`ComparatorChallenges/PolycyclicRecognition.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. 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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

23 declarations, 3 contain sorry, 0 declared as axioms

Quasi-isometric recognition and uniform bounded height

A polycyclic group has a finite subnormal series with cyclic factors. The height quotient below is the quotient of the Lie algebra by the common kernel of the diagonal adjoint characters.

structure OAI.ChromaticCyclic.CyclicStep line 16
Also declared: OAI/GroupTheory/CyclicSeries/Length.lean:9 (same declaration text).
structure OAI.ChromaticCyclic.CyclicSeries line 21
Also declared: OAI/GroupTheory/CyclicSeries/Length.lean:14 (same declaration text).
def OAI.PolycyclicRecognition.GroupBornologous Uniform control of finite sets of group differences line 39
Source documentation
Uniform control of finite sets of group differences.
L4DEL explanation
Uniform control of finite sets of group differences.
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/GroupTheory/PolycyclicRecognition/Analysis/CoarseEquivalence.lean:24 (same declaration text).
```
def GroupBornologous (f : G → H) : Prop :=
…
```
structure OAI.PolycyclicRecognition.GroupCoarseEquivalence Coarse inverse maps with errors contained in fixed finite sets line 44
Source documentation
Coarse inverse maps with errors contained in fixed finite sets.
L4DEL explanation
Coarse inverse maps with errors contained in fixed finite sets.
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/GroupTheory/PolycyclicRecognition/Analysis/CoarseEquivalence.lean:43 (same declaration text).
```
structure GroupCoarseEquivalence (G H : Type*) [Group G] [Group H] where
…
```
def OAI.PolycyclicRecognition.IsPolycyclic Polycyclicity expressed by an actual cyclic subnormal series line 61
Source documentation
Polycyclicity expressed by an actual cyclic subnormal series.
L4DEL explanation
Polycyclicity expressed by an actual cyclic subnormal series.
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/GroupTheory/PolycyclicRecognition/Algebra/Polycyclic.lean:22 (same declaration text).
```
def IsPolycyclic : Prop :=
…
```
def OAI.PolycyclicRecognition.IsVirtuallyPolycyclic Virtual polycyclicity means a polycyclic subgroup of finite index line 65
Source documentation
Virtual polycyclicity means a polycyclic subgroup of finite index.
L4DEL explanation
Virtual polycyclicity means a polycyclic subgroup of finite index.
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)
Distinct definitions of this name: OAI/GroupTheory/PolycyclicRecognition/Algebra/Polycyclic.lean:26 (same statement text; different proof or body). Qualify references as OAI.PolycyclicRecognition.IsVirtuallyPolycyclic@path.
```
def IsVirtuallyPolycyclic : Prop :=
…
```
def OAI.PolycyclicRecognition.IsUniformLattice A uniform lattice is discrete and has a compact fundamental cover line 75
Source documentation
A uniform lattice is discrete and has a compact fundamental cover.
L4DEL explanation
A uniform lattice is discrete and has a compact fundamental cover.
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)
Distinct definitions of this name: OAI/GroupTheory/PolycyclicRecognition/Lattices/UniformLatticeMetric.lean:28 (same statement text; different proof or body). Qualify references as OAI.PolycyclicRecognition.IsUniformLattice@path.
```
def IsUniformLattice [TopologicalSpace G] (Λ : Subgroup G) : Prop :=
…
```
structure OAI.PolycyclicRecognition.SimplyConnectedSolvableLieModel A connected simply connected finite-dimensional real solvable Lie group line 82
Source documentation
A connected simply connected finite-dimensional real solvable Lie group.
L4DEL explanation
A connected simply connected finite-dimensional real solvable Lie 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/GroupTheory/PolycyclicRecognition/Lattices/SolvableLatticeModel.lean:31 (same declaration text).
```
structure SimplyConnectedSolvableLieModel where
…
```
def OAI.PolycyclicRecognition.IsFiniteCovolumeLattice A discrete subgroup with a normalized invariant measure on its coset space line 106
Source documentation
A discrete subgroup with a normalized invariant measure on its coset space.
