= Solution
Conditional on an event and the immediately preceding history, <probabilities> are proportional to the three instantaneous hazards. Put $w=e^{\beta_0}$. The two zero-covariate subjects each have weight one, and the one-covariate subject has weight $w$. The common <baseline hazard> cancels. Thus
$$
\boxed{P(z_{\mathrm{event}}=0\mid\text{event, history})=\frac2{2+w},\qquad P(z_{\mathrm{event}}=1\mid\text{event, history})=\frac w{2+w}.}
$$
Each individual with zero covariate has <probability> $1/(2+w)$; the first boxed <probability> is their combined <probability>. Conditioning on an event at a specified continuous time can be understood by the limiting conditional event <probabilities> in a short interval.
The hazard-weighted covariate mean is $w/(2+w)$, so the <Schoenfeld function> at the true coefficient is
$$
s(\beta_0)=\begin{cases}-w/(2+w),&z_{\mathrm{event}}=0,\\2/(2+w),&z_{\mathrm{event}}=1.\end{cases}
$$
Multiplying by the two <conditional probabilities> gives
$$
\boxed{E\{s(\beta_0)\mid\text{event, history}\}=\frac2{2+w}\frac{-w}{2+w}+\frac w{2+w}\frac2{2+w}=0.}
$$
This verifies the score-centering property directly for this <risk set>.
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