Lemma: Fixed compact coefficient continuity

For each fixed two-edge distribution $E$, the map

is continuous in the strong compact-distribution topology. Fixed differentiation, proper projection, and the fixed field-slot transpositions used above preserve this continuity.

Proof. For a bounded family $B$ of smooth functions, a finite-order distribution estimate on the common compact gives

for some finite $N$. This proves strong continuity. Differentiation and projection follow by transposition of continuous maps on smooth tests. The same argument applies to the fixed $L$-slot operation, whose smooth joint-test map is continuous and maps bounded sets to bounded sets.