= Solution
In <potential outcome> notation, a valid <instrumental variable> $Z$ must satisfy:
* <Instrument relevance>: the conditional law of $A$ changes with $Z$.
* <Instrumental-variable independence>: $Z$ is independent of the joint collection of potential treatments and outcomes, such as $\{A(z),Y(a):z,a\}$.
* The <exclusion restriction>: $Y(z,a)=Y(a)$, so $Z$ can affect $Y$ only through $A$.
* <Consistency of potential outcomes> and no <interference in causal inference> connect the potential variables to the observations.
In the graph, $Z_1$ has open noncausal paths to $Y$ that do not pass through $A$, including
$$
Z_1\leftarrow M_1\to C\to Y
\qquad\text{and}\qquad
Z_1\leftarrow S\to Z_3\to Y.
$$
Equivalently, these paths remain after deleting the causal edge $A\to Y$. Thus $Z_1$ is associated with potential outcomes through the latent variables $C$ and $S$, violating <instrumental-variable independence>; it is not a valid marginal instrument.
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