= Solution
$Y_{j,k}$ is the number in group $k$ still observed and event-free immediately before event time $a_j$: its group-specific <risk set> size. Set $Y_j=Y_{j,0}+Y_{j,1}$. Under the common <hazard function> and conditional on exactly one event at that time, each at-risk individual has the same infinitesimal event probability. Therefore
$$
\boxed{p(j,k)=\frac{Y_{j,k}}{Y_j}.}
$$
The indicator $g_{\pi(j)}$ records whether the event came from group 1, while $p(j,1)$ is its conditional expectation under equal hazards. Thus $z_j$ is the observed-minus-expected group-1 event count at that time. Under the null these are <martingale differences>, with conditional <variance>
$$
\operatorname{Var}(z_j\mid\mathcal H_{a_j-})=p_j(1-p_j),
\qquad p_j=Y_{j,1}/Y_j.
$$
The <log-rank statistic> $z=\sum_jz_j$ has null mean zero. Its predictable variance estimate is $V=\sum_jp_j(1-p_j)$, so \b[the equal-hazards test uses]
$$
\boxed{Z=\frac{z}{\sqrt V}\ \dot\sim N(0,1),\qquad
\frac{z^2}{V}\ \dot\sim\chi^2_1.}
$$
Use a two-sided large-sample <log-rank test> if no direction was prespecified; $V>0$ and adequate information are needed. The variance is the no-ties form. A positive $z$ indicates more group-1 events than expected. <Independent censoring> conditional on the model is required for these at-risk event intensities to describe the data.
Under the <proportional hazards model>, the total instantaneous group-$k$ rate is $Y_{j,k}e^{\beta k}h_0(a_j)$. Conditioning on the next event cancels the <baseline hazard>, giving
$$
\boxed{p_\beta(j,k)=\frac{Y_{j,k}e^{\beta k}}
{Y_{j,0}+Y_{j,1}e^\beta}.}
$$
Keep $z_j$ defined using its null probability $p(j,1)$, as in the question. Its conditional expectation under the alternative is
$$
\boxed{\mathbb E_\beta[z_j\mid\mathcal H_{a_j-}]
=\frac{Y_{j,1}e^\beta}{Y_{j,0}+Y_{j,1}e^\beta}
-\frac{Y_{j,1}}{Y_j}
=\frac{Y_{j,0}Y_{j,1}(e^\beta-1)}
{Y_j(Y_{j,0}+Y_{j,1}e^\beta)}.}
$$
It is zero at $\beta=0$, as required, and has the sign of $\beta$ when both groups remain at risk. Replacing $p(j,1)$ inside the definition of $z_j$ with the alternative probability would give a different residual, whose mean would instead be zero under that alternative.
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