[proofplan] The two parts are independent assertions about the same geometric object $N^*S$. For Part 1 we choose a slice chart in which $S = \{z = 0\}$ via the [Implicit Function Theorem](/theorems/52); the bilinear phase $\phi_0(y,z,\zeta) = z\cdot\zeta$ then has critical set $\{z = 0\}$, and $\lambda_{\phi_0}$ sends $(y,0,\zeta)$ to the covector $\sum_j \zeta_j\, dz_j$ at $(y,0)$, sweeping out exactly $N^*S \setminus 0$; nondegeneracy is the [linear independence](/page/Linear%20Independence) of $dz_1,\dots,dz_\ell$. For Part 2 we first make the oscillatory integral meaningful by cutting off in $\theta$ and integrating by parts to remove the $\chi$-dependence in the limit. We then compute the stationary set of the total phase $\phi(x,\theta) - x\cdot\xi$ governing $\widehat{\psi u}$: stationarity forces $(x,\theta) \in C_\phi$ and $\xi = d_x\phi$, i.e. $(x,\xi) \in \lambda_\phi(C_\phi)$. Off this set the phase is nonstationary, and [integration by parts](/theorems/210) (the [Non-Stationary Phase Lemma](/theorems/635), packaged microlocally in [citetheorem:8200]) gives rapid decay, yielding the wave front bound. Finally, the [equivalence of nondegenerate phase parametrisations](#) reduces a general phase to the model $z\cdot\zeta$, identifying $u$ as a conormal distribution. [/proofplan] [step:Choose a slice chart in which $S$ is a coordinate plane] Let $(x_0,\xi_0) \in N^*S \setminus 0$, so $x_0 \in S$ and $\xi_0 \in T_{x_0}^*X$ annihilates $T_{x_0}S$ with $\xi_0 \neq 0$. Since $S \subset X$ is an embedded submanifold of codimension $\ell$, by the local description of embedded submanifolds (a consequence of the [Implicit Function Theorem](/theorems/52) applied to defining functions of $S$), there is a coordinate chart $\varphi: U \to \varphi(U) \subset \mathbb{R}^k \times \mathbb{R}^\ell$ around $x_0$, $\varphi(x) = (y,z)$ with $y \in \mathbb{R}^k$, $z \in \mathbb{R}^\ell$, such that \begin{align*} S \cap U = \{x \in U : z(x) = 0\}, \qquad \varphi(x_0) = (y_0, 0). \end{align*} We use the induced cotangent coordinates: a covector at $x \in U$ is written $\eta \cdot dy + \zeta \cdot dz$ with $\eta \in \mathbb{R}^k$, $\zeta \in \mathbb{R}^\ell$, so that $T^*U \cong \varphi(U) \times (\mathbb{R}^k \times \mathbb{R}^\ell)$ via $(y,z,\eta,\zeta)$. Because $T_xS = \operatorname{span}(\partial_{y_1},\dots,\partial_{y_k})$ at points $x = (y,0) \in S \cap U$, a covector $\eta\cdot dy + \zeta\cdot dz$ annihilates $T_xS$ if and only if $\eta = 0$. Hence \begin{align*} N^*S \cap (T^*U \setminus 0) = \{(y, 0, 0, \zeta) : y \in \mathbb{R}^k,\ \zeta \in \mathbb{R}^\ell \setminus 0\}. \end{align*} In these coordinates $\xi_0$ has the form $(0,\zeta_0)$ with $\zeta_0 \in \mathbb{R}^\ell \setminus 0$, i.e. $(x_0,\xi_0) \leftrightarrow (y_0,0,0,\zeta_0)$. [/step] [step:Verify that the model phase $z\cdot\zeta$ is nondegenerate and parametrizes $N^*S$] Define the model phase \begin{align*} \phi_0: \varphi(U) \times (\mathbb{R}^\ell \setminus 0) &\to \mathbb{R}, \end{align*} \begin{align*} (y,z,\zeta) &\mapsto z \cdot \zeta = \sum_{j=1}^\ell z_j \zeta_j, \end{align*} so here the fibre variable is $\theta = \zeta$ and $N = \ell$. The function $\phi_0$ is smooth, real-valued, and positively homogeneous of degree $1$ in $\zeta$, since $\phi_0(y,z,r\zeta) = r\,\phi_0(y,z,\zeta)$ for $r > 0$. We compute the differentials. The partial derivatives are \begin{align*} \partial_{\zeta_j}\phi_0 = z_j \quad (1 \le j \le \ell), \qquad \partial_{z_j}\phi_0 = \zeta_j \quad (1 \le j \le \ell), \qquad \partial_{y_i}\phi_0 = 0 \quad (1 \le i \le k). \end{align*} Thus $d_\zeta\phi_0 = (z_1,\dots,z_\ell)$ and $d_x\phi_0 = \sum_{j=1}^\ell \zeta_j\, dz_j$, i.e. in cotangent coordinates $d_x\phi_0 \leftrightarrow (\eta,\zeta) = (0,\zeta)$. *Nondegeneracy of the phase gradient.