Lean statement

Parabolic intersections, closures, and structural consequences

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.

Theorems selected for verification: OAI.HarmonicArtin.ParabolicIntersections.unconditional_parabolic_intersections OAI.HarmonicArtin.ParabolicIntersections.unconditional_arbitrary_intersections OAI.HarmonicArtin.ParabolicIntersections.unconditional_parabolic_closure OAI.HarmonicArtin.ParabolicIntersections.unconditional_nonclique_dynamics

Lean statement file: ArtinParabolicIntersections.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 254 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.

theorem unconditional_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) := by  sorry⋮theorem unconditional_arbitrary_intersections    (F : Set (Subgroup (Artin M))) (hF : ∀ P ∈ F, IsParabolic M P) :    (∃ F₀ : Finset (Subgroup (Artin M)),      (↑F₀ : Set (Subgroup (Artin M))) ⊆ F ∧ F₀.card ≤ Nat.card S ∧        sInf F = sInf (↑F₀ : Set (Subgroup (Artin M)))) ∧ IsParabolic M (sInf F) := by  sorry⋮theorem unconditional_parabolic_closure (E : Set (Artin M)) :    ∃! P : Subgroup (Artin M), IsParabolic M P ∧ E ⊆ P ∧      ∀ Q : Subgroup (Artin M), IsParabolic M Q → E ⊆ Q → P ≤ Q := by  sorry⋮theorem unconditional_nonclique_dynamics    (hIrred : IsIrreducible M) (hInf : ∃ s t : S, s ≠ t ∧ M s t = 0) :    AcylindricallyHyperbolic (Artin M) ∧      ∀ P : Subgroup (Artin M), IsParabolic M P → P ≠ ⊤ → WeaklyMalnormal P := 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.

definition OAI.HarmonicArtin.alternating line 13

def alternating (s t : S) : ℕ → FreeGroup S  | 0 => 1  | n + 1 => FreeGroup.of s * alternating t s n

definition OAI.HarmonicArtin.braidRelators line 17

def braidRelators (M : CoxeterMatrix S) : Set (FreeGroup S) :=  Set.range fun st : S × S =>    alternating st.1 st.2 (M st.1 st.2) *      (alternating st.2 st.1 (M st.1 st.2))⁻¹

abbreviation OAI.HarmonicArtin.Artin line 22

abbrev Artin (M : CoxeterMatrix S) := PresentedGroup (braidRelators M)

definition OAI.HarmonicArtin.generator line 24

def generator (M : CoxeterMatrix S) (s : S) : Artin M := PresentedGroup.of s

definition OAI.HarmonicArtin.artinParabolic line 26

def artinParabolic (M : CoxeterMatrix S) (T : Set S) : Subgroup (Artin M) :=  Subgroup.closure (generator M '' T)

definition OAI.HarmonicArtin.ParabolicIntersections.conjugate line 36

def conjugate (g : G) (P : Subgroup G) : Subgroup G :=  P.comap (MulAut.conj g⁻¹).toMonoidHom

definition OAI.HarmonicArtin.ParabolicIntersections.IsParabolic line 42

def IsParabolic (P : Subgroup (Artin M)) : Prop :=  ∃ (X : Set S) (g : Artin M), P = conjugate g (artinParabolic M X)

definition OAI.HarmonicArtin.ParabolicIntersections.gromovProduct line 49

noncomputable def gromovProduct {X : Type} [MetricSpace X] (o x y : X) : ℝ :=  (dist o x + dist o y - dist x y)/2

definition OAI.HarmonicArtin.ParabolicIntersections.IsGromovHyperbolic line 52

def IsGromovHyperbolic (X : Type) [MetricSpace X] : Prop :=  ∃ δ : ℝ, 0 ≤ δ ∧ ∀ o x y z : X,    min (gromovProduct o x y) (gromovProduct o y z) ≤ gromovProduct o x z + δ

