Five-vertex localization and exact 2-torsion in raw type-D Coxeter kernels

Let D̂_n^raw be the full-reflection Coxeter cover of W(D_n) with only thepairwise Coxeter-order relations, and let K_n^raw = ker(D̂_n^raw → W(D_n)).We determine the integral first homology of K_n^raw for every n ≥ 5: H₁(K_n^raw; Z) ≅ Z^((n−2)(3n² − 2n + 1)/2) ⊕ (Z/2)^(n(n−2)). The proof separates an integral localization theorem from a modular countingtheorem. First, one label orbit of the standard R₁ fork relation saturates the full forkdefect in first homology; the resulting target is torsion-free, so the rawtorsion is elementary 2-torsion and is generated by five-coordinate D₅subsystems. Second, the torsion is presented by a permutation module on formal R₁ classesmodulo three local relation orbits T, M₀, M₁. A finite exact D₅ rewritecertificate gives an all-rank normal-form induction, while a signed-reflectiondetector leaves one universal formal class κ_n unresolved. An explicit all-ncrossed cocycle evaluates nontrivially on κ_n, proving that no hidden relationremains. The only computer-assisted proof inputs are finite exact D₅ certificates; everypassage from D₅ to arbitrary D_n is symbolic. The rational multiplicities are quoted from the companion paper(doi:10.5281/zenodo.21966004) and are not reproved here.