Lean statement

Ergodicity of triangular billiards with an irrational angle

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorem selected for verification: OAI.TriangularBilliards.irrational_triangle_billiard

Lean statement file: IrrationalTriangleBilliard.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 150 link it.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

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

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

structure OAI.TriangularBilliards.Triangle line 20

structure Triangle where  vertex : Fin 3 → ℂ  nondegenerate : AffineIndependent ℝ vertex

definition OAI.TriangularBilliards.Triangle.table line 27

def table (Q : Triangle) : Set ℂ := interior (convexHull ℝ (range Q.vertex))

definition OAI.TriangularBilliards.Triangle.side line 30

def side (Q : Triangle) (i : Fin 3) : Set ℂ :=  openSegment ℝ (Q.vertex i) (Q.vertex (i + 1))

definition OAI.TriangularBilliards.Triangle.tangent line 34

def tangent (Q : Triangle) (i : Fin 3) : ℂ := Q.vertex (i + 1) - Q.vertex i

definition OAI.TriangularBilliards.Triangle.angle line 37

def angle (Q : Triangle) (i : Fin 3) : ℝ :=  InnerProductGeometry.angle (Q.vertex (i + 1) - Q.vertex i)    (Q.vertex (i + 2) - Q.vertex i)

definition OAI.TriangularBilliards.Triangle.HasIrrationalAngle line 41

def HasIrrationalAngle (Q : Triangle) : Prop :=  ∃ i : Fin 3, Irrational (Q.angle i / Real.pi)

definition OAI.TriangularBilliards.reflect line 49

def reflect (e v : ℂ) : ℂ := (e / star e) * star v

abbreviation OAI.TriangularBilliards.Phase line 55
```
abbrev Phase := ℂ × Circle
```

structure OAI.TriangularBilliards.FlightChain line 61

structure FlightChain (Q : Triangle) (z : Phase) where

Show the remaining 18 lines

  time : ℤ → ℝ  point : ℤ → ℂ  direction : ℤ → Circle  wall : ℤ → Fin 3  increasing : StrictMono time  unbounded_below : ∀ t : ℝ, ∃ n : ℤ, time n < t  unbounded_above : ∀ t : ℝ, ∃ n : ℤ, t < time n  zero_before : time 0 < 0  zero_after : 0 < time 1  on_side : ∀ n : ℤ, point n ∈ Q.side (wall n)  flight : ∀ n : ℤ,    point (n + 1) = point n + (time (n + 1) - time n) • (direction n : ℂ)  inside : ∀ (n : ℤ) (t : ℝ), time n < t → t < time (n + 1) →    point n + (t - time n) • (direction n : ℂ) ∈ Q.table  start_point : z.1 = point 0 + (-time 0) • (direction 0 : ℂ)  start_direction : z.2 = direction 0  specular : ∀ n : ℤ,    (direction n : ℂ) = reflect (Q.tangent (wall n)) (direction (n - 1) : ℂ)

definition OAI.TriangularBilliards.FlightChain.at line 82

def FlightChain.at {Q : Triangle} {z : Phase} (c : FlightChain Q z) (t : ℝ) : Phase := by  classical  exact if h : ∃ n : ℤ, c.time n ≤ t ∧ t < c.time (n + 1) then    let n := Classical.choose h    (c.point n + (t - c.time n) • (c.direction n : ℂ), c.direction n)  else z

definition OAI.TriangularBilliards.billiardFlow line 92

def billiardFlow (Q : Triangle) (t : ℝ) (z : Phase) : Phase := by  classical  exact if h : Nonempty (FlightChain Q z) then (Classical.choice h).at t else z

definition OAI.TriangularBilliards.angularMeasure line 97

def angularMeasure : Measure Circle :=  Measure.map Circle.exp    ((ENNReal.ofReal (2 * Real.pi))⁻¹ • volume.restrict (Ico 0 (2 * Real.pi)))

definition OAI.TriangularBilliards.phaseMeasure line 102

def phaseMeasure (Q : Triangle) : Measure Phase :=  ((volume Q.table)⁻¹ • volume.restrict Q.table).prod angularMeasure

definition OAI.TriangularBilliards.MainConclusion line 108

def MainConclusion (Q : Triangle) : Prop :=  phaseMeasure Q univ = 1 ∧  (∀ᵐ z ∂phaseMeasure Q, Nonempty (FlightChain Q z)) ∧  (∀ (z : Phase) (c d : FlightChain Q z), c = d) ∧  (∀ t : ℝ, MeasurePreserving (billiardFlow Q t) (phaseMeasure Q) (phaseMeasure Q)) ∧  (∀ s t : ℝ, billiardFlow Q (s + t) =ᵐ[phaseMeasure Q]    (billiardFlow Q s ∘ billiardFlow Q t)) ∧  (∀ A : Set Phase, MeasurableSet A →    (∀ t : ℝ, phaseMeasure Q (((billiardFlow Q t) ⁻¹' A) ∆ A) = 0) →    phaseMeasure Q A = 0 ∨ phaseMeasure Q A = 1)

Possible paper match, with no recorded review

Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

Words of the Lean statement, its names split into words, that also appear in the theorem: time, trajectories, vertex, specular, reflection, probability, direction, interior and twelve more. Found in the paper’s title, which its main theorem states: ergodicity. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: ergodic, speed, least, one and unit.

