Theorem: Compact-kernel obstruction

For fixed $x$, no linear distributional functional ${w_x}^{\mu }[h]$ of compact spacetime support satisfies

for every compact smooth $\xi$. The same holds for a compact spatially supported time-slice functional using only finitely many time derivatives of $h$.

Proof. Choose a compact set containing the distributional support and $x$. Take a compact vector field constant on its neighborhood, with a nonzero $\mu$-component. Then $K\xi$ vanishes near the support, so its pairing is zero, while the required value is nonzero.

For a time-slice functional, make the field constant in time as well near the relevant support. All finitely many tested derivatives vanish there.