* On $\varphi(U) \times (\mathbb{R}^\ell\setminus 0)$ we have $\zeta \neq 0$, so $d_x\phi_0 = \sum_j \zeta_j\,dz_j \neq 0$, whence $d_{x,\zeta}\phi_0 \neq 0$ everywhere. *Critical set.* By the formula for $d_\zeta\phi_0$, \begin{align*} C_{\phi_0} = \{(y,z,\zeta) : d_\zeta\phi_0 = 0\} = \{(y,z,\zeta) : z = 0\} = \{(y,0,\zeta) : \zeta \in \mathbb{R}^\ell\setminus 0\}. \end{align*} *Nondegeneracy condition.* For each $j$, $d_{x,\zeta}(\partial_{\zeta_j}\phi_0) = d_{x,\zeta}(z_j) = dz_j$. The covectors $dz_1,\dots,dz_\ell$ on $\varphi(U)\times(\mathbb{R}^\ell\setminus 0)$ are linearly independent (they are part of the coordinate coframe), and this independence holds in particular at every point of $C_{\phi_0}$. Hence $\phi_0$ is a nondegenerate conic phase. *Parametrization of $N^*S$.* The map $\lambda_{\phi_0}$ restricted to $C_{\phi_0}$ is \begin{align*} \lambda_{\phi_0}(y,0,\zeta) = \big((y,0),\, d_x\phi_0(y,0,\zeta)\big) = (y,0,0,\zeta). \end{align*} Comparing with the formula for $N^*S \cap (T^*U\setminus 0)$ from the previous step, $\lambda_{\phi_0}$ maps $C_{\phi_0}$ bijectively onto $N^*S \cap (T^*U \setminus 0)$, with smooth inverse $(y,0,0,\zeta) \mapsto (y,0,\zeta)$. In particular $\lambda_{\phi_0}$ is an injective immersion onto an open conic (indeed the full) subset $N^*S \cap (T^*U \setminus 0)$ of $N^*S \setminus 0$. Taking $\Gamma := T^*U \setminus 0$, which is a conic neighbourhood of $(x_0,\xi_0)$, we obtain that $\phi_0$ parametrizes $N^*S \cap \Gamma$, and $(x_0,\xi_0) = \lambda_{\phi_0}(y_0,0,\zeta_0) \in \lambda_{\phi_0}(C_{\phi_0})$. This proves Part 1. [guided] The goal of Part 1 is to exhibit, near any prescribed conormal direction $(x_0,\xi_0)$, an explicit nondegenerate phase whose associated Lagrangian is the conormal bundle. Why is the bilinear phase $z\cdot\zeta$ the right choice? Because the conormal bundle is the set of covectors that vanish in the $y$-directions and are arbitrary in the $z$-directions, and $z\cdot\zeta$ is precisely the [generating function](/page/Generating%20Function) whose $x$-differential is $\zeta\,dz$ along $\{z=0\}$. Let us re-derive everything carefully. The phase is $\phi_0(y,z,\zeta) = \sum_{j=1}^\ell z_j\zeta_j$. Three facts must be checked. First, homogeneity: scaling $\zeta \mapsto r\zeta$ multiplies each term $z_j\zeta_j$ by $r$, so $\phi_0$ is homogeneous of degree $1$ in $\zeta$. This is required so that $\lambda_{\phi_0}(C_{\phi_0})$ is a **conic** subset of $T^*U$, matching the conic nature of $N^*S\setminus 0$. Second, the critical set. The defining condition $d_\zeta\phi_0 = 0$ asks where the phase is stationary in the fibre variable $\zeta$. Since $\partial_{\zeta_j}\phi_0 = z_j$, stationarity is exactly $z = 0$ — the equation of $S$. This is the mechanism by which the phase "sees" $S$: its fibre-critical set projects onto $S$. Third, what covector does the phase produce there? On $C_{\phi_0}$ we evaluate $d_x\phi_0$. The $y$-derivatives vanish ($\phi_0$ does not depend on $y$), and $\partial_{z_j}\phi_0 = \zeta_j$, so $d_x\phi_0 = \sum_j \zeta_j\,dz_j$. As a covector this is $(0,\zeta)$ in the $(\eta,\zeta)$-coordinates: zero in the $y$-cotangent slots, free in the $z$-cotangent slots. That is the description of $N^*S$ we found in Step 1. Thus $\lambda_{\phi_0}(y,0,\zeta) = (y,0,0,\zeta)$ runs over all of $N^*S \cap (T^*U\setminus 