`ComparatorChallenges/ForestSpace.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

44 declarations, 1 contain sorry, 0 declared as axioms

def OAI.ForestSpace.Cofinite The closed finite-codimensional subspaces line 19
Source documentation
The closed finite-codimensional subspaces.
L4DEL explanation
The closed finite-codimensional subspaces.
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/Analysis/ForestSpace/Moduli.lean:17 (same declaration text).
```
def Cofinite : Type :=
…
```
def OAI.ForestSpace.aucValues The inner one-sided excess values and their infimum line 23
Source documentation
The inner one-sided excess values and their infimum.
L4DEL explanation
The inner one-sided excess values and their infimum.
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/Analysis/ForestSpace/Moduli.lean:24 (same declaration text).
```
def aucValues (N : E → ℝ) (t : ℝ) (x : E) (F : Cofinite E) : Set ℝ :=
…
```
Mentions: OAI.ForestSpace.Cofinite
def OAI.ForestSpace.aucInner line 26
Also declared: OAI/Analysis/ForestSpace/Moduli.lean:27 (same declaration text).
def OAI.ForestSpace.aucCenter line 29
Also declared: OAI/Analysis/ForestSpace/Moduli.lean:30 (same declaration text).
def OAI.ForestSpace.aucModulus The one-sided asymptotic uniform convexity modulus, with the order of infimum, supremum and infimum line 34
Source documentation
The one-sided asymptotic uniform convexity modulus, with the order of infimum, supremum and infimum.
L4DEL explanation
The one-sided asymptotic uniform convexity modulus, with the order of infimum, supremum and infimum.
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/Analysis/ForestSpace/Moduli.lean:35 (same declaration text).
```
noncomputable def aucModulus (N : E → ℝ) (t : ℝ) : ℝ :=
…
```
def OAI.ForestSpace.averageValues The inner averaged signed excess; it is not replaced by a maximum line 38
Source documentation
The inner averaged signed excess; it is not replaced by a maximum.
L4DEL explanation
The inner averaged signed excess; it is not replaced by a maximum.
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/Analysis/ForestSpace/Moduli.lean:39 (same declaration text).
```
def averageValues (N : E → ℝ) (t : ℝ) (x : E) (F : Cofinite E) : Set ℝ :=
…
```
Mentions: OAI.ForestSpace.Cofinite
def OAI.ForestSpace.averageInner line 42
Also declared: OAI/Analysis/ForestSpace/Moduli.lean:43 (same declaration text).
def OAI.ForestSpace.averageCenter line 45
Also declared: OAI/Analysis/ForestSpace/Moduli.lean:46 (same declaration text).
def OAI.ForestSpace.averageModulus The averaged asymptotic midpoint modulus line 49
Source documentation
The averaged asymptotic midpoint modulus.
L4DEL explanation
The averaged asymptotic midpoint modulus.
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/Analysis/ForestSpace/Moduli.lean:50 (same declaration text).
```
noncomputable def averageModulus (N : E → ℝ) (t : ℝ) : ℝ :=
…
```
structure OAI.ForestSpace.EquivalentNorm An equivalent norm is kept as its exact real seminorm, with explicit positive comparison constants line 56
Source documentation
An equivalent norm is kept as its exact real seminorm, with explicit positive comparison constants. The lower inequality makes it a norm.
L4DEL explanation
An equivalent norm is kept as its exact real seminorm, with explicit positive comparison constants. The lower inequality makes it a norm.
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/Analysis/ForestSpace/Moduli.lean:57 (same declaration text).
```
structure EquivalentNorm (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E]
    (α β : ℝ) where
…
```
def OAI.ForestSpace.Reflexive Canonical (not merely abstract) reflexivity line 65
Source documentation
Canonical (not merely abstract) reflexivity.
L4DEL explanation
Canonical (not merely abstract) reflexivity.
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/Analysis/ForestSpace/Moduli.lean:65 (same declaration text).
```
def Reflexive (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] : Prop :=
…
```
def OAI.ForestSpace.IsAUC AUC requires positivity at every strictly positive parameter line 69
Source documentation
AUC requires positivity at every strictly positive parameter.
L4DEL explanation
AUC requires positivity at every strictly positive parameter.
