= Uniformly narrow regular-height tree branch theorem
{title2=$|T_\alpha|<\kappa<\lambda\ \Longrightarrow\ T\text{ has a cofinal branch}$}
For infinite <regular cardinals> $\kappa<\lambda$, a $\lambda$-tree whose levels all have size less than $\kappa$ has a <cofinal branch>. At levels of <cofinality> $\kappa$, bound all heights distinguishing distinct predecessor chains. The <Fodor lemma> makes this bound constant on a stationary set; one predecessor at that height occurs stationarily often. The resulting predecessor chains agree below their common heights and unite into a branch through every level. Distinct limit-level nodes with identical predecessor chains do not affect this argument.
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