Proposition: Compact Hessian admission

For finite $R$, $K_{f,R}$ is a compactly supported distributional kernel satisfying

The same exclusion is locally valid for the native kernel $K_f$, but $K_f$ is generally not compactly supported.

Proof. Away from $\mathbf{x}=\mathbf{y}$, the Newton kernel is smooth. The only wavefront covectors from (eq:Kform) are

They have opposite time orientations.

At coincidence, use difference coordinates $(t_u-t_v,\mathbf{y}-\mathbf{x})$. The singular factors are a time delta and a spatial distribution in those independent variables. Their wavefront covectors, transferred to the two slots, have the form

They cannot be simultaneously future or simultaneously past causal. The local terms are derivatives of a full diagonal delta and have the same opposite-slot property.

Differentiation and smooth multiplication do not enlarge these wavefront upper bounds. Both slots are compact for finite $R$. For the native kernel only the local slot is confined by $f$; the other slot has the dipole tail of $P$.