[proofplan] The proof is local near the common Lagrangian subset $\Lambda_0$. We first discard amplitude terms supported away from the fibre-critical sets, because nonstationary phase makes their oscillatory integrals smooth. Near the critical sets, we stabilize the two phase functions by adding nondegenerate quadratic variables until they have the same number of fibre variables, then use the homogeneous parametrized Morse lemma to identify the stabilized phases by a smooth fibre change of variables. The desired map on classical symbol quotients is obtained by applying this change of variables and stationary phase in the auxiliary quadratic variables; the leading stationary-phase coefficient gives exactly the canonical half-density principal symbol and the stated Maslov factor. [/proofplan]

[step:Localize both oscillatory integrals near the common fibre-critical locus]For $j \in \{\phi,\psi\}$, write the corresponding phase as $\Phi_j$, with $\Phi_\phi=\phi$ and $\Phi_\psi=\psi$, and write the fibre variable as $\vartheta_j$, with $\vartheta_\phi=\theta \in \Theta_\phi$ and $\vartheta_\psi=\eta \in \Theta_\psi$. Define \begin{align*} \lambda_j: C_j \to \Lambda_0 \end{align*} by $\lambda_j(x,\vartheta_j)=(x,\nabla_x\Phi_j(x,\vartheta_j))$. By hypothesis, $\lambda_\phi$ and $\lambda_\psi$ are diffeomorphisms onto the same conic [open set](/page/Open%20Set) $\Lambda_0$. Hence \begin{align*} \Gamma_{\phi\psi}:=\lambda_\psi^{-1}\circ \lambda_\phi:C_\phi \to C_\psi \end{align*} is a smooth conic diffeomorphism satisfying \begin{align*} \nabla_x\phi(x,\theta)=\nabla_x\psi(\Gamma_{\phi\psi}(x,\theta)) \end{align*} for every $(x,\theta)\in C_\phi$. Choose conic neighbourhoods $W_\phi \subset U\times\Theta_\phi$ and $W_\psi \subset U\times\Theta_\psi$ of $C_\phi$ and $C_\psi$, respectively, small enough that all constructions below are defined. Let \begin{align*} \beta_\phi: U\times\Theta_\phi \to [0,1] \end{align*} and \begin{align*} \beta_\psi: U\times\Theta_\psi \to [0,1] \end{align*} be smooth conic cutoffs, homogeneous of degree $0$ for large fibre norm, equal to $1$ on smaller conic neighbourhoods of $C_\phi$ and $C_\psi$, and supported in $W_\phi$ and $W_\psi$. If $a \in S^\mu_{\mathrm{cl}}(U\times\Theta_\phi)$, then \begin{align*} a=\beta_\phi a+(1-\beta_\phi)a. \end{align*} On the conic support of $(1-\beta_\phi)a$, the vector $\nabla_\theta\phi(x,\theta)$ is nonzero. The homogeneous nonstationary phase theorem for oscillatory integrals with classical symbols therefore gives \begin{align*} I_\phi((1-\beta_\phi)a)\in C^\infty(U;\Omega_U^{1/2}). \end{align*} Thus the class of $I_\phi(a)$ modulo smooth half-densities depends only on the symbol class of $a$ in a conic neighbourhood of $C_\phi$. The same assertion holds for $\psi$.[/step]

[guided]The first reduction is to ignore every part of an amplitude that cannot contribute to the singularity. For $j \in \{\phi,\psi\}$, the phase $j$ has fibre-critical set \begin{align*} C_j=\{(x,\vartheta_j):\nabla_{\vartheta_j}j(x,\vartheta_j)=0\}. \end{align*} Only points of $C_j$ can produce a nonsmooth oscillatory integral, because away from $C_j$ the fibre gradient is nonzero and [integration by parts](/theorems/210) in the fibre variable gains arbitrary powers of the fibre norm. The two critical sets live in different fibre spaces, so they are not literally the same set. The correct comparison is through the Lagrangian parametrizations. Since \begin{align*} \lambda_\phi:C_\phi\to\Lambda_0 \end{align*} and \begin{align*} \lambda_\psi:C_\psi\to\Lambda_0 \end{align*} are diffeomorphisms, their composition \begin{align*} \Gamma_{\phi\psi}:=\lambda_\psi^{-1}\circ\lambda_\phi:C_\phi\to C_\psi \end{align*} is the precise identification of the two fibre-critical sets. It sends the critical point of $\phi$ representing a covector in $\Lambda_0$ to the critical point of $\psi$ representing the same covector. Now choose smooth conic cutoffs \begin{align*} \beta_\phi:U\times\Theta_\phi\to[0,1] \end{align*} and \begin{align*} \beta_\psi:U\times\Theta_\psi\to[0,1] \end{align*} which equal $1$ near $C_\phi$ and $C_\psi$, respectively. For a properly supported symbol $a\in S^\mu_{\mathrm{cl}}(U\times\Theta_\phi)$, decompose \begin{align*} a=\beta_\phi a+(1-\beta_\phi)a. \end{align*} On the support of $(1-\beta_\phi)a$, the defining equation for $C_\phi$ fails, so $\nabla_\theta\phi$ is nonzero. The homogeneous nonstationary phase theorem applies to the phase $\phi$ and the symbol $(1-\beta_\phi)a$: repeated application of a fibre differential operator whose transpose kills $e^{i\phi}$ lowers the symbol order arbitrarily. Therefore \begin{align*} I_\phi((1-\beta_\phi)a)\in C^\infty(U;\Omega_U^{1/2}). \end{align*} This proves that, modulo smooth half-densities, $I_\phi(a)$ only depends on the germ of $a$ near $C_\phi$. The identical argument applies to $\psi$. This localization is what allows the rest of the proof to work entirely in small conic neighbourhoods of the inverse images of $\Lambda_0$.[/guided]

[step:Stabilize the phase with fewer fibre variables by a nondegenerate quadratic form] Let \begin{align*} r:=\max\{N_\phi,N_\psi\}. \end{align*} Choose smooth positive functions $\rho_\phi:\Theta_\phi\to(0,\infty)$ and $\rho_\psi:\Theta_\psi\to(0,\infty)$, homogeneous of degree $1$ in the fibre variables. If $N_\phi<r$, choose an integer $q_\phi:=r-N_\phi$ and a nondegenerate real quadratic form $Q_\phi:\mathbb{R}^{q_\phi}\to\mathbb{R}$. Let $H(Q_\phi)\in\mathbb{R}^{q_\phi\times q_\phi}$ denote the constant Hessian matrix of $Q_\phi$ with respect to the standard coordinates on $\mathbb{R}^{q_\phi}$. If $N_\phi=r$, set $q_\phi:=0$ and omit the auxiliary variable. For a sufficiently small number $\varepsilon_\phi>0$, define the conic neighbourhood \begin{align*} \widetilde{\Theta}_\phi:=\{(\theta,z_\phi)\in\Theta_\phi\times\mathbb{R}^{q_\phi}: |z_\phi|<\varepsilon_\phi\rho_\phi(\theta)\} \end{align*} and define the homogeneous stabilized phase \begin{align*} \widetilde{\phi}:U\times\widetilde{\Theta}_\phi\to\mathbb{R},\qquad (x,\theta,z_\phi)\mapsto \phi(x,\theta)+\rho_\phi(\theta)^{-1}Q_\phi(z_\phi). \end{align*} Define $q_\psi$, $Q_\psi$, $H(Q_\psi)$, $\widetilde{\Theta}_\psi$, and $\widetilde{\psi}:U\times\widetilde{\Theta}_\psi\to\mathbb{R}$ in the same way. Then both stabilized phases have $r$ fibre variables and are positively homogeneous of degree $1$ under $(\theta,z_j)\mapsto (t\theta,tz_j)$. The fibre-critical set of $\widetilde{\phi}$ is \begin{align*} \widetilde{C}_\phi=C_\phi\times\{0\}\subset U\times\widetilde{\Theta}_\phi, \end{align*} because $\nabla_{z_\phi}(\rho_\phi(\theta)^{-1}Q_\phi(z_\phi))=\rho_\phi(\theta)^{-1}H(Q_\phi)z_\phi$ and $H(Q_\phi)$ is invertible. The induced Lagrangian map is still $\lambda_\phi$, since \begin{align*} \nabla_x\widetilde{\phi}(x,\theta,0)=\nabla_x\phi(x,\theta). \end{align*} The analogous statements hold for $\widetilde{\psi}$. By the [stationary phase theorem](/theorems/8198) for a nondegenerate quadratic form with the normalization $(2\pi)^{-q/2}$, integration in the auxiliary variable $z_j$ multiplies the principal symbol by the constant \begin{align*} m(Q_j,\rho_j):=e^{i\pi \operatorname{sgn}(Q_j)/4}\rho_j(\vartheta_j)^{q_j/2}|\det H(Q_j)|^{-1/2}, \end{align*} where $H(Q_j)$ is the constant Hessian matrix of $Q_j$, $\operatorname{sgn}(Q_j)$ is its signature, and $\vartheta_j$ denotes the original fibre variable. This factor is the local Maslov contribution of the homogeneous stabilization; the power of $\rho_j$ is the determinant contribution of the scaled Hessian $\rho_j(\vartheta_j)^{-1}H(Q_j)$. [/step]

