= Solution
No. The property
$$
\mathcal P(f)\iff|\operatorname{dom}f|=9
$$
is <extensional property of programs>[extensional]: it depends only on the partial function computed, not on its program index. It is nontrivial, since some partial recursive functions have a nine-element domain and others do not. The <Rice theorem> therefore implies that its <index set>
$$
\{n:|W_n|=9\}
$$
is undecidable. Hence
$$
\boxed{\text{there is no algorithm that decides whether }|W_n|=9.}
$$
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