Proposition: Root prescription

The function $F$ is the advanced zero-future solution of $({\square }_{\eta }+1)F=f$. On $|x{|}_{\mathrm{E}}<4$,

Proof. The equation holds by definition. Compact support gives zero Cauchy data on every sufficiently late slice. The homogeneous difference of two zero-future solutions vanishes by Cauchy uniqueness. This uniqueness follows locally from the energy identity on domains of dependence, so no spatial decay assumption on an arbitrary smooth difference is required. The plateau identities are immediate.