[step:Identify the stabilized phases by the homogeneous parametrized Morse lemma] We now use the homogeneous parametrized Morse lemma for nondegenerate phase functions parametrizing the same conic Lagrangian, applied to the two homogeneous stabilized phases. The lemma applies to two real smooth degree-$1$ homogeneous nondegenerate phase functions with the same number of fibre variables whose fibre-critical sets are mapped diffeomorphically to the same conic Lagrangian and whose critical values agree under that identification. These hypotheses hold here: $\widetilde{\phi}$ and $\widetilde{\psi}$ have $r$ fibre variables, the previous step identified their critical sets with $C_\phi\times\{0\}$ and $C_\psi\times\{0\}$, and their induced critical maps are $\lambda_\phi$ and $\lambda_\psi$. The critical values also agree. Indeed, Euler's identity for a degree-$1$ homogeneous function gives $\phi(x,\theta)=\theta\cdot\nabla_\theta\phi(x,\theta)=0$ on $C_\phi$ and $\psi(x,\eta)=\eta\cdot\nabla_\eta\psi(x,\eta)=0$ on $C_\psi$; the quadratic stabilization terms vanish at $z_\phi=0$ and $z_\psi=0$. By the stable homogeneous Morse normal-form branch compatibility assumed in the theorem statement, after shrinking to connected conic neighbourhoods of $\widetilde{C}_\phi$ and $\widetilde{C}_\psi$ there is a smooth fibre-preserving conic diffeomorphism \begin{align*} F:U\times\widetilde{\Theta}_\psi\to U\times\widetilde{\Theta}_\phi,\qquad (x,\zeta)\mapsto (x,F_x(\zeta)) \end{align*} such that $F$ maps $\widetilde{C}_\psi$ onto $\widetilde{C}_\phi$ and \begin{align*} \widetilde{\phi}(F(x,\zeta))=\widetilde{\psi}(x,\zeta) \end{align*} for all $(x,\zeta)$ in the chosen conic neighbourhood of $\widetilde{C}_\psi$. The same branch compatibility includes the vanishing of the possible additive base term. This is consistent with Euler's identity, since $\widetilde{\phi}=0$ on $\widetilde{C}_\phi$ and $\widetilde{\psi}=0$ on $\widetilde{C}_\psi$. [/step]

[step:Transfer amplitudes by change of variables and stationary phase]Let \begin{align*} J_F:U\times\widetilde{\Theta}_\psi\to (0,\infty) \end{align*} denote the absolute value of the Jacobian determinant of the fibre map $F_x$ with respect to [Lebesgue measure](/page/Lebesgue%20Measure) in the fibre variables. For a representative $a\in S^\mu_{\mathrm{cl}}(U\times\Theta_\phi)$, choose $\gamma_\phi\in C_c^\infty(\mathbb{R}^{q_\phi})$ equal to $1$ near $0$. For each $\alpha\in\mathbb{R}$, let $S^\alpha_{\mathrm{cl}}(U\times\widetilde{\Theta}_\phi)$ denote the properly supported classical symbols on the conic set $\widetilde{\Theta}_\phi$, classical with respect to the joint dilation $(\theta,z_\phi)\mapsto(t\theta,tz_\phi)$. The stationary-phase expansion in the auxiliary variable defines an elliptic classical operator \begin{align*} \mathcal{S}_\phi:S^\alpha_{\mathrm{cl}}(U\times\widetilde{\Theta}_\phi)/S^{-\infty}\to S^{\alpha+q_\phi/2}_{\mathrm{cl}}(U\times\Theta_\phi)/S^{-\infty} \end{align*} by requiring \begin{align*} (2\pi)^{-r/2}\int e^{i\widetilde{\phi}(x,\theta,z_\phi)}c(x,\theta,z_\phi)\,d\mathcal{L}^{q_\phi}(z_\phi)d\mathcal{L}^{N_\phi}(\theta) \end{align*} to equal $I_\phi(\mathcal{S}_\phi c)$ modulo smooth half-densities. Its leading