Theorem: Finite observation-compatible feedback model

The model (eq:example-labels)–(eq:example-instrument) has:

1. a finite set of completely specified, admissible geometric and controller labels;

2. a fixed qubit matter space and state-independent completely positive trace-preserving hybrid evolution;

3. matter-dependent successor probabilities determined by the declared source algebra;

4. an admissibility certificate for every nonzero branch, without postselection;

5. branchwise compositional observation compatibility;

6. exact spectator consistency for entangled extensions and complete histories;

7. two full-instrument-equivalent readout implementations;

8. an exact permission balance and finite pointer/history resource reserve;

9. assisted implementation stability and a quantitative approximation to the declared diagonal classical model.

For $0<\eta <1$, both source sensitivity and geometry-dependent matter phase operations are nontrivial. The ambient cone, particle worldline, deficit, and peripheral affine holonomy remain fixed.

Proof. Lemma lem:example-certificate proves (i) and (iv). Theorem thm:structure and Proposition prop:unsharp prove (ii) and (iii). Commutation with (eq:example-algebra) gives the branch inclusions, proving (v) by Theorem thm:histories. The specified controller and separate reserves satisfy Assumption ass:blind; Theorem thm:spectators gives (vi).

Proposition prop:deferred gives (vii). Equation (eq:example-resource) and Proposition prop:environment give (viii). Theorem thm:stability, Lemma lem:choi, Proposition prop:pilot-stability, and Proposition prop:classical-approx give (ix). The final statements follow from the explicit source effects, distinct successor phases, and the fixed-cone certificate.