[proofplan] We prove the result in local phase charts and then check that the local construction is invariant under the allowed changes of phase. In a nondegenerate conic parametrization, a Lagrangian distribution is represented by an oscillatory integral with a classical amplitude, and its principal symbol is the leading homogeneous amplitude transported to $\Lambda$ by the stationary-phase half-density and Maslov factor. The invariance of this transport under changes of auxiliary variables and stabilization is precisely the transition law defining $\mathcal{M}_\Lambda\otimes\Omega_\Lambda^{1/2}$. The kernel statement is the elementary fact that a classical symbol has zero leading homogeneous component exactly when its order drops by one, and surjectivity follows by choosing local phase charts, prescribing compatible leading amplitudes, and patching them with a locally finite conic [partition of unity](/page/Partition%20of%20Unity). [/proofplan]

[step:Represent the distribution in a nondegenerate conic phase chart] Let $(U,x)$ be a coordinate chart on $X$, where $U\subset X$ is open and $x:U\to x(U)\subset\mathbb{R}^n$ is a smooth coordinate map. Let $\Gamma\subset\mathbb{R}^N\setminus 0$ be an open conic set, and let \begin{align*} \phi:U\times\Gamma\to\mathbb{R} \end{align*} be a smooth real-valued function, homogeneous of degree $1$ in $\theta\in\Gamma$, which is nondegenerate in the standard conic sense. Define the critical set \begin{align*} C_\phi:=\{(p,\theta)\in U\times\Gamma:\partial_{\theta_j}\phi(p,\theta)=0 \text{ for every } j\in\{1,\dots,N\}\}. \end{align*} The phase parametrizes an open conic subset $\Lambda_\phi\subset\Lambda$ by the smooth map \begin{align*} \kappa_\phi:C_\phi\to\Lambda_\phi,\qquad (p,\theta)\mapsto (p,\partial_x\phi(p,\theta)). \end{align*} Let $\mathcal{L}^N: \mathcal{B}(\mathbb{R}^N)\to[0,\infty]$ denote the $N$-dimensional [Lebesgue measure](/page/Lebesgue%20Measure) on $\mathbb{R}^N$, so that $d\mathcal{L}^N(\theta)$ is the integration measure in the auxiliary-variable oscillatory integral below. By the definition of $I^m(X,\Lambda;\Omega_X^{1/2})$, every $u\in I^m(X,\Lambda;\Omega_X^{1/2})$ is locally, modulo a smooth half-density, a finite sum of oscillatory integrals of the form \begin{align*} u_\phi(p)=(2\pi)^{-(N+2n)/4}\int_\Gamma e^{i\phi(p,\theta)}a(p,\theta)\,d\mathcal{L}^N(\theta)\otimes |dx|^{1/2}, \end{align*} where \begin{align*} a:U\times\Gamma\to\mathbb{C} \end{align*} is a classical symbol of order \begin{align*} \mu:=m+\frac{n}{4}-\frac{N}{2}. \end{align*} Thus $a$ has a classical asymptotic expansion \begin{align*} a\sim \sum_{j=0}^{\infty}a_{\mu-j}, \end{align*} where each \begin{align*} a_{\mu-j}:U\times\Gamma\to\mathbb{C} \end{align*} is smooth and homogeneous of degree $\mu-j$ in the variable $\theta$ for $|\theta|$ sufficiently large on the conic support under consideration. [/step]

[step:Define the local principal symbol from the leading amplitude] Let \begin{align*} a_\mu:U\times\Gamma\to\mathbb{C} \end{align*} denote the leading homogeneous component of the classical amplitude $a$. Define $\mathcal{Q}_\phi^\mu$ to be the quotient of homogeneous degree-$\mu$ smooth functions on the chosen conic support in $U\times\Gamma$ by the following null-symbol relation: two leading terms are equivalent when their difference is a finite sum of a homogeneous term whose support is contained in the nonstationary region $\{\partial_\theta\phi\ne 0\}$, a term in the ideal generated by $\partial_{\theta_1}\phi,\dots,\partial_{\theta_N}\phi$, and a term of homogeneous degree at most $\mu-1$. By the null-symbol part of the local calculus included in the statement, each such difference changes the oscillatory integral by an element of $I^{m-1}(X,\Lambda;\Omega_X^{1/2})$. The stationary-phase construction for the nondegenerate phase $\phi$ gives a canonical local isomorphism \begin{align*} \tau_\phi:\mathcal{Q}_\phi^\mu\to