Proposition: Finite probes do not cover unrestricted tensors

Let $O$ be a nonempty open spacetime region and let ${\ell }_1,\dots ,{\ell }_m$ be any finite family of linear compact-test pairings on smooth symmetric tensors. There is a nonzero smooth compactly supported symmetric conserved tensor $H_{\mu \nu }$, supported in $O$, with nonzero trace and

Proof. Choose $m+1$ disjoint small balls in $O$, and nonzero real bumps $u_1,\dots ,u_{m+1}$ supported in them. Define

These tensors are symmetric and conserved, with traces $-3\square u_r$. No nonzero compact spacetime bump solves the homogeneous wave equation: its Cauchy data vanish sufficiently far in the past, so uniqueness would force it to vanish. Thus every displayed trace is nonzero.

The map

has nontrivial kernel. Select a nonzero vector in that kernel. Disjoint supports make the traces linearly independent, so the resulting tensor has nonzero trace and is nonzero.