Assumption: Uniform map-limit hypotheses

Fix a closed cone $\Gamma \subset T^{\ast }U\setminus 0$. For the complete family suppose:

1. Every pairing ${\mathsf{F}}_{\delta }(\mathsf{Rec}A)(\Phi )$ converges.

2. For each compact $L$, there are $C_L,m_L$ and a continuous input seminorm $p_L$ such that

   $$|{\mathsf{F}}_{\delta }(\mathsf{Rec}A)(\Phi )|\le C_L{p}_L(A)\Vert \Phi {\Vert }_{m_L}$$

   uniformly in $\delta$.

3. Every Fourier seminorm admissible for $\Gamma$ has a bound

   $$q_{r,\chi ,V}({\mathsf{F}}_{\delta }(\mathsf{Rec}A))\le Cp(A)$$

   with a continuous input seminorm, uniformly in $\delta$.