Finite sample logic

If defined functions $q,n_1,\dots ,n_L$ are sampled at finitely many points, yielding a vector $q_S$ and a column matrix $N_S$, then exactly one alternative holds:

To prove this, project $q_S$ orthogonally onto $\mathrm{im}{N}_S$. If the residual is nonzero, use it as $z$.

A finite fit is not a global identity. For example, $q(x)=x$ and $n_1(x)=1$ fit at $x=0$ but not globally. Injective evaluation on the stated finite-dimensional function span is sufficient to promote a fit to an identity, because it annihilates no nonzero difference. It is not necessary: $q=1,n_1=1,n_2=x$ has a global fit even though evaluation at $0$ is noninjective.

These are finite-table distinctions comparable with II and II-B CheungCertificates. They do not provide a complete quantum counterterm catalogue or a renormalization separator.