definition OAI.HarmonicArtin.ParabolicIntersections.IsGeodesicMetric line 56

def IsGeodesicMetric (X : Type) [MetricSpace X] : Prop :=  ∀ x y : X, ∃ γ : Set.Icc (0 : ℝ) (dist x y) → X,    Isometry γ ∧ γ ⟨0,le_rfl,dist_nonneg⟩=x ∧ γ ⟨dist x y,dist_nonneg,le_rfl⟩=y

definition OAI.HarmonicArtin.ParabolicIntersections.IsGromovSequence line 60

def IsGromovSequence {X : Type} [MetricSpace X] (o : X) (u : ℕ → X) : Prop :=  ∀ R : ℝ, ∃ N : ℕ, ∀ m, N ≤ m → ∀ n, N ≤ n → R ≤ gromovProduct o (u m) (u n)

definition OAI.HarmonicArtin.ParabolicIntersections.IsAcylindrical line 63

def IsAcylindrical (G X : Type) [Group G] [MetricSpace X] [MulAction G X] : Prop :=  ∀ ε : ℝ, 0 < ε → ∃ R : ℝ, ∃ N : ℕ, ∀ x y : X, R ≤ dist x y →    {g : G | dist x (g • x) ≤ ε ∧ dist y (g • y) ≤ ε}.Finite ∧      {g : G | dist x (g • x) ≤ ε ∧ dist y (g • y) ≤ ε}.ncard ≤ N

definition OAI.HarmonicArtin.ParabolicIntersections.HasThreeOrbitLimitPoints line 68

def HasThreeOrbitLimitPoints (G X : Type) [Group G] [MetricSpace X] [MulAction G X] : Prop :=  ∃ o : X, ∃ u : Fin 3 → ℕ → G,    (∀ i, IsGromovSequence o (fun n => u i n • o)) ∧    ∀ i j, i ≠ j → ∃ B : ℝ, ∀ m n, gromovProduct o (u i m • o) (u j n • o) ≤ B

definition OAI.HarmonicArtin.ParabolicIntersections.AcylindricallyHyperbolic line 75

def AcylindricallyHyperbolic (G : Type) [Group G] : Prop :=  ∃ (X : Type) (metric : MetricSpace X) (action : MulAction G X),    @IsGeodesicMetric X metric ∧ @IsGromovHyperbolic X metric ∧    (∀ g : G, @Isometry X X metric.toPseudoMetricSpace.toPseudoEMetricSpace metric.toPseudoMetricSpace.toPseudoEMetricSpace      (fun x => @SMul.smul G X action.toSMul g x)) ∧    @IsAcylindrical G X _ metric action ∧ @HasThreeOrbitLimitPoints G X _ metric action

definition OAI.HarmonicArtin.ParabolicIntersections.WeaklyMalnormal line 82

def WeaklyMalnormal {G : Type} [Group G] (P : Subgroup G) : Prop :=  ∃ g : G, (↑(P ⊓ conjugate g P) : Set G).Finite

definition OAI.HarmonicArtin.ParabolicIntersections.IsIrreducible line 88

def IsIrreducible : Prop :=  ∀ Y : Set S, Y.Nonempty → Yᶜ.Nonempty → ∃ s ∈ Y, ∃ t ∉ Y, M s t ≠ 2

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: dynamics, hyperbolic, conjugate, nontrivial, clique, non, partition, acylindrically and seven more.

Statements in Parabolic intersections in Artin groups come first because the release’s name for this Lean statement shares words with that paper’s title.

Corollary 1.4 of Parabolic intersections in Artin groups

Corollary (Non-clique Artin-group dynamics)

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\}.

Section 1, Introduction Read it in the paper TeX source, line 77

Show the TeX

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 $P\cap gPg^{-1}=\{1\}$.

