Proposition: Fixed-support continuity of the first stage

For fixed compact observation supports, both maps in (eq:first) depend jointly continuously on their tests, with target ${{\mathcal{V}}_{\Sigma }}^{[3]}$ carrying the inherited topology. This is a test-family statement, not a quantum hypocontinuity theorem on ${\mathcal{P}}_{\Sigma }$.

Proof. The pointwise terms have the fixed-support test-family continuity proved in IX-A IXA (thm:admission).

For the contraction term, its compact coefficient kernel is obtained from $f(x)g(y)K(x,y)$ by a fixed scalar-slot derivative and bundle placement. On fixed compact supports, multiplication of a fixed distribution by varying smooth coefficients and fixed finite differentiation are continuous into the strong compact-distribution topology. This follows directly by applying the distribution to bounded smooth tests and using its finite-order estimate on the common compact set.

The proof of IX-A's selected-output admission shows that every first metric derivative of this contraction, at a smooth configuration and after directional differentiation, is a smooth compact kernel IXA (prop:knownoutputs, item 3). Its boundary coordinates therefore vanish. The coefficient continuity consequently gives continuity in ${\tau }_{\Sigma }$. Conjugation gives the same result for the reverse contraction.