0)$ as $(y,\zeta)$ varies, and is a bijection with smooth inverse. Finally, why is the nondegeneracy condition satisfied? Nondegeneracy demands that the $\ell$ functions $\partial_{\zeta_j}\phi_0 = z_j$ have linearly independent differentials on $C_{\phi_0}$. But $d(z_j) = dz_j$, and the coordinate differentials $dz_1,\dots,dz_\ell$ are independent by construction of the chart. Geometrically, this is the statement that $\lambda_{\phi_0}$ is an immersion — without it the image could fail to be a smooth Lagrangian. Here the check is immediate because the phase is exactly linear in the fibre variable, the simplest nondegenerate situation. Since the whole punctured conormal bundle over $U$ is covered, we may take $\Gamma = T^*U\setminus 0$ rather than a smaller cone. [/guided] [/step] [step:Define the oscillatory integral as a distribution by cutting off in the phase variable] We now turn to Part 2. Fix $\chi \in C_c^\infty(\mathbb{R}^N)$ with $\chi \equiv 1$ on a neighbourhood of $0$. For $\varepsilon > 0$ define \begin{align*} u_\varepsilon: U &\to \mathbb{C}, \end{align*} \begin{align*} x &\mapsto \int_{\mathbb{R}^N} e^{i\phi(x,\theta)}\, a(x,\theta)\, \chi(\varepsilon\theta)\, d\mathcal{L}^N(\theta). \end{align*} Because $\theta \mapsto \chi(\varepsilon\theta)$ has compact support and $a$ is smooth with $x$-support contained in a fixed compact set $K \subset U$, each $u_\varepsilon \in C_c^\infty(U)$, with $\operatorname{supp} u_\varepsilon \subset K$. To pass to the limit, fix $\varphi \in C_c^\infty(U)$ and consider \begin{align*} I_\varepsilon(\varphi) := \int_U u_\varepsilon(x)\,\varphi(x)\, d\mathcal{L}^n(x) = \int_U \int_{\mathbb{R}^N} e^{i\phi(x,\theta)}\, a(x,\theta)\, \chi(\varepsilon\theta)\, \varphi(x)\, d\mathcal{L}^N(\theta)\, d\mathcal{L}^n(x). \end{align*} Since $\phi$ is a nondegenerate conic phase, $d_{x,\theta}\phi \neq 0$ on $\operatorname{supp} a$; by homogeneity of $\phi$ (degree $1$, so $d_x\phi$ is homogeneous of degree $1$ and $d_\theta\phi$ of degree $0$ in $\theta$), on the conic support of $a \cdot (\varphi \otimes 1)$ there is a constant $c > 0$ with \begin{align*} |\theta|\,|d_x\phi(x,\theta)| + |d_\theta\phi(x,\theta)| \ge c\,|\theta| \qquad (|\theta| \ge 1). \end{align*} Define the first-order differential operator \begin{align*} L := \frac{1}{i\big(|\theta|^2|d_x\phi|^2 + |d_\theta\phi|^2\big)}\Big( |\theta|^2 \sum_{k=1}^n \partial_{x_k}\phi\, \partial_{x_k} + \sum_{j=1}^N \partial_{\theta_j}\phi\, \partial_{\theta_j} \Big), \end{align*} which satisfies $L\,e^{i\phi} = e^{i\phi}$. Its transpose $L^{t}$ maps $S^m_{\mathrm{cl}} \to S^{m-1}_{\mathrm{cl}}$ in $\theta$ (each application improves decay by one order in $|\theta|$, by the [Oscillatory Integration by Parts Identity](/theorems/7678)). Choosing an integer $M > m + N$ and integrating by parts $M$ times, \begin{align*} I_\varepsilon(\varphi) = \int_U \int_{\mathbb{R}^N} e^{i\phi(x,\theta)}\, (L^{t})^{M}\!\big[ a(x,\theta)\, \chi(\varepsilon\theta)\, \varphi(x) \big]\, d\mathcal{L}^N(\theta)\, d\mathcal{L}^n(x). \end{align*} The integrand is now bounded by $C\,(1+|\theta|)^{m - M}\,|\varphi|$, which is absolutely integrable on $K \times \mathbb{R}^N$ uniformly in $\varepsilon \in (0,1]$, and the $\theta$-derivatives of $\chi(\varepsilon\theta)$ are uniformly bounded with support escaping to infinity. By the [Dominated Convergence Theorem](/theorems/19), $I_\varepsilon(\varphi) \to I_0(\varphi)$ as $\varepsilon \downarrow 0$, where $I_0(\varphi)$ is the absolutely convergent integral with $\chi(\varepsilon\theta)$ replaced