Other possible paper matches, closest wording first

Corollary 1.2 of Parabolic intersections in Artin groups. Words of the Lean statement, its names split into words, that also appear in the theorem: unique, closure, smallest, intersection, subgroup, contained and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: family.
Corollary 1.3 of Parabolic intersections in Artin groups. Words of the Lean statement, its names split into words, that also appear in the theorem: membership, rank, standard, generators, matrix, parabolic and four more.
Theorem 1.1 of Parabolic intersections in Artin groups. Words of the Lean statement, its names split into words, that also appear in the theorem: intersections, matrix, parabolic, coxeter and finite. Found in the paper’s title, which its main theorem states: groups and artin.
Corollary 3.11 of Parabolic intersections in Artin groups. Words of the Lean statement, its names split into words, that also appear in the theorem: without, obtained, arbitrary, coxeter and spherical.
Corollary 8.4 of Parabolic intersections in Artin groups. Words of the Lean statement, its names split into words, that also appear in the theorem: subgroup, standard, parabolic, group and finite.
Proposition 4.5 of Parabolic intersections in Artin groups. Words of the Lean statement, its names split into words, that also appear in the theorem: point, two, nonempty, integer and spherical.

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

two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. 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

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 254’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: 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).
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 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.”
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

129 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: 47 to its own declarations, 80 to Mathlib’s documentation. 20 declared names also link to the same declaration in OpenAI’s library. 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 unconditional_parabolic_intersections 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 all 16 other declarations of the file; with the theorem itself they span 64 lines. 7 of them are named in the statement itself.

Names in the namespace OAI.HarmonicArtin.ParabolicIntersections are listed without it.

def OAI.HarmonicArtin.alternatingline 13library
def OAI.HarmonicArtin.braidRelatorsline 17library
abbrev OAI.HarmonicArtin.Artinline 22in the statementlibrary
def OAI.HarmonicArtin.generatorline 24library
def OAI.HarmonicArtin.artinParabolicline 26in the statementlibrary
def conjugateline 36in the statementlibrary
def IsParabolicline 42in the statementlibrary
def gromovProductline 49library
def IsGromovHyperbolicline 52library
def IsGeodesicMetricline 56library
def IsGromovSequenceline 60library
def IsAcylindricalline 63library
def HasThreeOrbitLimitPointsline 68library
def AcylindricallyHyperbolicline 75in the statementlibrary
def WeaklyMalnormalline 82in the statementlibrary
def IsIrreducibleline 88in the statementlibrary
theorem unconditional_parabolic_intersectionsline 98selected theoremlibrary
theorem unconditional_arbitrary_intersectionsline 106selected theoremlibrary
theorem unconditional_parabolic_closureline 114selected theoremlibrary
theorem unconditional_nonclique_dynamicsline 121selected theoremlibrary

