Proposition: Supported patch agreement

All retained pairings depend only on the backgrounds and sources near the contributing compact subset of $D_f$. Completion extensions agreeing there give the same supported result. Transport through $\iota$ preserves the pairings, their continuity, and the transported wavefront bounds.

Proof. Every edge in the displayed trees is retarded towards the observation root or advanced after transposition. All retained source times are nonnegative, and all vertices remain in $J_{\eta }^{-}(K_f)$. Finite propagation therefore restricts the calculation to $D_f$. Nested scalar and coordinate responses obey the same causal restriction.

Distribution transport is defined by transposition on compact tests and densities. The chart and its inverse have bounded derivatives of every required finite order on the compact reserve. The ordinary test topologies and transported microlocal stages give the same local pairings.