Proposition: Calibration and recovery limits

For the instrument (eq:instrument):

1. Distinct labels $t_j$ required to equal one fixed reading-event time on every positive-probability branch permit at most one occurring label.

2. There is no channel left inverse on the full clock–survivor input space.

3. The classical record marginal does not determine the surviving quantum state.

4. No finite clock matrices satisfy $[T_C,H_C]=\mathrm{i}\hslash I_C$ with $\hslash \ne 0$.

Proof. Distinctness proves the first assertion. For the second, the two clock states

with the same surviving preparation have identical destructive outputs. A left inverse cannot recover both. Inputs $|{\tau }_0⟩⟨{\tau }_0|\otimes {\rho }_A$ and $|{\tau }_0⟩⟨{\tau }_0|\otimes {\tilde{\rho }}_A$ give the same deterministic record but different residuals when the two states differ. Finally, the trace of a commutator is zero, whereas $\mathrm{Tr}(\mathrm{i}\hslash I_C)=\mathrm{i}\hslash D\ne 0$.