Integral Structure of the Quaternionic Block: Hecke Orders of W(H₄) Parabolic Permutation Modules and Archimedean--Catalan Hom Spaces

Overview: This work establishes the exact Z-integral structure of Hecke orders associated with parabolic permutation modules for the Coxeter group W(H4) specialized at q = 1. The analysis focuses on the unique 48-dimensional rational-valued irreducible character phi with Schur index m_Q(phi) = 2, corresponding to the simple component M_24(D) where D = (2,3)_Q is the quaternion algebra ramified at 2 and 3. Main Results: Exact Indices: The index [Gamma : O_J] of the projected Hecke order inside a maximal order is computed for all seven active parabolic classes: Edge (1), Cell (2), Dihedral (5^3), Face (2^3 * 5^3), A_2 (2^12 * 5^22), A_1 x A_1 (2^20 * 5^56), and A_1 (2^80 * 5^248). For the non-parabolic rotational subgroup Alt_4, the index is 2^4 * 5^8. 5-adic Hereditary Orders: For the dihedral and face modules (k_J = 2), the 5-adic order is conjugate to the (1,3)-type hereditary order in M_4(Z_5). Mod 2 Artin-Wedderburn Structure: Reductions modulo the trace radical yield the central simple quotient algebras F/R isomorphic to M_2(F_4) for A_2, M_4(F_4) for A_1 x A_1, and M_8(F_4) for A_1, realizing the matrix degree sequence d in {2, 4, 8}. Archimedean-Catalan Spaces: Catalan Hom dimensions are classified for 15 Wythoff polytopes, and the algebraic metric spectrum and order-400 isometry group of the Grand Antiprism over Q(sqrt(5)) are certified. All results are certified via exact integer arithmetic and reproducible scripts.