`ComparatorChallenges/WLIdentification.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

33 declarations, 1 contain sorry, 0 declared as axioms

structure OAI.WLIdentification.Graph line 7
Also declared: OAI/Computability/WLIdentification/Model.lean:7 (same declaration text).
def OAI.WLIdentification.Isomorphic line 13
Also declared: OAI/Computability/WLIdentification/Model.lean:13 (same declaration text).
def OAI.WLIdentification.NeighborColor Colors retain their full history, and hence are shared between graphs line 17
Source documentation
Colors retain their full history, and hence are shared between graphs.
L4DEL explanation
Colors retain their full history, and hence are shared between graphs.
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/Computability/WLIdentification/Model.lean:17 (same declaration text).
```
def NeighborColor : ℕ → Type
  | 0 => Unit
  | s + 1 => NeighborColor s × Multiset (NeighborColor s)
```
def OAI.WLIdentification.neighborColor line 21
Also declared: OAI/Computability/WLIdentification/Model.lean:21 (same declaration text).
def OAI.WLIdentification.neighborHistogram line 26
Also declared: OAI/Computability/WLIdentification/Model.lean:26 (same declaration text).
def OAI.WLIdentification.TupleColor line 29
Also declared: OAI/Computability/WLIdentification/Model.lean:29 (same declaration text).
def OAI.WLIdentification.tupleColor Joint update: one multiset of whole replacement vectors, not independent marginals line 34
Source documentation
Joint update: one multiset of whole replacement vectors, not independent marginals.
L4DEL explanation
Joint update: one multiset of whole replacement vectors, not independent marginals.
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/Computability/WLIdentification/Model.lean:34 (same declaration text).
```
def tupleColor (G : Graph) (k : ℕ) : (s : ℕ) → (Fin k → Fin G.order) → TupleColor k s
  | 0, v => (fun i j => decide (v i = v j), fun i j => G.edge (v i) (v j))
  | s + 1, v => (tupleColor G k s v,
      Finset.univ.val.map (fun z => fun i => tupleColor G k s (Function.update v i z)))
```
Mentions: OAI.WLIdentification.Graph, OAI.WLIdentification.TupleColor
def OAI.WLIdentification.tupleHistogram line 39
Also declared: OAI/Computability/WLIdentification/Model.lean:39 (same declaration text).
def OAI.WLIdentification.Equivalent line 42
Also declared: OAI/Computability/WLIdentification/Model.lean:42 (same declaration text).
def OAI.WLIdentification.Identifies There is no restriction to mates from a graph construction or to a selected order line 47
Source documentation
There is no restriction to mates from a graph construction or to a selected order.
L4DEL explanation
There is no restriction to mates from a graph construction or to a selected 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/Computability/WLIdentification/Model.lean:47 (same declaration text).
```
def Identifies (k : ℕ) (G : Graph) : Prop :=
…
```
Mentions: OAI.WLIdentification.Graph
def OAI.WLIdentification.WLdim The infimum of the positive identifying dimensions; this set is nonempty for finite G line 51
Source documentation
The infimum of the positive identifying dimensions; this set is nonempty for finite G.
L4DEL explanation
The infimum of the positive identifying dimensions; this set is nonempty for finite G.
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/Computability/WLIdentification/Model.lean:51 (same declaration text).
```
noncomputable def WLdim (G : Graph) : ℕ :=
…
```
Mentions: OAI.WLIdentification.Graph
abbrev OAI.WLIdentification.Word line 54
Also declared: OAI/Computability/WLIdentification/Model.lean:54 (same declaration text).
abbrev OAI.WLIdentification.BinaryLanguage line 55
Also declared: OAI/Computability/WLIdentification/Model.lean:55 (same declaration text).
def OAI.WLIdentification.binaryValue line 57
Also declared: OAI/Computability/WLIdentification/Model.lean:57 (same declaration text).
def OAI.WLIdentification.matrixWord line 60
Also declared: OAI/Computability/WLIdentification/Model.lean:60 (same declaration text).
def OAI.WLIdentification.inputWord Unary order header, row-major binary matrix, then most-significant-first binary k line 64
Source documentation
Unary order header, row-major binary matrix, then most-significant-first binary k.
L4DEL explanation
Unary order header, row-major binary matrix, then most-significant-first binary k.
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/Computability/WLIdentification/Model.lean:64 (same declaration text).
```
def inputWord (G : Graph) (bits : Word) : Word :=
…
```
Mentions: OAI.WLIdentification.Graph, OAI.WLIdentification.Word
def OAI.WLIdentification.dimensionLanguage A malformed word is rejected: membership requires a valid nonempty simple matrix and k>0 line 68
Source documentation
A malformed word is rejected: membership requires a valid nonempty simple matrix and k>0.
L4DEL explanation
A malformed word is rejected: membership requires a valid nonempty simple matrix and k>0.
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/Computability/WLIdentification/Model.lean:68 (same declaration text).
```
def dimensionLanguage : BinaryLanguage :=
…
```
Mentions: OAI.WLIdentification.BinaryLanguage
inductive OAI.WLIdentification.Move line 76
Source comment
The deterministic local machine model is explicit; no oracle or arbitrary state-transition on whole inputs is included. Both alphabets are fixed finite types for each machine. Extra symbols and states allow arbitrary finite tables.

