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The computational complexity of Weisfeiler–Leman refinement

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Proves unconditional \(n^{\Omega(k)}\) deterministic time lower bounds for joint and separate k-dimensional Weisfeiler–Leman equivalence, for sufficiently large fixed k in the specified sequential adjacency-matrix models. With dimension as input, joint equivalence is EXPTIME-complete even on subcubic graphs; deciding whether refinement identifies a graph is also EXPTIME-complete. (OpenAI’s Lean note)

Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.

Both independent assessments find that the Lean statements express or imply the whole result described above. Read the assessments

The paper states

Theorem 1.1

In Unconditional time lower bounds for Weisfeiler–Leman equivalence, 25 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Four other possible matches are listed below the card, among them Theorem 1.1 in The complexity of identifying a graph by Weisfeiler–Leman refinement and Theorem 1.3 (Fixed-dimension ETH consequence) in Parity lifts and bounded-treewidth witnesses for Weisfeiler–Leman equivalence.

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.

Four more possible matches

  • Theorem 1.1 in The complexity of identifying a graph by Weisfeiler–Leman refinement, 25 September 2026.
  • Theorem 1.3 (Fixed-dimension ETH consequence) in Parity lifts and bounded-treewidth witnesses for Weisfeiler–Leman equivalence, 25 September 2026.
  • Theorem 1.2 (Parity reduction) in Parity lifts and bounded-treewidth witnesses for Weisfeiler–Leman equivalence, 25 September 2026.
  • Theorem 1.1 in Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs, 25 September 2026.
Reported proof verification
Four Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
Papers and explanations
Four current papers, each readable here as a page beside its PDF.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

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

25 September 2026, current edition.

Read it herePDF

For k ≥ 4, we construct two uncolored graphs that are k-dimensional Weisfeiler–Leman equivalent exactly when a prescribed finite-domain choice system has no compatible choice. The system has \(k+1\) domains for joint refinement and k for separate-coordinate refinement. A successful choice is detected after two joint rounds or one separate round. Applied to sparse satisfiability, the reduction gives fixed-dimension \(n^{\Omega(k)}\) time exclusions under positive-rate ETH, even for deciding equality of these early histograms.

Unconditional time lower bounds for Weisfeiler–Leman equivalence

25 September 2026, current edition.

Read it herePDF

For every sufficiently large fixed k, deciding whether two n-vertex graphs are k-Weisfeiler–Leman equivalent requires \(n^{\Omega(k)}\) deterministic sequential time in the worst case. The bound holds at every sufficiently large graph order, even for simple connected uncolored graphs of diameter at most two, without a complexity assumption. Inputs are explicit adjacency matrices; the models are multitape Turing machines and sequential logarithmic-word RAMs with fixed polynomial-bit-time instructions. Both joint and separate replacement conventions are covered.

Four Lean statements in this family

Choose one to see it beside the paper theorem it is paired with.

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

ParityLifts

The paper states

Theorem 1.2 (Parity reduction)

In Parity lifts and bounded-treewidth witnesses for Weisfeiler–Leman equivalence, 25 September 2026

Fix k\ge4, either refinement convention, and the corresponding t in (1.1). 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.

The words they share: star, histograms, rounds, bag, successful, parity, pure, system, color, eq, parameter, separate, size, convention, joint, choice, domain, reduction, order, simple, graphs, equal, construct, two, uncolored.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Parity-lift characterization of Weisfeiler–Leman equivalence

ParityLifts.lean, the release lists it as a supporting statement.

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

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.
  • Neal McBurnett, openai-math-lean-tooling records it as “accepted”.

Read the complete Lean statement

VariableWL

The paper states

No paper theorem is paired with this Lean statement.

no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.

Statements it shares words with:

  • Theorem 1.1 in Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs, 25 September 2026.
  • Proposition 8.2 (Uniform construction) in Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs, 25 September 2026.
  • Corollary 3.5 (The polarity of the reduction) in Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs, 25 September 2026.
  • Corollary 8.3 (Unary dimension) in Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs, 25 September 2026.
  • Theorem 1.1 in The complexity of identifying a graph by Weisfeiler–Leman refinement, 25 September 2026.
  • Proposition 8.1 (Uniform general-graph compiler) in Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs, 25 September 2026.
  • Theorem 1.1 in Unconditional time lower bounds for Weisfeiler–Leman equivalence, 25 September 2026.
  • Proposition 7.1 (Subcubic realization) in Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs, 25 September 2026.

The Lean statement

No paper theorem is paired with it.

EXPTIME-completeness of variable-dimension Weisfeiler–Leman equivalence

VariableWL.lean, the release lists it as a supporting statement.

theorem main : EXPTIMEComplete WL ∧ EXPTIMEComplete SubWL := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

WLIdentification

The paper states

Theorem 1.1

In The complexity of identifying a graph by Weisfeiler–Leman refinement, 25 September 2026

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.

The words they share: neighbor, ldim, color, matrix, update, complete, many, joint, finite, exptime, binary, positive, integer, nonempty, dimension, simple, one, graph, identifying, refinement, adjacency, reductions, polynomial, uncolored, time.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

EXPTIME-completeness of Weisfeiler–Leman graph identification

WLIdentification.lean, the release lists it as a supporting statement.

theorem main : EXPTIMEComplete dimensionLanguage :=

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

WeisfeilerLeman

The paper states

Theorem 1.1

In Unconditional time lower bounds for Weisfeiler–Leman equivalence, 25 September 2026

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.

