Proposition: Calibration and recovery limitations

1. If distinct calibrated labels $t_k$ must all equal the time of one fixed readout event whenever their outcomes occur, at most one code has positive probability.

2. The channel has no left inverse on all clock inputs.

3. Its record marginal does not determine the residual.

4. No finite clock matrices satisfy $[T_C,H_C]=i{I}_C$.

Proof. Distinctness gives (i). The clock inputs $(|{\tau }_0⟩\pm |{\tau }_1⟩)/\sqrt{2}$, with the same surviving state, have identical outputs, proving (ii). Inputs $P_0\otimes \rho$ and $P_0\otimes \tilde{\rho }$ have the same record and different residuals, proving (iii). The trace of a commutator is zero, while $\mathrm{Tr}(i{I}_C)\ne 0$, proving (iv).