Synchronization sectors of finite Coxeter systems: classification and equivariant structure

Let (W, S) be a finite Coxeter system with set of reflections R. We weight thecomplete graph on R by assigning to the edge {r, t} the order n_rt = ord(rt),and study the torsion cokernel S(W, S) = D / ⟨ n_rt (e_r − e_t) ⟩, D = { x ∈ Z^R : Σ_r x_r = 0 }. We show that S(W, S) represents the functor sending an abelian group A to thegroup of non-uniform phase assignments θ : R → A satisfying all pairwisedihedral closure relations n_rt (θ_r − θ_t) = 0, modulo the diagonal. Two general lemmas on weighted incidence matrices — a mod p rank formula and athreshold refinement determining the full p-primary structure — reduce thecomputation of S(W, S) to the connectivity of the graphs G(p,k) whose edges arethose with v_p(n_rt) < k. The graphs G(p,1) are instances of the π-productinvolution graphs of Rowley and Ward, and for p = 2 they are the local fusiongraphs studied by Ballantyne for finite Coxeter groups. We then determine S(W, S) for every finite Coxeter system, and describe itsstructure as a W-module. The dihedral case is given by a closed formula, and theexceptional types are settled by exact integer computation; all tables arereproducible from the supplementary scripts. This is the companion paper to "The noncommutative synchronization kernel and adichotomy for full-reflection Coxeter covers" (doi:10.5281/zenodo.21965977),where S(W, S) is recovered as the abelianisation of the even full-reflectioncover.