Proposition: Finite tests can miss a conserved symmetric perturbation

For any finite family of compact linear test evaluations, there is a nonzero smooth compactly supported symmetric conserved tensor perturbation invisible to that family but with nonzero trace. This assertion concerns the unrestricted smooth tensor-distribution class, not the full Family 282 framework.

Proof. Choose a nonempty relatively compact open set $U\subset M$. For $h\in {C_c}^{\infty }(U)$, put

It is symmetric and conserved. Every declared finite evaluation of $S[h]$, including a tested current divergence, is a linear functional of $h$. The common kernel of finitely many such functionals in the infinite-dimensional space ${C_c}^{\infty }(U)$ contains a nonzero $h$.

Its trace is $-3\square h$. A compactly supported smooth solution of $\square h=0$ is zero: it has zero Cauchy data before its compact time support, and uniqueness of the wave equation forces zero. Thus the chosen perturbation has nonzero trace.