Lean statement

Parity-lift characterization of Weisfeiler–Leman equivalence

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorem selected for verification: OAI.ParityWL.parity_reduction

Lean statement file: ParityLifts.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 133 link it.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

noncomputable def ParityReductionStatement : Prop :=  ∀ (k : ℕ), 4 ≤ k → ∀ c : Convention, ∀ C : ChoiceSystem (bagParameter c k),    let T := template (bagParameter c k)    let G := baseGraph C    let τ := baseProjection C    let V₀ := LiftVertex T G τ (fun _ => 0)    let V₁ := LiftVertex T G τ (starParity (bagParameter c k))    Fintype.card V₀ = Fintype.card V₁ ∧    (∀ A : ℕ, 1 ≤ A → C.SizeAtMost A → Fintype.card V₀ ≤ A * orderFactor (bagParameter c k)) ∧    (WLEquivalent (zeroLift C) (starLift C) c k ↔ ¬ C.Successful) ∧    (PureWLEquivalent (zeroLift C) (starLift C) c k ↔ ¬ C.Successful) ∧    (C.Successful → ¬ sameHistogram (zeroLift C) (starLift C) c k (detectionRound c) ∧      ¬ samePureHistogram (zeroLift C) (starLift C) c k (detectionRound c))⋮theorem parity_reduction : ParityReductionStatement := by  sorry

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

abbreviation OAI.ParityWL.Bit line 11
```
abbrev Bit := ZMod 2
```

abbreviation OAI.ParityWL.MainPair line 12

abbrev MainPair (t : ℕ) := {p : Fin t × Fin t // p.1 < p.2}

abbreviation OAI.ParityWL.Helper line 13

abbrev Helper (t : ℕ) :=  {q : MainPair t × Fin t // q.2 ≠ q.1.1.1 ∧ q.2 ≠ q.1.1.2}

abbreviation OAI.ParityWL.TemplateVertex line 15

abbrev TemplateVertex (t : ℕ) := Fin t ⊕ Helper t

structure OAI.ParityWL.ChoiceSystem line 17

structure ChoiceSystem (t : ℕ) where  Domain : Fin t → Type  domainFintype : ∀ i, Fintype (Domain i)  Label : MainPair t → Type  labelFintype : ∀ p, Fintype (Label p)  leftLabel : (p : MainPair t) → Domain p.1.1 → Label p  rightLabel : (p : MainPair t) → Domain p.1.2 → Label p

definition OAI.ParityWL.ChoiceSystem.Successful line 27

def ChoiceSystem.Successful {t : ℕ} (C : ChoiceSystem t) : Prop :=  ∃ d : (i : Fin t) → C.Domain i,    ∀ p : MainPair t, C.leftLabel p (d p.1.1) = C.rightLabel p (d p.1.2)

definition OAI.ParityWL.ChoiceSystem.SizeAtMost line 31

def ChoiceSystem.SizeAtMost {t : ℕ} (C : ChoiceSystem t) (A : ℕ) : Prop :=  (∀ i, Fintype.card (C.Domain i) ≤ A) ∧ (∀ p, Fintype.card (C.Label p) ≤ A)

definition OAI.ParityWL.helperIncident line 34

def helperIncident {t : ℕ} (h : Helper t) (i : Fin t) : Prop :=  i = h.1.1.1.1 ∨ i = h.1.1.1.2 ∨ i = h.1.2

definition OAI.ParityWL.templateAdj line 37

def templateAdj {t : ℕ} : TemplateVertex t → TemplateVertex t → Prop  | .inl i, .inl j => i ≠ j  | .inl i, .inr h => helperIncident h i  | .inr h, .inl i => helperIncident h i  | .inr _, .inr _ => False

definition OAI.ParityWL.template line 43

def template (t : ℕ) : SimpleGraph (TemplateVertex t) where  Adj := templateAdj  symm := ⟨by    intro x y h    cases x <;> cases y <;> simp_all [templateAdj, ne_comm]⟩  loopless := ⟨by    intro x    cases x <;> simp [templateAdj]⟩

