Proposition: Fixed event and recovery limits

If distinct calibrated values $t_k$ must all equal the time of one fixed event whenever their outcomes occur, at most one outcome has positive probability. The clock channel has no left inverse on all clock states.

Proof. Distinctness proves the first assertion. For the second, the clock superpositions $(|{\tau }_0⟩\pm |{\tau }_1⟩)/\sqrt{2}$, with identical surviving input, give the same output in (eq:clock). They cannot both be recovered by a left inverse.