Lemma: Exact response reduction and finite order

All displayed internal distributions exist. For $D=R,B$,

Every coefficient derivative generated by this formula is retained. For $L⋐W$,

Proof. Move the derivatives on the two incidence deltas to their external variables. The undifferentiated integration is

which gives (eq:response).

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

The source incidence is the differentiated pullback under the submersion $(v,a,z)\mapsto (v-z,v-a)$. The response integrations are locally proper because the incidence forces $v=a=z$ on compact external supports.

Both $R$ and $B$ are locally finite measures. Each row is a finite sum of at most four derivatives of such tensor-product seeds, multiplied by smooth coefficients. Transferring derivatives to coefficient–test products proves the estimate.