abbreviation OAI.ParityWL.BaseVertex line 52

abbrev BaseVertex {t : ℕ} (C : ChoiceSystem t) :=  ((i : Fin t) × C.Domain i) ⊕ ((h : Helper t) × C.Label h.1.1)

definition OAI.ParityWL.baseType line 55

def baseType {t : ℕ} {C : ChoiceSystem t} : BaseVertex C → TemplateVertex t  | .inl x => .inl x.1  | .inr x => .inr x.1

inductive type OAI.ParityWL.BaseArc line 59

inductive BaseArc {t : ℕ} (C : ChoiceSystem t) : BaseVertex C → BaseVertex C → Prop  | main (p : MainPair t) (d : C.Domain p.1.1) (e : C.Domain p.1.2)      (h : C.leftLabel p d = C.rightLabel p e) :      BaseArc C (.inl ⟨p.1.1, d⟩) (.inl ⟨p.1.2, e⟩)  | left (h : Helper t) (d : C.Domain h.1.1.1.1) :      BaseArc C (.inr ⟨h, C.leftLabel h.1.1 d⟩) (.inl ⟨h.1.1.1.1, d⟩)  | right (h : Helper t) (e : C.Domain h.1.1.1.2) :      BaseArc C (.inr ⟨h, C.rightLabel h.1.1 e⟩) (.inl ⟨h.1.1.1.2, e⟩)  | third (h : Helper t) (q : C.Label h.1.1) (f : C.Domain h.1.2) :      BaseArc C (.inr ⟨h, q⟩) (.inl ⟨h.1.2, f⟩)

definition OAI.ParityWL.baseGraph line 70

def baseGraph {t : ℕ} (C : ChoiceSystem t) : SimpleGraph (BaseVertex C) :=  SimpleGraph.fromRel (BaseArc C)

definition OAI.ParityWL.baseProjection line 73

def baseProjection {t : ℕ} (C : ChoiceSystem t) : baseGraph C →g template t where

Show the remaining 13 lines

  toFun := baseType  map_rel' := by    intro x y h    have arc : ∀ u v, BaseArc C u v → (template t).Adj (baseType u) (baseType v) := by      intro u v huv      cases huv with      | main p d e he => exact ne_of_lt p.property      | left h d => exact Or.inl rfl      | right h e => exact Or.inr (Or.inl rfl)      | third h q f => exact Or.inr (Or.inr rfl)    rcases h.2 with hxy | hyx    · exact arc x y hxy    · exact (arc y x hyx).symm

definition OAI.ParityWL.Tag line 92

noncomputable def Tag (b : A → Bit) (a : A) := by  classical  exact {z : T.neighborSet a → Bit // ∑ v, z v = b a}

abbreviation OAI.ParityWL.LiftVertex line 101

abbrev LiftVertex (τ : G →g T) (b : A → Bit) := (u : V) × Tag T b (τ u)

definition OAI.ParityWL.liftGraph line 103

noncomputable def liftGraph (τ : G →g T) (b : A → Bit) :    SimpleGraph (LiftVertex T G τ b) where

Show the remaining 9 lines

  Adj u v := ∃ h : G.Adj u.1 v.1,    u.2.1 ⟨τ v.1, τ.map_adj h⟩ = v.2.1 ⟨τ u.1, τ.map_adj h.symm⟩  symm := ⟨by    intro u v h    rcases h with ⟨h, e⟩    exact ⟨h.symm, e.symm⟩⟩  loopless := ⟨by    intro u h    exact G.irrefl h.1⟩

definition OAI.ParityWL.starParity line 117

noncomputable def starParity (t : ℕ) : TemplateVertex t → Bit :=  fun a => if ∃ i : Fin t, i.val = 0 ∧ a = .inl i then 1 else 0

definition OAI.ParityWL.zeroLift line 120

noncomputable def zeroLift {t : ℕ} (C : ChoiceSystem t) :=  liftGraph (template t) (baseGraph C) (baseProjection C) (fun _ => 0)