import Mathlibnamespace OAI/-! Parabolic intersections, parabolic closure, and non-clique Artin-group dynamics. -/namespace HarmonicArtinuniverse uvariable {S : Type u}def alternating (s t : S) : ℕ → FreeGroup S  | 0 => 1  | n + 1 => FreeGroup.of s * alternating t s ndef braidRelators (M : CoxeterMatrix S) : Set (FreeGroup S) :=  Set.range fun st : S × S =>    alternating st.1 st.2 (M st.1 st.2) *      (alternating st.2 st.1 (M st.1 st.2))⁻¹abbrev Artin (M : CoxeterMatrix S) := PresentedGroup (braidRelators M)def generator (M : CoxeterMatrix S) (s : S) : Artin M := PresentedGroup.of sdef artinParabolic (M : CoxeterMatrix S) (T : Set S) : Subgroup (Artin M) :=  Subgroup.closure (generator M '' T)end HarmonicArtinnamespace HarmonicArtin.ParabolicIntersectionsvariable {G : Type*} [Group G]/-- `g P g⁻¹`, with membership expressed without existential witnesses. -/def conjugate (g : G) (P : Subgroup G) : Subgroup G :=  P.comap (MulAut.conj g⁻¹).toMonoidHomvariable {S : Type*} (M : CoxeterMatrix S)/-- Conjugates of standard subgroups, with no spherical-type restriction. -/def IsParabolic (P : Subgroup (Artin M)) : Prop :=  ∃ (X : Set S) (g : Artin M), P = conjugate g (artinParabolic M X)end HarmonicArtin.ParabolicIntersectionsnamespace HarmonicArtin.ParabolicIntersectionsnoncomputable def gromovProduct {X : Type} [MetricSpace X] (o x y : X) : ℝ :=  (dist o x + dist o y - dist x y)/2def IsGromovHyperbolic (X : Type) [MetricSpace X] : Prop :=  ∃ δ : ℝ, 0 ≤ δ ∧ ∀ o x y z : X,    min (gromovProduct o x y) (gromovProduct o y z) ≤ gromovProduct o x z + δdef IsGeodesicMetric (X : Type) [MetricSpace X] : Prop :=  ∀ x y : X, ∃ γ : Set.Icc (0 : ℝ) (dist x y) → X,    Isometry γ ∧ γ ⟨0,le_rfl,dist_nonneg⟩=x ∧ γ ⟨dist x y,dist_nonneg,le_rfl⟩=ydef IsGromovSequence {X : Type} [MetricSpace X] (o : X) (u : ℕ → X) : Prop :=  ∀ R : ℝ, ∃ N : ℕ, ∀ m, N ≤ m → ∀ n, N ≤ n → R ≤ gromovProduct o (u m) (u n)def IsAcylindrical (G X : Type) [Group G] [MetricSpace X] [MulAction G X] : Prop :=  ∀ ε : ℝ, 0 < ε → ∃ R : ℝ, ∃ N : ℕ, ∀ x y : X, R ≤ dist x y →    {g : G | dist x (g • x) ≤ ε ∧ dist y (g • y) ≤ ε}.Finite ∧      {g : G | dist x (g • x) ≤ ε ∧ dist y (g • y) ≤ ε}.ncard ≤ Ndef HasThreeOrbitLimitPoints (G X : Type) [Group G] [MetricSpace X] [MulAction G X] : Prop :=  ∃ o : X, ∃ u : Fin 3 → ℕ → G,    (∀ i, IsGromovSequence o (fun n => u i n • o)) ∧    ∀ i j, i ≠ j → ∃ B : ℝ, ∀ m n, gromovProduct o (u i m • o) (u j n • o) ≤ B/-- An isometric non-elementary acylindrical action on a geodesicGromov-hyperbolic metric space. -/def AcylindricallyHyperbolic (G : Type) [Group G] : Prop :=  ∃ (X : Type) (metric : MetricSpace X) (action : MulAction G X),    @IsGeodesicMetric X metric ∧ @IsGromovHyperbolic X metric ∧    (∀ g : G, @Isometry X X metric.toPseudoMetricSpace.toPseudoEMetricSpace metric.toPseudoMetricSpace.toPseudoEMetricSpace      (fun x => @SMul.smul G X action.toSMul g x)) ∧    @IsAcylindrical G X _ metric action ∧ @HasThreeOrbitLimitPoints G X _ metric actiondef WeaklyMalnormal {G : Type} [Group G] (P : Subgroup G) : Prop :=  ∃ g : G, (↑(P ⊓ conjugate g P) : Set G).Finitevariable {S : Type} (M : CoxeterMatrix S)/-- No nontrivial partition has every cross label equal to two. -/def IsIrreducible : Prop :=  ∀ Y : Set S, Y.Nonempty → Yᶜ.Nonempty → ∃ s ∈ Y, ∃ t ∉ Y, M s t ≠ 2end HarmonicArtin.ParabolicIntersectionsnamespace HarmonicArtin.ParabolicIntersectionsvariable {S : Type} [Fintype S] [LinearOrder S] (M : CoxeterMatrix S)/-- The intersection of two conjugates of standard parabolics is parabolic. -/theorem unconditional_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) := by  sorry/-- An arbitrary parabolic intersection is obtained from at most the rank many members. -/theorem unconditional_arbitrary_intersections    (F : Set (Subgroup (Artin M))) (hF : ∀ P ∈ F, IsParabolic M P) :    (∃ F₀ : Finset (Subgroup (Artin M)),      (↑F₀ : Set (Subgroup (Artin M))) ⊆ F ∧ F₀.card ≤ Nat.card S ∧        sInf F = sInf (↑F₀ : Set (Subgroup (Artin M)))) ∧ IsParabolic M (sInf F) := by  sorry/-- Every subset is contained in a unique smallest parabolic subgroup. -/theorem unconditional_parabolic_closure (E : Set (Artin M)) :    ∃! P : Subgroup (Artin M), IsParabolic M P ∧ E ⊆ P ∧      ∀ Q : Subgroup (Artin M), IsParabolic M Q → E ⊆ Q → P ≤ Q := by  sorry/-- An irreducible non-clique Artin group has an acylindrical non-elementaryhyperbolic action and all its proper parabolics are weakly malnormal. -/theorem unconditional_nonclique_dynamics    (hIrred : IsIrreducible M) (hInf : ∃ s t : S, s ≠ t ∧ M s t = 0) :    AcylindricallyHyperbolic (Artin M) ∧      ∀ P : Subgroup (Artin M), IsParabolic M P → P ≠ ⊤ → WeaklyMalnormal P := by  sorryend HarmonicArtin.ParabolicIntersectionsend 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.

