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.

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.