Proposition: Complete-process equality

Consider two finite Markov models with identical initial accessible states, interval channels for every permitted setting and history, and complete instruments. Then every permitted adaptive experiment, including an untouched reference, has the same record law in both models. For a randomized binary test between record laws $P_0,P_1$,

If the laws coincide, $\alpha +\beta =1$.

Proof. Induct on the record history. Equal unnormalized conditional states, followed by equal conditional channels and instruments, give equal next conditional states and branch probabilities. For a test function $\phi \in [0,1]$, with output one interpreted as selecting model one,