definition OAI.ParityWL.starLift line 123

noncomputable def starLift {t : ℕ} (C : ChoiceSystem t) :=  liftGraph (template t) (baseGraph C) (baseProjection C) (starParity t)

inductive type OAI.ParityWL.Convention line 126

inductive Convention | joint | separate  deriving DecidableEq

definition OAI.ParityWL.bagParameter line 129

def bagParameter (c : Convention) (k : ℕ) : ℕ :=  match c with | .joint => k + 1 | .separate => k

definition OAI.ParityWL.detectionRound line 132

def detectionRound : Convention → ℕ | .joint => 2 | .separate => 1

11 more definitions are reached the same way; the full file lists them.

Possible paper match, with no recorded review

Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

Words of the Lean statement, its names split into words, that also appear in the theorem: star, histograms, rounds, bag, successful, parity, pure, system and 13 more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: equal, construct, two and uncolored.

Statements in Parity lifts and bounded-treewidth witnesses for Weisfeiler–Leman equivalence come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.2 of Parity lifts and bounded-treewidth witnesses for Weisfeiler–Leman equivalence

Theorem (Parity reduction)

Fix k\ge4, either refinement convention, and the corresponding t in (eq:bag-parameter). From every choice system on t domains one can explicitly construct two uncolored simple graphs X_0,X_\star of equal order such that

X_0\equiv_k X_\star \quad\Longleftrightarrow\quad \text{the choice system has no successful choice}.

The assertion holds both with common color names on separate graphs and with pure-tuple comparison in their disjoint union.

If every domain and label set has size at most A, where A\ge1, their common order is at most A M_t, where

M_t=t\,2^{\,t-2+3\binom{t-1}{2}}+12\binom t3.

When a successful choice exists, the two graphs already have different histograms after at most two joint rounds, or one separate round.

Section 1, Introduction Read it in the paper TeX source, line 76

Show the TeX

Fix \(k\ge4\), either refinement convention, and the corresponding
\(t\) in \eqref{eq:bag-parameter}. From every choice system on \(t\)
domains one can explicitly construct two uncolored simple graphs
\(X_0,X_\star\) of equal order such that
\[
 X_0\equiv_k X_\star
 \quad\Longleftrightarrow\quad
 \text{the choice system has no successful choice}.
\]
The assertion holds both with common color names on separate graphs
and with pure-tuple comparison in their disjoint union.

If every domain and label set has size at most \(A\), where \(A\ge1\),
their common order is at most \(A M_t\), where
\[
 M_t=t\,2^{\,t-2+3\binom{t-1}{2}}+12\binom t3.
\]
When a successful choice exists, the two graphs already have different
histograms after at most two joint rounds, or one separate round.

Other possible paper matches, closest wording first

Theorem 1.3 of Parity lifts and bounded-treewidth witnesses for Weisfeiler–Leman equivalence. Words of the Lean statement, its names split into words, that also appear in the theorem: decides, separate, vertex, joint, wl, integer and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: equivalence, two, uncolored and dimension.
Theorem 1.1 of Unconditional time lower bounds for Weisfeiler–Leman equivalence. Words of the Lean statement, its names split into words, that also appear in the theorem: multiset, separate, pairs, convention, joint, property and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: leman, weisfeiler, equivalence, two and uncolored.
Theorem 1.1 of The complexity of identifying a graph by Weisfeiler–Leman refinement. Words of the Lean statement, its names split into words, that also appear in the theorem: neighbor, color, update, joint, finite, reductions and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: two, uncolored and dimension.
Theorem 3.2 of Unconditional time lower bounds for Weisfeiler–Leman equivalence. Words of the Lean statement, its names split into words, that also appear in the theorem: arc, pair, reduction, order, simple and graphs. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: equal, equivalence, construction, two and uncolored.
Proposition 5.1 of Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: zero, base, vertex, maps, order, simple and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: uncolored.
Theorem 4.3 of Unconditional time lower bounds for Weisfeiler–Leman equivalence. Words of the Lean statement, its names split into words, that also appear in the theorem: update, joint, property, wl, reduction, integer and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constructed, construction, two, uncolored and dimensions.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.

