Aionics III: Constraints on Realizable Processes: Ordering, Equilibrium, and Representation without Primitive Time

We develop a minimal mathematical framework in which physicaldescriptions are characterized by constraints on realizable processesrather than by primitive time evolution. Starting from comparabilityon restricted domains, composability, and stability under refinement,we prove the existence of an additive ordering functional on processes.We show that equilibrium states stable under all cyclic operations nec-essarily take Gibbs form, that regular Euclidean continuation enforcesa universal 2π periodicity in the associated flow parameter, and thatstable interference together with reversibility implies a unitary expo-nential representation. Finally, we demonstrate that in a sector ad-mitting exactly three independent local deformation generators, theassociated boundary redundancy group is Z3. All results are derivedin finite or discrete settings, with explicitly stated assumptions, andwithout assuming spacetime geometry, quantum fields, or microscopicdynamics. The framework isolates structural features that any empir-ical physical theory supporting ordering, equilibrium, and interferencemust possess.