Theorem: Complete-process overlap

Two specified Markov models with the same initial accessible state, conditional interval channels after every allowed history and setting, and complete instruments have equal record laws for all allowed adaptive experiments, including untouched references.

For equal laws, every randomized binary test satisfies $\alpha +\beta =1$. In general,

Proof. Induction in (eq:history) gives equality of every unnormalized branch, also after tensoring with a reference identity. For a test $\varphi \in [0,1]$,

Bound the expectation difference by total variation.