Carlos Toledo’s cert-machine

How they ran it, in their words

Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory; the run was still in progress at this snapshot (192 accepted, 224 pending).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 45 s; peak memory 6.1 GiB).
Reported 10 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 133’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: None; all three headline claims are stated. The conditional running-time exclusions are not (docs).
Definitions the group asks readers to compare with the paper: Hand-rolled computation models: TM (multi-tape, fixed finite tables, halting states 1/2), a RAM with logarithmic word width (WordWidth.logarithmic), memory Word w → Word w, compute (op : WordOperation R) where a whole word operation (any map implemented by a TM within constant * (w+1)^exponent) counts as one instruction and one step runs a finite instruction tree; InEXPTIME (2^(c (n+1)^d) steps), PolytimeReduces/PolytimeManyOne, EXPTIMEComplete; WL colors are canonical nested multisets (Color c k t) with Equivalent := ∀ t, histogram G t = histogram H t
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization gives the paper's parity-lift graph construction for every Weisfeiler–Leman dimension $k\ge4$, under both joint and separate conventions. From a finite choice system it constructs two equal-size uncolored graphs that are Weisfeiler–Leman equivalent exactly when the system has no successful compatible choice. The same equivalence holds for the pure variant.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

209 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 127 to its own declarations, 90 to Mathlib’s documentation. 38 declared names also link to the same declaration in OpenAI’s library. 2 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names the scan of the pinned sources did not find. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves parity_reduction exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches 36 of the file’s 37 other declarations; with the theorem itself they span 149 lines. 1 of them is named in the statement itself.

Names in the namespace OAI.ParityWL are listed without it.

abbrev Bitline 11library
abbrev MainPairline 12library
abbrev Helperline 13library
abbrev TemplateVertexline 15library
structure ChoiceSystemline 17library
def ChoiceSystem.Successfulline 27library
def ChoiceSystem.SizeAtMostline 31library
def helperIncidentline 34library
def templateAdjline 37library
def templateline 43library
abbrev BaseVertexline 52library
def baseTypeline 55library
inductive BaseArcline 59library
def baseGraphline 70library
def baseProjectionline 73library
def Tagline 92library
abbrev LiftVertexline 101library
def liftGraphline 103library
def starParityline 117library
def zeroLiftline 120library
def starLiftline 123library
inductive Conventionline 126library
def bagParameterline 129library
def detectionRoundline 132library
def orderFactorline 134library
abbrev AtomicTypeline 137library
def JointColorline 139library
def SeparateColorline 143library
def atomicTypeline 147library
def jointColorline 152library
def separateColorline 158library
def sameHistogramline 164library
def samePureHistogramline 172library
def WLEquivalentline 182library
def PureWLEquivalentline 186library
def ParityReductionStatementline 190in the statementlibrary
theorem parity_reductionline 204selected theoremlibrary

Other Lean rules that may affect the statement 2

Lean can use instances, notation and attributes without naming them in the statement. These are rules for interpreting or finding mathematical objects. Atlas’s scan of names cannot follow all such uses; read the listed source as well.

instance tagFintype line 96
attribute [instance] ChoiceSystem.domainFintype ChoiceSystem.labelFint… line 25

