= Solution
A <forcing> <preserves cofinalities> over $M$ if every <generic extension> has
$$
\boxed{\operatorname{cf}^{M[G]}(\alpha)=\operatorname{cf}^{M}(\alpha)
\quad\text{for every ordinal }\alpha\in M.}
$$
The two values are compared as <ordinals>, using preservation of <ordinals> by <forcing>. Equivalently the <forcing> asserts that no ground <ordinal> acquires a smaller <cofinality>. The assertion concerns all <ordinal> <cofinalities>, not just that one specified <cardinal> remains uncountable.
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