= Solution
For every infinite $\alpha<\kappa$, the generic union at coordinate $\alpha$ gives a <function> $g_\alpha:\omega\to\alpha$. The requirement to assign $(\alpha,n)$ is dense for each $n$, and the requirement to use any specified value $\beta<\alpha$ is dense by assigning it at a fresh natural-number position. Therefore $g_\alpha$ is a <surjection>, and every <ordinal> below $\kappa$ becomes countable.
The $\kappa$-chain condition preserves the regularity of $\kappa$. For a direct verification, a <forcing name> for a <function> from some $\mu<\kappa$ into $\kappa$ has fewer than $\kappa$ possible values at each coordinate, using a maximal deciding <forcing antichain>. The union of these possible-value <sets> has size less than $\kappa$ by regularity and is bounded in $\kappa$. No such <function> can be cofinal. In particular $\kappa$ remains uncountable, while every smaller <ordinal> is countable. Hence
$$
\boxed{\kappa=\aleph_1^{M[H]}.}
$$
This is the <finite Lévy collapse to omega-one>.
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