Proposition: Calibration and recovery limits

At one fixed event, exact agreement of distinct calibrated values with that event's time allows at most one positive-probability code. The clock readout has no left inverse on all clock states.

Proof. The first claim is distinctness of the calibration. The states $(|{\tau }_0⟩\pm |{\tau }_1⟩)/\sqrt{2}$ give identical readout outputs with the same surviving input, so both cannot be recovered.