A Finite Generator Characterization of Monotonicity in Probabilistic Cellular Automata

We give a canonical finite characterization of stochastic monotonicity (attractiveness) for homogeneous binary synchronous probabilistic cellular automata (PCA) of arbitrary finite radius. We prove that global monotonicity of the PCA Markov operator is equivalent to monotonicity of the one-site update kernel on the Boolean neighborhood lattice. Equivalently, monotonicity is characterized by a finite family of local inequalities indexed precisely by the cover relations of that lattice. We organize the equivalence through a finite-generator viewpoint, consisting of a minimal cover-generator together with a dominance-chain closure principle. As a consequence, stochastic monotonicity becomes decidable by finitely many explicit local checks with an exact bound depending only on the neighborhood radius. Our formulation isolates the precise local content of monotonicity without appealing to global coupling constructions.