`ComparatorChallenges/HyperbolicObstruction.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.

CAT: Cartan-Alexandrov-Toponogov
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

24 declarations, 1 contain sorry, 0 declared as axioms

Finite complexes are standard barycentric realizations of finite abstract simplicial complexes. Area-counted elementary edge homotopies encode the usual triangular disk diagrams, allowing spurs and collapsed edges. Null edge loops (equivalently closed lifts to the universal cover) are those admitting such a disk, without any area bound. Asphericity is vanishing of all higher homotopy groups of this finite polyhedron. The CAT conditions use the standard squared-distance / hyperbolic-cosh comparison formulations.

structure OAI.HyperbolicObstruction.FiniteComplex line 26
Also declared: OAI/Geometry/HyperbolicGroups/Basic.lean:26 (same declaration text).
abbrev OAI.HyperbolicObstruction.FiniteComplex.Vertex line 32
Also declared: OAI/Geometry/HyperbolicGroups/Basic.lean:32 (same declaration text).
def OAI.HyperbolicObstruction.FiniteComplex.Carrier line 34
Also declared: OAI/Geometry/HyperbolicGroups/Basic.lean:34 (same declaration text).
instance OAI.HyperbolicObstruction.FiniteComplex.(instance at line 38) line 38
def OAI.HyperbolicObstruction.FiniteComplex.Edge line 41
Also declared: OAI/Geometry/HyperbolicGroups/Basic.lean:41 (same declaration text).
def OAI.HyperbolicObstruction.FiniteComplex.EdgePath line 44
Also declared: OAI/Geometry/HyperbolicGroups/Basic.lean:44 (same declaration text).
def OAI.HyperbolicObstruction.FiniteComplex.Closed line 49
Also declared: OAI/Geometry/HyperbolicGroups/Basic.lean:49 (same declaration text).
inductive OAI.HyperbolicObstruction.FiniteComplex.DiskMove An elementary disk move line 54
Source documentation
An elementary disk move. A noncollapsed triangular face costs one; a spur or a collapsed edge costs zero. Contexts are unchanged.
L4DEL explanation
An elementary disk move. A noncollapsed triangular face costs one; a spur or a collapsed edge costs zero. Contexts are unchanged.
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/HyperbolicGroups/Basic.lean:54 (same declaration text).
```
inductive DiskMove (K : FiniteComplex) : List K.Vertex → List K.Vertex → ℕ → Prop
  | stutter (l r) (a) :
      DiskMove K (l ++ [a, a] ++ r) (l ++ [a] ++ r) 0
  | spur (l r) (a b) (h : K.Edge a b) :
      DiskMove K (l ++ [a, b, a] ++ r) (l ++ [a] ++ r) 0
  | triangle (l r) (a b c) (h : {a, b, c} ∈ K.complex.faces)
      (hne : ({a, b, c} : Finset K.Vertex).card = 3) :
      DiskMove K (l ++ [a, b, c] ++ r) (l ++ [a, c] ++ r) 1

/-- A disk certificate obtained by elementary triangle attachments/removals,
    with spurs/collapses allowed. The index is the number of triangular faces. -/
```
Mentions: OAI.HyperbolicObstruction.FiniteComplex
inductive OAI.HyperbolicObstruction.FiniteComplex.DiskHomotopy A disk certificate obtained by elementary triangle attachments/removals, with spurs/collapses allowed line 65
Source documentation
A disk certificate obtained by elementary triangle attachments/removals, with spurs/collapses allowed. The index is the number of triangular faces.
L4DEL explanation
A disk certificate obtained by elementary triangle attachments/removals, with spurs/collapses allowed. The index is the number of triangular faces.
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/HyperbolicGroups/Basic.lean:65 (same declaration text).
```
inductive DiskHomotopy (K : FiniteComplex) : List K.Vertex → List K.Vertex → ℕ → Prop
  | refl (p) : DiskHomotopy K p p 0
  | move {p q n} : DiskMove K p q n → DiskHomotopy K p q n
  | symm {p q n} : DiskHomotopy K p q n → DiskHomotopy K q p n
  | trans {p q r m n} : DiskHomotopy K p q m → DiskHomotopy K q r n →
      DiskHomotopy K p r (m + n)
```
Mentions: OAI.HyperbolicObstruction.FiniteComplex, OAI.HyperbolicObstruction.FiniteComplex.DiskMove
def OAI.HyperbolicObstruction.FiniteComplex.Disk line 72
Also declared: OAI/Geometry/HyperbolicGroups/Basic.lean:72 (same declaration text).
def OAI.HyperbolicObstruction.FiniteComplex.LinearDiskFilling line 75
Also declared: OAI/Geometry/HyperbolicGroups/Basic.lean:75 (same declaration text).
def OAI.HyperbolicObstruction.FiniteComplex.Aspherical line 79
Also declared: OAI/Geometry/HyperbolicGroups/Basic.lean:79 (same declaration text).
def OAI.HyperbolicObstruction.IsSegment A constant-speed metric segment, parametrized on [0,1] line 85
Source documentation
A constant-speed metric segment, parametrized on [0,1].
L4DEL explanation
A constant-speed metric segment, parametrized on [0,1].
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/HyperbolicGroups/Basic.lean:85 (same declaration text).
