Source authenticated

Ergodicity of billiards in every triangle with an irrational angle

Theorem 1.1 (September 25): the unit-speed billiard flow in every nondegenerate triangle with at least one angle irrational relative to $\pi$ is ergodic for normalised area times uniform angle, off the null set of vertex-hitting trajectories; consequences include quantum ergodicity and growth of nodal domain counts. The October 5 companion proves the stronger weak mixing (the product flow is ergodic). Neither proves mixing, unique ergodicity, or any statement about individual orbits (for example existence of periodic orbits in obtuse triangles), and polygons with more sides are not treated.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 25, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Ergodicity of billiards in every triangle with an irrational angle · Irrational triangle billiards are ergodic

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Ergodicity of billiards in every triangle with an irrational angle

specialist registry · passed

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the ergodicity manuscript was read against the question; it gives ergodicity for every nondegenerate triangle with at least one irrational angle, no genericity or Diophantine condition. The challenge lean/ComparatorChallenges/IrrationalTriangleBilliard.json and its solution module OAI.Dynamics.TriangleBilliards.Main exist at the pinned commit (found via lean/docs/150.md) but are not in the formalization catalogue. The statement was read here: for every triangle with an irrational angle it asserts a normalised phase measure, almost-everywhere existence and uniqueness of complete flight chains avoiding vertices with specular reflection, measure preservation and the flow law almost everywhere, and that every flow-invariant measurable set has measure 0 or 1. That states the headline ergodicity claim. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice. Weak mixing is not formalised.

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Ergodicity of billiards in every triangle with an irrational angle

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Lean proof (OAI.TriangularBilliards.irrational_triangle_billiard)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Comparator statement: IrrationalTriangleBilliard.lean

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.