Lemma: Order covariance

Every matched equivalence preserves strengths, directed edges, reachability, and components. It induces an order isomorphism $\bar{f}:E_D\to E_{D^{\prime }}$.

Proof. Matched laws have the same total-variation distances. The bijection of tests gives identical suprema. Bijections preserve finite paths in both directions, hence mutual reachability and the quotient order.