coefficient is multiplication by $m(Q_\phi,\rho_\phi)$, which is elliptic. The full stationary-phase expansion is triangular in homogeneous degree, preserves proper support over $U$ because all fibre cutoffs are chosen over compact subsets of the base projection, and is independent of the auxiliary cutoff modulo $S^{-\infty}$. Hence the usual recursive stationary-phase construction gives a unique inverse on symbol quotients. Choose $a_\mathrm{st}\in S^{\mu-q_\phi/2}_{\mathrm{cl}}(U\times\widetilde{\Theta}_\phi)$ supported where $\gamma_\phi(z_\phi/\rho_\phi(\theta))=1$ near $z_\phi=0$ and satisfying $\mathcal{S}_\phi[a_\mathrm{st}]=[a]$. Then the stabilized oscillatory integral with amplitude $a_\mathrm{st}$ represents $I_\phi(a)$ modulo $C^\infty(U;\Omega_U^{1/2})$. Define the transferred stabilized amplitude \begin{align*} \widetilde{b}:U\times\widetilde{\Theta}_\psi\to\mathbb{C},\qquad (x,\zeta)\mapsto a_\mathrm{st}(F(x,\zeta))J_F(x,\zeta). \end{align*} On the localized conic neighbourhoods, $F_x$ is a diffeomorphism from the chosen fibre domain in $\widetilde{\Theta}_\psi$ onto the chosen fibre domain in $\widetilde{\Theta}_\phi$. Under the substitution $\omega=F_x(\zeta)$, the change-of-variables formula for Lebesgue measure in the fibre variables gives \begin{align*} (2\pi)^{-r/2}\int e^{i\widetilde{\phi}(x,\omega)}a_\mathrm{st}(x,\omega)\,d\mathcal{L}^r(\omega)=(2\pi)^{-r/2}\int e^{i\widetilde{\psi}(x,\zeta)}\widetilde{b}(x,\zeta)\,d\mathcal{L}^r(\zeta) \end{align*} modulo a smooth half-density. The equality is used only where the cutoffs place the amplitudes inside these domains, and the complementary terms are smoothing by nonstationary phase. If $q_\psi>0$, apply stationary phase in the auxiliary variable $z_\psi\in\mathbb{R}^{q_\psi}$ to $\widetilde{b}$. This produces a classical symbol \begin{align*} b:U\times\Theta_\psi\to\mathbb{C} \end{align*} of order \begin{align*} \mu+\frac{N_\phi-N_\psi}{2} \end{align*} such that \begin{align*} I_\phi(a)-I_\psi(b)\in C^\infty(U;\Omega_U^{1/2}). \end{align*} If $q_\psi=0$, this same definition is read without the final stationary phase step. Set \begin{align*} T_{\phi\to\psi}[a]:=[b]. \end{align*} The construction depends only on the class of $a$ modulo $S^{-\infty}$. Indeed, a smoothing change in $a$ remains smoothing after composition with the smooth conic diffeomorphism $F$ and after stationary phase. Proper support over $U$ is preserved because each operation is performed after shrinking to properly supported conic neighbourhoods over the same base coordinate set and because $F$ is fibre-preserving. Hence $T_{\phi\to\psi}$ is a well-defined [linear map](/page/Linear%20Map) on symbol quotient spaces.[/step]

[guided]This is the step where the symbol map is actually built. The stabilized phases have the same number $r$ of fibre variables, so the equality \begin{align*} \widetilde{\phi}\circ F=\widetilde{\psi} \end{align*} allows us to transfer an oscillatory integral for $\widetilde{\phi}$ into one for $\widetilde{\psi}$ by an ordinary change of variables. Let \begin{align*} J_F:U\times\widetilde{\Theta}_\psi\to(0,\infty) \end{align*} be the absolute value of the determinant of the fibre derivative of $F_x$. This is the factor required by the change-of-variables theorem for Lebesgue measure: \begin{align*} d\mathcal{L}^r(\omega)=J_F(x,\zeta)\,d\mathcal{L}^r(\zeta) \end{align*} under the substitution $\omega=F_x(\zeta)$. Starting with $a\in S^\mu_{\mathrm{cl}}(U\times\Theta_\phi)$, we do not merely extend $a$ constantly in the new variable. Stationary phase in $z_\phi$ has a nontrivial elliptic leading factor