S_{\mathrm{hom}}^m(\Lambda_\phi;\mathcal{M}_\Lambda\otimes\Omega_\Lambda^{1/2}). \end{align*} This isomorphism is obtained by restricting the class of $a_\mu$ to $C_\phi$, transporting it by $\kappa_\phi$, and multiplying by the half-density factor induced by the critical-set parametrization together with the Maslov factor determined by the signature of the nondegenerate Hessian block in the auxiliary variables normal to the critical set being eliminated. For the local oscillatory integral $u_\phi$, define \begin{align*} \sigma_{m,\phi}(u_\phi):=\tau_\phi(a_\mu). \end{align*} The order convention $\mu=m+n/4-N/2$ is exactly the convention built into $\tau_\phi$: the half-density factor from stationary phase contributes the compensating degree $N/2-n/4$, so the resulting section on $\Lambda_\phi$ is homogeneous of degree \begin{align*} \mu-\frac{n}{4}+\frac{N}{2}=m. \end{align*} Hence $\sigma_{m,\phi}(u_\phi)$ lies in \begin{align*} S_{\mathrm{hom}}^m(\Lambda_\phi;\mathcal{M}_\Lambda\otimes\Omega_\Lambda^{1/2}). \end{align*} [/step]

[step:Check that equivalent phase functions give the same symbol]Suppose $\phi_1:U_1\times\Gamma_1\to\mathbb{R}$ and $\phi_2:U_2\times\Gamma_2\to\mathbb{R}$ are two nondegenerate conic phase functions parametrizing the same open conic subset of $\Lambda$. Let $N_1$ and $N_2$ be their numbers of auxiliary variables, and let \begin{align*} a_1:U_1\times\Gamma_1\to\mathbb{C},\qquad a_2:U_2\times\Gamma_2\to\mathbb{C} \end{align*} be classical amplitudes of orders \begin{align*} m+\frac{n}{4}-\frac{N_1}{2},\qquad m+\frac{n}{4}-\frac{N_2}{2} \end{align*} which define the same Lagrangian half-density modulo $I^{m-1}$ on the overlap. The phase-equivalence theorem for nondegenerate conic parametrizations applies under the following verified hypotheses: each $\phi_i$ is homogeneous of degree $1$ in its auxiliary variable, each critical set $C_{\phi_i}$ is a smooth conic manifold by nondegeneracy, each map $\kappa_{\phi_i}:C_{\phi_i}\to\Lambda$ has full rank onto the same open conic Lagrangian subset, and the amplitudes have localized proper conic support on the chosen overlap. Its conclusion is that, after possibly restricting to smaller conic neighborhoods, $\phi_1$ and $\phi_2$ are related by a finite composition of three operations: a change of base coordinates on $X$, a homogeneous change of auxiliary variables with nonvanishing Jacobian, and stabilization by adding a nondegenerate quadratic form in extra auxiliary variables. For each operation, the [stationary phase theorem](/theorems/8198) with half-density normalization applies because the variables being eliminated have nondegenerate Hessian and because the localized amplitudes are classical symbols with proper conic support, so all remainders one homogeneous degree lower define elements of $I^{m-1}$. The conclusion of stationary phase in this setting is the leading-order transformation law for the leading amplitude classes in the quotients $\mathcal{Q}_{\phi_1}^{m+n/4-N_1/2}$ and $\mathcal{Q}_{\phi_2}^{m+n/4-N_2/2}$. The Jacobian factors from coordinate and auxiliary-variable changes transform the half-density component, and the determinant and signature factors from the quadratic stabilization transform the Maslov component. These factors are precisely the transition functions defining $\mathcal{M}_\Lambda\otimes\Omega_\Lambda^{1/2}$. Therefore \begin{align*} \tau_{\phi_1}((a_1)_{m+n/4-N_1/2})=\tau_{\phi_2}((a_2)_{m+n/4-N_2/2}) \end{align*} as sections of $\mathcal{M}_\Lambda\otimes\Omega_\Lambda^{1/2}$ over the common subset of $\Lambda$.[/step]

