`ComparatorChallenges/DiamondDistortion.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

37 declarations, 1 contain sorry, 0 declared as axioms

structure OAI.DiamondDistortion.SegmentSystem line 15
Also declared: OAI/Analysis/DiamondDistortion/SegmentNorm.lean:13 (same declaration text).
structure OAI.DiamondDistortion.SegmentSystem.Family The represented disjoint families in the definition of the primal segment norm line 24
Source documentation
The represented disjoint families in the definition of the primal segment norm.
L4DEL explanation
The represented disjoint families in the definition of the primal 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/DiamondDistortion/SegmentNorm.lean:21 (same declaration text).
```
structure Family where
…
```
instance OAI.DiamondDistortion.SegmentSystem.(instance at line 30) line 30
def OAI.DiamondDistortion.SegmentSystem.Family.test Segment testing as a Euclidean-valued linear map line 39
Source documentation
Segment testing as a Euclidean-valued linear map.
L4DEL explanation
Segment testing as a Euclidean-valued linear map.
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/DiamondDistortion/SegmentNorm.lean:36 (same declaration text).
```
def Family.test (F : T.Family) : (ι →₀ ℝ) →ₗ[ℝ] EuclideanSpace ℝ (Fin F.n) where
…
```
def OAI.DiamondDistortion.SegmentSystem.Family.seminorm line 48
Distinct definitions of this name: OAI/Analysis/DiamondDistortion/SegmentNorm.lean:48 (same statement text; different proof or body). Qualify references as OAI.DiamondDistortion.SegmentSystem.Family.seminorm@path.
def OAI.DiamondDistortion.SegmentSystem.primalSeminorm line 53
Also declared: OAI/Analysis/DiamondDistortion/SegmentNorm.lean:91 (same declaration text).
def OAI.DiamondDistortion.SegmentSystem.singletonFamily Singleton families ensure that the segment seminorm separates points line 56
Source documentation
Singleton families ensure that the segment seminorm separates points.
L4DEL explanation
Singleton families ensure that the segment seminorm separates points.
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/DiamondDistortion/SegmentNorm.lean:108 (same declaration text).
```
def singletonFamily (v : ι) : T.Family where
…
```
def OAI.DiamondDistortion.SegmentSystem.Primal line 62
Also declared: OAI/Analysis/DiamondDistortion/SegmentNorm.lean:171 (same declaration text).
instance OAI.DiamondDistortion.SegmentSystem.(instance at line 64) line 64
instance OAI.DiamondDistortion.SegmentSystem.(instance at line 65) line 65
instance OAI.DiamondDistortion.SegmentSystem.(instance at line 67) line 67
instance OAI.DiamondDistortion.SegmentSystem.(instance at line 127) line 127
abbrev OAI.DiamondDistortion.SegmentSystem.JamesSpace line 130
Also declared: OAI/Analysis/DiamondDistortion/SegmentNorm.lean:272 (same declaration text).
abbrev OAI.DiamondDistortion.SegmentSystem.Dual Its full real continuous dual line 133
Source documentation
Its full real continuous dual.
L4DEL explanation
Its full real continuous dual.
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/DiamondDistortion/SegmentNorm.lean:275 (same declaration text).
```
abbrev Dual := StrongDual ℝ T.JamesSpace
```
def OAI.DiamondDistortion.SegmentSystem.coordinateFinite Coordinate evaluation on finite arrays, bounded by the singleton test line 136
Source documentation
Coordinate evaluation on finite arrays, bounded by the singleton test.
L4DEL explanation
Coordinate evaluation on finite arrays, bounded by the singleton 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/DiamondDistortion/SegmentNorm.lean:278 (same declaration text).
```
def coordinateFinite (v : ι) : T.Primal →L[ℝ] ℝ := by
…
```
def OAI.DiamondDistortion.SegmentSystem.coordinate line 189
Also declared: OAI/Analysis/DiamondDistortion/SegmentNorm.lean:331 (same declaration text).
def OAI.DiamondDistortion.SegmentSystem.unitFinite Primal unit coordinate and its canonical image in the completion line 193
Source documentation
Primal unit coordinate and its canonical image in the completion.
L4DEL explanation
Primal unit coordinate and its canonical image in the completion.
