Frequency-modulated Processes: Floquet Phases as Natural Transformations

We develop the third law of a four-paper modular framework for emergent phases of matter from quantum information through category theory. Building directly on Law I (categorical primitives) and Law II (functorial classification of equilibrium phases), Law III lifts the static phase-classification framework to a temporal setting in which a periodic Hamiltonian H(t+T)=H(t) defines a monoidal functor from the discrete circle category BZT to the dagger-monoidal category of quantum channels. We show that Floquet evolution is naturally a 2-categorical structure: a Floquet system is a 1-cell, micromotion data are 2-cells, and Floquet phase transitions are connected components of the space of natural transformations between Floquet functors.