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, andŴ: 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 ≅ Z¹26, torsion free, although S (H₃) = 0.