The classical residual-report instrument

Let the finite dictionary have $J$ entries. Choose $d_C=2^{n_C}\ge J$, $n_C\ge 1$, and a calibrated orthonormal clock basis $\{|{\tau }_j⟩{\}}_{j=1}^{d_C}$. Unused labels receive an explicit $\mathrm{UNUSED}$ report.

Let $A$ contain all unmeasured finite quantum registers. Let $q_j$ be a finite classical word containing the entry identifier, enclosure, status, and order tags. Put

Distinct output labels retain $j$, even when several entries have the same zero residual.

Define