```
def IsSegment {X : Type*} [MetricSpace X] (γ : ℝ → X) (x y : X) : Prop :=
…
```
def OAI.HyperbolicObstruction.IsGeodesic line 90
Also declared: OAI/Geometry/HyperbolicGroups/Basic.lean:90 (same declaration text).
def OAI.HyperbolicObstruction.CATZero The CN characterization of CAT(0), with whole metric segments line 94
Source documentation
The CN characterization of CAT(0), with whole metric segments.
L4DEL explanation
The CN characterization of CAT(0), with whole metric segments.
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/HyperbolicGroups/Basic.lean:94 (same declaration text).
```
def CATZero (X : Type*) [MetricSpace X] : Prop :=
…
```
def OAI.HyperbolicObstruction.CATNegOne The hyperbolic-cosh midpoint characterization of CAT(-1) line 101
Source documentation
The hyperbolic-cosh midpoint characterization of CAT(-1).
L4DEL explanation
The hyperbolic-cosh midpoint characterization of CAT(-1).
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/HyperbolicGroups/Basic.lean:101 (same declaration text).
```
def CATNegOne (X : Type*) [MetricSpace X] : Prop :=
…
```
structure OAI.HyperbolicObstruction.GeometricAction A proper, cocompact isometric action of a discrete group line 108
Source documentation
A proper, cocompact isometric action of a discrete group.
L4DEL explanation
A proper, cocompact isometric action of a discrete group.
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/HyperbolicGroups/Basic.lean:108 (same declaration text).
```
structure GeometricAction (G X : Type*) [Group G] [MetricSpace X] where
…
```
def OAI.HyperbolicObstruction.NoGeometricCATZeroAction line 116
Also declared: OAI/Geometry/HyperbolicGroups/Basic.lean:116 (same declaration text).
def OAI.HyperbolicObstruction.WordLength line 120
Also declared: OAI/Geometry/HyperbolicGroups/Basic.lean:120 (same declaration text).
def OAI.HyperbolicObstruction.WordDistance line 124
Also declared: OAI/Geometry/HyperbolicGroups/Basic.lean:124 (same declaration text).
def OAI.HyperbolicObstruction.WordHyperbolic Four-point word hyperbolicity, for some finite generating set line 128
Source documentation
Four-point word hyperbolicity, for some finite generating set.
L4DEL explanation
Four-point word hyperbolicity, for some finite generating set.
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/HyperbolicGroups/Basic.lean:128 (same declaration text).
```
def WordHyperbolic (G : Type*) [Group G] : Prop :=
…
```
def OAI.HyperbolicObstruction.CompatibleLocallyCATNegOne A metric on the given finite polyhedron, not on a different topology line 136
Source documentation
A metric on the given finite polyhedron, not on a different topology.
L4DEL explanation
A metric on the given finite polyhedron, not on a different topology.
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/HyperbolicGroups/Basic.lean:136 (same declaration text).
```
def CompatibleLocallyCATNegOne (K : FiniteComplex) : Prop :=
…
```
Mentions: OAI.HyperbolicObstruction.FiniteComplex
def OAI.HyperbolicObstruction.MainStatement A finite aspherical complex with hyperbolic fundamental group, no geometric CAT(0) action, and no homotopy equivalent finite complex admitting a locally CAT(-1) metric line 145
Source documentation
A finite aspherical complex with hyperbolic fundamental group, no geometric CAT(0) action, and no homotopy equivalent finite complex admitting a locally CAT(-1) metric.
L4DEL explanation
A finite aspherical complex with hyperbolic fundamental group, no geometric CAT(0) action, and no homotopy equivalent finite complex admitting a locally CAT(-1) 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/Geometry/HyperbolicGroups/Basic.lean:145 (same declaration text).
```
def MainStatement : Prop :=
…
```
theorem OAI.HyperbolicObstruction.main Challenge statement: A hyperbolic group without a geometric CAT(0) action sorry line 154
L4DEL explanation
The formalized result constructs a finite connected aspherical complex with a linear disk-filling bound whose fundamental group is word-hyperbolic but admits no geometric action on a nonempty proper complete $\mathrm{CAT}(0)$ space. It also excludes locally $\mathrm{CAT}(-1)$ geodesic metrics on every finite complex homotopy equivalent to the witness. The stronger claims that the witness is two-dimensional and that those models admit no locally $\mathrm{CAT}(0)$ metric are not included.
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.Geometry.HyperbolicGroups.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Geometry/HyperbolicGroups/Main.lean:15 (same statement text; different proof or body). Qualify references as OAI.HyperbolicObstruction.main@path.
```
theorem main : MainStatement.{u} := by sorry
…
```