`ComparatorChallenges/IrrationalTriangleBilliard.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 120 curated AI-written annotations and 1273 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 9 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. Those 9 annotations were revised on 2026-10-14 to bind all of them to exact page-one quotations from 6 pinned OpenAI manuscripts, the matching pinned formalization notes, and named historical primary sources; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. The 3 manuscript-grounded matrix-multiplication annotations were composed from pinned OpenAI manuscripts and named historical sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Six additional target-specific annotations for Catalan's constant, the irrationality exponent of pi, dyadic Erdos similarity, Unique Games, quintic Lienard systems, and Yau nodal growth were composed from pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for Seymour's second-neighborhood conjecture, Cannon's conjecture, Koebe circle-domain uniformization, and circle-domain rigidity were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical primary sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for simple Lebesgue spectrum on the three-torus, Rokhlin multiple mixing, the nine-dimensional Borsuk counterexample, and Tingley's sphere-isometry problem were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical primary sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for the Abhyankar-Sathaye counterexample, the circulant Hadamard conjecture, the Eilenberg-Ganea counterexample, and short Egyptian fractions were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and quotations from Sathaye, Turyn with Storer, Eilenberg with Ganea, and Erdos in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for the complete Crouzeix inequality, unrestricted Euclidean five-coloring, the general Mahler conjecture, and Thompson F nonamenability were composed on 2026-10-14 from exact Lean statements, 4 pinned OpenAI manuscripts, the matching formalization notes, and linked historical sources by Crouzeix, de Grey, Mahler, Cannon, Floyd, and Parry; no human review is claimed. Four additional target-specific annotations for algorithmic strong thin trees, Barnette's Hamiltonian-cycle conjecture, the explicit Artin CAT(0) counterexample, and generalized amplitude-damping capacity were composed on 2026-10-14 from exact Lean statements, 4 pinned OpenAI manuscripts, matching formalization notes, and linked historical sources by Anari with Goddyn, Arts with Hopman and Timmermans, Charney, and Khatri with Sharma and Wilde; no human review is claimed. Fourteen solution-level annotations for Seymour's second-neighborhood theorem were added on 2026-10-14 from the pinned 15-page manuscript, covering the exact-neighborhood model, minimal-counterexample reduction, subset deficit, transitive blowup, pruning, extremal contradiction, and proof-bearing final theorem. The subset and blowup annotations also bind exact quotations from Tyler Seacrest (2019) and Brantner, Brockman, Kay, and Snively (2009); no human review is claimed. Thirty-seven declaration-level annotations were added on 2026-10-14 for the rest of ComparatorChallenges/AlgorithmicThinTrees.lean, covering its multigraph semantics, stack-machine cost model, explicit and binary-multiplicity encodings, output certificate, and complete algorithmic proposition. They are bound to the pinned OpenAI manuscript and linked earlier thin-tree sources by Luis Goddyn, Nima Anari, and Shayan Oveis Gharan; the annotation explicitly preserves that the concluding comparator theorem still has body by sorry, and no human review is claimed. Thirty-one supporting declarations in ComparatorChallenges/UniqueGamesTheorem.lean were upgraded or added on 2026-10-14, completing curated coverage of all thirty-two declarations in that file. The annotations cover the finite game model, Boolean formula and parser, binary-vector translations, explicit output serialization, finite-alphabet machine contract, and gap theorem; they distinguish manuscript-backed mathematics from Lean-specific canonical parsing and unary game output encoding, bind exact OpenAI manuscript quotations, and credit Subhash Khot through his 2010 survey. The comparator theorem remains explicitly labeled as having body by sorry; no human review is claimed. The source-derived generator was hardened on 2026-10-14 so a formalization note covering multiple comparator results no longer copies its combined Scope prose onto each declaration; those targets now use declaration-specific source comments when available and otherwise remain explicitly bounded to the exact Lean statement. The remaining source-derived annotations were generated mechanically from pinned formalization notes and source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the source prose.

Credit: OpenAI's pinned repository says an unreleased internal model produced the manuscripts and supporting proof artifacts. The collection builds on the mathematics community's accumulated problems, results, language, and formal libraries; each manuscript's BibTeX and bibliography are the authority for individual scholarly credit.

