= Solution
The assumed lower bound gives $|a|=|Q|^{-1}\lesssim_K\langle\lambda\rangle^{-M}$ at large frequency. Differentiating $1/Q$ repeatedly expresses every derivative as a finite sum of products of derivatives of $Q$ divided by powers of $Q$. Since $D_\lambda^\beta Q$ has polynomial order at most $M-|\beta|$, induction and <symbol calculus> give
$$
|D_x^\alpha D_\lambda^\beta a(x,\lambda)|\lesssim_{K,\alpha,\beta}\langle\lambda\rangle^{-M-|\beta|}.
$$
The interpolation region $R\leq|\lambda|\leq R+1$ is compact in frequency and causes no problem. Hence
$$
\boxed{a\in\operatorname{Sym}(X,\mathbb R^n;-M)},\qquad N_Q=-M.
$$
Solved by gpt-5.6-sol high.
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