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Weak mixing of triangular billiards with an irrational angle

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Proves that the billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is weakly mixing for normalized area times uniform direction. This strengthens ergodicity on the entire irrational-angle class, with no genericity or Diophantine restrictions. (OpenAI’s Lean note)

Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.

The independent assessments find that the Lean statements express only part of the result described above, or a narrower statement. Read the assessments

What the papers state

Theorem 1.1

In Weak mixing of triangular billiards with an irrational angle, 5 October 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

The unit-speed billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to \pi is weakly mixing for normalized Liouville measure.

No Lean statement in the release is paired with this theorem.

Theorem 1

In Ergodicity of triangular billiards with an irrational angle, 25 September 2026

The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.

Let Q be the interior of a nondegenerate Euclidean triangle with angles \alpha,\beta,\gamma. Suppose that at least one of

\alpha/\pi,\qquad \beta/\pi,\qquad \gamma/\pi

is irrational. The unit-speed billiard flow \Phi_t, with specular reflection at the open sides, is ergodic on Q\times S^1 for the probability measure

d\mu=\frac{dA\,d\theta}{2\pi\operatorname{Area}(Q)}.

Here trajectories hitting a vertex in either time direction are discarded. In particular, if a measurable set A satisfies \mu(\Phi_t^{-1}A\mathbin\triangle A)=0 for every t\in\R, then \mu(A)\in\{0,1\}.

The release does not say which paper theorem its catalogue describes. Both statements above are shown whole, the closer match to the description’s wording first. The atlas has not reviewed either choice.

Reported proof verification
One Lean statement in the release; three independent groups report that Comparator accepted runs for this file at the version shown here.
Papers and explanations
Two current papers, each readable here as a page beside its PDF.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

Ergodicity of triangular billiards with an irrational angle

25 September 2026, current edition.

Read it herePDF

We prove that the unit-speed billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is ergodic for normalized area times uniform angular measure. No genericity or Diophantine condition is required. The flow is considered outside the null set of trajectories that hit a vertex.

One Lean statement in this family

Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.

For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.

IrrationalTriangleBilliard

The paper states

Theorem 1

In Ergodicity of triangular billiards with an irrational angle, 25 September 2026

Let Q be the interior of a nondegenerate Euclidean triangle with angles \alpha,\beta,\gamma. Suppose that at least one of

\alpha/\pi,\qquad \beta/\pi,\qquad \gamma/\pi

is irrational. The unit-speed billiard flow \Phi_t, with specular reflection at the open sides, is ergodic on Q\times S^1 for the probability measure

d\mu=\frac{dA\,d\theta}{2\pi\operatorname{Area}(Q)}.

Here trajectories hitting a vertex in either time direction are discarded. In particular, if a measurable set A satisfies \mu(\Phi_t^{-1}A\mathbin\triangle A)=0 for every t\in\R, then \mu(A)\in\{0,1\}.

The words they share: time, trajectories, vertex, specular, reflection, probability, direction, interior, sides, area, billiard, irrational, measurable, flow, times, angles, triangle, pi, nondegenerate, euclidean, ergodicity, ergodic, speed, least, one, unit.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Ergodicity of triangular billiards with an irrational angle

IrrationalTriangleBilliard.lean, the release lists it as a supporting statement.

theorem irrational_triangle_billiard (Q : Triangle) (hQ : Q.HasIrrationalAngle) :    MainConclusion Q := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.
  • Neal McBurnett, openai-math-lean-tooling records it as “accepted”.

Read the complete Lean statement

Other mathematical statements in OpenAI’s Lean library

Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.

MathVet finds that the Lean statements express a weaker statement than the advertised result (their term: “weaker-statement”).

Their note: MainConclusion states ergodicity (a.e.-invariant measurable sets have measure 0 or 1), plus a.e. existence and uniqueness of flight chains, measure preservation and the a.e. flow law; weak mixing, the summary’s headline, is strictly stronger and is not stated (docs also only say ‘ergodic’).

Definitions the group asks readers to compare with the paper: billiardFlow Q t z := if h : Nonempty (FlightChain Q z) then (Classical.choice h).at t else z (flow defined by choice on chains; sound because the same statement proves chain uniqueness ∀ z c d : FlightChain Q z, c = d; singular states are frozen)

“weaker-statement”, in their words: The challenge statements do not imply the headline claim: they state a nontrivially weaker statement.

Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNLINKED”).

The gap they name: Weak mixing (from the October-5 manuscript, which the scope note does not link); only ergodicity is stated.

“NARROWER — UNLINKED”, in their words: The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

The formalization proves that the unit-speed billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to $\pi$ is ergodic for normalized area times uniform angular measure. It constructs the flow outside the null set of exceptional trajectories, proves uniqueness of the flight chain there, and establishes measure preservation and the flow law almost everywhere. No genericity or Diophantine condition on the irrational angle is assumed.

AI-drafted comparisons, with no recorded human review

  • Ergodicity of triangular billiards with an irrational angle IrrationalTriangleBilliard

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: the billiard flow in every triangle with an angle irrational relative to π is ergodic, with the flow constructed off a null set. The weak mixing of the family’s later paper is not part of the statement.

Who ran the proof-verification program?

These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.

These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.

Ergodicity of triangular billiards with an irrational angle IrrationalTriangleBilliard

Three reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report
Neal McBurnett, openai-math-lean-tooling“accepted”Re-run through Comparator with Lean’s kernel: accepted (the check took 185 s; peak memory 6.5 GiB).11 October 2026Original report

Other proofs and exact calculations

These reports examine claims in the papers using HOL Light, a different proof assistant, independently of the release’s Lean proofs. Read each report for the exact claim it covers.

  • Counting step only: at most ⌈π/α⌉ multiples of α fit in an open interval of length π

    HOL Light card 150

    Reported by
    Ben Knill, openai-math-hol-cards
    Method
    Proof of a selected step in HOL Light
    Their verdict
    “PASS” The report records a proof of a supporting fact.
    What it covers
    Not the bounce claim; the unfolding geometry is not certified
    Original report
    Their report

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Within this family

Papers elsewhere in the release that this family’s papers cite

This family’s papers cite no paper elsewhere in the release.

Papers elsewhere in the release that cite this family’s papers

No paper elsewhere in the release cites this family’s papers.

References outside the release

Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release

Explain a result, compare statements or propose a correction

Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.

Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.

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What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.

    Author not stated. Source

Questions awaiting review

Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.

Read the Lean statement against the paper

  • An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded. Open the form

Identify the theorem the catalogue describes

  • Family 150 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

Address a question raised in an independent assessment

Compare the proposed paper and Lean matches

  • Family 150 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form