Project sources: Pinned openai/math documentation, Pinned formalization catalogue

17 declarations, 1 contain sorry, 0 declared as axioms

The phase space is represented in the ambient plane × circle, with measure supported on the interior of the triangle. Values on side-collision states at a fixed time are immaterial to the modulo-null-set conclusion. The trajectories themselves include the collisions and impose specular reflection.

structure OAI.TriangularBilliards.Triangle Three affinely independent vertices in the Euclidean plane line 20
Source documentation
Three affinely independent vertices in the Euclidean plane.
L4DEL explanation
Three affinely independent vertices in the Euclidean plane.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Dynamics/TriangleBilliards/Model.lean:20 (same declaration text).
```
structure Triangle where
…
```
def OAI.TriangularBilliards.Triangle.table The open table; the complex plane carries its ordinary real Euclidean metric line 27
Source documentation
The open table; the complex plane carries its ordinary real Euclidean metric.
L4DEL explanation
The open table; the complex plane carries its ordinary real Euclidean metric.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Dynamics/TriangleBilliards/Model.lean:27 (same declaration text).
```
def table (Q : Triangle) : Set ℂ := interior (convexHull ℝ (range Q.vertex))
```
Mentions: OAI.TriangularBilliards.Triangle
def OAI.TriangularBilliards.Triangle.side The open side from vertex i to the next vertex, indexed cyclically line 30
Source documentation
The open side from vertex i to the next vertex, indexed cyclically.
L4DEL explanation
The open side from vertex i to the next vertex, indexed cyclically.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Dynamics/TriangleBilliards/Model.lean:30 (same declaration text).
```
def side (Q : Triangle) (i : Fin 3) : Set ℂ :=
…
```
Mentions: OAI.TriangularBilliards.Triangle
def OAI.TriangularBilliards.Triangle.tangent A tangent vector to side i, with no choice of inward or outward normal line 34
Source documentation
A tangent vector to side i, with no choice of inward or outward normal.
L4DEL explanation
A tangent vector to side i, with no choice of inward or outward normal.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Dynamics/TriangleBilliards/Model.lean:34 (same declaration text).
```
def tangent (Q : Triangle) (i : Fin 3) : ℂ := Q.vertex (i + 1) - Q.vertex i
```
Mentions: OAI.TriangularBilliards.Triangle
def OAI.TriangularBilliards.Triangle.angle Interior angle at vertex i line 37
Source documentation
Interior angle at vertex i.
L4DEL explanation
Interior angle at vertex i.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Dynamics/TriangleBilliards/Model.lean:37 (same declaration text).
```
def angle (Q : Triangle) (i : Fin 3) : ℝ :=
…
```
Mentions: OAI.TriangularBilliards.Triangle
def OAI.TriangularBilliards.Triangle.HasIrrationalAngle line 41
Also declared: OAI/Dynamics/TriangleBilliards/Model.lean:41 (same declaration text).
def OAI.TriangularBilliards.reflect Reflection in the real line spanned by the nonzero tangent e line 49
Source documentation
Reflection in the real line spanned by the nonzero tangent e. In coordinates with e real this is complex conjugation, preserving the tangential component and reversing the normal component.
L4DEL explanation
Reflection in the real line spanned by the nonzero tangent e. In coordinates with e real this is complex conjugation, preserving the tangential component and reversing the normal component.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Dynamics/TriangleBilliards/Model.lean:49 (same declaration text).
```
def reflect (e v : ℂ) : ℂ := (e / star e) * star v
```
instance OAI.TriangularBilliards.circleMeasurableSpace line 51
Also declared: OAI/Dynamics/TriangleBilliards/Model.lean:51 (same declaration text).
instance OAI.TriangularBilliards.circleBorelSpace line 53
Also declared: OAI/Dynamics/TriangleBilliards/Model.lean:53 (same declaration text).
abbrev OAI.TriangularBilliards.Phase line 55
Also declared: OAI/Dynamics/TriangleBilliards/Model.lean:55 (same declaration text).
structure OAI.TriangularBilliards.FlightChain A complete nonsingular billiard trajectory, indexed by its collisions line 61
Source documentation
A complete nonsingular billiard trajectory, indexed by its collisions. Flight zero contains the initial time strictly. Both ends of the collision time sequence are unbounded; collisions occur only in open sides, never vertices. These are geometric conditions, not ergodicity/analytic assumptions.