Statements in Ergodicity of triangular billiards with an irrational angle come first because the release lists only that paper for this Lean statement.

Theorem 1 of Ergodicity of triangular billiards with an irrational angle

Theorem

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\}.

Section 1, Introduction Read it in the paper TeX source, line 88

Show the TeX

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
\(\mu(A)\in\{0,1\}\).

Other possible paper matches, closest wording first

Corollary 7 of Ergodicity of triangular billiards with an irrational angle. Words of the Lean statement, its names split into words, that also appear in the theorem: using, real, conclusions, outward, three, zero and 16 more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: unit and one.
Theorem 1.1 of Weak mixing of triangular billiards with an irrational angle. Words of the Lean statement, its names split into words, that also appear in the theorem: billiard, irrational, flow, normalized, measure, angle and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: relative, speed, least, one and unit.
Proposition 5.1 of Weak mixing of triangular billiards with an irrational angle. Words of the Lean statement, its names split into words, that also appear in the theorem: nonzero, irrational, times, angle, triangle, pi and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: relative, almost and everywhere.
Proposition 2.1 of Weak mixing of triangular billiards with an irrational angle. Words of the Lean statement, its names split into words, that also appear in the theorem: flow, regular, normalized, measure, triangle, nondegenerate and one more.
Proposition 5 of Ergodicity of triangular billiards with an irrational angle. Words of the Lean statement, its names split into words, that also appear in the theorem: regular. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: almost and everywhere.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

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

Carlos Toledo’s cert-machine

How they ran it, in their words

Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

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. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory; the run was still in progress at this snapshot (192 accepted, 224 pending).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 185 s; peak memory 6.5 GiB).
Reported 11 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 150’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

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

MathVet
Finds part of it stated, or a narrower or weaker statement Their term: “weaker-statement”.
The challenge statements do not imply the headline claim: they state a nontrivially weaker statement. Their record
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)
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — UNLINKED”.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link. Their record
Their note: Weak mixing (from the October-5 manuscript, which the scope note does not link); only ergodicity is stated.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement 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.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

126 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 39 to its own declarations, 48 to Mathlib’s documentation. 17 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves irrational_triangle_billiard exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches 14 of the file’s 16 other declarations; with the theorem itself they span 60 lines. 3 of them are named in the statement itself.

Names in the namespace OAI.TriangularBilliards are listed without it.

structure Triangleline 20in the statementlibrary
def Triangle.tableline 27library
def Triangle.sideline 30library
def Triangle.tangentline 34library
def Triangle.angleline 37library
def Triangle.HasIrrationalAngleline 41in the statementlibrary
def reflectline 49library
abbrev Phaseline 55library
structure FlightChainline 61library
def FlightChain.atline 82library
def billiardFlowline 92library
def angularMeasureline 97library
def phaseMeasureline 102library
def MainConclusionline 108in the statementlibrary
theorem irrational_triangle_billiardline 119selected theoremlibrary

Other Lean rules that may affect the statement 2

Lean can use instances, notation and attributes without naming them in the statement. These are rules for interpreting or finding mathematical objects. Atlas’s scan of names cannot follow all such uses; read the listed source as well.

