Theorem: Exact compatible operational refinement

Let $T_0:M_4\to M_4$ be a specified complete device, or a fixed finite complete recorded instrument, built from the gauge-preserving realizations above. Put

Then:

1. every $T_n$ is UCP and every ${\omega }_n$ is a state;

2. all complete outcome laws are identical at every scale;

3. the logical channel equality is exact at every independent reference size;

4. compatible regional containments are preserved;

5. matched pairs retain the same unchanged device actions, central data, resources, and report designations;

6. finite compositions and complete recorded continuations are preserved exactly.

The electric readout in Example ex:localread and closed-cycle unitaries generated by $R$ give nonconstant instances.

Proof. Conjugation by a $\ast$-isomorphism preserves complete positivity, unitality, and states. For every output observable,

The same identity holds componentwise for recorded instruments and after matrix amplification, proving (ii)–(iii).

Supported invariant realizations refine by the construction in Lemma lem:nestedregions. Their physical channels are (eq:exactrefinement), so containment passes under the same identifications. Choose the refinements once for a matched pair; unchanged realizations and resources remain common. Unitary conjugation respects composition and finite direct sums, giving (v)–(vi). The specified examples depend nontrivially on physical inputs.