L4DEL explanation
A complete nonsingular billiard trajectory, indexed by its collisions. Flight zero contains the initial time strictly. Both ends of the collision time sequence are unbounded; collisions occur only in open sides, never vertices. These are geometric conditions, not ergodicity/analytic assumptions.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Dynamics/TriangleBilliards/Model.lean:61 (same declaration text).
```
structure FlightChain (Q : Triangle) (z : Phase) where
…
```
Mentions: OAI.TriangularBilliards.Triangle, OAI.TriangularBilliards.Phase
def OAI.TriangularBilliards.FlightChain.at The geometric state on a chain, using outgoing velocity at a collision line 82
Source documentation
The geometric state on a chain, using outgoing velocity at a collision.
L4DEL explanation
The geometric state on a chain, using outgoing velocity at a collision.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Dynamics/TriangleBilliards/Model.lean:82 (same declaration text).
```
def FlightChain.at {Q : Triangle} {z : Phase} (c : FlightChain Q z) (t : ℝ) : Phase := by
…
```
Mentions: OAI.TriangularBilliards.Triangle, OAI.TriangularBilliards.Phase, OAI.TriangularBilliards.FlightChain
def OAI.TriangularBilliards.billiardFlow The actual specular flow on complete regular chains line 92
Source documentation
The actual specular flow on complete regular chains. Singular initial states are assigned their initial state; the theorem must prove these form a null set. No nonemptiness of the regular set is postulated.
L4DEL explanation
The actual specular flow on complete regular chains. Singular initial states are assigned their initial state; the theorem must prove these form a null set. No nonemptiness of the regular set is postulated.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Dynamics/TriangleBilliards/Model.lean:92 (same declaration text).
```
def billiardFlow (Q : Triangle) (t : ℝ) (z : Phase) : Phase := by
…
```
Mentions: OAI.TriangularBilliards.Triangle, OAI.TriangularBilliards.Phase
def OAI.TriangularBilliards.angularMeasure Uniform angular probability, parameterized by [0,2π) line 97
Source documentation
Uniform angular probability, parameterized by [0,2π).
L4DEL explanation
Uniform angular probability, parameterized by [0,2π).
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Dynamics/TriangleBilliards/Model.lean:97 (same declaration text).
```
def angularMeasure : Measure Circle :=
…
```
def OAI.TriangularBilliards.phaseMeasure Exactly normalized area in the open table times uniform angular measure line 102
Source documentation
Exactly normalized area in the open table times uniform angular measure.
L4DEL explanation
Exactly normalized area in the open table times uniform angular measure.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Dynamics/TriangleBilliards/Model.lean:102 (same declaration text).
```
def phaseMeasure (Q : Triangle) : Measure Phase :=
…
```
Mentions: OAI.TriangularBilliards.Triangle, OAI.TriangularBilliards.Phase
def OAI.TriangularBilliards.MainConclusion The main conclusion, together with the geometric/nonvacuity guarantees needed by the concrete definition of the actual two-sided billiard line 108
Source documentation
The main conclusion, together with the geometric/nonvacuity guarantees needed by the concrete definition of the actual two-sided billiard.
L4DEL explanation
The main conclusion, together with the geometric/nonvacuity guarantees needed by the concrete definition of the actual two-sided billiard.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Dynamics/TriangleBilliards/Model.lean:108 (same declaration text).
```
def MainConclusion (Q : Triangle) : Prop :=
…
```
Mentions: OAI.TriangularBilliards.Triangle
theorem OAI.TriangularBilliards.irrational_triangle_billiard Challenge statement: Weak mixing of triangular billiards with an irrational angle sorry line 119
L4DEL explanation
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.
1. This is an exact comparator target linked by the pinned repository's formalization note for this result.
2. The repository's comparator configuration selects OAI.Dynamics.TriangleBilliards.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Dynamics/TriangleBilliards/Main.lean:418 (same statement text; different proof or body). Qualify references as OAI.TriangularBilliards.irrational_triangle_billiard@path.
```
theorem irrational_triangle_billiard (Q : Triangle) (hQ : Q.HasIrrationalAngle) :
    MainConclusion Q := by
…
```
Mentions: OAI.TriangularBilliards.Triangle, OAI.TriangularBilliards.MainConclusion