Liouville measure, after deleting trajectories that hit a vertex?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2683-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2683

Manuscript 1 (family 150, 2026-10-05): 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.

Weak mixing of every triangular billiard with an irrational angle

OpenAI's claim

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

Mathematical statement of the claimed result

Mathematical claim 1.
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, 2026-10-05): 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. Manuscript 1 is not among the papers selected by that page, so the family link does not formalize its main conclusion.

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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