[proofplan] We choose canonical cotangent coordinates near $\lambda_0$ and split the variables so that mixed variables $(x',\xi'')$ give local coordinates on $\Lambda$. In these coordinates the Lagrangian condition makes the one-form $\xi'\,dx' - x''\,d\xi''$ closed, so the Poincare lemma gives a [generating function](/page/Generating%20Function) $S(x',\xi'')$, chosen homogeneous of degree one by using conicity. The phase $\phi(x,\theta)=S(x',\theta)+x''\cdot\theta$ has critical equations exactly equal to the generating-function equations for $\Lambda$, and its parametrization map is precisely the local parametrization of $\Lambda$. Finally we shrink the phase domain and the target conic neighbourhood to isolate this branch and obtain the stated diffeomorphism. [/proofplan]

[step:Choose mixed canonical coordinates on $\Lambda$] Choose a coordinate chart on $X$ near $x_0$ with local coordinates $x=(x_1,\dots,x_n)$, and use the induced canonical cotangent coordinates $(x,\xi)$ on $T^*X$, where $\xi=(\xi_1,\dots,\xi_n)$. Let $L:=T_{\lambda_0}\Lambda\subset T_{\lambda_0}(T^*X)$ be the tangent space. Since $\Lambda$ is Lagrangian, $L$ is a Lagrangian subspace for the canonical symplectic form. We use the following mixed-coordinate linear algebra fact. In a symplectic [vector space](/page/Vector%20Space) with canonical coordinates $(x_1,\dots,x_n,\xi_1,\dots,\xi_n)$ and symplectic form $\sum_{i=1}^n d\xi_i\wedge dx_i$, every Lagrangian subspace $L$ admits a choice of exactly one coordinate from each pair $(x_i,\xi_i)$ such that the corresponding coordinate projection is injective on $L$. Indeed, if no such choice existed, the coordinate restrictions on $L$ would be linearly dependent for every transversal choice; applying the elementary exchange argument for the symplectic basis would produce a nonzero vector in $L\cap L^\omega{}^\perp$ annihilating all chosen coordinates, contradicting $L=L^\omega$ and $\dim L=n$. Since the target coordinate space has dimension $n$, injectivity is then an isomorphism onto that coordinate space. Because $\Lambda$ is conic and $\xi_0\neq 0$, the radial vector $(0,\xi_0)$ belongs to $L$ and is nonzero. Hence the selected coordinate space cannot consist only of base coordinates, since all base coordinate differentials vanish on the radial vector. After relabelling the canonical coordinate pairs, the selected variables may therefore be written as $(x',\xi'')$, where $x'=(x_1,\dots,x_k)$ and $\xi''=(\xi_{k+1},\dots,\xi_n)$ with $k+N=n$ and $N\geq 1$. Equivalently, writing \begin{align*} x=(x',x'')\in \mathbb{R}^k\times\mathbb{R}^N,\qquad \xi=(\xi',\xi'')\in \mathbb{R}^k\times\mathbb{R}^N, \end{align*} the map \begin{align*} \rho:\Lambda \to \mathbb{R}^k\times \mathbb{R}^N,\qquad (x',x'',\xi',\xi'')\mapsto (x',\xi'') \end{align*} has rank $n$ at $\lambda_0$. By the [inverse function theorem](/theorems/51) applied to $\rho|_\Lambda$, after shrinking $\Lambda$ around $\lambda_0$, the map $\rho$ is a coordinate map onto an [open set](/page/Open%20Set). Since $\Lambda$ is invariant under positive fibre dilations and the variables $(x',\xi'')$ scale by $(x',\xi'')\mapsto (x',t\xi'')$, the shrink can be taken invariant under this scaling. Thus the image is an open conic set $V\subset \mathbb{R}^k\times\mathbb{R}^N_0$. Thus there are smooth maps \begin{align*} a:V\to\mathbb{R}^N \end{align*} and \begin{align*} b:V\to\mathbb{R}^k \end{align*} such that the corresponding local parametrization of $\Lambda$ is \begin{align*} F:V\to T^*X\setminus 0,\qquad (x',\eta)\mapsto (x',a(x',\eta),b(x',\eta),\eta), \end{align*} where $\eta$ denotes the mixed fibre variable $\xi''$. Since $\Lambda$ is conic and $F(x',\eta)$ has fibre component $(b(x',\eta),\eta)$, after shrinking $V$ to a conic neighbourhood we have \begin{align*} a(x',t\eta)=a(x',\eta) \end{align*} and \begin{align*} b(x',t\eta)=t\,b(x',\eta) \end{align*} for all $(x',\eta)\in V$ and all $t>0$ for which $(x',t\eta)\in V$. [/step]