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/Analysis/ForestSpace/Moduli.lean:68 (same declaration text).
```
def IsAUC {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] (N : E → ℝ) : Prop :=
…
```
def OAI.ForestSpace.IsSegment A finite, order-convex chain line 73
Source documentation
A finite, order-convex chain. The empty chain is a harmless zero test.
L4DEL explanation
A finite, order-convex chain. The empty chain is a harmless zero test.
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/Analysis/ForestSpace/Segments.lean:18 (same declaration text).
```
def IsSegment {V : Type} [PartialOrder V] (S : Finset V) : Prop :=
…
```
def OAI.ForestSpace.segSum The sum of coordinates over a segment line 78
Source documentation
The sum of coordinates over a segment.
L4DEL explanation
The sum of coordinates over a segment.
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/Analysis/ForestSpace/Segments.lean:23 (same declaration text).
```
def segSum {V : Type} (u : V → ℝ) (S : Finset V) : ℝ := ∑ v ∈ S, u v
```
def OAI.ForestSpace.TestBound Squared bound on every disjoint family of segment tests line 81
Source documentation
Squared bound on every disjoint family of segment tests.
L4DEL explanation
Squared bound on every disjoint family of segment tests.
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/Analysis/ForestSpace/Segments.lean:26 (same declaration text).
```
def TestBound {V : Type} [PartialOrder V] (u : V → ℝ) (C : ℝ) : Prop :=
…
```
def OAI.ForestSpace.testValues All squared disjoint-segment tests line 93
Source documentation
All squared disjoint-segment tests. Quantifying over finite index types is equivalent to families S₁,...,Sₘ; a zero-sized family adds only zero to the set of test values.
L4DEL explanation
All squared disjoint-segment tests. Quantifying over finite index types is equivalent to families S₁,...,Sₘ; a zero-sized family adds only zero to the set of test values.
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/Analysis/ForestSpace/Norm.lean:19 (same declaration text).
```
def testValues (u : V → ℝ) : Set ℝ :=
…
```
def OAI.ForestSpace.squareSup line 98
Also declared: OAI/Analysis/ForestSpace/Norm.lean:24 (same declaration text).
def OAI.ForestSpace.rho Exactly the disjoint-segment norm, written as the square root of its squared testing supremum line 102
Source documentation
Exactly the disjoint-segment norm, written as the square root of its squared testing supremum.
L4DEL explanation
Exactly the disjoint-segment norm, written as the square root of its squared testing supremum.
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/Analysis/ForestSpace/Norm.lean:28 (same declaration text).
```
noncomputable def rho (u : V → ℝ) : ℝ := Real.sqrt (squareSup u)
```
Mentions: OAI.ForestSpace.squareSup
def OAI.ForestSpace.l1size line 104
Distinct definitions of this name: OAI/Analysis/ForestSpace/Norm.lean:106 (same statement text; different proof or body). Qualify references as OAI.ForestSpace.l1size@path.
def OAI.ForestSpace.segmentSeminorm The disjoint-segment seminorm on finite coordinate vectors line 109
Source documentation
The disjoint-segment seminorm on finite coordinate vectors.
L4DEL explanation
The disjoint-segment seminorm on finite coordinate vectors.
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/Analysis/ForestSpace/Completion.lean:17 (same declaration text).
```
def segmentSeminorm (V : Type) [PartialOrder V] : Seminorm ℝ (V →₀ ℝ) := by
…
```
def OAI.ForestSpace.TestVector Finite coordinate vectors with the segment norm (a type synonym prevents clashes with any other norm on finitely supported functions) line 226
Source documentation
Finite coordinate vectors with the segment norm (a type synonym prevents clashes with any other norm on finitely supported functions).
L4DEL explanation
Finite coordinate vectors with the segment norm (a type synonym prevents clashes with any other norm on finitely supported functions).
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/Analysis/ForestSpace/Completion.lean:15 (same declaration text).