[guided]We must prove that the symbol does not depend on the phase chart used to describe the same distribution. Let \begin{align*} \phi_1:U_1\times\Gamma_1\to\mathbb{R} \end{align*} and \begin{align*} \phi_2:U_2\times\Gamma_2\to\mathbb{R} \end{align*} be nondegenerate conic phases parametrizing the same open conic part of $\Lambda$. Let $N_1$ and $N_2$ be the corresponding numbers of auxiliary variables. Let \begin{align*} a_1:U_1\times\Gamma_1\to\mathbb{C} \end{align*} and \begin{align*} a_2:U_2\times\Gamma_2\to\mathbb{C} \end{align*} be classical amplitudes of orders $m+n/4-N_1/2$ and $m+n/4-N_2/2$ which define the same Lagrangian distribution modulo $I^{m-1}$ on the overlap. The issue is that the leading term of $a_1$ and the leading term of $a_2$ need not be equal as ordinary functions, because they live on different parameter spaces. The point of the Maslov and half-density factors is exactly to record how these leading functions transform. The equivalence theorem for nondegenerate phase functions says that, after shrinking to conic neighborhoods if necessary, any two such parametrizations differ by a composition of the standard moves: a coordinate change in $X$, a homogeneous change of auxiliary variables, and stabilization by a nondegenerate quadratic form in additional variables. Consider first a homogeneous change of auxiliary variables. If $\theta=\Theta(p,\vartheta)$ is the change of variables, then the measure transforms by the Jacobian factor \begin{align*} d\mathcal{L}^{N_1}(\theta)=|\det \partial_\vartheta\Theta(p,\vartheta)|\,d\mathcal{L}^{N_1}(\vartheta). \end{align*} Because the distribution is a half-density on $X$, the square-root density induced on the parametrized critical set transforms by the corresponding square-root Jacobian factor. Thus the leading amplitude is not invariant as a scalar function, but its product with the stationary-phase half-density factor is invariant as a half-density on $\Lambda$. Now consider stabilization. Stabilizing a phase means replacing $\phi(p,\theta)$ by \begin{align*} \widetilde{\phi}(p,\theta,\eta):=\phi(p,\theta)+q(\eta), \end{align*} where $r\in\mathbb{N}$ is the number of stabilization variables and \begin{align*} q:\mathbb{R}^r\to\mathbb{R} \end{align*} is a nondegenerate real quadratic form. Thus $r$ is fixed before the stabilized phase $\widetilde{\phi}:U\times\Gamma\times\mathbb{R}^r\to\mathbb{R}$ is formed. The [stationary phase lemma](/theorems/636) applied in the $\eta$ variables contributes the factor determined by $|\det q''|^{-1/2}$ and the signature phase $e^{i\pi\operatorname{sgn}(q'')/4}$, with the normalization fixed by the definition of $I^m(X,\Lambda;\Omega_X^{1/2})$. This is exactly the local transition rule for the Maslov line. The shift in the number of auxiliary variables is also accounted for by the order convention $m+n/4-N/2$, so the resulting homogeneous degree on $\Lambda$ remains $m$. Finally, coordinate changes on $X$ transform the displayed half-density $|dx|^{1/2}$ by the usual square-root Jacobian. Since the target symbol takes values in $\Omega_\Lambda^{1/2}$, the same square-root Jacobian is built into the transported density on $\Lambda$. Combining these three checks gives \begin{align*} \tau_{\phi_1}((a_1)_{m+n/4-N_1/2})=\tau_{\phi_2}((a_2)_{m+n/4-N_2/2}) \end{align*} on the overlap. Thus the local principal symbols glue to a single global section of $\mathcal{M}_\Lambda\otimes\Omega_\Lambda^{1/2}$.[/guided]

[step:Construct the global principal symbol map] Let $u\in I^m(X,\Lambda;\Omega_X^{1/2})$. Choose a locally finite family of phase charts \begin{align*} (\phi_\alpha:U_\alpha\times\Gamma_\alpha\to\mathbb{R})_{\alpha\in A} \end{align*} whose associated conic subsets $(\Lambda_{\phi_\alpha})_{\alpha\in A}$ cover the conic support of $u$ in $\Lambda$. Write $u$ locally as a finite sum of oscillatory integrals with classical amplitudes \begin{align*} a_\alpha:U_\alpha\times\Gamma_\alpha\to\mathbb{C}. \end{align*} On each $\Lambda_{\phi_\alpha}$, define \begin{align*} s_\alpha:=\tau_{\phi_\alpha}((a_\alpha)_{m+n/4-N_\alpha/2}). \end{align*} The preceding step shows that $s_\alpha=s_\beta$ on every overlap $\Lambda_{\phi_\alpha}\cap\Lambda_{\phi_\beta}$. If a different