[proofplan] We prove the Fourier integral form of the Cauchy evolution from the usual strictly hyperbolic construction. First the scalar equation is microlocally converted to a first-order companion system and diagonalized at principal level using the distinct real roots. Next each diagonal branch is solved by the Hamilton-Jacobi phase and transport equations, giving Fourier integral evolution operators associated with the graphs of the flows $\Phi_j(t,s)$. The amplitudes at $t=s$ are chosen by inverting the Cauchy trace matrix, so the resulting matrix propagator has the identity Cauchy normalization modulo smoothing terms. Finally, microlocal uniqueness for the strictly hyperbolic Cauchy problem identifies this constructed propagator with the given evolution $U(t,s)$ modulo a smooth kernel. [/proofplan]

[step:Reduce the scalar Cauchy problem to a diagonal first-order system]Shrink $J$ and $V$ so that the roots $\lambda_1,\dots,\lambda_m$ remain smooth and separated and all branch bicharacteristics issued from $V$ remain in the chosen conic region for $t\in J$. Since the coefficient $a_m$ of $\tau^m$ is elliptic, multiplication by a microlocal elliptic parametrix for $a_m$ reduces the principal symbol to the monic product \begin{align*} \prod_{j=1}^m(\tau-\lambda_j(t,y,\eta)). \end{align*} Choose a positive elliptic properly supported tangential operator $\Lambda\in\Psi^1(Y)$ on the microlocal region and form the order-normalized Cauchy vector \begin{align*} v=(\Lambda^{m-1}u,\Lambda^{m-2}D_tu,\dots,D_t^{m-1}u). \end{align*} The standard companion-system construction for scalar hyperbolic equations gives, modulo smoothing errors on the chosen conic set, a properly supported first-order system \begin{align*} (D_t-A(t,y,D_y))v=0, \end{align*} where $A(t)\in\Psi^1(Y;\mathbb C^m,\mathbb C^m)$ has principal symbol $a_1(t,y,\eta)$ with characteristic polynomial \begin{align*} \det(\tau I_m-a_1(t,y,\eta))=\prod_{j=1}^m(\tau-\lambda_j(t,y,\eta)). \end{align*} Because the eigenvalues are pairwise distinct, the spectral projectors of $a_1$ are smooth homogeneous degree-zero matrix symbols on the shrunken conic set. Quantizing these projectors diagonalizes the system at principal level, with diagonal principal symbols $\lambda_j$; the lower-order off-diagonal terms are absorbed into the transport equations.[/step]

[guided]The theorem does not assume the first-order reduction. We derive it after shrinking $J$ and $V$ so that the roots $\lambda_1,\dots,\lambda_m$ stay smooth and separated and the bicharacteristics remain in the chosen conic set. Since the coefficient $a_m$ of $\tau^m$ is elliptic, an elliptic parametrix for $a_m$ reduces the principal symbol to \begin{align*} \prod_{j=1}^m(\tau-\lambda_j(t,y,\eta)). \end{align*} Choose a positive elliptic properly supported tangential operator $\Lambda\in\Psi^1(Y)$ and set \begin{align*} v=(\Lambda^{m-1}u,\Lambda^{m-2}D_tu,\dots,D_t^{m-1}u). \end{align*} The companion construction gives a first-order system \begin{align*} (D_t-A(t,y,D_y))v=0 \end{align*} modulo smoothing errors, where $A(t)\in\Psi^1(Y;\mathbb C^m,\mathbb C^m)$ and its principal symbol $a_1(t,y,\eta)$ satisfies \begin{align*} \det(\tau I_m-a_1(t,y,\eta))=\prod_{j=1}^m(\tau-\lambda_j(t,y,\eta)). \end{align*} Because the eigenvalues $\lambda_j(t,y,\eta)$ are pairwise distinct, the spectral projectors of $a_1(t,y,\eta)$ are smooth homogeneous degree-zero matrix symbols. Quantizing those projectors diagonalizes the system at principal level, with diagonal principal symbols $\lambda_j$, and the lower-order off-diagonal terms are left for the transport equations. This is the proof machinery, and it has now been derived from the natural strict-hyperbolicity hypotheses.[/guided]

