Proposition: Independent translation check

The compact distribution ${\Omega }_{\Phi }$ exists, and

Proof. Compact coefficient–test factors make the first pushforwards proper; both factors in the final tensor product are compact. Transpose the relative vertex derivatives, using symmetry in $\mu ,\nu$. The resulting pairing is against ${\rho }_{\delta }(x-v)$, which is precisely the convolution pairing.