= Solution
Let $V=S^1\times D^2$ contain the <pattern of a satellite knot> $P$. Its class in $H_1(V;\mathbb Z)\cong\mathbb Z$ is the <winding number of a satellite pattern>. Winding number zero therefore makes $P$ null-homologous in $V$, so $P$ bounds an oriented embedded surface $F_P\subset V$. Let
$$
g_P=g(F_P).
$$
For any companion <knot> $K$, an embedding $V\hookrightarrow S^3$ as a tubular neighborhood of $K$ carries $F_P$ to a <Seifert surface> for the <satellite knot> $P(K)$. Hence
$$
g_s(P(K))\leq g(F_P)=g_P,
$$
and the bound depends only on the pattern.
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