instance circleMeasurableSpace line 51
instance circleBorelSpace line 53

import Mathlibnamespace OAI/-!The phase space is represented in the ambient plane × circle, with measuresupported on the interior of the triangle. Values on side-collision statesat a fixed time are immaterial to the modulo-null-set conclusion. Thetrajectories themselves include the collisions and impose specular reflection.-/open MeasureTheory Setopen scoped ENNReal symmDiffnoncomputable sectionnamespace TriangularBilliards/-- Three affinely independent vertices in the Euclidean plane. -/structure Triangle where  vertex : Fin 3 → ℂ  nondegenerate : AffineIndependent ℝ vertexnamespace Triangle/-- The open table; the complex plane carries its ordinary real Euclidean metric. -/def table (Q : Triangle) : Set ℂ := interior (convexHull ℝ (range Q.vertex))/-- The open side from vertex `i` to the next vertex, indexed cyclically. -/def side (Q : Triangle) (i : Fin 3) : Set ℂ :=  openSegment ℝ (Q.vertex i) (Q.vertex (i + 1))/-- A tangent vector to side `i`, with no choice of inward or outward normal. -/def tangent (Q : Triangle) (i : Fin 3) : ℂ := Q.vertex (i + 1) - Q.vertex i/-- Interior angle at vertex `i`. -/def angle (Q : Triangle) (i : Fin 3) : ℝ :=  InnerProductGeometry.angle (Q.vertex (i + 1) - Q.vertex i)    (Q.vertex (i + 2) - Q.vertex i)def HasIrrationalAngle (Q : Triangle) : Prop :=  ∃ i : Fin 3, Irrational (Q.angle i / Real.pi)end Triangle/-- Reflection in the real line spanned by the nonzero tangent `e`.In coordinates with `e` real this is complex conjugation, preserving thetangential component and reversing the normal component. -/def reflect (e v : ℂ) : ℂ := (e / star e) * star vinstance circleMeasurableSpace : MeasurableSpace Circle := borel Circleinstance circleBorelSpace : BorelSpace Circle := ⟨rfl⟩abbrev Phase := ℂ × Circle/-- A complete nonsingular billiard trajectory, indexed by its collisions.Flight zero contains the initial time strictly. Both ends of the collisiontime sequence are unbounded; collisions occur only in open sides, neververtices. These are geometric conditions, not ergodicity/analytic assumptions. -/structure FlightChain (Q : Triangle) (z : Phase) where  time : ℤ → ℝ  point : ℤ → ℂ  direction : ℤ → Circle  wall : ℤ → Fin 3  increasing : StrictMono time  unbounded_below : ∀ t : ℝ, ∃ n : ℤ, time n < t  unbounded_above : ∀ t : ℝ, ∃ n : ℤ, t < time n  zero_before : time 0 < 0  zero_after : 0 < time 1  on_side : ∀ n : ℤ, point n ∈ Q.side (wall n)  flight : ∀ n : ℤ,    point (n + 1) = point n + (time (n + 1) - time n) • (direction n : ℂ)  inside : ∀ (n : ℤ) (t : ℝ), time n < t → t < time (n + 1) →    point n + (t - time n) • (direction n : ℂ) ∈ Q.table  start_point : z.1 = point 0 + (-time 0) • (direction 0 : ℂ)  start_direction : z.2 = direction 0  specular : ∀ n : ℤ,    (direction n : ℂ) = reflect (Q.tangent (wall n)) (direction (n - 1) : ℂ)/-- The geometric state on a chain, using outgoing velocity at a collision. -/def FlightChain.at {Q : Triangle} {z : Phase} (c : FlightChain Q z) (t : ℝ) : Phase := by  classical  exact if h : ∃ n : ℤ, c.time n ≤ t ∧ t < c.time (n + 1) then    let n := Classical.choose h    (c.point n + (t - c.time n) • (c.direction n : ℂ), c.direction n)  else z/-- The actual specular flow on complete regular chains. Singular initialstates are assigned their initial state; the theorem must prove these forma null set. No nonemptiness of the regular set is postulated. -/def billiardFlow (Q : Triangle) (t : ℝ) (z : Phase) : Phase := by  classical  exact if h : Nonempty (FlightChain Q z) then (Classical.choice h).at t else z/-- Uniform angular probability, parameterized by [0,2π). -/def angularMeasure : Measure Circle :=  Measure.map Circle.exp    ((ENNReal.ofReal (2 * Real.pi))⁻¹ • volume.restrict (Ico 0 (2 * Real.pi)))/-- Exactly normalized area in the open table times uniform angular measure. -/def phaseMeasure (Q : Triangle) : Measure Phase :=  ((volume Q.table)⁻¹ • volume.restrict Q.table).prod angularMeasure/-- The main conclusion, together with the geometric/nonvacuity guaranteesneeded by the concrete definition of the actual two-sided billiard. -/def MainConclusion (Q : Triangle) : Prop :=  phaseMeasure Q univ = 1 ∧  (∀ᵐ z ∂phaseMeasure Q, Nonempty (FlightChain Q z)) ∧  (∀ (z : Phase) (c d : FlightChain Q z), c = d) ∧  (∀ t : ℝ, MeasurePreserving (billiardFlow Q t) (phaseMeasure Q) (phaseMeasure Q)) ∧  (∀ s t : ℝ, billiardFlow Q (s + t) =ᵐ[phaseMeasure Q]    (billiardFlow Q s ∘ billiardFlow Q t)) ∧  (∀ A : Set Phase, MeasurableSet A →    (∀ t : ℝ, phaseMeasure Q (((billiardFlow Q t) ⁻¹' A) ∆ A) = 0) →    phaseMeasure Q A = 0 ∨ phaseMeasure Q A = 1)theorem irrational_triangle_billiard (Q : Triangle) (hQ : Q.HasIrrationalAngle) :    MainConclusion Q := by  sorryend TriangularBilliardsendend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.