An Algebraic Roadmap of Particle Theories, Part III: Intersections

In this article, we bypass the detailed symmetry breaking pathways established in [1]. 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 $\mathfrak{s}\mathfrak{o}(10)$ actions coincide on a fixed space of multi-vector fermions. This $\mathfrak{s}\mathfrak{o}(10)\mapsto \mathfrak{s}\mathfrak{u}(3{)}_C\oplus \mathfrak{s}\mathfrak{u}(2{)}_L\oplus u(1{)}_Y$ breaking (from a three-way intersection) mirrors, in certain ways, the $\mathfrak{s}\mathfrak{o}(8)\mapsto {\mathfrak{g}}_2$ 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 $\mathfrak{s}\mathfrak{o}(10)\mapsto \mathfrak{s}\mathfrak{u}(3{)}_C\oplus \mathfrak{u}(1{)}_Q$. These are the Standard Model’s unbroken gauge symmetries, post-Higgs-mechanism.