Classifying spaces and geometric obstructions for Artin groups
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
The Salvetti complex of every finite-rank Artin group is aspherical, proving the Artin \(K(\pi,1)\) conjecture. Arbitrary intersections of its parabolic subgroups are parabolic, proving the Parabolic Intersection Conjecture. An explicit Artin group admits no proper cocompact isometric action on any nonempty proper CAT\((0)\) space. (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
Here the intersection of an empty family is A. In particular, arbitrary intersections of parabolic subgroups are parabolic, and every subset E\subseteq A is contained in a unique smallest parabolic subgroup.
No Lean statement in the release is paired with this theorem.
The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.
Four more possible matches
Theorem 1.1in An Artin group with no geometric CAT(0) action, 23 September 2026.
Corollary 1.3(Word and parabolic membership problems)in Parabolic intersections in Artin groups, 23 September 2026.
Corollary 1.4(Non-clique Artin-group dynamics)in Parabolic intersections in Artin groups, 23 September 2026.
Theorem 1.1(Parabolic intersections)in Parabolic intersections in Artin groups, 23 September 2026.
Reported proof verification
Three Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
We prove that the standard Salvetti complex of every Artin group with finitely many standard generators is aspherical. This resolves the Artin \(K(\pi,1)\) conjecture in finite rank.
We construct an Artin group on 116 generators that admits no proper, cocompact isometric action on a nonempty proper CAT(0) space. This refutes the CAT(0) conjecture for Artin groups.
We prove that every intersection of parabolic subgroups of any finite-rank Artin group, with arbitrary finite or infinite Coxeter labels, is parabolic. This resolves the Parabolic Intersection Conjecture affirmatively.
Three 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.
There is an Artin matrix M on 116 generators, with every off-diagonal label in \{2,3,\infty\}, such that A_M admits no geometric action. One explicit matrix is given in Definition 7.1.
The words they share: explicit, geometric, action, matrix, artin, cat, group, diagonal, off, generators.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
ArtinCAT0.lean, the release lists it as a main result.
theoremmain : (∀ s t : Letters, explicitMatrix s t = explicitMatrix t s) ∧ (∀ s : Letters, explicitMatrix s s = 1) ∧ (∀ s t : Letters, s ≠ t → explicitMatrix s t = 2 ∨ explicitMatrix s t = 3 ∨ explicitMatrix s t = ⊤) ∧ (∀ (X : Type u) [MetricSpace X] [ProperSpace X] [Nonempty X] [MulAction (ArtinGroup explicitMatrix) X], ¬ GeometricAction (ArtinGroup explicitMatrix) X) := bysorry
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”.
Suppose M is irreducible, meaning that there is no nontrivial partition S=S_1\sqcup S_2 with m_{st}=2 for every s\in S_1,t\in S_2, and suppose m_{st}=\infty for some pair. Then A is acylindrically hyperbolic. Every proper parabolic subgroup P has a conjugate meeting it trivially: there is g\in A such that P\cap gPg^{-1}=\{1\}.
The words they share: dynamics, hyperbolic, conjugate, nontrivial, clique, non, partition, acylindrically, proper, irreducible, st, subgroup, parabolic, group, artin.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
ArtinParabolicIntersections.lean, the release lists it as a supporting statement.
theoremunconditional_parabolic_intersections (X Y : Set S) (g h : Artin M) : ∃ (Z : Set S) (k : Artin M), conjugate g (artinParabolic M X) ⊓ conjugate h (artinParabolic M Y) = conjugate k (artinParabolic M Z) := bysorry
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.
There is an algorithm which, given a finite Coxeter matrix M, a subset X\subseteq S, and words u,v in the standard generators of A_M and their inverses, decides whether the element represented by v belongs to uA_Xu^{-1}. In particular, the word problem is decidable for every finite-rank Artin group.
The words they share: artin, coxeter, finite, generators, group, matrix, parabolic, rank, standard, subset, whether.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
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.
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:All three claims are stated: HarmonicArtin.salvetti_cover_contractible [Finite S] (M : CoxeterMatrix S) : ContractibleSpace (SalvettiCover M) (K(π,1): the cover is the realization of the nerve of the lifted spherical-cell poset), ArtinCAT0.main (the explicit 116-generator matrix has entries in {1,2,3,∞} and ¬ GeometricAction (ArtinGroup explicitMatrix) X for every nonempty proper metric space X with a MulAction, CAT0 via the CN inequality), and ArtinParabolicIntersections (unconditional_arbitrary_intersections: sInf F of any family of parabolics is parabolic and equals sInf of at most Nat.card S members; unique parabolic closure; acylindrical hyperbolicity and weak malnormality for irreducible infinite-label groups). Config notes: ArtinParabolicIntersections.json has enable_nanoda: true (an extra independent-kernel check, not a weakening); tmp/import_cones.json flags external OAI for 254 only because it mis-parses the module MulBlockDiagonal'Entry (apostrophe), so the true cone is Mathlib-only and ~32 lines larger.
