The Noncommutative Synchronization Kernel and a Dichotomy for Full-Reflection Coxeter Covers

Let (W, S) be a finite Coxeter system with reflection set R, and let Ŵ = ⟨ r̂ (r ∈ R) | r̂² = 1, (r̂ t̂)^ord(rt) = 1 ⟩ be the Coxeter group generated by all reflections subject only to the pairwiseorders. The canonical map ρ : Ŵ → W, r̂ ↦ r, has kernel K(W, S) := ker ρ, and[Ŵ : K] = |W|. We prove a dichotomy: for irreducible (W, S) of rank at least two, K(W, S) isabelian if and only if W ≅ A₂, in which case K(A₂) ≅ Z²; in every other caseK(W, S) is an infinite nonabelian group containing a free subgroup of rank two.The proof reduces to the signature of the Gram matrix of Ŵ, and we show by aninterlacing argument together with a circulant row-sum estimate that this matrixis positive semidefinite exactly for A₂. We then sharpen the signature statement to an exact inertia classification: forevery irreducible finite type we determine the triple (n₊, n₀, n₋) of the Gramform, together with canonical W-submodule structures on the null and negativeeigenspaces. For the dihedral systems we determine the cohomological dimension: K(I₂(m)) istorsion free with cd K(I₂(m)) = 2 for every m ≥ 3, hence is never free. Thiscloses a line of enquiry suggested by the Euler characteristic alone. The abelianisation of the even cover recovers the synchronization sectorS(W, S) studied in the companion paper(doi:10.5281/zenodo.21965921). In particular, for irreducible systems of rank atleast two, S(W, S) = 0 if and only if the even cover is perfect — which does notmake K perfect: a coset enumeration gives K(H₃)^ab ≅ Z^126, torsion free,although S(H₃) = 0.