Theorem: Completed compact root

For fixed $f,g,\vartheta$, the map

is continuous and linear, with range in a fixed compact support contained in

It reproduces the completed insertion:

The effective roots ${\sigma }_{\epsilon }{a}_{f,g}$, ${\sigma }_{\epsilon }^{\prime }{a}_{f,g}$, and ${\sigma }_{\epsilon }^{\prime \prime }{a}_{f,g}$ are compact and supported in $C_K$. The insertion is independent of the collar cutoff.

Proof. The first root (eq:alpha) is supported in $C_K$. Advanced support gives $\mathrm{supp}\beta \subset J^{-}(C_K)\subset J^{-}(K)$. Intersecting this past with the bounded time collar makes its support compact.

Multiplication by fixed smooth coefficients, differentiation, and wave propagation on fixed causal compact sets are continuous smooth-function operations. This proves continuity of the root map into its fixed compact support stage.

Transpose the metric response in (eq:CD). The resulting integral is $\int \beta {\mathcal{J}}_{\mathrm{D}}$. Only nonnegative times below the latest observation time contribute: the source vanishes for $t\le 0$, and the advanced root vanishes above the observation. Hence insertion of $\vartheta$ changes nothing and gives (eq:root-identity). All three effective roots vanish smoothly at nonpositive times and have support in $C_K$.