Lemma: Continuity of the smooth derivative sources

At each fixed smooth configuration and each fixed direction, the maps

used in the metric admission proof are continuous into their fixed-support smooth topologies.

Proof. They are linear proper pushforwards of $VE$ multiplied by fixed smooth configuration fields. Lemma lem:forest shows their values are smooth.

Their graphs are closed: if $V_n\to V$ in ${\mathcal{D}}_K$ and the corresponding source functions converge smoothly, Lemma lem:strongcontinuity identifies the smooth limit with the distributional pushforward at $V$. The closed graph theorem for Fréchet spaces gives continuity. Restriction to the fixed-support subspaces preserves the conclusion.