Lemma: Internal seeds

The seeds underlying $K_{47}$, $K_{5,+}$, and $K_{5,-}$ are defined tensor-product pullbacks, including at $a=z$. The source incidence (eq:Q) is defined.

On the retained patch, the supported sector

is a finite sum of at most three derivatives of smooth coefficients times $\delta (q_R){\delta }^{(4)}(a-z)$.

Proof. The coordinate maps

are invertible linear maps. They identify the three seeds with tensor products of existing retarded distributions.

The map $(v,a,z)\mapsto (v-z,v-a)$ is a submersion. Its pullback of the delta at the origin defines the source incidence before differentiation.

The convolution in (eq:R) is support-proper in the intermediate variable on compact endpoint sets. Its spatial transform is

with value $\Theta (t)t^3/6$ at $\rho =0$, obtained by causal time convolution. The displayed formula for $B$ has the same spatial transform.

On the separated chronological patch,

The source contributes at most two derivatives and the output divergence at most one. Thus $K_8,K_9$ have the asserted form; $K_6$ needs at most two derivatives.