Proposition: When a finite fit is global

Let

If evaluation on $S$ is injective on $U$, then $q_S=N_{S}c$ implies $q={\sum }_a{c}_a{n}_a$ on $\mathcal{B}$.

There exists some injective evaluation family of size at most $\dim U$, but this existence statement supplies neither explicit smearings nor rational evaluations.

Proof. The difference $q-{\sum }_a{c}_a{n}_a$ belongs to $U$ and vanishes on $S$. Injectivity makes it zero.

For existence, start with no evaluation rows. If the common evaluation kernel in $U$ is nonzero, select a nonzero function in that kernel and a label where it is nonzero. Adding that row strictly decreases the kernel dimension. At most $\dim U$ steps give zero kernel.