Proposition: Complete process equality and testing obstruction

Suppose two models have a common complete state description, identical initial states, and identical conditional complete instruments and interval maps after every admitted setting and history. Then their complete record laws are equal, including with an untouched reference.

For any two complete record laws $P_0,P_1$, a randomized test satisfies

For equal laws, $\alpha +\beta =1$.

Proof. Induct through the equal unnormalized conditional states. Their traces give equal probabilities for every record history.

For a test $\phi \in [0,1]$ declaring model 1,

Apply Lemma lem:tv.