[proofplan] We define the global principal symbol by localizing an operator in finitely many coordinate charts, taking the usual Euclidean semiclassical principal symbols, and using the coordinate invariance of the principal part to glue these local classes on $T^*M$. The kernel is identified by observing that vanishing of all localized principal symbols forces every localized full symbol to start in $hS^{m-1}$, so a finite partition-of-unity decomposition writes the operator as $h$ times an operator of order $m-1$, up to residual terms, which are harmless. Surjectivity follows by quantizing local representatives of a given global symbol and summing them with a partition of unity. Exactness is then the combination of injectivity of the first inclusion, the kernel computation, and surjectivity of the principal symbol map. [/proofplan] [step:Define the principal symbol by gluing local principal symbols] Choose a finite coordinate cover $(U_j,\kappa_j)_{j=1}^N$ of $M$, where each map $\kappa_j: U_j \to V_j \subset \mathbb{R}^n$ is a smooth coordinate chart. For every point $p \in U_j$, choose functions $\rho,\theta \in C_c^\infty(U_j)$ such that $\theta=1$ on an open neighbourhood of $\operatorname{supp}\rho$ and $\rho=1$ on an open neighbourhood of $p$. For $A \in \Psi_h^m(M)$, define the localized operator $A_{j,\rho,\theta} \in \Psi_h^m(V_j)$ by transporting $\rho A\theta$ through the chart $\kappa_j$. Let $a_{j,\rho,\theta} \in S^m(T^*V_j)$ be any full semiclassical symbol of $A_{j,\rho,\theta}$ in local coordinates. On the open set over which $\rho=1$, the Euclidean semiclassical calculus determines a local principal class $[a_{j,\rho,\theta}] \in S^m(T^*V_j)/hS^{m-1}(T^*V_j)$ independent of the auxiliary cutoffs. On an overlap $U_i \cap U_j$, choose cutoffs in both charts equal to $1$ on a neighbourhood of the point being compared. The two localized descriptions then represent the same operator microlocally near that point, and the coordinate invariance of the semiclassical principal class identifies their symbols through the cotangent lift of the transition map modulo $hS^{m-1}$. Therefore these pointwise local classes agree on overlaps after the natural identification of cotangent coordinates. They glue to a unique global class in $S^m(T^*M)/hS^{m-1}(T^*M)$. Define \begin{align*} \sigma_h^m(A) := \text{the global class obtained by gluing these cutoff-independent local classes}. \end{align*} This definition is independent of the choices of full symbols, cutoffs, and finite atlas: changing a full symbol changes $a_j$ by an element of $hS^{m-1}$ modulo residual symbols, while changing the localization changes only lower-order or residual contributions. Hence $\sigma_h^m$ is a well-defined map \begin{align*} \sigma_h^m: \Psi_h^m(M) \to S^m(T^*M)/hS^{m-1}(T^*M). \end{align*} [guided] The goal is to define a principal symbol on a manifold by reducing to the coordinate-space definition, but the reduction must be local near the point whose symbol is being read. Since $M$ is compact, we may choose a finite coordinate cover $(U_j,\kappa_j)_{j=1}^N$. Fix a point $p \in U_j$, and choose cutoffs $\rho,\theta \in C_c^\infty(U_j)$ such that $\rho=1$ near $p$ and $\theta=1$ on an open neighbourhood of $\operatorname{supp}\rho$. The purpose of these cutoffs is to isolate $A$ inside one chart while keeping the localized operator unchanged microlocally near $p$. For an operator $A \in \Psi_h^m(M)$, transport $\rho A\theta$ through the chart $\kappa_j: U_j \to V_j \subset \mathbb{R}^n$. This gives an operator $A_{j,\rho,\theta} \in \Psi_h^m(V_j)$ in the Euclidean semiclassical calculus. Choose a full local symbol $a_{j,\rho,\theta} \in S^m(T^*V_j)$ for $A_{j,\rho,\theta}$. The local principal symbol near $p$ is not the full symbol itself, but its class \begin{align*} [a_{j,\rho,\theta}] \in S^m(T^*V_j)/hS^{m-1}(T^*V_j) \end{align*} restricted to the region where $\rho=1$. Why do these local classes glue? If two charts $U_i$ and $U_j$ both contain $p$, choose the corresponding cutoffs equal to $1$ near $p$. Then both localized operators represent the same original operator microlocally near $p$. The transformation law for semiclassical symbols says that, under the cotangent