L4DEL explanation
A discrete subgroup with a normalized invariant measure on its coset 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/GroupTheory/PolycyclicRecognition/FiniteCovolumeLattice.lean:20 (same declaration text).
```
def IsFiniteCovolumeLattice {G : Type} [Group G] [TopologicalSpace G]
    (Λ : Subgroup G) : Prop :=
…
```
structure OAI.PolycyclicRecognition.LinearMetricChains Explicit chains with bounded steps and a linear bound on their length line 113
Source documentation
Explicit chains with bounded steps and a linear bound on their length.
L4DEL explanation
Explicit chains with bounded steps and a linear bound on their length.
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)
Distinct definitions of this name: OAI/GroupTheory/PolycyclicRecognition/Analysis/LinearMetricChains.lean:23 (same statement text; different proof or body). Qualify references as OAI.PolycyclicRecognition.LinearMetricChains@path.
```
structure LinearMetricChains (X : Type*) [PseudoMetricSpace X] where
…
```
structure OAI.PolycyclicRecognition.ProperChainMetric A compatible proper left-invariant metric with large-scale-geodesic chains line 125
Source documentation
A compatible proper left-invariant metric with large-scale-geodesic chains.
L4DEL explanation
A compatible proper left-invariant metric with large-scale-geodesic chains.
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/GroupTheory/PolycyclicRecognition/Geometry/LieModelMetric.lean:30 (same declaration text).
```
structure ProperChainMetric (G : Type*) [Group G] [TopologicalSpace G] where
…
```
abbrev OAI.PolycyclicRecognition.ProperChainMetric.compatibleMetric The compatible metric retains the original topology definitionally line 133
Source documentation
The compatible metric retains the original topology definitionally.
L4DEL explanation
The compatible metric retains the original topology definitionally.
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)
Distinct definitions of this name: OAI/GroupTheory/PolycyclicRecognition/Geometry/LieModelMetric.lean:43 (different declaration text). Qualify references as OAI.PolycyclicRecognition.ProperChainMetric.compatibleMetric@path.
```
abbrev ProperChainMetric.compatibleMetric
    {G : Type*} [Group G] [TopologicalSpace G] (d : ProperChainMetric G) : MetricSpace G :=
…
```
Mentions: OAI.PolycyclicRecognition.ProperChainMetric
def OAI.PolycyclicRecognition.heightKernel The common kernel, including the empty family of characters line 142
Source documentation
The common kernel, including the empty family of characters.
L4DEL explanation
The common kernel, including the empty family of characters.
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/GroupTheory/PolycyclicRecognition/Geometry/CanonicalHeight.lean:23 (same declaration text).
```
def heightKernel (χ : I → L →ₗ[ℝ] ℝ) : Submodule ℝ L := ⨅ i, (χ i).ker
```
abbrev OAI.PolycyclicRecognition.HeightSpace The canonical linear height quotient line 145
Source documentation
The canonical linear height quotient.
L4DEL explanation
The canonical linear height quotient.
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/GroupTheory/PolycyclicRecognition/Geometry/CanonicalHeight.lean:26 (same declaration text).
```
abbrev HeightSpace (χ : I → L →ₗ[ℝ] ℝ) := L ⧸ heightKernel χ
```
Mentions: OAI.PolycyclicRecognition.heightKernel
def OAI.PolycyclicRecognition.heightProjection The quotient map defining infinitesimal height line 148
Source documentation
The quotient map defining infinitesimal height.
L4DEL explanation
The quotient map defining infinitesimal height.
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/GroupTheory/PolycyclicRecognition/Geometry/CanonicalHeight.lean:29 (same declaration text).
```
def heightProjection (χ : I → L →ₗ[ℝ] ℝ) : L →ₗ[ℝ] HeightSpace χ :=
…
```
Mentions: OAI.PolycyclicRecognition.HeightSpace
structure OAI.PolycyclicRecognition.TriangularAdjoint A simultaneous real upper triangular basis for the adjoint representation line 154
Source documentation
A simultaneous real upper triangular basis for the adjoint representation.
L4DEL explanation
A simultaneous real upper triangular basis for the adjoint representation.
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/GroupTheory/PolycyclicRecognition/Geometry/TriangularHeight.lean:59 (same declaration text).
