Lemma: Response reduction and local order

The rows above are defined distributions. For $D_0=R,B$,

On every compact joint support, the retained components have order at most four.

Proof. The source deltas are a differentiated tensor-product pullback under the submersion $(v,a,z)\mapsto (v-z,v-a)$. Including $D_0(b-v)$ introduces another independent relative variable. Thus the response product is defined in independent relative coordinates before integration.

Move the $v$-derivatives on the deltas to $z,a$. The undifferentiated integration is

The incidence gives local properness over compact external supports, proving (eq:response). All derivatives of the smooth coefficient generated by this identity remain in the expression.

The nonsupported seeds are tensor-product pullbacks under the invertible linear coordinate maps

The supported response seed is likewise a tensor product in independent relative variables. These seeds have local order zero. Each retained row uses at most four derivatives of such seeds with smooth multipliers. Transposition gives the order-four compact-local estimate.