lift of the transition map $\kappa_i \circ \kappa_j^{-1}$, the leading class is invariant modulo $hS^{m-1}$. Thus the two representatives determine the same element of the quotient over the overlap. Consequently the cutoff-independent pointwise local classes glue to a global class on $T^*M$, and we define $\sigma_h^m(A)$ to be that class. The quotient by $hS^{m-1}$ is exactly what makes the definition independent of auxiliary choices. A different full local symbol for the same localized operator differs from $a_j$ by a term in $hS^{m-1}$, up to residual symbols. A different choice of cutoffs or atlas changes the localized operator only by terms whose local symbols have lower semiclassical order, again invisible in the quotient. Therefore the map \begin{align*} \sigma_h^m: \Psi_h^m(M) \to S^m(T^*M)/hS^{m-1}(T^*M) \end{align*} is well-defined. [/guided] [/step] [step:Show that every operator in $h\Psi_h^{m-1}(M)$ has zero principal symbol] Let $\Psi_h^{-\infty}(M)$ denote the residual class of semiclassical smoothing operators whose kernels are $O(h^N)$ in all smooth seminorms for every $N \in \mathbb{N}$. Let $A \in h\Psi_h^{m-1}(M)$. Then there exists $B \in \Psi_h^{m-1}(M)$ such that $A = hB$. For each chart localization above, let $b_j \in S^{m-1}(T^*V_j)$ be a local full symbol of the localized operator associated to $B$. Then the corresponding localized full symbol of $A$ is $hb_j$ modulo residual terms. Since $hb_j \in hS^{m-1}(T^*V_j)$, its class in $S^m(T^*V_j)/hS^{m-1}(T^*V_j)$ is zero. Thus every local principal class of $A$ vanishes, and hence \begin{align*} \sigma_h^m(A) = 0. \end{align*} Therefore $h\Psi_h^{m-1}(M) \subset \ker \sigma_h^m$. [/step] [step:Recover an $h\Psi_h^{m-1}$ operator from vanishing local principal symbols] Assume $A \in \Psi_h^m(M)$ and $\sigma_h^m(A)=0$. With the finite coordinate cover, partition of unity, and auxiliary cutoffs fixed above, let $A_j \in \Psi_h^m(V_j)$ be the localization of $\chi_j A\psi_j$ and let $a_j \in S^m(T^*V_j)$ be a full local symbol of $A_j$. The condition $\sigma_h^m(A)=0$ means precisely that, for every $j$, the local principal class $[a_j]$ is zero in $S^m(T^*V_j)/hS^{m-1}(T^*V_j)$. Hence \begin{align*} a_j \in hS^{m-1}(T^*V_j) \end{align*} modulo residual symbols. Therefore each localized operator $A_j$ belongs to $h\Psi_h^{m-1}(V_j)$ modulo $\Psi_h^{-\infty}(V_j)$. Transporting back to $U_j$, there exist operators $B_j \in \Psi_h^{m-1}(M)$, supported in the coordinate patch localization, and residual operators $R_j \in \Psi_h^{-\infty}(M)$ such that \begin{align*} \chi_j A\psi_j = hB_j + R_j. \end{align*} Since $\psi_j=1$ near $\operatorname{supp}\chi_j$ and $\sum_{j=1}^N \chi_j=1$ on $M$, the standard properly supported localization identity gives \begin{align*} A = \sum_{j=1}^N \chi_j A\psi_j + R_0 \end{align*} for some residual operator $R_0 \in \Psi_h^{-\infty}(M)$. Substituting the previous decomposition yields \begin{align*} A = h\sum_{j=1}^N B_j + \sum_{j=1}^N R_j + R_0. \end{align*} Because the cover is finite and each $B_j$ lies in $\Psi_h^{m-1}(M)$, the sum $B := \sum_{j=1}^N B_j$ lies in $\Psi_h^{m-1}(M)$. Also, by the stated residual convention, if a residual kernel is $O(h^N)$ in every smooth seminorm for every $N \in \mathbb{N}$, then multiplying it by $h^{-1}$ is still $O(h^N)$ in every such seminorm after replacing $N$ by $N+1$. Thus residual operators lie in every semiclassical order class and remain residual after multiplication by $h^{-1}$; hence there exists $R \in \Psi_h^{-\infty}(M) \subset \Psi_h^{m-1}(M)$ such that \begin{align*} \sum_{j=1}^N R_j + R_0 = hR. \end{align*} Thus \begin{align*} A = h(B+R), \end{align*} with $B+R \in \Psi_h^{m-1}(M)$. Therefore $A \in h\Psi_h^{m-1}(M)$, proving \begin{align*} \ker \sigma_h^m \subset h\Psi_h^{m-1}(M). \end{align*} [guided] Now suppose the global principal symbol of $A$ vanishes. We must prove that $A$ is not merely lower order, but actually has the precise form $hB$ with $B \in \Psi_h^{m-1}(M)$. Fix the same finite coordinate cover $(U_j,\kappa_j)_{j=1}^N$, partition of unity $\chi_j \in C_c^\infty(U_j)$, and auxiliary cutoffs $\psi_j \in C_c^\infty(U_j)$ with $\psi_j=1$ near $\operatorname{supp}\chi_j$. For each $j$, transport $\chi_j A\psi_j$ to an operator $A_j \in \Psi_h^m(V_j)$, where $V_j=\kappa_j(U_j) \subset \mathbb{R}^n$, and let $a_j \in S^m(T^*V_j)$ be a full local symbol. The condition $\sigma_h^m(A)=0$ says exactly that each local class $[a_j]$ is zero in the quotient \begin{align*} S^m(T^*V_j)/hS^{m-1}(T^*V_j). \end{align*} Being zero in this quotient means that the representative lies in the subspace being quotiented out. Therefore \begin{align*} a_j \in hS^{m-1}(T^*V_j) \end{align*} modulo residual symbols. Equivalently, the localized operator $A_j$ has the form $h$ times an order $m-1$ operator, up to a residual operator. After transporting this statement back to $M$, there are operators $B_j \in \Psi_h^{m-1}(M)$ and $R_j \in \Psi_h^{-\infty}(M)$ such that \begin{align*} \chi_j A\psi_j = hB_j + R_j. \end{align*} We now reconstruct $A$ from these localized pieces. Since $\sum_{j=1}^N \chi_j = 1$ and $\psi_j=1$ near $\operatorname{supp}\chi_j$, inserting $\psi_j$ after $A$ does not change the part of the kernel selected by $\chi_j$ except by a residual term. Thus the properly supported localization identity gives \begin{align*} A = \sum_{j=1}^N \chi_j A\psi_j + R_0 \end{align*} for some $R_0 \in \Psi_h^{-\infty}(M)$. Substituting the local decompositions gives \begin{align*} A = h\sum_{j=1}^N B_j + \sum_{j=1}^N R_j + R_0. \end{align*} Here compactness is used in a concrete way: the cover is finite, so the sum $\sum_{j=1}^N B_j$ is still an element of $\Psi_h^{m-1}(M)$. The residual remainder is also harmless. Under the stated residual convention, a residual kernel is $O(h^N)$ in every smooth seminorm for every $N \in \mathbb{N}$. Dividing by one power of $h$ preserves this property because $h^{-1}O(h^{N+1})=O(h^N)$ for each fixed $N$. Hence the residual sum can be written as $hR$ for some $R \in \Psi_h^{-\infty}(M) \subset \Psi_h^{m-1}(M)$. Therefore \begin{align*} A = h\left(\sum_{j=1}^N B_j + R\right). \end{align*} The operator inside parentheses belongs to $\Psi_h^{m-1}(M)$, so $A \in h\Psi_h^{m-1}(M)$. This proves the reverse inclusion \begin{align*} \ker \sigma_h^m \subset h\Psi_h^{m-1}(M). \end{align*} [/guided] [/step] [step:Quantize a global symbol class to prove surjectivity] Let $[a] \in S^m(T^*M)/hS^{m-1}(T^*M)$ be arbitrary, and choose a representative $a \in S^m(T^*M)$. Using the same finite chart cover, choose local representatives $a_j \in S^m(T^*V_j)$ for the restrictions of $a$ under the cotangent lifts of the charts. Let $\operatorname{Op}_{h,j}(a_j)$ denote the local semiclassical quantization of $a_j$ on $V_j$. Choose cutoffs $\chi_j,\psi_j \in C_c^\infty(U_j)$ as above. Let $M_{\chi_j}: C^\infty(M) \to C_c^\infty(U_j)$ and $M_{\psi_j}: C^\infty(M) \to C_c^\infty(U_j)$ denote multiplication by $\chi_j$ and $\psi_j$, respectively. Let $(\kappa_j^{-1})^*: C_c^\infty(U_j) \to C_c^\infty(V_j)$ be pullback by $\kappa_j^{-1}$, and let $\kappa_j^*: C_c^\infty(V_j) \to C_c^\infty(U_j)$ be pullback by $\kappa_j$. Define $A \in \Psi_h^m(M)$ by summing the transported local quantizations with cutoffs: \begin{align*} A := \sum_{j=1}^N M_{\chi_j}\, \kappa_j^*\, \operatorname{Op}_{h,j}(a_j)\, (\kappa_j^{-1})^*\, M_{\psi_j}. \end{align*} The sum is finite, so $A \in \Psi_h^m(M)$. In the chart $U_j$, the principal symbol of the $j$th summand is $\chi_j a_j$ modulo $hS^{m-1}$, and the coordinate invariance of the principal part identifies these local symbols with the corresponding pieces of the global symbol $a$. Summing over $j$ and using $\sum_{j=1}^N \chi_j=1$ on $M$, the global principal class is \begin{align*} \sigma_h^m(A) = [a]. \end{align*} Thus $\sigma_h^m$ is surjective. [/step] [step:Assemble the kernel computation and surjectivity into the exact sequence] The inclusion map $h\Psi_h^{m-1}(M) \to \Psi_h^m(M)$ is injective because it is the set-theoretic inclusion of a linear subspace. The previous two kernel inclusions give \begin{align*} \ker \sigma_h^m = h\Psi_h^{m-1}(M). \end{align*} The quantization construction proves that $\sigma_h^m$ is surjective. Therefore the sequence \begin{align*} 0 \to h\Psi_h^{m-1}(M) \to \Psi_h^m(M) \xrightarrow{\sigma_h^m} S^m(T^*M)/hS^{m-1}(T^*M) \to 0 \end{align*} is exact. This proves the theorem. [/step]