Direct application versus polarization

The reported XII-D compact-root theorem supplies, for fixed $f,g$ and collar $\vartheta$, a compact smooth root $a_{f,g}[\varphi ]$ and the diagonal identity

on its stated domain. The root has support in a fixed compact subset of $J^{-}(\mathrm{supp}f)\cap \{t\ge -\delta \}$ XIID.

A polarized source $S_{\lambda }$ is not itself a single metric argument $\gamma$ of this theorem. Thus substituting $S_{\lambda }$ for $\gamma$ is not a direct application of (eq:root-identity). This does not prohibit polarization of valid diagonal values.