import Mathlibnamespace OAInamespace ParityWLopen scoped BigOperatorsabbrev Bit := ZMod 2abbrev MainPair (t : ℕ) := {p : Fin t × Fin t // p.1 < p.2}abbrev Helper (t : ℕ) :=  {q : MainPair t × Fin t // q.2 ≠ q.1.1.1 ∧ q.2 ≠ q.1.1.2}abbrev TemplateVertex (t : ℕ) := Fin t ⊕ Helper tstructure ChoiceSystem (t : ℕ) where  Domain : Fin t → Type  domainFintype : ∀ i, Fintype (Domain i)  Label : MainPair t → Type  labelFintype : ∀ p, Fintype (Label p)  leftLabel : (p : MainPair t) → Domain p.1.1 → Label p  rightLabel : (p : MainPair t) → Domain p.1.2 → Label pattribute [instance] ChoiceSystem.domainFintype ChoiceSystem.labelFintypedef ChoiceSystem.Successful {t : ℕ} (C : ChoiceSystem t) : Prop :=  ∃ d : (i : Fin t) → C.Domain i,    ∀ p : MainPair t, C.leftLabel p (d p.1.1) = C.rightLabel p (d p.1.2)def ChoiceSystem.SizeAtMost {t : ℕ} (C : ChoiceSystem t) (A : ℕ) : Prop :=  (∀ i, Fintype.card (C.Domain i) ≤ A) ∧ (∀ p, Fintype.card (C.Label p) ≤ A)def helperIncident {t : ℕ} (h : Helper t) (i : Fin t) : Prop :=  i = h.1.1.1.1 ∨ i = h.1.1.1.2 ∨ i = h.1.2def templateAdj {t : ℕ} : TemplateVertex t → TemplateVertex t → Prop  | .inl i, .inl j => i ≠ j  | .inl i, .inr h => helperIncident h i  | .inr h, .inl i => helperIncident h i  | .inr _, .inr _ => Falsedef template (t : ℕ) : SimpleGraph (TemplateVertex t) where  Adj := templateAdj  symm := ⟨by    intro x y h    cases x <;> cases y <;> simp_all [templateAdj, ne_comm]⟩  loopless := ⟨by    intro x    cases x <;> simp [templateAdj]⟩abbrev BaseVertex {t : ℕ} (C : ChoiceSystem t) :=  ((i : Fin t) × C.Domain i) ⊕ ((h : Helper t) × C.Label h.1.1)def baseType {t : ℕ} {C : ChoiceSystem t} : BaseVertex C → TemplateVertex t  | .inl x => .inl x.1  | .inr x => .inr x.1inductive BaseArc {t : ℕ} (C : ChoiceSystem t) : BaseVertex C → BaseVertex C → Prop  | main (p : MainPair t) (d : C.Domain p.1.1) (e : C.Domain p.1.2)      (h : C.leftLabel p d = C.rightLabel p e) :      BaseArc C (.inl ⟨p.1.1, d⟩) (.inl ⟨p.1.2, e⟩)  | left (h : Helper t) (d : C.Domain h.1.1.1.1) :      BaseArc C (.inr ⟨h, C.leftLabel h.1.1 d⟩) (.inl ⟨h.1.1.1.1, d⟩)  | right (h : Helper t) (e : C.Domain h.1.1.1.2) :      BaseArc C (.inr ⟨h, C.rightLabel h.1.1 e⟩) (.inl ⟨h.1.1.1.2, e⟩)  | third (h : Helper t) (q : C.Label h.1.1) (f : C.Domain h.1.2) :      BaseArc C (.inr ⟨h, q⟩) (.inl ⟨h.1.2, f⟩)def baseGraph {t : ℕ} (C : ChoiceSystem t) : SimpleGraph (BaseVertex C) :=  SimpleGraph.fromRel (BaseArc C)def baseProjection {t : ℕ} (C : ChoiceSystem t) : baseGraph C →g template t where  toFun := baseType  map_rel' := by    intro x y h    have arc : ∀ u v, BaseArc C u v → (template t).Adj (baseType u) (baseType v) := by      intro u v huv      cases huv with      | main p d e he => exact ne_of_lt p.property      | left h d => exact Or.inl rfl      | right h e => exact Or.inr (Or.inl rfl)      | third h q f => exact Or.inr (Or.inr rfl)    rcases h.2 with hxy | hyx    · exact arc x y hxy    · exact (arc y x hyx).symmsection Liftsvariable {A V : Type} [Fintype A] (T : SimpleGraph A) (G : SimpleGraph V)noncomputable def Tag (b : A → Bit) (a : A) := by  classical  exact {z : T.neighborSet a → Bit // ∑ v, z v = b a}noncomputable instance tagFintype (b : A → Bit) (a : A) : Fintype (Tag T b a) := by  classical  unfold Tag  infer_instanceabbrev LiftVertex (τ : G →g T) (b : A → Bit) := (u : V) × Tag T b (τ u)noncomputable def liftGraph (τ : G →g T) (b : A → Bit) :    SimpleGraph (LiftVertex T G τ b) where  Adj u v := ∃ h : G.Adj u.1 v.1,    u.2.1 ⟨τ v.1, τ.map_adj h⟩ = v.2.1 ⟨τ u.1, τ.map_adj h.symm⟩  symm := ⟨by    intro u v h    rcases h with ⟨h, e⟩    exact ⟨h.symm, e.symm⟩⟩  loopless := ⟨by    intro u h    exact G.irrefl h.1⟩end Liftsnoncomputable def starParity (t : ℕ) : TemplateVertex t → Bit :=  fun a => if ∃ i : Fin t, i.val = 0 ∧ a = .inl i then 