Largest Acylindrical Actions for Artin Groups under a Strict Triple Condition

Let A_S be a finitely generated Artin group such that the sum of the reciprocals of the three labels on every triple of generators is strictly less than one, with 1/infinity = 0. This note records a direct consequence of the Martin-Przytycki coned-off Deligne construction and the domination criterion of Abbott-Balasubramanya-Osin. The coned-off action dominates every acylindrical action on a hyperbolic space precisely when every standard generator has a finite-labelled neighbour. In that case it is a largest acylindrical hyperbolic structure. Isolated generators obstruct this particular action, but the known free-product combination theorem still gives a largest action for the group. The argument includes mixed label-2 presentations; the large-type overlap is already covered by existing work.

This is an attributed corollary note, not a solution of the general AIM Question 4.5 for arbitrary nonspherical irreducible Artin groups (dataset record AIM-GEOMETRIC_GROUP_THEORY-0059). The general domination criterion, Deligne construction and free-product combination are credited, not claimed as new. This AI-assisted preprint is self-audited and unrefereed. No independent review, formal proof-assistant verification, originality certificate or absolute priority is claimed. Exact finite-presentation regression checks accompany the source; they are not a verification of the infinite-group proof.