= Solution
Suppose a ground <cardinal> $\lambda\ge\kappa$ became smaller in the extension. There would be a <surjection> $f:\alpha\to\lambda$ for some <ordinal> $\alpha<\lambda$. A finite domain cannot map onto an infinite <ordinal>, so use part (a) to cover $f$ by a ground <function> $y$ of small value sets. Its range is contained in $U=\bigcup_{\zeta<\alpha}y(\zeta)$.
If $\lambda>\kappa$, the ground cardinality of $U$ is at most $|\alpha|^M\cdot\kappa<\lambda$. If $\lambda=\kappa$, regularity bounds a union of fewer than $\kappa$ sets of size less than $\kappa$ by a <cardinal> less than $\kappa$. In either case $U$ is a proper ground <subset> of $\lambda$, and the same missing <ordinal> remains missing in the extension, contradicting surjectivity.
Hence \b[all ground cardinalities at least $\kappa$ are preserved]. Ground bijections also remain bijections, so old noncardinals cannot turn into <cardinals>. The assertion does not prevent collapse of smaller <cardinals>, and consequently does not by itself preserve the aleph index of $\kappa$.
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