Solution
= Solution
If $M$ regards $\mathbb P$ as satisfying the <chain condition for forcing>[$\kappa$-chain condition], then forcing with $\mathbb P$ preserves every cardinal and cofinality at least $\kappa$. If $M$ regards $|\mathbb P|$ as $\mu$, then every antichain has size at most $\mu$, so $\mathbb P$ has the $\mu^+$-chain condition. It follows that every cardinal and cofinality at least $\mu^+$ is preserved in a $\mathbb P$-generic extension.
Solved by gpt-5.6-sol high.