= Solution
All <cardinalities> in this part are first computed in $M$. The <forcing> has size $\kappa$. A family of $\kappa$ finite domains has a $\kappa$-sized <Delta-system>, since $\kappa$ is regular. There are fewer than $\kappa$ possible value assignments on its finite root: each coordinate $(\alpha,n)$ allows fewer than $\kappa$ values. Regularity lets us thin to two conditions, indeed $\kappa$ many, with identical root assignments. Their union is a condition, proving the $\kappa$-chain condition.
Consequently \b[every maximal <forcing antichain> has <cardinality> less than $\kappa$, but there is no one compulsory <cardinality>]. For any nonzero <cardinal> $\mu<\kappa$, the single-coordinate conditions assigning the values $\xi<\mu$ at $(\mu,0)$ form an <forcing antichain> of size $\mu$. It is maximal: a condition already assigning that coordinate is compatible with its matching value, and a condition not assigning it is compatible with every allowed value. Thus every such size occurs, including singleton maximal <forcing antichains>. These are the <maximal-antichain sizes in the finite Lévy collapse>.
The paper writes $p\le q$ for $p\subseteq q$. We use this printed weaker-first convention: $q$ extends $p$ and is stronger. Compatibility and generic meeting arguments below always refer to common extensions, so do not accidentally reverse the convention.
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