Bounded rational extension property ($\ell (s)<q⟹(\exists t\in T_{\alpha })s{<}_{T}t,\ell (t)<q$)

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.