A fixed input for which the jet is empty

The same obstruction can use compact scalar Cauchy data. Choose a spatial cutoff equal to one on a ball containing the initial causal shadows of $\mathrm{supp}g$, and prescribe ${\varphi }_0(0)=0$, ${\dot{\varphi }}_0(0)$ equal to that cutoff. Finite propagation and uniqueness imply that this compact-data solution equals (eq:input) on a neighbourhood of $\mathrm{supp}g$. Thus its stress gives the same witness. This uses the classical Cauchy theory already admitted in X-A, not a gravitational propagator.