```
structure TriangularAdjoint (L : Type*) [LieRing L] [LieAlgebra ℝ L] (n : ℕ) where
…
```
def OAI.PolycyclicRecognition.TriangularAdjoint.character The diagonal entries of the adjoint representation line 164
Source documentation
The diagonal entries of the adjoint representation.
L4DEL explanation
The diagonal entries of the adjoint representation.
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)
Distinct definitions of this name: OAI/GroupTheory/PolycyclicRecognition/Geometry/TriangularHeight.lean:69 (same statement text; different proof or body). Qualify references as OAI.PolycyclicRecognition.TriangularAdjoint.character@path.
```
noncomputable def character (i : Fin n) : L →ₗ[ℝ] ℝ where
…
```
structure OAI.PolycyclicRecognition.IsGroupExponential The exponential is characterized by its one-parameter subgroups and by their tangent vectors at the identity line 181
Source documentation
The exponential is characterized by its one-parameter subgroups and by their tangent vectors at the identity.
L4DEL explanation
The exponential is characterized by its one-parameter subgroups and by their tangent vectors at the identity.
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)
Distinct definitions of this name: OAI/GroupTheory/PolycyclicRecognition/Geometry/LieExponential.lean:26 (same statement text; different proof or body). Qualify references as OAI.PolycyclicRecognition.IsGroupExponential@path.
```
structure IsGroupExponential (Exp : GroupLieAlgebra (𝓘(ℝ, E)) G → G) : Prop where
…
```
def OAI.PolycyclicRecognition.ModelIsUnimodular Unimodularity for the canonical Borel Haar measure of the Lie group line 192
Source documentation
Unimodularity for the canonical Borel Haar measure of the Lie group.
L4DEL explanation
Unimodularity for the canonical Borel Haar measure of the Lie 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/GroupTheory/PolycyclicRecognition/ConditionalBoundedHeight.lean:30 (same declaration text).
```
def ModelIsUnimodular (R : SimplyConnectedSolvableLieModel) : Prop :=
…
```
Mentions: OAI.PolycyclicRecognition.SimplyConnectedSolvableLieModel
structure OAI.PolycyclicRecognition.SelfQuasiIsometry The upper, lower, and coarse-surjectivity bounds of a self quasi-isometry line 202
Source documentation
The upper, lower, and coarse-surjectivity bounds of a self quasi-isometry.
L4DEL explanation
The upper, lower, and coarse-surjectivity bounds of a self quasi-isometry.
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)
Distinct definitions of this name: OAI/GroupTheory/PolycyclicRecognition/Analysis/SelfQuasiIsometry.lean:15 (different declaration text). Qualify references as OAI.PolycyclicRecognition.SelfQuasiIsometry@path.
```
structure SelfQuasiIsometry (G : Type*) [PseudoMetricSpace G] (K C : ℝ) where
…
```
theorem OAI.PolycyclicRecognition.group_recognition Challenge statement: Quasi-isometric recognition of virtually polycyclic groups sorry line 209
Source documentation
Quasi-isometric recognition, including the finite-index uniform-lattice realization.
L4DEL explanation
The formalization proves quasi-isometric recognition of virtually polycyclic groups: every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is itself virtually polycyclic. It also realizes a finite-index subgroup of the recognized group as a uniform lattice in a simply connected solvable Lie group, which may differ from the original ambient group. The linked structural estimate controls the height-coordinate behavior of self quasi-isometries of the stated unimodular solvable Lie models, up to bounded error and one of finitely many linear height symmetries.
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.GroupTheory.PolycyclicRecognition.Recognition as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/GroupTheory/PolycyclicRecognition/Recognition.lean:24 (same statement text; different proof or body). Qualify references as OAI.PolycyclicRecognition.group_recognition@path.
```
theorem group_recognition
    {P J : Type} [Group P] [Group J] [Group.FG P] [Group.FG J]
    (e : GroupCoarseEquivalence P J) (hP : IsVirtuallyPolycyclic P) :
    IsVirtuallyPolycyclic J ∧
      ∃ Γ : Subgroup J, Γ.FiniteIndex ∧
        ∃ (S : SimplyConnectedSolvableLieModel) (Λ : Subgroup S.carrier),
          IsUniformLattice Λ ∧ Nonempty (Γ ≃* Λ) := by
…
```
Mentions: OAI.PolycyclicRecognition.GroupCoarseEquivalence, OAI.PolycyclicRecognition.SimplyConnectedSolvableLieModel
theorem OAI.PolycyclicRecognition.main_uniform_bounded_height Challenge statement: Quasi-isometric recognition of virtually polycyclic groups sorry line 219
Source documentation
One finite slope group controls the height of every self quasi-isometry.
