`ComparatorChallenges/CannonGeometricAction.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

structure OAI.CannonRelease.CayleyData line 10
Also declared: OAI/Geometry/Cannon/Model.lean:10 (same declaration text).
def OAI.CannonRelease.CayleyData.graph line 15
Also declared: OAI/Geometry/Cannon/Model.lean:15 (same declaration text).
def OAI.CannonRelease.CayleyData.ThinTriangles A uniform thinness constant for geodesic triangles in the Cayley graph line 19
Source documentation
A uniform thinness constant for geodesic triangles in the Cayley graph.
L4DEL explanation
A uniform thinness constant for geodesic triangles in the Cayley graph.
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/Geometry/Cannon/Model.lean:19 (same declaration text).
```
def CayleyData.ThinTriangles {G : Type u} [Group G] (D : CayleyData G) : Prop :=
…
```
Mentions: OAI.CannonRelease.CayleyData
def OAI.CannonRelease.BasedRay Based geodesic rays, parametrized at integer times line 27
Source documentation
Based geodesic rays, parametrized at integer times.
L4DEL explanation
Based geodesic rays, parametrized at integer times.
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/Geometry/Cannon/Model.lean:27 (same declaration text).
```
def BasedRay {G : Type u} [Group G] (D : CayleyData G) :=
…
```
Mentions: OAI.CannonRelease.CayleyData
instance OAI.CannonRelease.(instance at line 30) line 30
def OAI.CannonRelease.raySetoid Finite synchronous distance, equivalent to asymptoticity for based rays line 35
Source documentation
Finite synchronous distance, equivalent to asymptoticity for based rays.
L4DEL explanation
Finite synchronous distance, equivalent to asymptoticity for based rays.
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/Geometry/Cannon/Model.lean:35 (same declaration text).
```
def raySetoid {G : Type u} [Group G] (D : CayleyData G) : Setoid (BasedRay D) where
…
```
Mentions: OAI.CannonRelease.CayleyData, OAI.CannonRelease.BasedRay
def OAI.CannonRelease.Boundary line 48
Also declared: OAI/Geometry/Cannon/Model.lean:48 (same declaration text).
instance OAI.CannonRelease.(instance at line 50) line 50
abbrev OAI.CannonRelease.SphereTwo line 54
Also declared: OAI/Geometry/Cannon/Model.lean:54 (same declaration text).
def OAI.CannonRelease.HyperbolicThree The upper-half-space model of hyperbolic three-space line 57
Source documentation
The upper-half-space model of hyperbolic three-space.
L4DEL explanation
The upper-half-space model of hyperbolic three-space.
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/Geometry/Cannon/Model.lean:57 (same declaration text).
```
def HyperbolicThree := {x : Fin 3 → ℝ // 0 < x 2}
```
instance OAI.CannonRelease.(instance at line 59) line 59
def OAI.CannonRelease.hyperbolicDist line 62
Also declared: OAI/Geometry/Cannon/Model.lean:62 (same declaration text).
def OAI.CannonRelease.hyperbolicIsometries The full isometry group, including orientation-reversing isometries line 66
Source documentation
The full isometry group, including orientation-reversing isometries.
L4DEL explanation
The full isometry group, including orientation-reversing isometries.
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/Geometry/Cannon/Model.lean:66 (same declaration text).
```
def hyperbolicIsometries : Subgroup (HyperbolicThree ≃ₜ HyperbolicThree) where
…
```
Mentions: OAI.CannonRelease.HyperbolicThree
abbrev OAI.CannonRelease.H3Isom line 79
Also declared: OAI/Geometry/Cannon/Model.lean:79 (same declaration text).
def OAI.CannonRelease.ProperAction line 81
Also declared: OAI/Geometry/Cannon/Model.lean:81 (same declaration text).
def OAI.CannonRelease.CocompactAction line 85
Also declared: OAI/Geometry/Cannon/Model.lean:85 (same declaration text).
theorem OAI.CannonRelease.cannon Challenge statement: Cannon's conjecture sorry line 89
L4DEL explanation
every hyperbolic group whose boundary is homeomorphic to the two-sphere admits a proper cocompact isometric action on hyperbolic three-space with finite kernel
— OpenAI Cannon manuscript (abstract)
We characterize those discrete groups G which can act properly discontinuously, isometrically, and cocompactly on hyperbolic 3-space H3
— James W. Cannon and Eric L. Swenson (1998), abstract
A hyperbolic group whose boundary is a topological two-sphere admits an isometric action on hyperbolic three-space that is proper and cocompact and has finite kernel.
1. CayleyData chooses a finite symmetric generating set whose Cayley graph is connected. ThinTriangles supplies one uniform natural-number bound for the distance from every point on one geodesic side to the other two sides.
2. The boundary is a quotient of based integer-time geodesic rays by bounded synchronous distance. The hypothesis Nonempty (Boundary D ≃ₜ SphereTwo) records the boundary homeomorphism to S².
3. The conclusion produces a homomorphism into the full isometry group of the upper-half-space model of H³. ProperAction, CocompactAction, and finite kernel are three separate conjuncts; orientation preservation is not required.
4. The manuscript's route is an upper combinatorial-modulus bound, boundary uniformization, and straightening of the boundary action. This comparator body is sorry; the paired solution module is OAI.Geometry.Cannon.Main, so the prose does not substitute for its Lean proof.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI Cannon manuscript (abstract), Pinned OpenAI formalization note 246, James W. Cannon and Eric L. Swenson (1998), abstract
Distinct definitions of this name: OAI/Geometry/Cannon/Main.lean:23 (same statement text; different proof or body). Qualify references as OAI.CannonRelease.cannon@path.

geometric action: an isometric group action that is proper and cocompact
hyperbolic-group boundary: asymptotic classes of geodesic rays in a Cayley graph, equipped with its boundary topology
```
theorem cannon (G : Type u) [Group G] [TopologicalSpace G] [DiscreteTopology G]
    (D : CayleyData G) (hthin : D.ThinTriangles)
    (hsphere : Nonempty (Boundary D ≃ₜ SphereTwo)) :
    ∃ ρ : G →* H3Isom, ProperAction ρ ∧ CocompactAction ρ ∧
      (ρ.ker : Set G).Finite := by
…
```
Mentions: OAI.CannonRelease.CayleyData, OAI.CannonRelease.Boundary, OAI.CannonRelease.SphereTwo, OAI.CannonRelease.H3Isom, OAI.CannonRelease.ProperAction, OAI.CannonRelease.CocompactAction