Lemma: Finite response reduction and internal order

For $D=R$ or $D=B$,

All derivatives, including those of smooth coefficients, are retained.

The internal distributions are defined. For every $L⋐W$,

Proof. Use

and the analogous identity for $a$. External derivatives commute with the distributional integration. The undifferentiated integration is

This proves (eq:response-reduction).

The source incidence is a differentiated pullback under the submersion $(v,a,z)\mapsto (v-z,v-a)$. Adding $b-v$ supplies independent propagation variables. Its integration is locally proper because the incidence forces $v=a=z$ on compact external test supports.

The $K_{47}$ and $K_5$ seeds are tensor-product pullbacks under the invertible maps

The response seeds in (eq:response-reduction) are likewise tensor-product pullbacks.

Both $R$ and $B$ are locally finite measures. For $R$, the future-cone radial measure is

Every internal row has at most four derivatives of order-zero seeds, multiplied by smooth coefficients. For $K_8$, this bound uses two source derivatives and two displayed output derivatives. Transferring derivatives to the coefficient–test product proves (eq:internal-order).