Let $n\in\mathbb{N}$, let $m \in \mathbb{R}$, let $h_0>0$, and write $\langle \xi\rangle=(1+|\xi|^2)^{1/2}$ for $\xi\in\mathbb{R}^n$. Let $a: T^*\mathbb{R}^n\times(0,h_0]\to\mathbb{C}$ be a semiclassical symbol in $S^m_{1,0}(T^*\mathbb{R}^n)$. Define the semiclassical Weyl quantization by the oscillatory integral
for $u \in \mathcal{S}(\mathbb{R}^n)$, $x\in\mathbb{R}^n$, and $0<h\le h_0$, and define the left Kohn--Nirenberg quantization by the oscillatory integral
A residual left symbol means a family $c:T^*\mathbb{R}^n\times(0,h_0]\to\mathbb{C}$ such that, for every $N\in\mathbb{N}$, every multi-indices $\alpha,\beta\in\mathbb{N}_0^n$, there is a constant $C_{N\alpha\beta}>0$ with
for all $x,\xi\in\mathbb{R}^n$ and $0<h\le h_0$. Then there exists a semiclassical symbol $b:T^*\mathbb{R}^n\times(0,h_0]\to\mathbb{C}$ with $b \in S^m_{1,0}(T^*\mathbb{R}^n)$ such that