A strict one-register observable program

If one additionally wants a finite program whose observable output is only ${Q_C}^{\mathrm{out}}$, the residual must be consumed explicitly:

(R,B) := CoherentBody_t(C); x := Enter(B); dropBit(x); Cout := Decode(R); returnQ(Cout);

Its channel is again $\mathcal{D}{\hat{\mathcal{N}}}_t$, by Theorem thm:drop below.

This readout-and-drop is an external interface implementation. It is not a claim that the autonomous Hamiltonian measures $B$, nor that the omitted physical apparatus has disappeared. The complete dilation retains its apparatus outputs.