A Cubic Matrix Action for the Standard Model and Beyond

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.