= Solution
Let $R(t^*)$ be the <risk set> immediately before $t^*$, conditional on the preceding history. Only its members can experience the next event. In a short interval $[t^*,t^*+dt)$, the conditional probability that subject $j$ fails is $h_0(t^*)e^{\beta z_j}dt+o(dt)$, whereas the probability of one event from the <risk set> is $h_0(t^*)\sum_{i\in R(t^*)}e^{\beta z_i}dt+o(dt)$. Dividing and taking the small-interval limit gives
$$
\boxed{P_\beta\{\pi(t^*)=j\mid\mathcal F_{t^*-},\text{one event}\}
=\frac{e^{\beta z_j}}{\sum_{i\in R(t^*)}e^{\beta z_i}},\qquad j\in R(t^*).}
$$
It is zero outside the <risk set>. The common <baseline hazard> cancels. This is the conditional event-label probability underlying the <Cox partial likelihood>. Conditioning at an exact continuous event time is understood by this limiting conditional-intensity argument, rather than as division by the zero unconditional probability of an event at a fixed time. We use the standard simple event process with no simultaneous events and <independent censoring>.
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