Observation, feedback, and implementation

Choose the source observation algebra

Both $M_a$ and $U_{k,q}$ commute with this algebra. Therefore

on every branch, including absorbing ones. Theorems thm:histories and thm:spectators apply to every input and every entangled spectator extension.

For $0<\eta <1$, the two feedback unitaries at the successors of one active label act differently on surviving off-diagonal matter blocks. Thus geometry affects the subsequent matter channel. That phase dependence is not observable on $\mathcal{A}$, which records only populations. A continuation measuring coherence must declare an enlarged observation boundary; it is not licensed to treat coherence as both inaccessible and later observable.

The source dilation

uses one fresh pointer qubit. Immediate readout followed by (eq:example-unitaries), and coherent controlled feedback followed by readout, agree as full transition instruments by Proposition prop:deferred. Their complete residual outputs, not merely their geometric probabilities, coincide.

The classical label update (eq:example-successors) is a finite table and has a reversible completion with retained input history: an injective map $z\mapsto (f(z),z)$ can be extended to a permutation on a finite enlarged code space by exchanging disjoint input–output pairs. This does not claim discrete-gate exact synthesis of the unsharp quantum rotations for arbitrary $\eta$.