An Algebraic Roadmap of Particle Theories

Abstract In this study, the detailed symmetry breaking pathways established in Furey et al. (2024) are bypassed. Instead, a direct route from the Spin(10) model to the Standard Model is enabled via a single algebraic constraint. This single constraint, however, may be reconfigured as a requirement that three actions coincide on a fixed space of multi‐vector fermions. This breaking (from a three‐way intersection) mirrors, in certain ways, the breaking (from a three‐way intersection) in the context of octonionic triality. By extending this result to include quaternions and complex numbers, we find that a five‐way intersection breaks . These are the Standard Model's unbroken gauge symmetries, post‐Higgs‐mechanism.