Proposition: Fixed-event and recovery limits

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

The clock channel has no left inverse on all inputs, and its record alone does not determine its residual. No finite clock matrices satisfy $[T_C,H_C]=i{I}_C$.

Proof. Distinctness proves the first assertion. The inputs

with the same surviving state have identical outputs, so no left inverse recovers both. Inputs $P_0\otimes \rho$ and $P_0\otimes \tilde{\rho }$ have the same record and different residuals. Finally, the trace of a finite matrix commutator is zero, whereas $\mathrm{Tr}(i{I}_C)\ne 0$.