L4DEL explanation
The formalization proves quasi-isometric recognition of virtually polycyclic groups: every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is itself virtually polycyclic. It also realizes a finite-index subgroup of the recognized group as a uniform lattice in a simply connected solvable Lie group, which may differ from the original ambient group. The linked structural estimate controls the height-coordinate behavior of self quasi-isometries of the stated unimodular solvable Lie models, up to bounded error and one of finitely many linear height symmetries.
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.GroupTheory.PolycyclicRecognition.Recognition as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/GroupTheory/PolycyclicRecognition/Recognition.lean:80 (same statement text; different proof or body). Qualify references as OAI.PolycyclicRecognition.main_uniform_bounded_height@path.
```
theorem main_uniform_bounded_height
    (R : SimplyConnectedSolvableLieModel) (d : ProperChainMetric R.carrier)
    {a : ℕ} (T : TriangularAdjoint (GroupLieAlgebra (𝓘(ℝ, R.modelSpace)) R.carrier) a)
    (huni : ModelIsUnimodular R)
    {W : Type} [NormedAddCommGroup W] [NormedSpace ℝ W] [FiniteDimensional ℝ W]
    (q : HeightSpace T.character ≃ₗ[ℝ] W)
    (Exp : GroupLieAlgebra (𝓘(ℝ, R.modelSpace)) R.carrier → R.carrier)
    (hExp : IsGroupExponential Exp) (π : R.carrier →* Multiplicative W)
    (hπExp : ∀ x, (π (Exp x)).toAdd = q (heightProjection T.character x)) :
    letI := d.compatibleMetric
…
```
Mentions: OAI.PolycyclicRecognition.SimplyConnectedSolvableLieModel, OAI.PolycyclicRecognition.ProperChainMetric, OAI.PolycyclicRecognition.TriangularAdjoint, OAI.PolycyclicRecognition.ModelIsUnimodular, OAI.PolycyclicRecognition.HeightSpace, OAI.PolycyclicRecognition.heightProjection
theorem OAI.PolycyclicRecognition.lattice_recognition Challenge statement: Quasi-isometric recognition of virtually polycyclic groups sorry line 236
Source documentation
Finite-covolume lattice recognition, allowing a different ambient Lie group.
L4DEL explanation
The formalization proves quasi-isometric recognition of virtually polycyclic groups: every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is itself virtually polycyclic. It also realizes a finite-index subgroup of the recognized group as a uniform lattice in a simply connected solvable Lie group, which may differ from the original ambient group. The linked structural estimate controls the height-coordinate behavior of self quasi-isometries of the stated unimodular solvable Lie models, up to bounded error and one of finitely many linear height symmetries.
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.GroupTheory.PolycyclicRecognition.Recognition as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/GroupTheory/PolycyclicRecognition/Recognition.lean:63 (same statement text; different proof or body). Qualify references as OAI.PolycyclicRecognition.lattice_recognition@path.
```
theorem lattice_recognition
    (S : SimplyConnectedSolvableLieModel) (Λ : Subgroup S.carrier)
    (hΛ : IsFiniteCovolumeLattice Λ)
    {J : Type} [Group J] [Group.FG J] (e : GroupCoarseEquivalence Λ J) :
    ∃ Γ : Subgroup J, Γ.FiniteIndex ∧
      ∃ (R : SimplyConnectedSolvableLieModel) (Δ : Subgroup R.carrier),
        IsUniformLattice Δ ∧ Nonempty (Γ ≃* Δ) := by
…
```
Mentions: OAI.PolycyclicRecognition.SimplyConnectedSolvableLieModel, OAI.PolycyclicRecognition.IsFiniteCovolumeLattice, OAI.PolycyclicRecognition.GroupCoarseEquivalence