Coarse records, failure flags, and local POVMs

Failure or rejection is treated in the same way: its flag is part of the complete output alphabet. If the rejected residual is discarded or replaced, that action is an additional declared channel. A successful normalized branch alone is not the complete experiment.

For a local clock instrument with Kraus operators $\{F_{y,a}:{\mathcal{H}}_C\to {\mathcal{H}}_{C^{\prime }}{\}}_a$, suppose the entire output $C^{\prime }$ is also discarded or consumed, while $S$ is untouched. Its residual system branch is

The equality follows by pairing both sides with every system observable and using trace cyclicity. Thus a represented local POVM can provide a further destructive conditioning interface.

Equation (eq:destructive-povm) does not say that effects determine arbitrary residual instruments. It uses the special assumptions that only the clock is acted on and that all quantum clock output is removed. Measuring an ancilla pointer while returning a quantum clock is a different interface.

The supplied X-Enter selective-readout theorem is an appropriate comparison for retaining failure branches and proving selective-map, rather than effect-only, agreement CheungXEnter (Theorem thm:relational). Its encoded phase model is not a premise or a specialization of the clock Hamiltonian construction here.