Proposition: Patch agreement

Every retained coordinate, scalar-response, and matter-tree pairing depends only on the backgrounds and sources near the relevant compact subset of $D_f$. Completion extensions agreeing there give the same supported result. Transport through $\iota$ preserves the pairings and their stated continuity.

Proof. Every displayed edge is retarded towards the observation root, or advanced after transposition from that root. All vertices remain in $J_{\eta }^{-}(K)$, and the zero-data coordinate problems stop at $\Sigma$. Thus finite propagation restricts their dependence to $D_f$. The explicit tree formula has both scalar vertices in $C$.

Distribution transport through a smooth embedding is defined by the corresponding test-density pairing. On the fixed compact reserve, the chart and its inverse have bounded derivatives of every needed finite order. The smooth test topologies and transported microlocal bounds are consequently the corresponding equivalent local descriptions.