Let $M$ be a smooth $n$-dimensional manifold, let $h_0>0$, and for $h \in (0,h_0]$ fix a standard properly supported semiclassical left quantization $\operatorname{Op}_h$ constructed from a locally finite coordinate cover, a subordinate partition of unity, and compactly supported coordinate cutoffs. In each coordinate chart $U \subset M$ identified with an open subset of $\mathbb{R}^n$, the coordinate-local quantization of a local symbol has kernel

\begin{align*} K_a(x,y;h)=(2\pi h)^{-n}\int_{\mathbb{R}^n} e^{i(x-y)\cdot \xi/h} a(x,\xi;h)\,d\mathcal{L}^n(\xi). \end{align*}

Suppose $a \in S^{-\infty}(T^*M)$ is rapidly vanishing in $h$ in every smoothing symbol seminorm: for every coordinate chart $U \subset M$, every compact set $K \subset U$, every pair of multi-indices $\alpha,\beta \in \mathbb{N}_0^n$, every integer $m \geq 0$, and every $N \in \mathbb{N}$, there exists $C_{\alpha,\beta,m,N,K}>0$ such that

\begin{align*} \sup_{x \in K}\sup_{\xi \in \mathbb{R}^n} \langle \xi\rangle^m |\partial_x^\alpha \partial_\xi^\beta a(x,\xi;h)| \leq C_{\alpha,\beta,m,N,K} h^N \end{align*}

for all $h \in (0,h_0]$, where $\langle \xi\rangle=(1+|\xi|^2)^{1/2}$.

Then $\operatorname{Op}_h(a)$ is residual: in every pair of coordinate charts and for every compact set $K \subset U \times U$, every pair of multi-indices $\alpha,\beta \in \mathbb{N}_0^n$, and every $N \in \mathbb{N}$, its local Schwartz kernel $K_a$ satisfies

\begin{align*} \sup_{(x,y)\in K} |\partial_x^\alpha \partial_y^\beta K_a(x,y;h)| \leq C_{\alpha,\beta,N,K}h^N. \end{align*}

Conversely, if $R_h$ is a properly supported residual operator for $h \in (0,h_0]$, then, relative to the same fixed coordinate cover, partition of unity, cutoffs, and local left-quantization convention, each localized coordinate kernel determines a local symbol by semiclassical Fourier transform in the difference variable. The resulting locally finite family of rapidly vanishing smoothing symbols represents $R_h$ modulo residual equality: the quantization formed by summing these fixed coordinate-local symbol pieces differs from $R_h$ by a residual operator. Equivalently, $R_h$ is represented modulo residual equality by a symbol $r \in S^{-\infty}(T^*M)$ whose smoothing symbol seminorms satisfy the same $O(h^N)$ estimates for every $N$.