The words they share: instruction, multiset, memory, log, diameter, applies, separate, pairs, turing, convention, joint, word, bit, constants, bound, integer, computable, machines, two, length, time, graphs, unconditional, lower, required, sequential, ck, dimensional, absolute, leman, multitape, weisfeiler, equivalence, deterministic.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Unconditional sequential time lower bounds for Weisfeiler–Leman equivalence

WeisfeilerLeman.lean, the release lists it as a supporting statement.

def MainClaim : Prop :=  ∃ c : ℝ, 0 < c ∧ ∃ k₀ : ℕ, ∀ k : ℕ, k₀ ≤ k →    ∀ convention : Convention, ∀ p : InputClass, ∀ A : Model,      Decides A convention k p → ∃ n₀ : ℕ, ∀ n : ℕ, n₀ ≤ n →        (n : ℝ) ^ (c * (k : ℝ)) ≤ (worstTime A p n : ℝ)theorem WLTime.unconditional_time_lower_bound : WLTime.MainClaim := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.
  • Neal McBurnett, openai-math-lean-tooling records it as “accepted”.

Read the complete Lean statement

Other mathematical statements in OpenAI’s Lean library

Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

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

MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.

MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).

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

“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.

Their method: 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.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express or imply the whole advertised result (their term: “MATCHES”).

“MATCHES”, in their words: The challenge statements state the headline as OpenAI wrote it.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean 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.

A successful choice is detected by the specified early round: round two in the joint convention and round one in the separate convention. The graph-size bound is linear in the choice-system size with a factor depending on $k$. The paper’s conditional running-time exclusions are outside these selected construction statements. The formalization proves that deciding whether Weisfeiler–Leman refinement of a supplied dimension identifies a graph is EXPTIME-complete under polynomial-time many-one reductions. The input is a nonempty finite simple uncolored graph encoded by its adjacency matrix together with a positive binary-encoded dimension. Identification quantifies over every comparison graph, and the statement includes both the exponential-time upper bound and hardness. The formalization proves an unconditional deterministic time lower bound for deciding $k$-dimensional Weisfeiler–Leman equivalence. For every sufficiently large fixed $k$, every deciding algorithm in the stated multitape Turing-machine or sequential logarithmic-word RAM model requires at least $n^{ck}$ worst-case time for all sufficiently large graph orders $n$, for one absolute $c>0$. The bound covers both joint and separate replacement conventions and remains valid for explicit adjacency-matrix inputs restricted to simple connected uncolored graphs of diameter at most two. The formalization proves EXPTIME-completeness of deciding joint-update $k$-dimensional Weisfeiler–Leman equivalence when $k\ge2$ is part of the input in binary and the graphs are encoded by explicit adjacency matrices. It proves both the general result and the restriction to connected simple uncolored graphs of equal positive order and maximum degree at most three. Completeness uses polynomial-time many-one reductions.

AI-drafted comparisons, with no recorded human review

  • Parity-lift characterization of Weisfeiler–Leman equivalence ParityLifts

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the paper’s parity-reduction theorem for every dimension k ≥ 4 under both conventions. The ETH-conditional running-time consequence is outside the statement.

  • EXPTIME-completeness of variable-dimension Weisfeiler–Leman equivalence VariableWL

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: EXPTIME-completeness of variable-dimension Weisfeiler–Leman equivalence, in general and on connected subcubic graphs.

  • EXPTIME-completeness of Weisfeiler–Leman graph identification WLIdentification

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: deciding whether Weisfeiler–Leman refinement of a given dimension identifies a graph is EXPTIME-complete under polynomial-time many-one reductions.

  • Unconditional sequential time lower bounds for Weisfeiler–Leman equivalence WeisfeilerLeman

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: an unconditional n^{ck} deterministic time lower bound for deciding k-dimensional Weisfeiler–Leman equivalence, for all large fixed k.

Who ran the proof-verification program?

These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.

These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.

Parity-lift characterization of Weisfeiler–Leman equivalence ParityLifts

Three reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report
Neal McBurnett, openai-math-lean-tooling“accepted”Re-run through Comparator with Lean’s kernel: accepted (the check took 45 s; peak memory 6.1 GiB).10 October 2026Original report

EXPTIME-completeness of variable-dimension Weisfeiler–Leman equivalence VariableWL

Two reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report

EXPTIME-completeness of Weisfeiler–Leman graph identification WLIdentification

Two reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report

Unconditional sequential time lower bounds for Weisfeiler–Leman equivalence WeisfeilerLeman

Three reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report
Neal McBurnett, openai-math-lean-tooling“accepted”Re-run through Comparator with Lean’s kernel: accepted (the check took 238 s; peak memory 6.2 GiB).11 October 2026Original report

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Within this family

Papers elsewhere in the release that this family’s papers cite

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Offer to write an explanationCompare one Lean statement with its paper: ParityLifts, WLIdentification, WeisfeilerLeman, VariableWLSuggest a source attributionSuggest a change

What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.

    Author not stated. Source

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  • Family 133 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

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  • Family 133 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form