Theorem: Conditional continuous map

Under Assumption ass:map-limit, the limits define a continuous linear map

and each fixed-input family converges in the normal topology. The map has exact reconstruction descent and agrees with the ordinary canonical contraction on its verified domain.

Proof. The local finite-order bounds make the pointwise limits distributions and imply strong convergence by the finite-net argument of Theorem thm:fixed-limit. The Fourier bounds pass to the limit and give the wavefront inclusion.

The same inequalities bound every strong and Fourier seminorm of the lifted limit map by continuous input seminorms. It is therefore continuous. Every finite regularization annihilates $\ker \mathsf{Rec}$, so the limit does too; Proposition prop:descent gives the descended map.

For convergence of a degree-$r$ Fourier seminorm, the uniform degree-$r+1$ bound makes the tail of the difference at most $C_A/(1+R)$ above radius $R$. On the frequency ball the tests $\chi e^{-i\cdot \xi }$ form a bounded test set, and strong convergence gives uniform convergence there. This proves normal convergence. Canonical agreement follows from Corollary cor:canonical.