Theorem: Destructive clock and combined instrument

The clock channel is CPTP and, with every reference,

Composing it with the task-and-signal instrument gives a complete finite instrument. If all its records are discarded and the intervening operations act only on the consumed clock, task, and signal factors, the surviving output is their joint partial trace.

Clock, task, and measured signal names are not quantum outputs of that instrument.

Proof. The clock Kraus maps are $|k{⟩}_{R_C}\otimes D_k$. Their adjoint products sum to the identity. Clock-basis expansion proves the partial-trace identity. Composition with Theorem thm:dictionaryTV proves completeness; composing the displayed partial-trace identities proves the final statement.