Relative Salvetti models and the finiteness qualification

PDF p. 13: Remark 3.27

Let $(W,S)$ be a Coxeter system with finite $S$ and $J\subseteq S$. For the relative complement ${\mathcal{X}}_J^{\mathbb{C}}$ of Relative complexified hyperplane complement, assume its cone interior is nonempty. The locally finite relative arrangement of Local finiteness of the interior relative arrangement has a Salvetti CW model homotopy equivalent to ${\mathcal{X}}_J^{\mathbb{C}}$, by Independent Salvetti homotopy-model input. This model is finite whenever the relative arrangement is finite; in particular it is finite when $W$ is finite. Local finiteness alone does not imply finite CW homotopy type: for $W=⟨s,t\mid s^2=t^2=1⟩$ and $J=\mathrm{\varnothing }$, there is no finite CW model, as established in The dihedral product complement admits no finite CW model.

Remark 3.27 of the original article explicitly says “has a finite CW-model”; the independent Paris theorem does not assert finiteness for infinite arrangements. The preceding sentence on spherical type is a conditional identification, not a standing assumption for the remark: §2.2 explicitly imposes spherical type only in Sections 5 and 6. Therefore the counterexample addresses the original article, rather than only a generated reconstruction. The correction preserves the cited Salvetti method and its finite spherical case; it does not claim finite type is necessary for every relative arrangement, or address equivariance under arbitrary normalizer actions. Exact PDF anchors and the explicit counterexample are retained in run-local evidence for the final assessor, who must independently compare the finding against the original PDF.