Proposition: A random reading at one event is not several event times

Exact event-time calibration at a fixed event, with distinct $t_k$, forces a deterministic outcome. If no $t_k=T(e)$, no normalized exactly calibrated input exists.

Proof. At most one index $j$ can satisfy $t_j=T(e)$. Calibration and normalization force $p_j=1$. For $Q=I-P_j\otimes I_X$, positivity gives

Thus $\rho =(P_j\otimes I_X)\rho (P_j\otimes I_X)$. The converse is immediate.