and lower-order differential corrections. Define the stationary-phase reduction map $\mathcal{S}_\phi$ by integrating the stabilized phase in $z_\phi$ and reading off the resulting classical symbol in the remaining variable $\theta$. Since the leading multiplier is $m(Q_\phi,\rho_\phi)$ and this multiplier never vanishes, solve recursively for a stabilized amplitude $a_\mathrm{st}$ satisfying $\mathcal{S}_\phi[a_\mathrm{st}]=[a]$. This is the standard triangular inversion of stationary phase: the top homogeneous term is divided by $m(Q_\phi,\rho_\phi)$, and each lower homogeneous term is then chosen after the previously fixed higher terms are known. Then define \begin{align*} \widetilde{b}:U\times\widetilde{\Theta}_\psi\to\mathbb{C},\qquad (x,\zeta)\mapsto a_\mathrm{st}(F(x,\zeta))J_F(x,\zeta). \end{align*} Because $F$ is a smooth conic diffeomorphism, composition with $F$ preserves classical symbol expansions, and multiplication by $J_F$ also preserves classicality. Therefore $\widetilde{b}$ is a classical symbol of the corresponding stabilized order. The equality of phases now gives \begin{align*} e^{i\widetilde{\phi}(F(x,\zeta))}=e^{i\widetilde{\psi}(x,\zeta)}. \end{align*} Combining this identity with the fibre change of variables yields equality of the localized stabilized oscillatory integrals. The only terms not covered by this equality are those cut off away from the critical sets, and the previous step already proved that such terms are smooth by nonstationary phase. It remains to remove the auxiliary variables introduced during stabilization. This is exactly the role of stationary phase. If $z_\psi\in\mathbb{R}^{q_\psi}$ is an auxiliary variable for $\psi$, the phase is quadratic and nondegenerate in $z_\psi$ at $z_\psi=0$. Stationary phase in $z_\psi$ converts the stabilized amplitude $\widetilde{b}$ into a classical amplitude \begin{align*} b:U\times\Theta_\psi\to\mathbb{C}. \end{align*} The inverse stationary-phase choice of $a_\mathrm{st}$ has order $\mu-q_\phi/2$ in the stabilized variables. Stationary integration over $q_\psi$ variables raises the remaining symbol order by $q_\psi/2$. Since $q_\phi=r-N_\phi$ and $q_\psi=r-N_\psi$, the net change is \begin{align*} \frac{q_\psi-q_\phi}{2}=\frac{N_\phi-N_\psi}{2}. \end{align*} Thus \begin{align*} b\in S^{\mu+(N_\phi-N_\psi)/2}_{\mathrm{cl}}(U\times\Theta_\psi). \end{align*} By construction, \begin{align*} I_\phi(a)-I_\psi(b)\in C^\infty(U;\Omega_U^{1/2}). \end{align*} Finally, if $a$ is changed by an element of $S^{-\infty}(U\times\Theta_\phi)$, then the transferred amplitude is also rapidly decreasing to infinite order after composition with $F$ and after stationary phase. Hence the class $[b]$ depends only on $[a]$, so the formula defines a linear map \begin{align*} T_{\phi\to\psi}:S^\mu_{\mathrm{cl}}(U\times\Theta_\phi)/S^{-\infty}\to S^{\mu+(N_\phi-N_\psi)/2}_{\mathrm{cl}}(U\times\Theta_\psi)/S^{-\infty}. \end{align*}[/guided]

[step:Construct the inverse map by reversing the phase comparison] Apply the same construction with the roles of $\phi$ and $\psi$ reversed. The inverse fibre-critical identification is \begin{align*} \Gamma_{\psi\phi}:=\lambda_\phi^{-1}\circ\lambda_\psi:C_\psi\to C_\phi. \end{align*} The homogeneous parametrized Morse lemma gives the inverse fibre change of variables, and stationary phase in the reversed auxiliary variables defines \begin{align*} T_{\psi\to\phi}:S^{\mu+(N_\phi-N_\psi)/2}_{\mathrm{cl}}(U\times\Theta_\psi)/S^{-\infty}\to S^\mu_{\mathrm{cl}}(U\times\Theta_\phi)/S^{-\infty}. \end{align*} The construction can be made with $F^{-1}$ as the reverse fibre