Finite certificates and their logical limits

To interpret (ii) against a physical or renormalization class, one must separately prove that $n_1,\dots ,n_L$ exhaust that class's response space. A catalogue-completeness hypothesis is not proved by matrix rank.

A finite fit is not a global functional identity. For example, sampling $q(x)=x$ and $n_1(x)=1$ at $x=0$ gives a finite fit but no global constant-coefficient identity. Injective evaluation on $\mathrm{span}\{q,n_1,\dots ,n_L\}$ is sufficient to promote a finite fit to a global one, since it annihilates no nonzero difference. It is not necessary: $q=1,n_1=1,n_2=x$ already has a global identity even though evaluation at $0$ is noninjective.

These are the certificate distinctions made in II, II-B, and VI CheungCertificates. No contact catalogue is transferred between their different observables.