by $1$. The bound also gives $|I_0(\varphi)| \le C \sum_{|\alpha|\le M}\sup|\partial_x^\alpha\varphi|$, so \begin{align*} u: C_c^\infty(U) \to \mathbb{C}, \qquad u(\varphi) := \lim_{\varepsilon \downarrow 0} I_\varepsilon(\varphi) = I_0(\varphi), \end{align*} is a continuous linear functional, i.e. $u \in \mathcal{D}'(U)$. Because the value $I_0(\varphi)$ does not involve $\chi$, the distribution $u$ is independent of the choice of cutoff. (This well-definedness is exactly the content of the oscillatory-integral construction in [citetheorem:8200], whose hypotheses we verify in the next step.) [/step] [step:Locate the stationary set and bound the wave front set by nonstationary phase] We show \begin{align*} \operatorname{WF}(u) \subset \lambda_\phi\big(C_\phi \cap \operatorname{supp} a\big). \end{align*} Let $(x_1,\xi_1) \in (T^*U \setminus 0)$ with $(x_1,\xi_1) \notin \lambda_\phi(C_\phi \cap \operatorname{supp} a)$; we prove $(x_1,\xi_1) \notin \operatorname{WF}(u)$. Choose $\psi \in C_c^\infty(U)$ with $\psi \equiv 1$ near $x_1$ and support in a small ball, and a closed conic neighbourhood $V \subset \mathbb{R}^n \setminus 0$ of $\xi_1$. Using the symmetric Fourier convention, for $\xi \in \mathbb{R}^n$, \begin{align*} \widehat{\psi u}(\xi) = (2\pi)^{-n/2}\int_U \int_{\mathbb{R}^N} e^{i\Phi(x,\theta;\xi)}\, \psi(x)\, a(x,\theta)\, d\mathcal{L}^N(\theta)\, d\mathcal{L}^n(x), \qquad \Phi(x,\theta;\xi) := \phi(x,\theta) - x\cdot\xi, \end{align*} the integral interpreted as the regularized oscillatory integral of the previous step. *Stationary set.* The total phase $\Phi$ is stationary in $(x,\theta)$ precisely where \begin{align*} d_\theta\Phi = d_\theta\phi(x,\theta) = 0 \qquad\text{and}\qquad d_x\Phi = d_x\phi(x,\theta) - \xi = 0. \end{align*} The first equation says $(x,\theta) \in C_\phi$; the second says $\xi = d_x\phi(x,\theta)$. Together they say $(x,\xi) = \lambda_\phi(x,\theta)$ for some $(x,\theta) \in C_\phi$, i.e. $(x,\xi) \in \lambda_\phi(C_\phi)$. *Nonstationarity off the parametrized set.* Since $(x_1,\xi_1) \notin \lambda_\phi(C_\phi \cap \operatorname{supp} a)$ and $\lambda_\phi(C_\phi \cap \operatorname{supp} a)$ is a closed conic set (the image of the closed conic $C_\phi \cap \operatorname{supp} a$ under the proper, homogeneous map $\lambda_\phi$), we may shrink $\psi$ and $V$ so that for $x \in \operatorname{supp}\psi$, $\theta \in \mathbb{R}^N$ with $(x,\theta) \in \operatorname{supp} a$, and $\xi \in V$ there is no common stationary point; quantitatively, by homogeneity there exist $c_1 > 0$ and a conic neighbourhood such that \begin{align*} |d_{x,\theta}\Phi(x,\theta;\xi)| = |d_x\phi(x,\theta) - \xi| + |d_\theta\phi(x,\theta)| \gtrsim |\xi| + |\theta| \end{align*} on the relevant support, with the implicit constant depending only on $\phi$, $\psi$, and $V$. *Rapid decay.* Apply the [Non-Stationary Phase Lemma](/theorems/635): on the region where $|d_{x,\theta}\Phi| \gtrsim |\xi| + |\theta|$, repeated [integration by parts](/theorems/2098) with the transpose of \begin{align*} \widetilde{L} := \frac{1}{i\,|d_{x,\theta}\Phi|^2}\, \overline{d_{x,\theta}\Phi} \cdot \nabla_{x,\theta}, \qquad \widetilde{L}\,e^{i\Phi} = e^{i\Phi}, \end{align*} yields, for every $M \in \mathbb{N}$, a constant $C_M > 0$ with \begin{align*} |\widehat{\psi u}(\xi)| \le C_M\,(1+|\xi|)^{-M} \qquad (\xi \in V). \end{align*} The classical symbol estimates $|\partial_x^\beta \partial_\theta^\gamma a(x,\theta)| \le C_{\beta\gamma}(1+|\theta|)^{m-|\gamma|}$ control the amplitude factors produced by each integration by parts, and