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/DiamondDistortion/SegmentNorm.lean:340 (same declaration text).
```
def unitFinite (v : ι) : T.Primal := Finsupp.single v 1
```
def OAI.DiamondDistortion.SegmentSystem.unitVector line 195
Also declared: OAI/Analysis/DiamondDistortion/SegmentNorm.lean:342 (same declaration text).
def OAI.DiamondDistortion.SegmentSystem.headDual line 197
Also declared: OAI/Analysis/DiamondDistortion/SegmentNorm.lean:394 (same declaration text).
def OAI.DiamondDistortion.SegmentSystem.CoordinatePredual The norm-closed coordinate span, not the full infinite-tree dual line 201
Source documentation
The norm-closed coordinate span, not the full infinite-tree dual.
L4DEL explanation
The norm-closed coordinate span, not the full infinite-tree dual.
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/DiamondDistortion/SegmentNorm.lean:1080 (same declaration text).
```
def CoordinatePredual : Submodule ℝ T.Dual :=
…
```
def OAI.DiamondDistortion.intervalSystem A finite interval in an ancestral order line 207
Source documentation
A finite interval in an ancestral order. This keeps the root in every tree.
L4DEL explanation
A finite interval in an ancestral order. This keeps the root in every tree.
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/DiamondDistortion/Ancestry.lean:14 (same declaration text).
```
def intervalSystem {ι : Type*} (r : ι → ι → Prop)
    (hrefl : ∀ a, r a a) (hanti : ∀ a b, r a b → r b a → a = b) : SegmentSystem ι where
…
```
Mentions: OAI.DiamondDistortion.SegmentSystem
class OAI.DiamondDistortion.Ancestry An ancestral partial order: the predecessors of any vertex form a chain line 221
Source documentation
An ancestral partial order: the predecessors of any vertex form a chain.
L4DEL explanation
An ancestral partial order: the predecessors of any vertex form a chain.
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/DiamondDistortion/Ancestry.lean:28 (same declaration text).
```
class Ancestry (ι : Type*) extends PartialOrder ι where
…
```
def OAI.DiamondDistortion.Ancestry.segments Nonempty finite intervals for an ancestral order line 231
Source documentation
Nonempty finite intervals for an ancestral order.
L4DEL explanation
Nonempty finite intervals for an ancestral order.
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/DiamondDistortion/Ancestry.lean:37 (same declaration text).
```
def segments : SegmentSystem ι :=
…
```
Mentions: OAI.DiamondDistortion.SegmentSystem
def OAI.DiamondDistortion.ForestNode line 236
Also declared: OAI/Analysis/DiamondDistortion/ForestModels.lean:13 (same declaration text).
def OAI.DiamondDistortion.ForestAncestor line 238
Also declared: OAI/Analysis/DiamondDistortion/ForestModels.lean:15 (same declaration text).
def OAI.DiamondDistortion.forestAncestry line 241
Also declared: OAI/Analysis/DiamondDistortion/ForestModels.lean:18 (same declaration text).
def OAI.DiamondDistortion.forestSegments line 254
Also declared: OAI/Analysis/DiamondDistortion/ForestModels.lean:31 (same declaration text).
def OAI.DiamondDistortion.infiniteAncestry Prefix order on all finite words, explicitly distinct from their lexicographic order line 258
Source documentation
Prefix order on all finite words, explicitly distinct from their lexicographic order.
L4DEL explanation
Prefix order on all finite words, explicitly distinct from their lexicographic order.
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/DiamondDistortion/ForestModels.lean:35 (same declaration text).
```
@[instance_reducible] def infiniteAncestry : Ancestry (List ℕ) where
…
```
Mentions: OAI.DiamondDistortion.Ancestry
def OAI.DiamondDistortion.infiniteSegments The infinite-height tree of all finite words; this includes the empty root line 267
Source documentation
The infinite-height tree of all finite words; this includes the empty root.
L4DEL explanation
The infinite-height tree of all finite words; this includes 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/DiamondDistortion/ForestModels.lean:44 (same declaration text).
