= Solution
The difference $s(t^*,\beta)$ is the observed event subject's covariate minus its conditional expected value under the working coefficient. It is the <Schoenfeld function>; at a fitted coefficient it becomes a <Schoenfeld residual>. If the conditional event-label distribution is generated by coefficient $\beta$, part (b) gives
$$
\boxed{\mathbb E_\beta[s(t^*,\beta)\mid\mathcal F_{t^*-},\text{one event}]
=\mathbb E_\beta[z_{\pi(t^*)}\mid\cdots]-\overline z(t^*,\beta)=0.}
$$
Hence its unconditional expectation is zero whenever it is integrable. Centering is at the true coefficient: data generated under $\beta_0$ do not generally make $s(t^*,\beta)$ mean zero at an arbitrary trial value $\beta\ne\beta_0$.
The same quantity is the <score function> of one conditional event factor, since
$$
\frac{\partial}{\partial\beta}\log\frac{e^{\beta z_{\pi(t^*)}}}{S_0(t^*,\beta)}
=z_{\pi(t^*)}-\frac{S_1(t^*,\beta)}{S_0(t^*,\beta)}.
$$
This connects its conditional centering to the usual zero-mean likelihood-score identity.
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