No categorical cardinal for the complete named rational diagram

For any infinite cardinal $\kappa$, insert a dense order without endpoints of size $\kappa$ into an irrational cut of the named rationals. Compare it with an order of the same size having a dense tail above every named rational. Both models satisfy the complete named rational diagram by quantifier elimination, but only the second realizes the type of being above every named rational. This is an isomorphism invariant over the constants, even though it is an infinite type rather than one defining sentence. Thus the diagram is categorical in no cardinal with a model.