Harmonic heights and the Artin K(pi,1) conjecture

Theorem

For every Coxeter matrix on a finite set S, the universal cover of X(W,S) is contractible. Equivalently,

\pi_n\bigl(X(W,S)\bigr)=0\qquad(n\ge2).

The conclusion allows arbitrary finite labels, infinite labels, and connected or disconnected diagrams.

Section 1, Introduction Read it in the paper TeX source, line 22

Show the TeX

For every Coxeter matrix on a finite set \(S\), the universal cover
of \(X(W,S)\) is contractible.  Equivalently,
\[
  \pi_n\bigl(X(W,S)\bigr)=0\qquad(n\ge2).
\]
The conclusion allows arbitrary finite labels, infinite labels,
and connected or disconnected diagrams.

An Artin group with no geometric CAT(0) action

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 def:matrix.

Section 1, Introduction Read it in the paper TeX source, line 61

Show the TeX

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~\ref{def:matrix}.

Parabolic intersections in Artin groups

Theorem (Parabolic intersections)

For every Coxeter matrix M on a finite set S, subsets X,Y\subseteq S, and g,h\in A_M, there are Z\subseteq S and k\in A_M such that

gA_Xg^{-1}\cap hA_Yh^{-1}=kA_Zk^{-1}.

Section 1, Introduction Read it in the paper TeX source, line 21

Show the TeX

For every Coxeter matrix $M$ on a finite set $S$, subsets $X,Y\subseteq S$, and
$g,h\in A_M$, there are $Z\subseteq S$ and $k\in A_M$ such that
\[
 gA_Xg^{-1}\cap hA_Yh^{-1}=kA_Zk^{-1}.
\]

Corollary (Arbitrary intersections and parabolic closure)

For every family \mathcal F of parabolic subgroups of A, there is a subfamily \mathcal F_0\subseteq\mathcal F with |\mathcal F_0|\le |S| such that

\bigcap_{P\in\mathcal F}P=\bigcap_{P\in\mathcal F_0}P.

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.

Section 1, Introduction Read it in the paper TeX source, line 37

Show the TeX

For every family $\mathcal F$ of parabolic subgroups of $A$, there is a
subfamily $\mathcal F_0\subseteq\mathcal F$ with $|\mathcal F_0|\le |S|$ such
that
\[
 \bigcap_{P\in\mathcal F}P=\bigcap_{P\in\mathcal F_0}P.
\]
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.

Corollary (Word and parabolic membership problems)

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.

Section 1, Introduction Read it in the paper TeX source, line 59

Show the TeX

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.

Corollary (Non-clique Artin-group dynamics)

Section 1, Introduction Read it in the paper TeX source, line 77

Show the TeX

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 $P\cap gPg^{-1}=\{1\}$.

Configuration and file facts

File: lean/ComparatorChallenges/ArtinParabolicIntersections.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 254 link it.
Solution module: OAI.Topology.ArtinGroups.ParabolicIntersections.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: 16 local vocabulary declarations
Lines of Lean, without comments or blank lines: 64
Second proof-verification program (nanoda): Enabled 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