Proposition: Finite-test limitation

A finite family of compact linear evaluations need not determine a conserved smooth symmetric tensor on a regular flat region. If an independently specified finite-dimensional residual class $\mathcal{V}$ contains every residual admissible for a claim, an evaluation map injective on $\mathcal{V}$ does determine residuals within that class.

Proof. For $h\in {C_c}^{\infty }$ in a flat chart set

It is symmetric and conserved. Finitely many evaluations impose finitely many linear conditions on the infinite-dimensional space of $h$'s. Choose a nonzero $h$ in their common kernel. The map $h\mapsto S[h]$ is injective on compact supports: if $S[h]=0$, tracing gives $\square h=0$, then its Hessian is zero, so $h$ is affine and compact support forces $h=0$. Thus the evaluations miss a nonzero conserved tensor. The finite-dimensional statement is injectivity itself.