Proposition: Degree-four relation kernels

The kernel $\mathsf{P}[v]$ is defined and compact in its field variables, and

Consequently

The right relation is the exchanged equation with $v(x,y)=v_R(y,x)$.

Proof. Before smooth multiplication, $J(x,z_1)$ and $\delta (z_2-z_3)$ are tensor factors in disjoint variables. Observation integration by parts gives the positive derivative in (eq:Prelationkernel). Saturated finite propagation gives compact field support. One scalar derivative and three metric derivatives recover the symmetric kernel with factor $3!$. Polarization and product-test density prove the equivalence.