[step:Integrate the closed mixed one-form to a homogeneous generating function]Let $\alpha$ be the smooth one-form on $V$ defined by \begin{align*} \alpha := \sum_{i=1}^k b_i(x',\eta)\,dx_i - \sum_{j=1}^N a_j(x',\eta)\,d\eta_j. \end{align*} Let $\vartheta$ denote the canonical one-form on $T^*X$, so in these coordinates \begin{align*} \vartheta = \sum_{i=1}^k \xi_i'\,dx_i + \sum_{j=1}^N \xi_j''\,dx_j''. \end{align*} Pulling back by $F$ gives \begin{align*} F^*\vartheta = \sum_{i=1}^k b_i(x',\eta)\,dx_i + \sum_{j=1}^N \eta_j\,d a_j(x',\eta). \end{align*} Since $d\left(\sum_{j=1}^N \eta_j a_j(x',\eta)\right)=\sum_{j=1}^N \eta_j\,d a_j(x',\eta)+\sum_{j=1}^N a_j(x',\eta)\,d\eta_j$, we have \begin{align*} \alpha = F^*\vartheta - d\left(\sum_{j=1}^N \eta_j a_j(x',\eta)\right). \end{align*} The Lagrangian condition says that the canonical symplectic form $d\vartheta$ restricts to zero on $\Lambda$, hence $d\alpha = F^*(d\vartheta) - d^2\left(\sum_{j=1}^N \eta_j a_j(x',\eta)\right)=0$. Write $x_0=(x_0',x_0'')$ and $\eta_0:=\xi_0''$ for the coordinates of $\lambda_0$ in the mixed coordinate system from the previous step. Choose a small open ball $B'$ around $x_0'$ in the $x'$ variables and a small open patch $A$ around the ray direction $\eta_0/|\eta_0|$ in the unit sphere $S^{N-1}$. After shrinking inside the mixed-coordinate chart, take \begin{align*} V:=\{(x',r\omega):x'\in B',\omega\in A,r>0\}. \end{align*} This set is open and conic. By choosing $B'$ and $A$ contractible, the map $(x',r\omega)\mapsto (x',\omega,r)$ identifies $V$ with $B'\times A\times(0,\infty)$, so $V$ is connected and contractible. On this contractible neighbourhood, the Poincare lemma for contractible open sets gives a smooth function \begin{align*} S:V\to\mathbb{R} \end{align*} such that $dS=\alpha$. The conic scaling map on $V$ is \begin{align*} r_t:V\to V,\qquad (x',\eta)\mapsto (x',t\eta). \end{align*} The conicity relations for $a$ and $b$ imply $r_t^*\alpha=t\alpha$. Let $E$ be the Euler vector field in the $\eta$ variables, \begin{align*} E:=\sum_{j=1}^N \eta_j\partial_{\eta_j}. \end{align*} Differentiating $r_t^*\alpha=t\alpha$ at $t=1$ gives $\mathcal{L}_E\alpha=\alpha$. Therefore $d(ES-S)=\mathcal{L}_E(dS)-dS=\mathcal{L}_E\alpha-\alpha=0$. On the connected neighbourhood $V$, the function $ES-S$ is constant. Replacing $S$ by $S+c$ for the unique constant $c$ equal to this value gives \begin{align*} E S=S. \end{align*} Thus $S$ is homogeneous of degree one in $\eta$. Since $dS=\alpha$, its partial derivatives satisfy $\partial_{x_i}S(x',\eta)=b_i(x',\eta)$ for $1\leq i\leq k$, and $\partial_{\eta_j}S(x',\eta)=-a_j(x',\eta)$ for $1\leq j\leq N$.[/step]

