= Bounded rational extension property
{title2=$\ell(s)<q\Longrightarrow(\exists t\in T_\alpha)\ s<_Tt,\ \ell(t)<q$}
For every earlier node and rational cap above its label, an extension at each later level remains below the cap. In the <rationally labelled Aronszajn tree construction> this reserves space for limit-stage unions. Choosing an intermediate rational and forcing successive values toward it ensures a rational supremum and strictly increasing labels even at limit levels.
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