Phase-bound Matter: Functorial Classification of Thermodynamic and Topological Phases

We present the second installment of the modular research series Emergent Spacetime Dynamics. Building strictly modularly on the categorical primitives established in Law I—the matter–information functor M:Phys→Info, the symmetric monoidal structure of physical composition, and the sheaf semantics of local observables—we classify phases of matter as functorial equivalence classes. We first formalise the Landau paradigm as the connected-component invariant π₀([BG, Ham₀]ᵉq), where Ham₀ is the groupoid of gapped local Hamiltonians under gap-preserving adiabatic equivalence. We then generalise beyond Landau by showing that topological order in (2+1)D corresponds to unitary modular tensor categories (MTCs), arising as Drinfeld centers Z(C) of unitary fusion categories.