Definitions the group asks readers to compare with the paper:SalvettiCover M := SSet.toTop.obj (nerve (LiftedCell M)) with LiftedCell M := Artin M × SphericalType M and the face order generated by LiftedFace (minimal-length Coxeter coset representatives lifted to the Artin group): a hand-built model of the universal cover of the Salvetti complex (nerve of its face poset). Nothing in the challenge statement ties this space to a Mathlib notion of Salvetti complex or to the action of the Artin group, so ‘asphericity of the Salvetti complex’ is only as faithful as this model (the docs call it the equivalent statement that the universal cover is contractible).
“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.
The Artin $K(\pi,1)$ conjecture asserts that the standard Salvetti complex of every finite-rank Artin group is aspherical. The formalized result proves the equivalent statement that its universal cover is contractible.
It applies to every Coxeter matrix on a finite generating set, including empty and disconnected diagrams and infinite edge labels. The associated Coxeter and Artin groups may be infinite. Later corollaries in the manuscript are not included.
The geometric-action question asks whether every Artin group acts properly and cocompactly by isometries on a proper CAT(0) space. The formalized result gives a counterexample: an Artin group defined by a symmetric matrix on $116$ generators, with diagonal entries $1$ and off-diagonal entries in $\{2,3,\infty\}$.
It admits no such action on any nonempty proper CAT(0) space, in any dimension.
The formalization proves that the intersection of any family of parabolic subgroups of a finite-rank Artin group is parabolic. Every such intersection is already the intersection of at most the rank many members, and every subset has a unique parabolic closure. Both finite and infinite Coxeter labels are allowed.
For irreducible Artin groups with an infinite Coxeter label, it also proves acylindrical hyperbolicity and weak malnormality of every proper parabolic subgroup. These are the intersection and structural consequences selected from the paper.
AI-drafted comparisons, with no recorded human review
Artin group with no geometric $\mathrm{CAT}(0)$ actionArtinCAT0
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: an Artin group on 116 generators with labels in {2, 3, ∞} that acts geometrically on no proper CAT(0) space.
Parabolic intersections, closures, and structural consequencesArtinParabolicIntersections
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem (intersections of parabolic subgroups are parabolic) with the corollaries on arbitrary intersections, parabolic closures and the dynamics of non-clique Artin groups. The membership-algorithm corollary is not part of it.
Contractibility of the Salvetti coverHarmonicArtin
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem, the K(π, 1) conjecture: the universal cover of the Salvetti complex is contractible for every Coxeter matrix on a finite set.
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.
Artin group with no geometric $\mathrm{CAT}(0)$ actionArtinCAT0
Three reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original 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.
The program that collects Atlas’s source records, Statement Map, could not obtain or interpret some source material. These notes describe those limits; they do not identify mathematical errors.
HarmonicArtin: Tracked source lean/OAI/Topology/ArtinGroups/Main.lean unavailable: bytes were not captured at fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb
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.
Papers elsewhere in the release that this family’s papers cite
This family’s papers cite no paper elsewhere in the release.
Papers elsewhere in the release that cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
References outside the release
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release
Explain a result, compare statements or propose a correction
Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.
Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.
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 “Career-defining publication”. The page says that nothing on it is verified.
Giulio Tiozzo reports that the Artin group example in 254 has 116 generators and that Astra, asked to try, returned a 12-generator example in about ten minutes, suggesting no human review.
Questions awaiting review
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Read the Lean statement against the paper
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
Identify the theorem the catalogue describes
Family 254The 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
Address a question raised in an independent assessment
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. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
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; 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…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
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.
Re-run through Comparator with Lean’s kernel: accepted (the check took 72 s; peak memory 6.2 GiB).
Date
10 October 2026
Version
at the release’s recorded version
Their method
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.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
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.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
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.
Artin group with no geometric $\mathrm{CAT}(0)$ action
Role
the release lists it as a main result
Paired with
Theorem 1.1 (candidate pairing)
Who ran the proof-verification program
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.