Source authenticated

The Parabolic Intersection Conjecture for Artin groups

Theorem 1.1: for every Coxeter matrix on a finite set, all $X,Y$ and $g,h$ , there are $Z,k$ with $gA_Xg^{-1}\cap hA_Yh^{-1}=kA_Zk^{-1}$ . Corollary 1.2 (via Moller-Paris-Varghese): arbitrary intersections of parabolics are parabolic, already attained by at most $|S|$ members, and every subset has a unique parabolic closure. Corollary 1.4: irreducible Artin groups with an infinite label are acylindrically hyperbolic (already known from Kato-Oguni) with weakly malnormal proper parabolics. The proof uses a relative harmonic height for a framed zigzag braid action and does not use asphericity of the Salvetti complex.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Parabolic Intersection Conjecture for Artin groups · Parabolic Intersection Conjecture

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

Open the source record ↗

Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: The Parabolic Intersection Conjecture for Artin groups

specialist registry · passed

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 with its corollaries, read against the conjecture as cited (Godelle 2023, Cumplido survey). The stabilizer and height arguments were not refereed. Lean-checked on the release's Comparator challenge ArtinParabolicIntersections, found through lean/docs/254.md; this challenge is not in the release's formalization catalogue, but its solution module OAI.Topology.ArtinGroups.ParabolicIntersections.Main exists at the pinned commit. Its statement was read here: theorem unconditional_parabolic_intersections asserts that for every finite Coxeter matrix the intersection of two conjugates of standard subgroups is a conjugate of a standard subgroup; further theorems state the arbitrary-intersection, closure and non-clique dynamics corollaries. That covers the headline. Not rebuilt here.

Source ↗

Parabolic intersections in Artin groups (companion, same harmonic-height method)

paper · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Lean Comparator challenge ArtinParabolicIntersections (OpenAI math release)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Godelle, On parabolic subgroups of Artin-Tits groups (arXiv)

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

Source ↗

Artifacts and verifiers

Lean Comparator challenge ArtinParabolicIntersections (OpenAI math release)

lean artifact · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

VibeMathed record: The Parabolic Intersection Conjecture for Artin groups evidence for this event

This event attributed to Unreleased internal OpenAI model

Godelle, On parabolic subgroups of Artin-Tits groups (arXiv) evidence for this event

Parabolic intersections in Artin groups (companion, same harmonic-height method) evidence for this event

Lean Comparator challenge ArtinParabolicIntersections (OpenAI math release) evidence for this event

The Parabolic Intersection Conjecture for Artin groups parent of this event

Act on this frontier

Verify, challenge, or extend the result.

Help verify Open canonical problem Explore evidence graph