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 nᵣt = ord (rt), and study the torsion cokernel S (W, S) = D / ⟨ nᵣt (eᵣ − eₜ) ⟩, D = x ∈ ZR: Σᵣ xᵣ = 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 nᵣt (θᵣ − θₜ) = 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 vₚ (nᵣt) < 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.