Also declared: OAI/Computability/WLIdentification/Model.lean:76 (same declaration text).
```
inductive Move where
…
```
def OAI.WLIdentification.Move.displacement line 80
Also declared: OAI/Computability/WLIdentification/Model.lean:80 (same declaration text).
structure OAI.WLIdentification.Machine line 85
Also declared: OAI/Computability/WLIdentification/Model.lean:85 (same declaration text).
structure OAI.WLIdentification.Configuration line 91
Also declared: OAI/Computability/WLIdentification/Model.lean:91 (same declaration text).
def OAI.WLIdentification.inputSymbol line 96
Also declared: OAI/Computability/WLIdentification/Model.lean:96 (same declaration text).
def OAI.WLIdentification.initial line 99
Also declared: OAI/Computability/WLIdentification/Model.lean:99 (same declaration text).
def OAI.WLIdentification.Halted line 104
Also declared: OAI/Computability/WLIdentification/Model.lean:104 (same declaration text).
def OAI.WLIdentification.step One read/write/head-movement transition line 108
Source documentation
One read/write/head-movement transition. A halted configuration is frozen.
L4DEL explanation
One read/write/head-movement transition. A halted configuration is frozen.
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/Computability/WLIdentification/Model.lean:108 (same declaration text).
```
def step (M : Machine) (c : Configuration M) : Configuration M :=
…
```
Mentions: OAI.WLIdentification.Machine, OAI.WLIdentification.Configuration
def OAI.WLIdentification.run line 114
Also declared: OAI/Computability/WLIdentification/Model.lean:114 (same declaration text).
def OAI.WLIdentification.Output Output is the binary word beginning at origin and ending at its first blank line 118
Source documentation
Output is the binary word beginning at origin and ending at its first blank.
L4DEL explanation
Output is the binary word beginning at origin and ending at its first blank.
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/Computability/WLIdentification/Model.lean:118 (same declaration text).
```
def Output {M : Machine} (c : Configuration M) (w : Word) : Prop :=
…
```
Mentions: OAI.WLIdentification.Machine, OAI.WLIdentification.Configuration, OAI.WLIdentification.Word
def OAI.WLIdentification.DecidesWithin Uniform genuine executions on every bit word, including malformed graph words line 123
Source documentation
Uniform genuine executions on every bit word, including malformed graph words.
L4DEL explanation
Uniform genuine executions on every bit word, including malformed graph words.
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/Computability/WLIdentification/Model.lean:123 (same declaration text).
```
def DecidesWithin (M : Machine) (L : BinaryLanguage) (bound : ℕ → ℕ) : Prop :=
…
```
Mentions: OAI.WLIdentification.Machine, OAI.WLIdentification.BinaryLanguage
def OAI.WLIdentification.InEXPTIME line 126
Also declared: OAI/Computability/WLIdentification/Model.lean:126 (same declaration text).
def OAI.WLIdentification.ComputesWithin line 130
Also declared: OAI/Computability/WLIdentification/Model.lean:130 (same declaration text).
def OAI.WLIdentification.PolytimeManyOne line 133
Also declared: OAI/Computability/WLIdentification/Model.lean:133 (same declaration text).
def OAI.WLIdentification.EXPTIMEComplete line 137
Also declared: OAI/Computability/WLIdentification/Model.lean:137 (same declaration text).
theorem OAI.WLIdentification.main Challenge statement: EXPTIME-completeness of Weisfeiler–Leman graph identification sorry line 140
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 4 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
2. The repository's comparator configuration selects OAI.Computability.WLIdentification.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Computability/WLIdentification/Main.lean:7 (same statement text; different proof or body). Qualify references as OAI.WLIdentification.main@path.
```
theorem main : EXPTIMEComplete dimensionLanguage :=
…
```
Mentions: OAI.WLIdentification.EXPTIMEComplete, OAI.WLIdentification.dimensionLanguage