An Algebraic Roadmap of Particle Theories

An optimal algebraic framework of particle physics has a number of checkpoints to pass. As a minimum, models should ⟨1⟩$\langle 1 \rangle$ conform to the Coleman‐Mandula theorem (or establish a loophole), ⟨2⟩$\langle 2 \rangle$ evade familiar fermion doubling problems, ⟨3⟩$\langle 3 \rangle$ naturally explain the Standard Model's chirality, ⟨4⟩$\langle 4 \rangle$ exclude B‐L gauge symmetry at low energy, and ⟨5⟩$\langle 5 \rangle$ explain the existence of three generations. The framework introduced in ref. [1] is shown to passes checkpoints ⟨1⟩,⟨2⟩,⟨3⟩,⟨4⟩$\langle 1 \rangle, \langle 2 \rangle, \langle 3 \rangle, \langle 4 \rangle$, and has yet to cross ⟨5⟩$\langle 5 \rangle$. The discussion is concluded by elucidating an unexpected appearance of spacetime symmetries.