Closed forcing adds no short ordinal sequences

If a ground-model order is $\lambda$-closed, it adds no ordinal-valued functions with domain $\mu <\lambda$. Recursively decide each name value inside the ground model, using closure at intermediate limit stages and once more at the end. Below each starting condition, conditions deciding the entire function are dense. A generic filter meets them by the dense-below generic meeting lemma. The recursion must be internal to the ground model, since its closure property only covers its own sequences.

A $\lambda$-closed forcing order preserves a ground-model cardinal number $\lambda$, even when it is singular. A proposed surjection from an ordinal below $\lambda$ to $\lambda$ would be an old function because closed forcing adds no short ordinal sequences, contradicting old cardinalhood. This concerns $\lambda$ and smaller cardinals; closure alone does not supply preservation of all larger cardinals.