Proposition: Supported transfer of the limit

The coordinate coefficients on $K$, the limiting tests, and their compact pairings depend only on the backgrounds near $D_f$, with the same initial data. Extensions on the declared flat completion agreeing there give the same supported result.

Transport through $\iota$ preserves these compact pairings and their convergence. No global conical propagator or gluing around the worldsheet is obtained.

Proof. The coordinate equations are solved successively by flat finite propagation. Every nested retarded dependence ending in $K$ remains in $D_f$. The limiting initial data are the traces of those same local coefficients on the relevant initial shadow.

For the response terms, all contributing interaction vertices lie in $\mathrm{supp}g\cap J_{\eta }^{-}(K)$. A second advanced test action remains in that causal past. The same local coefficients and fixed completion Green operators therefore give the same tests.

Distribution transport through a chart is defined by transposition on tests and densities. It rewrites the retained pairing. On a fixed compact reserve, the chart and its inverse have bounded derivatives of every required finite order, so the associated fixed-support test topologies are equivalent.