Proposition: Local smooth response

Suppose the local source is smooth and vanishes for $t\le 0$. On a finite slab, the response is the unique smooth zero-data solution of $L_{\mathrm{gr}}{u}_{\mathrm{D}}={\mathcal{J}}_{\mathrm{D}}$. On a compact observation region $K$, it is determined by the source on

Spatial cutoffs agreeing near $C_K$ give the same local response. Its support lies in $J^{+}(\mathrm{supp}{\mathcal{J}}_{\mathrm{D}})$, not necessarily in the ramp-transition layer.

Proof. The truncated causal past $C_K$ is compact. Cut off the input fields spatially outside a neighborhood of its projection; locality preserves the source near $C_K$. Wave Cauchy theory and finite propagation give equal responses near $K$. These local constructions agree on overlaps. Zero-data uniqueness and finite propagation prove the final claims BGP.