local decomposition of $u$ is chosen, subtracting the two decompositions gives local representatives of the zero distribution; by the same phase-equivalence argument, their leading amplitude classes vanish in the corresponding quotients and hence contribute zero local symbol. Hence the family $(s_\alpha)_{\alpha\in A}$ glues to a unique global section independent of the chosen phase cover and local representatives, \begin{align*} \sigma_m(u)\in S_{\mathrm{hom}}^m(\Lambda;\mathcal{M}_\Lambda\otimes\Omega_\Lambda^{1/2}). \end{align*} Linearity follows from the linearity of taking the leading homogeneous component of a classical amplitude and from the linearity of each transition map $\tau_{\phi_\alpha}$. Therefore \begin{align*} \sigma_m:I^m(X,\Lambda;\Omega_X^{1/2})\to S_{\mathrm{hom}}^m(\Lambda;\mathcal{M}_\Lambda\otimes\Omega_\Lambda^{1/2}) \end{align*} is a well-defined [linear map](/page/Linear%20Map). [/step]

[step:Identify the kernel with the lower-order Lagrangian distributions] First let $u\in I^{m-1}(X,\Lambda;\Omega_X^{1/2})$. In a phase chart with $N$ auxiliary variables, $u$ has a classical amplitude of order \begin{align*} m-1+\frac{n}{4}-\frac{N}{2}. \end{align*} Viewed as an element of $I^m(X,\Lambda;\Omega_X^{1/2})$, this representative has no homogeneous amplitude component of degree \begin{align*} m+\frac{n}{4}-\frac{N}{2}. \end{align*} Thus $\sigma_m(u)=0$, and \begin{align*} I^{m-1}(X,\Lambda;\Omega_X^{1/2})\subset\ker\sigma_m. \end{align*} Conversely, let $u\in I^m(X,\Lambda;\Omega_X^{1/2})$ satisfy $\sigma_m(u)=0$. In every phase chart $\phi:U\times\Gamma\to\mathbb{R}$ with $N$ auxiliary variables, write the corresponding amplitude as \begin{align*} a\sim \sum_{j=0}^{\infty}a_{\mu-j},\qquad \mu=m+\frac{n}{4}-\frac{N}{2}. \end{align*} The equality $\sigma_m(u)=0$ means \begin{align*} \tau_\phi(a_\mu)=0 \end{align*} on $\Lambda_\phi$. Since $\tau_\phi:\mathcal{Q}_\phi^\mu\to S_{\mathrm{hom}}^m(\Lambda_\phi;\mathcal{M}_\Lambda\otimes\Omega_\Lambda^{1/2})$ is a local vector-bundle isomorphism, the class of $a_\mu$ in $\mathcal{Q}_\phi^\mu$ is zero. We now use the symbolic division lemma for nondegenerate phases: if a homogeneous degree-$\mu$ leading amplitude has zero class in $\mathcal{Q}_\phi^\mu$, then, after shrinking the conic support, it can be written as an ideal term generated by the phase derivatives, plus a degree at most $\mu-1$ term, plus a term supported in the nonstationary region $\{\partial_\theta\phi\ne 0\}$. The hypotheses of this lemma are satisfied because $C_\phi$ is smooth by nondegeneracy, the differentials of the defining functions $\partial_{\theta_j}\phi$ have the required rank transverse to the critical set, and the amplitude is localized in a proper conic support. Thus there are homogeneous functions \begin{align*} q_j:U\times\Gamma\to\mathbb{C},\qquad j\in\{1,\dots,N\}, \end{align*} of degree $\mu$ on the conic support, and a homogeneous term $r_{\mu-1}:U\times\Gamma\to\mathbb{C}$ of degree at most $\mu-1$, such that \begin{align*} a_\mu=\sum_{j=1}^N q_j\partial_{\theta_j}\phi+r_{\mu-1} \end{align*} modulo a homogeneous term supported where $\partial_\theta\phi\ne 0$. The latter term gives a smooth half-density by the nonstationary phase integration-by-parts operator on that conic region; this operator is valid there because at least one component of $\partial_\theta\phi$ is bounded away from zero on each localized conic patch. For the displayed ideal term, use \begin{align*} e^{i\phi}\partial_{\theta_j}\phi=\frac{1}{i}\partial_{\theta_j}(e^{i\phi}). \end{align*} Integrating by parts in $\theta_j$ is legitimate because the amplitude has localized conic support and the oscillatory integral is interpreted with the standard conic cutoff regularization. Since $q_j$ is homogeneous of degree $\mu$, its auxiliary derivative $\partial_{\theta_j}q_j$ is homogeneous of degree $\mu-1$. Thus [integration by