= Solution
\b[True.] The valid <distributive law for ordinal multiplication> is
$$
\boxed{\alpha(\beta+\gamma)=\alpha\beta+\alpha\gamma.}
$$
The synthetic definition of <ordinal multiplication> takes a sequence of copies of the left factor, indexed by the right factor. A sequence indexed first by $\beta$ and then by $\gamma$ is exactly the concatenation of the two corresponding sequences. Taking <order types> gives the identity. Equivalently, use <transfinite induction> on $\gamma$, with the successor rule and the supremum rule at <limit ordinals>.
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