Corollary: Causal vanishing and supported agreement

If

both new maps vanish.

For $c$ supported in $\mathcal{O}$, all effective interaction vertices lie in $D_c\subset D_{\ast }$. Two extensions of the background data on the same declared flat completion that agree near $D_c$ give the same new output tests. The analogous coordinate-data agreement for $c[q]$ is the inherited X-A-ST completion contract.

Proof. The vanishing follows from (eq:new-support). The second-vertex expression is local in its background coefficients on $S\cap J_{\eta }^{-}(\mathrm{supp}c)$. For the iterated expression, every contributing chain has

Since $S$ lies in the open slab, these vertices and their relevant causal intervals lie in $D_c$. The Green kernels are the same fixed completion kernels. Thus agreement of the coefficients near $D_c$ gives agreement of the resulting tests.