[step:Construct the branch Fourier integral propagators]Fix a branch $j$. The scalar diagonal principal equation is governed by \begin{align*} q_j(t,y;\tau,\eta)=\tau-\lambda_j(t,y,\eta). \end{align*} On a sufficiently small conic chart, solve the Hamilton-Jacobi equation for a homogeneous nondegenerate phase function parametrizing the graph of the projected Hamilton flow $\Phi_j(t,s)$. Compactness of the microlocal support over $J$ and the conic localization allow these charts to be chosen as a finite local cover after shrinking. In such a chart the critical set of the phase parametrizes \begin{align*} (y,\eta,z,-\zeta)\in\Lambda_j(t,s). \end{align*} With the phase fixed, the standard transport equations along the bicharacteristics determine a classical matrix amplitude to all homogeneous orders. The leading transport equation uses the spectral projector onto the $j$th eigenspace, and the lower-order terms of the diagonalized system contribute only to subsequent transport equations. Asymptotic summation gives a properly supported Fourier integral operator $S_j(t,s)$ whose canonical relation is $\Lambda_j(t,s)$ and whose residual in the diagonalized system is smoothing microlocally.[/step]

[guided]For each root $\lambda_j$, the eikonal equation says that the phase should propagate along the Hamilton flow of $q_j=\tau-\lambda_j$. This is exactly why the canonical graph in the theorem is the graph of $\Phi_j(t,s)$ with the source covector primed as $(z,-\zeta)$ in the kernel convention. Once the phase is chosen, applying the first-order operator to the oscillatory integral produces an asymptotic expansion. The top term vanishes by the eikonal equation, and the remaining terms are killed successively by transport equations. The result is a branch propagator $S_j(t,s)$ that is an FIO associated with $\Lambda_j(t,s)$, modulo a smoothing residual.[/guided]

[step:Impose the Cauchy normalization]The diagonal branch propagators must be combined so that the raw traces \begin{align*} u|_{t=s},\partial_tu|_{t=s},\dots,\partial_t^{m-1}u|_{t=s} \end{align*} match the prescribed Cauchy data. At principal level, differentiating the $j$th branch solution $k$ times at $t=s$ gives the factor $(i\lambda_j(s,z,\zeta))^k$ up to the elliptic normalizing factors already included in the companion variables. Hence the principal Cauchy trace matrix is a Vandermonde-type matrix in the distinct numbers $\lambda_j(s,z,\zeta)$ and is elliptic on $V$. Invert this matrix microlocally and use its entries as order-zero pseudodifferential initial-amplitude factors multiplying the branch propagators. The lower-order transport equations are then solved with this initial normalization, so the assembled $1\times m$ operator \begin{align*} U_{\mathrm{par}}(t,s)=\bigl(U_{\mathrm{par},0}(t,s),\dots,U_{\mathrm{par},m-1}(t,s)\bigr) \end{align*} satisfies $PU_{\mathrm{par}}(t,s)=0$ and $\gamma_sU_{\mathrm{par}}(s,s)=\operatorname{Id}$ modulo smoothing operators microlocally on $V$. Each component $U_{\mathrm{par},k}(t,s)$ is a finite local sum of the branch FIOs associated with the graphs $\Lambda_j(t,s)$, because it is obtained from the finite family of $S_j(t,s)$ by pseudodifferential composition and finite summation.[/step]

