Let $U \subset \mathbb{R}^n$ be open, let $h_0>0$, and let $\mathcal{L}^n$ denote $n$-dimensional Lebesgue measure on $\mathbb{R}^n$. Let $(u_h)_{0<h\le h_0}$ be a semiclassically tempered family of distributions on $U$, meaning that for every compact set $K\Subset U$ there exist integers $m,L\geq 0$ and a constant $C_K>0$ such that

\begin{align*} |u_h(\varphi)|\leq C_K h^{-L}\sum_{|\alpha|\leq m}\sup_{x\in K}|\partial_x^\alpha\varphi(x)| \end{align*}

for every $\varphi\in C_c^\infty(U)$ with $\operatorname{supp}\varphi\subset K$ and every $0<h\leq h_0$. Let $(x_0,\xi_0)\in T^*U$ with $\xi_0\in\mathbb{R}^n$. For $\phi \in C_c^\infty(U)$, define the semiclassical Fourier transform of $\phi u_h$ by

\begin{align*} \mathcal{F}_h(\phi u_h)(\xi) = (\phi u_h)\bigl(x \mapsto e^{-i x\cdot \xi/h}\bigr). \end{align*}

Let $\operatorname{Op}_h(a)$ denote a properly supported semiclassical quantization in the standard properly supported semiclassical pseudodifferential calculus on $U$ for symbols $a\in C_c^\infty(T^*U)$, whose action on compactly supported functions is given by

\begin{align*} \operatorname{Op}_h(a)f(x)=(2\pi h)^{-n}\int_{\mathbb{R}^n}e^{ix\cdot\xi/h}a(x,\xi)\mathcal{F}_h f(\xi)\,d\mathcal{L}^n(\xi), \end{align*}

and whose extension to distributions is defined by its properly supported Schwartz kernel. Assume this calculus has asymptotic composition, microlocal elliptic parametrices for symbols bounded away from zero on compact subsets of $T^*U$, proper-support continuity on local distribution and smooth seminorms, and residual operators whose properly supported Schwartz kernels are $O(h^M)$ with all derivatives for every $M\geq0$. Say that $(x_0,\xi_0)\notin \operatorname{WF}_h(u_h)$ in the operator-theoretic finite-fiber sense if there exists $a\in C_c^\infty(T^*U)$ with $a(x_0,\xi_0)\neq 0$ such that $\operatorname{Op}_h(a)u_h=O(h^\infty)$ in $C^\infty_{\mathrm{loc}}(U)$. Then $(x_0,\xi_0)\notin \operatorname{WF}_h(u_h)$ in the operator-theoretic finite-fiber sense if and only if there exist $\chi\in C_c^\infty(U)$ with $\chi(x_0)\neq 0$ and an open neighbourhood $V\subset\mathbb{R}^n$ of $\xi_0$ such that, for every $N\geq 0$ and every $M\geq 0$, there exist constants $C_{N,M}>0$ and $h_{N,M}\in(0,h_0]$ satisfying

\begin{align*} |\mathcal{F}_h(\chi u_h)(\xi)| \leq C_{N,M} h^M \langle \xi\rangle^{-N} \end{align*}

for all $\xi\in V$ and all $0<h\leq h_{N,M}$.