Proposition: Supported patch agreement

For retained laboratory tests, the response and dressing tests agree for two auxiliary completion extensions that coincide near the complete causal support used by their calculation and have the same inherited coordinate initial data.

The resulting compact pairings are unchanged by transport through $\iota$. This is exact patch agreement, not a global conical representation.

Proof. Coordinate coefficients agree by the supplied X-A-ST completion contract. For the response, ${\mathsf{E}}_{\mathrm{A}}c$ at an interaction point uses only the causal region joining that point to $\mathrm{supp}c$. On a retained calculation this region lies in the declared saturated compact support. The Green equations there have the same source and terminal prescription, so uniqueness gives the same test action. Applying the same local $L_{1,g}[h]$ gives the same $d_c$.

Transport of a compact distributional pairing through a chart is defined by transposition on tests and densities. It rewrites, rather than changes, that pairing.