= Unbounded-extension kernel of a regular tree
{title2=$T^*=\{t:\sup\operatorname{ht}(T_{\ge t})=\kappa\}$}
For a <kappa-tree>, retain nodes whose extension heights are unbounded in $\kappa$. The retained nodes are predecessor-closed. Each level remains nonempty: otherwise regularity would bound the union of fewer than $\kappa$ bounded extension <sets>. The same argument above a retained node ensures retained extensions at every later level. Singular height invalidates this proof and the unrestricted conclusion.
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