Higher A-Homotopy Vanishing for Coxeter Chamber Graphs

Abstract

We prove that the standard Cayley graph of every finite-rank Coxeter system has trivial A-homotopy groups in all degrees at least two. The argument replaces the Coxeter group by the right-angled Coxeter group that retains exactly its commuting relations. All short closed walks lift to the resulting cover, while every finite image in its median Cayley graph admits a based contraction through graph maps. A direct lattice-lifting proof checks the exterior basepoint condition and explains why degree one is different. For a finite real reflection group, the theorem establishes the all-degree group comparison for the real 3-parabolic arrangement by combining higher-A vanishing with the previously known fundamental-group comparison and classical asphericity of the complement. The right-angled construction and degree-one kernel are credited to earlier work. The result concerns k=3 and does not settle the general k-parabolic comparison for k>3. This is an unrefereed manuscript; no absolute priority claim is made. AI-assisted tools supported research, finite regression checks, proof development and manuscript preparation. The result is self-audited; no independent peer review or formal proof-assistant verification is claimed. The accompanying finite checks do not replace the general written proof. Corpus reference: AIM-TOPOLOGY-0097 (UnsolvedMath v1.6.0); the broader source question remains partially resolved.