```
def TestVector (V : Type) := V →₀ ℝ
…
```
instance OAI.ForestSpace.TestVector.(instance at line 232) line 232
instance OAI.ForestSpace.TestVector.(instance at line 234) line 234
def OAI.ForestSpace.TestVector.toFinsupp line 236
Also declared: OAI/Analysis/ForestSpace/Completion.lean:136 (same declaration text).
instance OAI.ForestSpace.TestVector.(instance at line 238) line 238
instance OAI.ForestSpace.TestVector.(instance at line 242) line 242
instance OAI.ForestSpace.TestVector.(instance at line 244) line 244
instance OAI.ForestSpace.TestVector.(instance at line 326) line 326
abbrev OAI.ForestSpace.J J(T), completion of the finite-support disjoint-segment norm line 332
Source documentation
J(T), completion of the finite-support disjoint-segment norm.
L4DEL explanation
J(T), completion of the finite-support disjoint-segment norm.
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/Analysis/ForestSpace/Completion.lean:385 (same declaration text).
```
abbrev J (V : Type) [PartialOrder V] := UniformSpace.Completion (TestVector V)
```
Mentions: OAI.ForestSpace.TestVector
abbrev OAI.ForestSpace.FullDual The full continuous dual, used for the finite-height forest example line 335
Source documentation
The full continuous dual, used for the finite-height forest example.
L4DEL explanation
The full continuous dual, used for the finite-height forest example.
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/Analysis/ForestSpace/Completion.lean:388 (same declaration text).
```
abbrev FullDual (V : Type) [PartialOrder V] := StrongDual ℝ (J V)
…
```
Mentions: OAI.ForestSpace.J
instance OAI.ForestSpace.fullDualAddCommGroup line 341
Distinct definitions of this name: OAI/Analysis/ForestSpace/Polar.lean:169 (same statement text; different proof or body). Qualify references as OAI.ForestSpace.fullDualAddCommGroup@path.
structure OAI.ForestSpace.Word Finite words, ordered by extension, including the empty root line 347
Source documentation
Finite words, ordered by extension, including the empty root.
L4DEL explanation
Finite words, ordered by extension, including the empty root.
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/Analysis/ForestSpace/Trees.lean:16 (same declaration text).
```
structure Word where
…
```
instance OAI.ForestSpace.Word.(instance at line 353) line 353
structure OAI.ForestSpace.Vertex A disjoint forest: one copy of Nat^{≤h} for every h≥1 line 364
Source documentation
A disjoint forest: one copy of Nat^{≤h} for every h≥1.
L4DEL explanation
A disjoint forest: one copy of Nat^{≤h} for every h≥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/Analysis/ForestSpace/Trees.lean:118 (same declaration text).
```
structure Vertex where
…
```
instance OAI.ForestSpace.Vertex.(instance at line 373) line 373
structure OAI.ForestSpace.Diamond.Network An oriented edge enumeration; only the associated undirected unit-edge simple graph is used for distances line 388
Source documentation
An oriented edge enumeration; only the associated undirected unit-edge simple graph is used for distances.
L4DEL explanation
An oriented edge enumeration; only the associated undirected unit-edge simple graph is used for distances.
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/Analysis/ForestSpace/Diamonds.lean:18 (same declaration text).
```
structure Network where
…
```
def OAI.ForestSpace.Diamond.Network.graph line 399
Also declared: OAI/Analysis/ForestSpace/Diamonds.lean:28 (same declaration text).
def OAI.ForestSpace.Diamond.Network.refine Every edge is replaced by countably many internally disjoint two-edge paths line 413
Source documentation
Every edge is replaced by countably many internally disjoint two-edge paths. All old vertices remain, and each pair (edge,index) is a new midpoint.
L4DEL explanation
Every edge is replaced by countably many internally disjoint two-edge paths. All old vertices remain, and each pair (edge,index) is a new midpoint.
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/Analysis/ForestSpace/Diamonds.lean:45 (same declaration text).
```
@[reducible] def refine : Network where
…
```
Mentions: OAI.ForestSpace.Diamond.Network
def OAI.ForestSpace.Diamond.Network.iterate Iterating the very same edge replacement, starting from an arbitrary network line 422
Source documentation
Iterating the very same edge replacement, starting from an arbitrary network. This recursion groups the first replacement before the remaining ones.
L4DEL explanation
Iterating the very same edge replacement, starting from an arbitrary network. This recursion groups the first replacement before the remaining ones.