1 else 0noncomputable def zeroLift {t : ℕ} (C : ChoiceSystem t) :=  liftGraph (template t) (baseGraph C) (baseProjection C) (fun _ => 0)noncomputable def starLift {t : ℕ} (C : ChoiceSystem t) :=  liftGraph (template t) (baseGraph C) (baseProjection C) (starParity t)inductive Convention | joint | separate  deriving DecidableEqdef bagParameter (c : Convention) (k : ℕ) : ℕ :=  match c with | .joint => k + 1 | .separate => kdef detectionRound : Convention → ℕ | .joint => 2 | .separate => 1def orderFactor (t : ℕ) : ℕ :=  t * 2 ^ (t - 2 + 3 * (t - 1).choose 2) + 12 * t.choose 3abbrev AtomicType (k : ℕ) := Fin k → Fin k → Bool × Booldef JointColor (k : ℕ) : ℕ → Type  | 0 => AtomicType k  | r + 1 => JointColor k r × Multiset (Fin k → JointColor k r)def SeparateColor (k : ℕ) : ℕ → Type  | 0 => AtomicType k  | r + 1 => SeparateColor k r × (Fin k → Multiset (SeparateColor k r))noncomputable def atomicType {V : Type} (G : SimpleGraph V) {k : ℕ}    (x : Fin k → V) : AtomicType k := by  classical  exact fun i j => (decide (x i = x j), decide (G.Adj (x i) (x j)))noncomputable def jointColor {V : Type} [Fintype V] (G : SimpleGraph V) (k : ℕ) :    (r : ℕ) → (Fin k → V) → JointColor k r  | 0, x => atomicType G x  | r + 1, x => (jointColor G k r x,      Finset.univ.val.map (fun y => fun i => jointColor G k r (Function.update x i y)))noncomputable def separateColor {V : Type} [Fintype V] (G : SimpleGraph V) (k : ℕ) :    (r : ℕ) → (Fin k → V) → SeparateColor k r  | 0, x => atomicType G x  | r + 1, x => (separateColor G k r x,      fun i => Finset.univ.val.map (fun y => separateColor G k r (Function.update x i y)))noncomputable def sameHistogram {V W : Type} [Fintype V] [Fintype W]    (G : SimpleGraph V) (H : SimpleGraph W) (c : Convention) (k r : ℕ) : Prop :=  match c with  | .joint => Finset.univ.val.map (jointColor G k r) =      Finset.univ.val.map (jointColor H k r)  | .separate => Finset.univ.val.map (separateColor G k r) =      Finset.univ.val.map (separateColor H k r)noncomputable def samePureHistogram {V W : Type} [Fintype V] [Fintype W]    (G : SimpleGraph V) (H : SimpleGraph W) (c : Convention) (k r : ℕ) : Prop :=  match c with  | .joint =>      Finset.univ.val.map (fun x : Fin k → V => jointColor (G.sum H) k r (Sum.inl ∘ x)) =      Finset.univ.val.map (fun x : Fin k → W => jointColor (G.sum H) k r (Sum.inr ∘ x))  | .separate =>      Finset.univ.val.map (fun x : Fin k → V => separateColor (G.sum H) k r (Sum.inl ∘ x)) =      Finset.univ.val.map (fun x : Fin k → W => separateColor (G.sum H) k r (Sum.inr ∘ x))noncomputable def WLEquivalent {V W : Type} [Fintype V] [Fintype W]    (G : SimpleGraph V) (H : SimpleGraph W) (c : Convention) (k : ℕ) : Prop :=  ∀ r, sameHistogram G H c k rnoncomputable def PureWLEquivalent {V W : Type} [Fintype V] [Fintype W]    (G : SimpleGraph V) (H : SimpleGraph W) (c : Convention) (k : ℕ) : Prop :=  ∀ r, samePureHistogram G H c k rnoncomputable def ParityReductionStatement : Prop :=  ∀ (k : ℕ), 4 ≤ k → ∀ c : Convention, ∀ C : ChoiceSystem (bagParameter c k),    let T := template (bagParameter c k)    let G := baseGraph C    let τ := baseProjection C    let V₀ := LiftVertex T G τ (fun _ => 0)    let V₁ := LiftVertex T G τ (starParity (bagParameter c k))    Fintype.card V₀ = Fintype.card V₁ ∧    (∀ A : ℕ, 1 ≤ A → C.SizeAtMost A → Fintype.card V₀ ≤ A * orderFactor (bagParameter c k)) ∧    (WLEquivalent (zeroLift C) (starLift C) c k ↔ ¬ C.Successful) ∧    (PureWLEquivalent (zeroLift C) (starLift C) c k ↔ ¬ C.Successful) ∧    (C.Successful → ¬ sameHistogram (zeroLift C) (starLift C) c k (detectionRound c) ∧      ¬ samePureHistogram (zeroLift C) (starLift C) c k (detectionRound c))theorem parity_reduction : ParityReductionStatement := by  sorryend ParityWLend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