[guided]The point of the mixed coordinates is that a Lagrangian submanifold becomes the graph of two functions over the variables $(x',\eta)=(x',\xi'')$. We have therefore written the local branch of $\Lambda$ as \begin{align*} F(x',\eta)=(x',a(x',\eta),b(x',\eta),\eta). \end{align*} The desired generating function should satisfy $\partial_{x'}S=b$ and $\partial_\eta S=-a$. Equivalently, its differential should be the one-form \begin{align*} \alpha=\sum_{i=1}^k b_i(x',\eta)\,dx_i-\sum_{j=1}^N a_j(x',\eta)\,d\eta_j. \end{align*} So the first question is: why is this one-form closed? Let $\vartheta$ be the canonical one-form on $T^*X$. In the present canonical coordinates, \begin{align*} \vartheta=\sum_{i=1}^k \xi_i'\,dx_i+\sum_{j=1}^N \xi_j''\,dx_j''. \end{align*} Pulling $\vartheta$ back along the parametrization $F$ gives \begin{align*} F^*\vartheta=\sum_{i=1}^k b_i(x',\eta)\,dx_i+\sum_{j=1}^N \eta_j\,d a_j(x',\eta). \end{align*} The term involving $d a_j$ is not yet the form $\alpha$, but it differs from $-\sum_j a_j\,d\eta_j$ by an exact differential. Indeed, \begin{align*} d\left(\sum_{j=1}^N \eta_j a_j(x',\eta)\right)=\sum_{j=1}^N \eta_j\,d a_j(x',\eta)+\sum_{j=1}^N a_j(x',\eta)\,d\eta_j. \end{align*} Subtracting this exact differential from $F^*\vartheta$ gives exactly \begin{align*} \alpha=F^*\vartheta-d\left(\sum_{j=1}^N \eta_j a_j(x',\eta)\right). \end{align*} Now the Lagrangian hypothesis enters. Since $\Lambda$ is Lagrangian, the canonical symplectic form $d\vartheta$ restricts to zero on $\Lambda$. Therefore \begin{align*} d\alpha=F^*(d\vartheta)-d^2\left(\sum_{j=1}^N \eta_j a_j(x',\eta)\right)=0. \end{align*} The second term is zero because the [exterior derivative](/theorems/1525) squares to zero. Thus $\alpha$ is closed. Choose a small open ball $B'$ around $x_0'$ in the $x'$ variables and a small contractible open patch $A$ around the ray direction $\eta_0/|\eta_0|$ in the unit sphere $S^{N-1}$. Replacing $V$ by \begin{align*} V=\{(x',r\omega):x'\in B',\omega\in A,r>0\}, \end{align*} inside the mixed-coordinate chart keeps the neighbourhood conic. The coordinates $(x',\omega,r)$ identify this set with $B'\times A\times(0,\infty)$, so it is connected and contractible. On this contractible neighbourhood, the Poincare lemma for contractible open sets gives a smooth function \begin{align*} S:V\to\mathbb{R} \end{align*} with $dS=\alpha$. This already gives \begin{align*} \partial_{x_i}S(x',\eta)=b_i(x',\eta) \end{align*} and \begin{align*} \partial_{\eta_j}S(x',\eta)=-a_j(x',\eta). \end{align*} It remains to make the primitive exactly homogeneous of degree one. This is not automatic, because primitives are only determined up to an additive constant. Let \begin{align*} r_t(x',\eta)=(x',t\eta) \end{align*} be fibre scaling, and let \begin{align*} E=\sum_{j=1}^N \eta_j\partial_{\eta_j} \end{align*} be the Euler vector field. Conicity gives $a(x',t\eta)=a(x',\eta)$ and $b(x',t\eta)=t\,b(x',\eta)$, so $r_t^*\alpha=t\alpha$. Differentiating this identity at $t=1$ gives $\mathcal{L}_E\alpha=\alpha$. Since $dS=\alpha$, \begin{align*} d(E S-S)=\mathcal{L}_E(dS)-dS=\mathcal{L}_E\alpha-\alpha=0. \end{align*} Thus $ES-S$ is constant on the connected neighbourhood $V$. If this constant is $c$, replacing $S$ by $S+c$ changes $ES-S$ to $ES-S-c$, hence makes it zero. Therefore the adjusted $S$ satisfies \begin{align*} ES=S, \end{align*} which is Euler's homogeneous-function identity for degree-one homogeneity in $\eta$.[/guided]

[step:Build the homogeneous phase and compute its critical equations] Shrink the coordinate neighbourhood in $X$ if necessary so that the variables $x=(x',x'')$ range over an open set $U\subset X$, and let $\Theta\subset\mathbb{R}^N_0$ be a conic open neighbourhood of the chosen value $\eta_0=\xi_0''$ such that $(x',\theta)\in V$ whenever $(x',x'')\in U$ and $\theta\in\Theta$ in the smaller neighbourhood under consideration. Define \begin{align*} \phi:U\times\Theta\to\mathbb{R},\qquad (x',x'',\theta)\mapsto S(x',\theta)+\sum_{j=1}^N x_j''\theta_j. \end{align*} Because $S$ is homogeneous of degree one in $\theta$ and the second term is linear in $\theta$, the function $\phi$ is homogeneous of degree one in $\theta$. For $1\leq j\leq N$, $\partial_{\theta_j}\phi(x',x'',\theta)=\partial_{\theta_j}S(x',\theta)+x_j''$. Hence $C_\phi=\{(x',x'',\theta)\in U\times\Theta:x''=-\partial_\theta S(x',\theta)\}$. Using $\partial_\theta S=-a$, this becomes $C_\phi=\{(x',a(x',\theta),\theta):(x',\theta)\in V\cap(U'\times\Theta)\}$, where $U'$ denotes the $x'$-projection of $U$ in the chosen coordinates. [/step]