parts](/theorems/210) replaces $q_j\partial_{\theta_j}\phi$ by $-i\partial_{\theta_j}q_j$ plus cutoff-error terms, all of which are classical symbols of order at most $\mu-1$. Therefore the leading degree-$\mu$ term may be removed and $a$ is equivalent, modulo a smooth half-density, to a classical amplitude $a':U\times\Gamma\to\mathbb{C}$ of order $\mu-1$. Hence the local representative of $u$ lies in $I^{m-1}(X,\Lambda;\Omega_X^{1/2})$ in every phase chart. These local lower-order representatives agree on overlaps by the same phase-equivalence argument used above, so \begin{align*} u\in I^{m-1}(X,\Lambda;\Omega_X^{1/2}). \end{align*} Therefore \begin{align*} \ker\sigma_m=I^{m-1}(X,\Lambda;\Omega_X^{1/2}). \end{align*} [/step]

[step:Lift an arbitrary homogeneous section to a Lagrangian distribution] Let \begin{align*} s\in S_{\mathrm{hom}}^m(\Lambda;\mathcal{M}_\Lambda\otimes\Omega_\Lambda^{1/2}) \end{align*} be given. Choose a locally finite cover of the conic support of $s$ by phase charts \begin{align*} (\phi_\alpha:U_\alpha\times\Gamma_\alpha\to\mathbb{R})_{\alpha\in A}. \end{align*} Because $\Lambda$ is a smooth conic manifold and the cover is locally finite after restriction to the conic support of $s$, choose a smooth locally finite conic partition of unity \begin{align*} (\rho_\alpha:\Lambda\to[0,1])_{\alpha\in A} \end{align*} subordinate to $(\Lambda_{\phi_\alpha})_{\alpha\in A}$ on that support. Each $\rho_\alpha$ is homogeneous of degree $0$, has conic support contained in $\Lambda_{\phi_\alpha}$, and \begin{align*} \sum_{\alpha\in A}\rho_\alpha=1 \end{align*} on the conic support of $s$. For each $\alpha\in A$, define \begin{align*} s_\alpha:=\rho_\alpha s\big|_{\Lambda_{\phi_\alpha}}. \end{align*} Since $\tau_{\phi_\alpha}$ is a local isomorphism on the quotient $\mathcal{Q}_{\phi_\alpha}^{\mu_\alpha}$, where \begin{align*} \mu_\alpha:=m+\frac{n}{4}-\frac{N_\alpha}{2}, \end{align*} there is a unique quotient class $[b_{\alpha,\mu_\alpha}]\in\mathcal{Q}_{\phi_\alpha}^{\mu_\alpha}$ with the prescribed conic support such that \begin{align*} \tau_{\phi_\alpha}([b_{\alpha,\mu_\alpha}])=s_\alpha. \end{align*} Choose a homogeneous representative \begin{align*} b_{\alpha,\mu_\alpha}:U_\alpha\times\Gamma_\alpha\to\mathbb{C} \end{align*} of this class. Extend $b_{\alpha,\mu_\alpha}$ from the large-$|\theta|$ conic region to a classical symbol \begin{align*} b_\alpha:U_\alpha\times\Gamma_\alpha\to\mathbb{C} \end{align*} of order $\mu_\alpha$ by multiplying the homogeneous leading term by a degree-$0$ conic cutoff on the prescribed support and then choosing arbitrary lower homogeneous terms supported in the same conic set. This construction preserves the proper conic support convention required in the definition of $I^m(X,\Lambda;\Omega_X^{1/2})$. Let $u_\alpha$ be the corresponding local oscillatory half-density. Because the family is locally finite and the supports are conically proper in the defining sense of $I^m$, the sum \begin{align*} u:=\sum_{\alpha\in A}u_\alpha \end{align*} defines an element of $I^m(X,\Lambda;\Omega_X^{1/2})$. By linearity of the principal symbol map, \begin{align*} \sigma_m(u)=\sum_{\alpha\in A}\sigma_m(u_\alpha)=\sum_{\alpha\in A}s_\alpha=\sum_{\alpha\in A}\rho_\alpha s=s. \end{align*} Thus $\sigma_m$ is surjective. [/step]

[step:Conclude exactness of the principal symbol sequence] The inclusion \begin{align*} I^{m-1}(X,\Lambda;\Omega_X^{1/2})\to I^m(X,\Lambda;\Omega_X^{1/2}) \end{align*} is the natural inclusion of lower-order Lagrangian distributions into higher-order ones, so it is injective. The kernel of $\sigma_m$ is exactly $I^{m-1}(X,\Lambda;\Omega_X^{1/2})$, and $\sigma_m$ is surjective. Hence the sequence \begin{align*} 0\to I^{m-1}(X,\Lambda;\Omega_X^{1/2})\to I^m(X,\Lambda;\Omega_X^{1/2})\xrightarrow{\sigma_m} S_{\mathrm{hom}}^m(\Lambda;\mathcal{M}_\Lambda\otimes\Omega_\Lambda^{1/2})\to 0 \end{align*} is exact. [/step]