Parity lifts and bounded-treewidth witnesses for Weisfeiler–Leman equivalence

Theorem (Parity reduction)

X_0\equiv_k X_\star \quad\Longleftrightarrow\quad \text{the choice system has no successful choice}.

The assertion holds both with common color names on separate graphs and with pure-tuple comparison in their disjoint union.

If every domain and label set has size at most A, where A\ge1, their common order is at most A M_t, where

M_t=t\,2^{\,t-2+3\binom{t-1}{2}}+12\binom t3.

When a successful choice exists, the two graphs already have different histograms after at most two joint rounds, or one separate round.

Section 1, Introduction Read it in the paper TeX source, line 76

Show the TeX

Fix \(k\ge4\), either refinement convention, and the corresponding
\(t\) in \eqref{eq:bag-parameter}. From every choice system on \(t\)
domains one can explicitly construct two uncolored simple graphs
\(X_0,X_\star\) of equal order such that
\[
 X_0\equiv_k X_\star
 \quad\Longleftrightarrow\quad
 \text{the choice system has no successful choice}.
\]
The assertion holds both with common color names on separate graphs
and with pure-tuple comparison in their disjoint union.

If every domain and label set has size at most \(A\), where \(A\ge1\),
their common order is at most \(A M_t\), where
\[
 M_t=t\,2^{\,t-2+3\binom{t-1}{2}}+12\binom t3.
\]
When a successful choice exists, the two graphs already have different
histograms after at most two joint rounds, or one separate round.

Theorem (Fixed-dimension ETH consequence)

Assume positive exponential-rate ETH. There are constants c>0 and K\ge4 such that, for every fixed integer k\ge K, no deterministic algorithm decides k-WL equivalence of two n-vertex graphs in time O(n^{ck}). This holds for both joint and separate refinement. The inputs are explicit adjacency matrices of uncolored simple undirected graphs, and time is measured on a multitape Turing machine.

