Let $Y$ be a smooth $n$-dimensional manifold, let $I \subset \mathbb{R}$ be an open interval containing $0$, and let $\lambda:I \times (T^*Y \setminus 0) \to \mathbb{R}$ be a smooth real-valued function that is positively homogeneous of degree $1$ in the cotangent variable. Fix $(z_0,\eta_0) \in T^*Y \setminus 0$ and $T_0>0$ with $(-T_0,T_0)\subset I$. Choose a coordinate chart $(U,\kappa)$ with $z_0\in U$ and use the induced cotangent trivialisation to identify a conic neighbourhood of $(z_0,\eta_0)$ in $T^*U\setminus 0$ with a conic subset of $\kappa(U)\times\mathbb{R}^n_0$. Define $q:I \times T^*U \times \mathbb{R}_\tau \to \mathbb{R}$ in these coordinates by $q(t,y,\tau,\xi)=\tau+\lambda(t,y,\xi)$. Assume that the Hamilton flow of $q$, with respect to the canonical one-form $\Theta=\tau\,dt+\xi\cdot dy$ and the convention that the Hamilton vector field $H_q$ satisfies $d\Theta(H_q,\cdot)=-dq$, is defined for $|t|<T_0$ on a conic neighbourhood of $\{(0,z,-\lambda(0,z,\eta),\eta):(z,\eta)\in V\}$, where $V \subset T^*U\setminus 0$ is a conic neighbourhood of $(z_0,\eta_0)$. Let $(z,\eta)\mapsto (y(t;z,\eta),\xi(t;z,\eta))$ denote the spatial part of this flow, with initial condition $y(0;z,\eta)=z$ and $\xi(0;z,\eta)=\eta$. Assume that, for every $|t|<T_0$ and every $(z,\eta)$ in a neighbourhood of $(z_0,\eta_0)$, the map $\Pi_t:V \to \kappa(U) \times \mathbb{R}^n_0$ given by $\Pi_t(z,\eta)=(y(t;z,\eta),\eta)$ is a local diffeomorphism onto its image, equivalently $\det(\partial y/\partial z(t;z,\eta))\ne 0$ in the chosen coordinates. Then, after replacing $V$ by a smaller conic neighbourhood of $(z_0,\eta_0)$ and replacing $T_0$ by a smaller positive number, there exist an open conic set $\Omega \subset (-T_0,T_0)\times \kappa(U)\times\kappa(U)\times \mathbb{R}^n_0$ and a smooth real-valued phase function $\phi:\Omega \to \mathbb{R}$ such that, writing the variables of $\phi$ as $(t,y,z,\eta)$, the following hold. First, $\phi$ is positively homogeneous of degree $1$ in $\eta$. Second, the critical set $\Sigma_\phi := \{(t,y,z,\eta)\in \Omega:\partial_\eta\phi(t,y,z,\eta)=0\}$ is smooth, and the differential of the map $(t,y,z,\eta)\mapsto \partial_\eta\phi(t,y,z,\eta)$ has rank $n$ at every point of $\Sigma_\phi$. Third, on $\Sigma_\phi$, $\partial_t\phi(t,y,z,\eta)+\lambda(t,y,\partial_y\phi(t,y,z,\eta))=0$. Finally, for each fixed $t$ with $|t|<T_0$, the phase parametrises the shrunken local canonical graph $C_t=\{(y(t;z,\eta),\xi(t;z,\eta);z,\eta):(z,\eta)\in V\}\subset (T^*U\setminus 0)\times (T^*U\setminus 0)$ in the sense that $C_t=\{(y,\partial_y\phi(t,y,z,\eta);z,-\partial_z\phi(t,y,z,\eta)):(t,y,z,\eta)\in \Sigma_\phi\}$.