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/Analysis/ForestSpace/Diamonds.lean:186 (same declaration text).
```
@[reducible] def iterate : Network → ℕ → Network
  | A,0 => A
  | A,n+1 => iterate A.refine n

end Network

/-- D₀ is precisely one edge. -/
```
Mentions: OAI.ForestSpace.Diamond.Network
def OAI.ForestSpace.Diamond.base D₀ is precisely one edge line 429
Source documentation
D₀ is precisely one edge.
L4DEL explanation
D₀ is precisely one edge.
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/Analysis/ForestSpace/Diamonds.lean:234 (same declaration text).
```
@[reducible] def base : Network where
…
```
Mentions: OAI.ForestSpace.Diamond.Network
def OAI.ForestSpace.Diamond.diamond Countably branching diamonds, all current edges of length one line 437
Source documentation
Countably branching diamonds, all current edges of length one.
L4DEL explanation
Countably branching diamonds, all current edges of length one.
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/Analysis/ForestSpace/Diamonds.lean:251 (same declaration text).
```
@[reducible] def diamond (k : ℕ) := base.iterate k
```
def OAI.ForestSpace.Diamond.distance The shortest-path distance, via Mathlib's actual graph distance line 440
Source documentation
The shortest-path distance, via Mathlib's actual graph distance.
L4DEL explanation
The shortest-path distance, via Mathlib's actual graph distance.
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/Analysis/ForestSpace/Diamonds.lean:254 (same statement text; different proof or body). Qualify references as OAI.ForestSpace.Diamond.distance@path.
```
def distance (k : ℕ) (u v : (diamond k).V) : ℝ := ((diamond k).graph.dist u v : ℝ)
…
```
Mentions: OAI.ForestSpace.Diamond.diamond
def OAI.ForestSpace.Diamond.EmbeddingBound The distortion convention, including its arbitrary positive scale line 444
Source documentation
The distortion convention, including its arbitrary positive scale.
L4DEL explanation
The distortion convention, including its arbitrary positive scale.
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/Analysis/ForestSpace/Distortion.lean:139 (same declaration text).
```
def EmbeddingBound (k : ℕ) (f : (diamond k).V → E) (C : ℝ) : Prop :=
…
```
Mentions: OAI.ForestSpace.Diamond.diamond
theorem OAI.ForestSpace.main_theorem Challenge statement: Reflexive tree space, midpoint modulus, and diamond distortion sorry line 453
Source documentation
The space is the full continuous dual of the specified segment-norm completion, not the coordinate completion in place of that dual.
L4DEL explanation
The space is the full continuous dual of the specified segment-norm completion, not the coordinate completion in place of that dual.
1. This explanation is a condensed rendering of this declaration's own doc comment at the pinned source commit.
2. The linked formalization note covers 7 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
3. The repository's comparator configuration selects OAI.Analysis.ForestSpace.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/ForestSpace/Main.lean:16 (same statement text; different proof or body). Qualify references as OAI.ForestSpace.main_theorem@path.
```
theorem main_theorem :
    (¬FiniteDimensional ℝ (FullDual Vertex)) ∧
    TopologicalSpace.SeparableSpace (FullDual Vertex) ∧
    Reflexive (FullDual Vertex) ∧
    (∀ (α β : ℝ) (R : EquivalentNorm (FullDual Vertex) α β), ¬IsAUC R.N) ∧
    (∀ t : ℝ, 0 < t →
      Real.sqrt (1+t^2/12)-1 ≤ averageModulus (FullDual Vertex) norm t) ∧
    (∀ (k : ℕ) (f : (Diamond.diamond k).V → FullDual Vertex) (C : ℝ),
      Diamond.EmbeddingBound k f C → Real.sqrt (1+(k:ℝ)/12) ≤ C) := by
…
```
Mentions: OAI.ForestSpace.FullDual, OAI.ForestSpace.Vertex, OAI.ForestSpace.Reflexive, OAI.ForestSpace.EquivalentNorm, OAI.ForestSpace.IsAUC, OAI.ForestSpace.averageModulus, OAI.ForestSpace.Diamond.diamond, OAI.ForestSpace.Diamond.EmbeddingBound