Theorem: Local construction and cutoff independence

For every smooth $\gamma$, (eq:response) defines a unique smooth zero-initial-data solution on the slab. For every compact observation region $K\subset {M_T}^{\circ }$, it can be computed using a fixed spatial cutoff equal to one near

The result on a neighborhood of $K$ is independent of that cutoff. Moreover,

The support assertion is not a layer-support assertion.

Proof. The set $C_K$ is compact. Choose a spatial cutoff $\zeta$ equal to one near its spatial projection and replace $\gamma$ by $\zeta \gamma$ for this local calculation. The modified source is spatially compact on bounded time intervals and vanishes smoothly for $t\le 0$. Smooth wave Cauchy theory gives its zero-data response.

All derivatives of the two metric configurations agree near $C_K$, so their local differential sources agree there. Finite propagation implies equality of the resulting responses near $K$. Two admissible cutoffs therefore give the same result. Applying this argument on compact regions defines a consistent smooth response throughout the slab.

The source is past compact in the causal sense: its intersection with the causal past of any compact set is compact. Consequently the ambient retarded prescription agrees with the locally constructed zero-data solution. Finite propagation gives (eq:response-support); homogeneous zero-data uniqueness gives uniqueness.