Proposition: Numerical scaling and declared subtraction orders

For every $\mathsf{k}=\{p,q\}\in \mathcal{N}$,

For a nonzero kernel the scaling degree is exactly $4+N_{\mathsf{k}}$. The declared subtraction order is always

For a zero kernel, this declared upper bound is retained and its selected extension is zero; its scaling degree is not assigned a finite value.

Proof. Dilations are diffeomorphisms of the punctured domain and commute with the defined distributional product. Apply Lemma lem:homog to its two factors. A nonzero homogeneous distribution of degree $-4-N_{\mathsf{k}}$ has scaling degree $4+N_{\mathsf{k}}$. The zero distribution satisfies the upper bound.