Established framework and present contribution

Quantum instruments distinguish outcome probabilities from their selective residual maps DaviesLewis1970. Finite-dimensional complete positivity and its matrix certificates are standard Choi1975,Watrous2018. Linear completely positive classical–quantum dynamics is also an established framework, including stochastic trajectories and broader hybrid generators Oppenheim2023,OppenheimTrajectories2023, OppenheimClasses.

Accordingly, we do not claim novelty for the direct-sum state space, the Kraus construction, the exact recovery–complement criterion, or the general channel observation principle. In particular, VII-E already proves a composition criterion for block-diagonal state spaces CheungVIIE (Proposition prop:classical-composition). Our direct-sum criterion is its intrinsic specialization.

The purpose of VII-G is to organize these facts into a complete finite feedback specification, with:

1. a state-independent transition table and certified geometric output graph;

2. branch-sensitive observation boundaries and stochastic histories;

3. exact consistency with entangled spectators;

4. selective-map implementation tests;

5. resource accounting and assisted error certificates;

6. an explicit finite example satisfying the specification.

All results used in the assembly are proved below.