the gradient lower bound makes the resulting series of terms summable; this is precisely the microlocal estimate recorded in [citetheorem:8200], whose hypotheses — $\phi$ smooth, real, homogeneous of degree $1$ in $\theta$, with $d_{x,\theta}\phi \neq 0$ on $\operatorname{supp} a$, and $a \in S^m_{\mathrm{cl}}$ — were all verified above. Therefore $(x_1,\xi_1) \notin \operatorname{WF}(u)$. Since $(x_1,\xi_1)$ was an arbitrary point outside $\lambda_\phi(C_\phi \cap \operatorname{supp} a)$, we conclude $\operatorname{WF}(u) \subset \lambda_\phi(C_\phi \cap \operatorname{supp} a)$. Because $\phi$ parametrizes $N^*S$, the right-hand side is contained in $N^*S \setminus 0$, so $\operatorname{WF}(u) \subset N^*S \setminus 0$. *Smoothness off $S$.* By the [Projection of the Wave Front Set Equals the Singular Support](#) (same-run [citetheorem:8162]), $\operatorname{sing\,supp}(u) = \pi(\operatorname{WF}(u)) \subset \pi(N^*S \setminus 0) = S$, where $\pi: T^*X \setminus 0 \to X$ is the base projection. Hence $u$ is smooth on $U \setminus S$. [guided] The entire microlocal content of Part 2 is the computation of the stationary set of the total phase $\Phi(x,\theta;\xi) = \phi(x,\theta) - x\cdot\xi$. Everything else is bookkeeping with integration by parts. Let us see exactly why the wave front set is forced into $N^*S$. To detect whether $(x_1,\xi_1) \in \operatorname{WF}(u)$, we localize in space with $\psi$ (a bump equal to $1$ near $x_1$) and test the decay of the [Fourier transform](/page/Fourier%20Transform) $\widehat{\psi u}(\xi)$ as $\xi \to \infty$ inside a cone $V$ around $\xi_1$. A covector is **not** in the wave front set iff this Fourier transform decays faster than any polynomial in such a cone. So we must understand $\widehat{\psi u}(\xi)$, which after inserting the oscillatory-integral representation of $u$ becomes an oscillatory integral in $(x,\theta)$ with phase $\Phi = \phi - x\cdot\xi$. Where can such an integral fail to decay rapidly? Only near points where $\Phi$ is stationary in the integration variables $(x,\theta)$; away from stationary points, integration by parts produces arbitrarily fast decay. So compute the stationarity equations: - $\partial_\theta \Phi = \partial_\theta\phi = 0$. This is the defining equation of $C_\phi$. - $\partial_x \Phi = \partial_x\phi - \xi = 0$, i.e. $\xi = \partial_x\phi$. Read together: a stationary point can occur only when $(x,\theta) \in C_\phi$ **and** $\xi$ equals $\partial_x\phi$ at that point — that is, exactly when $(x,\xi) = \lambda_\phi(x,\theta)$ lies on the parametrized Lagrangian. This is the precise sense in which the phase "selects" the directions $\lambda_\phi(C_\phi)$: those are the only $\xi$ for which the oscillation can resonate with the test frequency $x\cdot\xi$. Now suppose $(x_1,\xi_1)$ is **not** of this form, i.e. not in $\lambda_\phi(C_\phi \cap \operatorname{supp} a)$. Since that set is closed and conic, we can find a genuine cone $V$ around $\xi_1$ and a small ball around $x_1$ that stay away from it, with room to spare. "Room to spare" is made quantitative by the lower bound $|d_{x,\theta}\Phi| \gtrsim |\xi| + |\theta|$: the homogeneity of $\phi$ (degree $1$ in $\theta$, so $d_x\phi$ scales like $|\theta|$ and $d_\theta\phi$ like $|\theta|^0$) guarantees the estimate is uniform along rays. The two scales $|\xi|$ and $|\theta|$ both appear because the integration is over $\theta$ while the decay we want is in $\xi$; the operator $\widetilde L$ exploits whichever is dominant. With this lower bound, the operator $\widetilde{L} = \frac{1}{i|d_{x,\theta}\Phi|^2}\,\overline{d_{x,\theta}\Phi}\cdot\nabla_{x,\theta}$ reproduces $e^{i\Phi}$ and, upon transposing, lowers the order of the amplitude by one each time — using the classical symbol bounds $|\partial^\gamma_\theta a| \lesssim (1+|\theta|)^{m-|\gamma|}$ to control the new terms. After $M$ applications the integrand is $O((1+|\xi|)^{-M})$ uniformly for $\xi \in V$, which is the rapid decay we wanted. This is the Non-[Stationary Phase Lemma](/theorems/636) in action; the same-run result [citetheorem:8200] packages exactly this estimate for homogeneous phases, so we invoke it after checking its hypotheses. Concluding, $(x_1,\xi_1) \notin \operatorname{WF}(u)$, and since this holds for every covector off $\lambda_\phi(C_\phi\cap\operatorname{supp} a)$, the wave front set is trapped inside that set, hence inside $N^*S \setminus 0$. Projecting to the base recovers that the singularities of $u$ live on $S$. [/guided] [/step] [step:Reduce a general nondegenerate phase to the model phase and conclude conormality] It remains to upgrade the wave front bound to the assertion that $u$ is **conormal** to $S$, i.e. that near each point of $S$, $u$ admits, in adapted coordinates, an oscillatory representation with the model phase $z\cdot\zeta$ and a classical symbol. Fix $p \in S \cap \operatorname{supp} u$ and adapted coordinates $(y,z)$ as in Step 1 with $S = \{z = 0\}$. Both the given phase $\phi$ (restricted to a conic neighbourhood of a point of $C_\phi$ over $p$) and the model phase $\phi_0(y,z,\zeta) = z\cdot\zeta$ are nondegenerate conic phases parametrizing the same open conic piece of $N^*S \setminus 0$ near the relevant covector, by Step 2 for $\phi_0$ and by hypothesis for $\phi$. By the [Local Equivalence of Nondegenerate Phase Parametrisations](#) (same-run [citetheorem:8201]), after shrinking $\Omega$ there is, microlocally near the common critical point, a fibre-preserving change of the phase variable together with the addition of a nondegenerate quadratic form in auxiliary variables carrying $\phi$ to $\phi_0$. Substituting this equivalence into the oscillatory integral and eliminating the auxiliary variables by the [Stationary Phase Theorem](#) (same-run [citetheorem:8198]) — whose nondegenerate Hessian hypothesis is exactly the nondegeneracy of the added quadratic form — transforms $u$, modulo a $C^\infty$ remainder, into \begin{align*} u(x) = \int_{\mathbb{R}^\ell} e^{i\, z\cdot\zeta}\, b(y,\zeta)\, d\mathcal{L}^\ell(\zeta) + C^\infty, \end{align*} where $b \in S^{m'}_{\mathrm{cl}}(\mathbb{R}^k;\mathbb{R}^\ell)$ is again a classical symbol, of the shifted order $m' = m + (N-\ell)/2$ produced by the stationary-phase elimination of the $N - \ell$ excess variables, and with $y$-support compactly contained in the chart. This is exactly the local oscillatory representation defining a distribution conormal to $S = \{z = 0\}$; hence $u$ is conormal to $S$. Consistency of the wave front bound is confirmed by the [Wave Front Set Inclusion for Conormal Distributions](#) (same-run [citetheorem:8185]), which gives $\operatorname{WF}(u) \subset N^*S \setminus 0$ for any such conormal distribution, and by [citetheorem:8204] identifying conormal distributions to $S$ with the Lagrangian distributions associated to $N^*S \setminus 0$. Combining the four preceding steps: Part 1 is the explicit model-phase parametrization of $N^*S$ established in Steps 1–2, and Part 2 — well-definedness (Step 3), the wave front bound and smoothness off $S$ (Step 4), and conormality (this step) — completes the proof. [/step]