Lemma: Exact smoothed crossed array

For sufficiently small scales on the relevant relative image,

Proof. The $\kappa$-marginal of $({\rho }_4{)}_{\delta }$ is ${\delta }^{-1}p(\cdot /\delta )$. Hence

The product profile makes the eight-variable convolution the tensor product of these convolutions.

The relative cutoff is one near the compact image needed by the test. Compact profile support gives exact local equality with the uncut convolution at sufficiently small scales, including its required derivatives.