Proposition: Number records do not determine residual coherence

Equation (eq:numberinstrument) is a complete trace-preserving instrument. Different packet states can produce the same number record while having different delayed-query or quantum-recovery performance.

Proof. The traces of its positive diagonal blocks sum to one, which proves convergence and completeness. For the second assertion, the states

have the same joint number distribution and are orthogonal. A coherent measurement distinguishes them perfectly; the destructive number record does not distinguish them.