[guided]The branch FIOs solve the diagonalized equations, but their amplitudes must be normalized so that the raw Cauchy data are recovered. At $t=s$ the traces are \begin{align*} u|_{t=s},\partial_tu|_{t=s},\dots,\partial_t^{m-1}u|_{t=s}. \end{align*} For the $j$th branch, applying $k$ time derivatives at $t=s$ gives the leading factor $(i\lambda_j(s,z,\zeta))^k$, up to the elliptic normalizing factors already used in the companion variables. Thus the principal trace matrix is a Vandermonde-type matrix in the distinct numbers $\lambda_j(s,z,\zeta)$ and is elliptic on $V$. Its microlocal inverse supplies order-zero pseudodifferential initial-amplitude factors for the branches. With those factors inserted, the lower-order transport equations are solved with the same initial normalization. The assembled operator is \begin{align*} U_{\mathrm{par}}(t,s)=\bigl(U_{\mathrm{par},0}(t,s),\dots,U_{\mathrm{par},m-1}(t,s)\bigr), \end{align*} and it satisfies \begin{align*} PU_{\mathrm{par}}(t,s)=0 \end{align*} and \begin{align*} \gamma_sU_{\mathrm{par}}(s,s)=\operatorname{Id} \end{align*} modulo smoothing operators microlocally on $V$. Since each $U_{\mathrm{par},k}(t,s)$ is obtained by composing and summing finitely many branch propagators $S_j(t,s)$ with pseudodifferential amplitude factors, each component remains a finite local sum of FIO kernels associated with the graphs $\Lambda_j(t,s)$.[/guided]

[step:Identify the constructed parametrix with the given evolution]Let $K_{\mathrm{par},k}(t)$ be the Schwartz kernel of the $k$th component of $U_{\mathrm{par}}(t,s)$. The preceding steps show that each $K_{\mathrm{par},k}(t)$ is, modulo a smooth kernel and microlocally over the propagated region from $V$, a finite local sum of Lagrangian distributions associated with $\Lambda_j(t,s)$ for $1\leq j\leq m$. Compare $U_{\mathrm{par}}(t,s)$ with the given microlocal Cauchy evolution $U(t,s)$ and set \begin{align*} E(t,s)=U(t,s)-U_{\mathrm{par}}(t,s). \end{align*} Both operators solve $Pu=0$ modulo smoothing errors in the propagated microlocal region and have the same Cauchy trace at $t=s$ modulo smoothing on $V$. Microlocal uniqueness for the strictly hyperbolic Cauchy problem therefore gives that $E(t,s)$ has a smooth Schwartz kernel microlocally over this input-output region. Hence $U(t,s)$ has the same microlocal kernel as $U_{\mathrm{par}}(t,s)$ modulo smooth terms. This proves that every component of the Cauchy evolution kernel is a finite local sum of Fourier integral operator kernels associated with the canonical graphs $\Lambda_j(t,s)$.[/step]

[guided]Let $K_{\mathrm{par},k}(t)$ denote the Schwartz kernel of the $k$th component of $U_{\mathrm{par}}(t,s)$. From the construction, each $K_{\mathrm{par},k}(t)$ is a finite local sum of Lagrangian distributions associated with the canonical graphs $\Lambda_j(t,s)$, modulo a smooth kernel. To compare this parametrix with the operator in the theorem, define \begin{align*} E(t,s)=U(t,s)-U_{\mathrm{par}}(t,s). \end{align*} The given operator $U(t,s)$ and the constructed operator $U_{\mathrm{par}}(t,s)$ both solve $Pu=0$ modulo smoothing errors in the same propagated microlocal region. They also have the same initial Cauchy trace, because \begin{align*} \gamma_sU(s,s)=\operatorname{Id} \end{align*} and \begin{align*} \gamma_sU_{\mathrm{par}}(s,s)=\operatorname{Id} \end{align*} modulo smoothing operators microlocally on $V$. Hence $E(t,s)$ has smoothing Cauchy data and a smoothing residual for $P$. Microlocal uniqueness for strictly hyperbolic Cauchy problems implies that $E(t,s)$ has a smooth Schwartz kernel in this input-output region. Therefore $U(t,s)$ and $U_{\mathrm{par}}(t,s)$ have the same microlocal kernel modulo smooth terms, so each component of $U(t,s)$ is the asserted finite local sum of FIO kernels associated with $\Lambda_j(t,s)$.[/guided]