Section 1, Introduction Read it in the paper TeX source, line 197

Show the TeX

Assume positive exponential-rate ETH. There are constants \(c>0\)
and \(K\ge4\) such that, for every fixed integer \(k\ge K\), no
deterministic algorithm decides \(k\)-WL equivalence of two
\(n\)-vertex graphs in time \(O(n^{ck})\).
This holds for both joint and separate refinement. The inputs are
explicit adjacency matrices of uncolored simple undirected graphs,
and time is measured on a multitape Turing machine.

The complexity of identifying a graph by Weisfeiler–Leman refinement

Theorem

The following problem is EXPTIME-complete under deterministic polynomial-time many-one reductions: given a nonempty finite simple undirected uncolored graph G by its adjacency matrix and a positive integer k in binary, decide whether \WLdim(G)\le k. Here dimension one uses ordinary neighbor color refinement, and dimensions at least two use the joint-update rule described above.

Section 1, Introduction Read it in the paper TeX source, line 36

Show the TeX

The following problem is EXPTIME-complete under deterministic
polynomial-time many-one reductions: given a nonempty finite simple
undirected uncolored graph $G$ by its adjacency matrix and a positive
integer $k$ in binary, decide whether $\WLdim(G)\le k$.
Here dimension one uses ordinary neighbor color refinement, and
dimensions at least two use the joint-update rule described above.

Unconditional time lower bounds for Weisfeiler–Leman equivalence

Theorem

There are absolute constants c>0 and k_0 with the following property. For every fixed integer k\ge k_0 and every correct deterministic decider A for k-Weisfeiler–Leman equivalence, there is an integer n_0=n_0(A,k) such that

T_A(n)\ge n^{ck} \qquad\text{for every integer }n\ge n_0.

This holds for multitape Turing machines and for sequential random-access machines with O(\log(n+2))-bit addresses and words, boundedly many memory accesses per instruction, and fixed instructions computable in time polynomial in the word length. The same bound holds if A is required to decide correctly only on pairs of simple connected uncolored graphs of diameter at most two, with T_A(n) maximized over that class. It applies both to the joint replacement-multiset convention for k-dimensional refinement and to the separate-multiset convention defined below.

Section 1, Introduction Read it in the paper TeX source, line 82

Show the TeX

There are absolute constants $c>0$ and $k_0$ with the following
property. For every fixed integer $k\ge k_0$ and every correct
deterministic decider $A$ for $k$-Weisfeiler--Leman equivalence, there
is an integer $n_0=n_0(A,k)$ such that
\[
                     T_A(n)\ge n^{ck}
             \qquad\text{for every integer }n\ge n_0.
\]
This holds for multitape Turing machines and for sequential
random-access machines with $O(\log(n+2))$-bit addresses and words,
boundedly many memory accesses per instruction, and fixed instructions
computable in time polynomial in the word length.
The same bound holds if $A$ is required to decide correctly only on
pairs of simple connected uncolored graphs of diameter at most two,
with $T_A(n)$ maximized over that class. It applies both to the
joint replacement-multiset convention for $k$-dimensional refinement
and to the separate-multiset convention defined below.

Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs

Theorem

Both \WL and \SubWL are \EXPTIME-complete under deterministic polynomial-time many-one reductions.

Section 1, Introduction Read it in the paper TeX source, line 28

Show the TeX

Both \(\WL\) and \(\SubWL\) are \(\EXPTIME\)-complete under deterministic
polynomial-time many-one reductions.

Configuration and file facts

File: lean/ComparatorChallenges/ParityLifts.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 133 link it.
Solution module: OAI.Combinatorics.ParityLifts.Main
Allowed axioms: propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib
Imports from outside Mathlib: None
Local vocabulary: 37 local vocabulary declarations
Lines of Lean, without comments or blank lines: 154
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

Plectis Atlas