[step:Verify nondegeneracy of the phase] For each $1\leq j\leq N$, define \begin{align*} G_j:U\times\Theta\to\mathbb{R},\qquad (x',x'',\theta)\mapsto \partial_{\theta_j}\phi(x',x'',\theta). \end{align*} Since \begin{align*} G_j(x',x'',\theta)=\partial_{\theta_j}S(x',\theta)+x_j'', \end{align*} the differential $dG_j$ contains the term $dx_j''$ with coefficient $1$. If [real numbers](/page/Real%20Numbers) $c_1,\dots,c_N$ satisfy \begin{align*} \sum_{j=1}^N c_j\,dG_j=0 \end{align*} at a point of $C_\phi$, then evaluating this covector on the coordinate vector $\partial_{x_\ell''}$ gives $c_\ell=0$ for every $1\leq \ell\leq N$. Therefore $dG_1,\dots,dG_N$ are linearly independent on $C_\phi$, and $\phi$ is a nondegenerate homogeneous phase function. [/step]

[step:Identify the parametrized Lagrangian with the original local branch] Let $\Omega\subset U\times\Theta$ be the conic neighbourhood of the point $(x_0,\theta_0)$ that will be fixed in the final shrinking step. Define the localized critical set by \begin{align*} C_\phi^{\mathrm{loc}}:=C_\phi\cap\Omega. \end{align*} The phase parametrization map is \begin{align*} j_\phi:C_\phi\to T^*U\setminus 0,\qquad (x,\theta)\mapsto (x,\partial_x\phi(x,\theta)). \end{align*} On $C_\phi$ we have $x''=a(x',\theta)$. Moreover, $\partial_{x_i'}\phi(x',x'',\theta)=\partial_{x_i'}S(x',\theta)=b_i(x',\theta)$ for $1\leq i\leq k$, and $\partial_{x_j''}\phi(x',x'',\theta)=\theta_j$ for $1\leq j\leq N$. Thus, for every $(x',\theta)$ in the localized parameter domain, $j_\phi(x',a(x',\theta),\theta)=(x',a(x',\theta),b(x',\theta),\theta)$. The right-hand side is exactly $F(x',\theta)$, the chosen local parametrization of $\Lambda$. Since $(x',\theta)$ are coordinates on $\Lambda$ through $F$, the map $j_\phi$ restricts to a diffeomorphism from the localized critical set $C_\phi^{\mathrm{loc}}$ onto the corresponding local branch of $\Lambda$. [/step]

[step:Shrink the neighbourhoods to exclude other branches] Choose $(x_0,\theta_0)\in C_\phi$ with $j_\phi(x_0,\theta_0)=\lambda_0$. Shrink $U\times\Theta$ to a conic neighbourhood $\Omega$ of $(x_0,\theta_0)$ inside the phase domain. On $C_\phi\cap\Omega$, the previous step identifies $j_\phi$ with the coordinate parametrization $F(x',\theta)$; hence $j_\phi$ is an immersion there and is one-to-one after shrinking $\Omega$ further around $(x_0,\theta_0)$. With $C_\phi^{\mathrm{loc}}:=C_\phi\cap\Omega$, the set $j_\phi(C_\phi^{\mathrm{loc}})$ is an open local branch of $\Lambda$ containing $\lambda_0$. Choose an open conic neighbourhood $W\subset T^*U\setminus 0$ of $\lambda_0$ contained in the coordinate neighbourhood and small enough that no point of $(\Lambda\setminus j_\phi(C_\phi^{\mathrm{loc}}))$ lies in $W$. Then $j_\phi(C_\phi^{\mathrm{loc}})=\Lambda\cap W$. By the definition of $\Lambda_\phi$ as the image of $C_\phi$ under $j_\phi$, and by the branch exclusion obtained from the shrink, this gives $\Lambda_\phi\cap W=\Lambda\cap W$. The same shrink preserves the diffeomorphism property of $j_\phi:C_\phi^{\mathrm{loc}}\to\Lambda\cap W$, because in the coordinates above it is the map induced by the coordinate parametrization $F(x',\theta)$. This proves both the local equality of conic Lagrangians and the asserted diffeomorphic parametrization of the localized critical set. [/step]