A Conditional Algebraic Framework for Charge, Chirality, Flavor and Cosmological Correspondences

This paper develops a conditional algebraic framework in which fields, equations and scale-flow relations are organized by the four normed division algebras R, C, H, O. The Hurwitz classification, quaternionic chirality, octonionic non-associativity, G2-representation theory, the Cartan decomposition of U (p, q), the Clifford algebra isomorphism Cl(1, 3) ∼= M2(H), and the heat-kernel expansion of the spectral action are presented as standard mathematical theorems. The QCD beta-function data β0(0) = 11, β0(3) = 9, β0(6) = 7, β1(6) = 26 are recalled from the QCD renormalization-group equations. The fermion level generation, CKM and PMNS matrices, mixing angles, CP phase, absolute neutrino mass, quantum-gravity effective action and anomaly cancellation are conditional on the structural axioms A1–A3 and the explicitly listed physical boundary conditions and mechanisms. The logarithmic and exponential mechanisms are derived from the one-loop renormalization-group solution and its higher-loop extensions. The framework provides a conditional algebraic, geometric and spectral foundation for the Standard Model and its extensions.