Unbounded-extension kernel of a regular tree ($T^{\ast }=\{t:\sup \mathrm{ht}(T_{\ge t})=\kappa \}$)

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.