change and with the same homogeneous quadratic stabilizations. In the forward construction the only operations on symbol quotients are: the inverse stationary-phase map $\mathcal{S}_\phi^{-1}$, pullback by the conic diffeomorphism $F$ with the fibre Jacobian factor, and the stationary-phase map $\mathcal{S}_\psi$. In the reverse construction the corresponding operations are $\mathcal{S}_\psi^{-1}$, pullback by $F^{-1}$ with the reciprocal fibre Jacobian factor, and $\mathcal{S}_\phi$. These are literal inverse operations on the quotient spaces because $\mathcal{S}_\phi$ and $\mathcal{S}_\psi$ were inverted by the recursive stationary-phase construction and because the change-of-variables Jacobians multiply to $1$. Therefore $T_{\psi\to\phi}T_{\phi\to\psi}=\operatorname{id}$ modulo $S^{-\infty}$. The same argument with $\phi$ and $\psi$ interchanged gives $T_{\phi\to\psi}T_{\psi\to\phi}=\operatorname{id}$. Hence $T_{\phi\to\psi}$ is a linear isomorphism. [/step]

[step:Identify the principal symbol and the Maslov factor] Let $a_\mu$ denote the leading homogeneous component of $a$ of degree $\mu$ in the $\theta$ variable. The principal symbol of $I_\phi(a)$ on $\Lambda_0$ is obtained by restricting $a_\mu$ to $C_\phi$, multiplying by the half-density induced from the nondegenerate fibre Hessian transverse to $C_\phi$, and transporting it to $\Lambda_0$ through $\lambda_\phi$. The same construction for $\psi$ uses $\lambda_\psi$. Under the diffeomorphism $F$, the restriction of the transferred leading amplitude to $\widetilde{C}_\psi$ is exactly the pullback of the restriction of $a_\mu$ to $\widetilde{C}_\phi$, multiplied by the fibre Jacobian factor required for the half-density transformation law. The stationary phase evaluations in the auxiliary quadratic variables use the transverse Hessian matrices $\rho_\phi(\theta)^{-1}H(Q_\phi)$ and $\rho_\psi(\eta)^{-1}H(Q_\psi)$. Hence the leading stationary-phase factors are \begin{align*} e^{i\pi\operatorname{sgn}(Q_\phi)/4}\rho_\phi(\theta)^{q_\phi/2}|\det H(Q_\phi)|^{-1/2} \end{align*} and the inverse of \begin{align*} e^{i\pi\operatorname{sgn}(Q_\psi)/4}\rho_\psi(\eta)^{q_\psi/2}|\det H(Q_\psi)|^{-1/2}. \end{align*} With the transverse Hessian convention fixed by $H(Q_j)=J(\nabla Q_j)$, the principal symbol formula for nondegenerate homogeneous oscillatory integrals gives the leading coefficient as the restriction of the leading amplitude to the critical set multiplied by the square root of the inverse transverse Hessian density and by $e^{i\pi\operatorname{sgn}/4}$. Under $F$, the fibre Jacobian converts this transverse Hessian density by the half-density transformation law. The factors $\rho_j^{q_j/2}|\det H(Q_j)|^{-1/2}$ are exactly the auxiliary transverse Hessian densities for the scaled Hessians $\rho_j^{-1}H(Q_j)$, and the powers of $(2\pi)$ cancel because the oscillatory integrals use the normalization $(2\pi)^{-N/2}$. After these density factors cancel, the remaining unit complex number is the constant Maslov factor attached to the chosen local stable normal-form branches; changing the auxiliary $Q_j$ or $\rho_j$ within the same chosen branch changes the intermediate Hessian factors and Jacobians inversely, so the resulting half-density on $\Lambda_0$ is unchanged. Consequently the principal symbol of $I_\psi(T_{\phi\to\psi}a)$ is the canonical image of the principal symbol of $I_\phi(a)$ as a half-density on $\Lambda_0$, including this Maslov factor. Together with the smoothing remainder proved above, this proves the asserted equivalence of the two nondegenerate homogeneous phase parametrizations. [/step]