Lemma: Continuity of the sharp–sharp one-edge family

For fixed $K_1,K_3$, the map

is jointly continuous from ${\mathcal{D}}_{K_1}\times {\mathcal{D}}_{K_3}$ to $({\mathcal{P}}_{\Sigma },{\tau }_{\Sigma })$.

Proof. Its compact coefficient kernels arise from the fixed distribution $S_{\mu \nu }$, multiplication by $f\otimes f^{\prime }$, and the fixed field-slot transpositions of $L$. The compact-distribution estimates and continuity of these transpositions are the same inherited operations used in IX-Free. They control all strong coefficient seminorms.

At a fixed smooth metric configuration, its metric derivative is $q_{a^{(1)}+a^{(2)}}$, where

These are compact smooth sources supported in $K_1$ and $K_3$, respectively. The fixed difference-variable distribution $S$, convolved against a compact smooth test times the fixed smooth field $w$, gives a smooth function continuously in that test. On compact sets this follows directly from finite-order distribution estimates after differentiating the smooth test.

Multiplication by $f$ or $f^{\prime }$ is continuous in the fixed smooth topologies. Thus both sources depend jointly continuously on $(f,f^{\prime })$, and the trace

does so as well by the smooth Green-operator mapping property. After one metric direction, replace $w$ by $L{\eta }_h$. Later metric derivative traces vanish by quadratic degree. Scalar directions have no effect.

These are all coefficient and trace coordinates required by ${\tau }_{\Sigma }$, proving the assertion.