The K(pi,1) conjecture for Artin groups of finite rank

Theorem 1.1: for every Coxeter matrix on a finite set, with arbitrary finite or infinite labels and connected or disconnected diagrams, the universal cover of the full Salvetti complex $X(W,S)$ is contractible, so $\pi_n(X)=0$ for $n\ge2$ . The proof contracts the poset of spherical cosets by a well-ordered vertex filtration, using a framed zigzag-algebra braid action whose layer multiplicities define a harmonic height. Consequences recorded: torsion-freeness, the center $Z(A)\cong\mathbb Z^k$ , and Dobrinskaya's monoid equivalence. Infinite rank is not covered.

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 K(pi,1) conjecture for Artin groups of finite rank · Artin K(pi,1) conjecture

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

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Named evidence

VibeMathed record: The K(pi,1) conjecture for Artin groups of finite rank

specialist registry · passed

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1, read against the conjecture as cited (Brieskorn 1973, Charney-Davis 1995). The categorical and harmonic-height arguments were not refereed. Lean-checked on the release's Comparator challenge HarmonicArtin (declaration OAI.HarmonicArtin.salvetti_cover_contractible, listed in lean/formalization.yaml). Its statement was read here: for every Mathlib CoxeterMatrix on a finite type it asserts ContractibleSpace of the geometric realization of the nerve of a face preorder on pairs (Artin element, spherical subset). That is a combinatorial model of the Salvetti universal cover; the identification of this model with the topological universal cover is classical and is not part of the formal statement. Not rebuilt here.

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The K(pi,1) conjecture for Artin groups of finite rank

witness · pending

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

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Lean Comparator challenge HarmonicArtin (OpenAI math release)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

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

paper · pending

paper link supplied by the VibeMathed registry.