Proposition: Clock limitations

1. If every occurring $t_k$ must equal the time of one fixed event, at most one code has positive probability.

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

3. Its record marginal does not determine its residual.

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

Proof. Distinctness of the labels proves (i). The two 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 deterministic record but arbitrary different residuals, proving (iii). Taking the trace of the proposed commutator proves (iv).