For the symmetric and demihyperoctahedral families we study the rawfull-reflection Coxeter cover: the Coxeter group Ŵᵣaw generated by allreflections of W subject only to the pairwise product orders, with kernelKᵣaw = ker (Ŵᵣaw → W). This is the group before the Y- and fork-relations ofRowen–Teicher–Vishne and Amram–Shwartz–Teicher are imposed. We determine the rational first homology in both classical families. For type A we prove H¹ (Kᵣaw (A_ (n−1) ) ; Q) ≅ tₙ · U_ (n−1) for n ≥ 5, wheretₙ = C (n, 2) − n + 1 is the cycle rank of the complete graph and U_ (n−1) thestandard module. The module shape coincides with the abelianisationZ^ (tₙ (n−1) ) of the Y-quotient kernel computed by Rowen–Teicher–Vishne, and weclaim no novelty for that target. What is new is the stability statement: forevery n ≥ 5 the surjection Kᵣaw ↠ KY induces an isomorphism on rational H₁, and integrally its kernel is exactly the torsion subgroup, so the Y-relationskill precisely the torsion. The invariant factors are (Z/2) ^ (tₙ−1) forn = 5, 6, 7 by exact Smith reduction. For type D we prove H¹ (Kᵣaw (Dₙ) ; Q) ≅ tₙ · U_ (n−1) ⊕ n (n−2) · Vₙ for n ≥ 5, with Vₙ the reflection module. Both multiplicities are graph-theoretic ranks: tₙ is the cycle rank of Kₙ, and n (n−2) is the rank of the flow space of thedoubled signed complete graph, which is unbalanced, so that|E| − |V| + b = n (n−1) − n + 0 = n (n−2). The summand n (n−2) Vₙ, with thismultiplicity, is specific to the raw cover, and we identify it canonically withthe relative residual of the sign-forgetting map Kᵣaw (Dₙ) → Kᵣaw (A_ (n−1) ). All statements are theorems with complete proofs; the base case n = 5 and theintegral torsion computations are computer-assisted, by exact integer Smithreduction with unit pivots, and are reproducible from the supplementary scripts.
Sana Kamiki (2026) studied this question.