  We propose a new framework for matrix theories that are equivalent to field
theories on a toroidal spacetime. The correspondence is accomplished via
infinite Toeplitz matrices whose entries match the field degrees of freedom on
an energy-momentum lattice, thereby replacing the background geometry with
matrix indices. These matrix theories can then be embedded into the purely
cubic action of a single matrix and combined into a common universality class.
We reconstruct the Standard Model action in this framework and discuss its
extensions within the same class.