```
def infiniteSegments : SegmentSystem (List ℕ) :=
…
```
Mentions: OAI.DiamondDistortion.SegmentSystem
abbrev OAI.DiamondDistortion.ForestDual line 270
Also declared: OAI/Analysis/DiamondDistortion/ForestModels.lean:271 (same declaration text).
abbrev OAI.DiamondDistortion.InfinitePredual line 271
Also declared: OAI/Analysis/DiamondDistortion/ForestModels.lean:272 (same declaration text).
def OAI.DiamondDistortion.Diamond.Edge Stage edges, retaining their parent, child number, and side line 276
Source documentation
Stage edges, retaining their parent, child number, and side.
L4DEL explanation
Stage edges, retaining their parent, child number, and side.
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/DiamondDistortion/DiamondGraph.lean:16 (same declaration text).
```
@[reducible] def Edge : ℕ → Type
  | 0 => Unit
  | n + 1 => Edge n × ℕ × Bool

/-- At every replacement every parent edge receives countably many distinct new vertices. -/
```
def OAI.DiamondDistortion.Diamond.Vertex At every replacement every parent edge receives countably many distinct new vertices line 281
Source documentation
At every replacement every parent edge receives countably many distinct new vertices.
L4DEL explanation
At every replacement every parent edge receives countably many distinct new vertices.
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/DiamondDistortion/DiamondGraph.lean:21 (same declaration text).
```
@[reducible] def Vertex : ℕ → Type
  | 0 => Bool
  | n + 1 => Vertex n ⊕ (Edge n × ℕ)
```
Mentions: OAI.DiamondDistortion.Diamond.Edge
def OAI.DiamondDistortion.Diamond.endpoint line 285
Also declared: OAI/Analysis/DiamondDistortion/DiamondGraph.lean:25 (same declaration text).
def OAI.DiamondDistortion.Diamond.graph The undirected replacement graph line 293
Source documentation
The undirected replacement graph.
L4DEL explanation
The undirected replacement 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/Analysis/DiamondDistortion/DiamondGraph.lean:33 (same declaration text).
```
def graph (n : ℕ) : SimpleGraph (Vertex n) :=
…
```
Mentions: OAI.DiamondDistortion.Diamond.Vertex
def OAI.DiamondDistortion.Diamond.distance Final edges have length 2⁻ᵏ: this is the shortest-path metric of the normalized graph line 297
Source documentation
Final edges have length 2⁻ᵏ: this is the shortest-path metric of the normalized graph.
L4DEL explanation
Final edges have length 2⁻ᵏ: this is the shortest-path metric of the normalized 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/Analysis/DiamondDistortion/DiamondGraph.lean:206 (same declaration text).
```
def distance (k : ℕ) (u v : Vertex k) : ℝ :=
…
```
Mentions: OAI.DiamondDistortion.Diamond.Vertex
theorem OAI.DiamondDistortion.headline_all_pairs Challenge statement: All-pairs diamond distortion bound sorry line 302
L4DEL explanation
This declaration is an exact comparator-selected statement. Read the Lean statement below for its mathematical content.
1. The linked formalization note covers 7 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
2. The repository's comparator configuration selects OAI.Analysis.DiamondDistortion.Descent as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/DiamondDistortion/Descent.lean:104 (same statement text; different proof or body). Qualify references as OAI.DiamondDistortion.headline_all_pairs@path.
```
theorem headline_all_pairs :
    (∀ (k : ℕ) (f : Diamond.Vertex k → ForestDual) (D s : ℝ),
      0 < s → (∀ u v,s*Diamond.distance k u v ≤ ‖f u-f v‖ ∧
        ‖f u-f v‖ ≤ D*s*Diamond.distance k u v) → 1+(k:ℝ)/4 ≤ D^2) ∧
    (∀ (k : ℕ) (f : Diamond.Vertex k → InfinitePredual) (D s : ℝ),
      0 < s → (∀ u v,s*Diamond.distance k u v ≤ ‖f u-f v‖ ∧
        ‖f u-f v‖ ≤ D*s*Diamond.distance k u v) → 1+(k:ℝ)/4 ≤ D^2) := by
…
```
Mentions: OAI.DiamondDistortion.Diamond.Vertex, OAI.DiamondDistortion.ForestDual, OAI.DiamondDistortion.Diamond.distance, OAI.DiamondDistortion.InfinitePredual