[proofplan] We first shrink the conic region so that the time-covariable roots remain smooth, real, and separated, and then replace the scalar equation by its microlocally diagonalized first-order Cauchy system. Each diagonal branch is quantized by a Fourier integral evolution operator obtained from the Hamilton-Jacobi phase and transport equations. The initial amplitudes are chosen by inverting the Vandermonde matrix determined by the distinct roots, and the inhomogeneous operator is obtained from the zero-Cauchy retarded Green operator for the diagonal system. Finally, the wave Cauchy evolution theorem, Sobolev mapping for the resulting Fourier integral operators, and propagation of microlocal smoothness give the asserted error, regularity, support, and uniqueness statements. [/proofplan]

[step:Shrink the conic region and normalize the principal symbol] Choose a conic neighbourhood $V_1\subset T^*Y\setminus 0$ of $(y_0,\eta_0)$ and a smaller time interval $I_1=(-T_1,T_1)\subset I$ such that every point \begin{align*} (0,y;\lambda_\ell(0,y,\eta),\eta) \end{align*} with $(y,\eta)\in V_1$ lies in $\Gamma$, and such that the roots $\lambda_\ell$ remain pairwise separated on $I_1\times V_1$. Let $a_m(t,y,\eta)$ be the coefficient of $\tau^m$ in $p_m$. Since $a_m$ is elliptic on $\Gamma$, multiplication by a microlocal elliptic parametrix for $a_m$ reduces the principal symbol to the normalized monic form \begin{align*} \widetilde p_m(t,y;\tau,\eta)=\prod_{\ell=1}^m q_\ell(t,y;\tau,\eta). \end{align*} Here \begin{align*} q_\ell(t,y;\tau,\eta)=\tau-\lambda_\ell(t,y,\eta). \end{align*} The noncharacteristic hypothesis for $\{0\}\times Y$ is the ellipticity of $a_m(0,y,\eta)$ on $\Gamma_0$. Separately, the pairwise distinctness of the roots implies that the Cauchy trace matrix formed from the numbers $\lambda_\ell(0,y,\eta)$ is invertible after shrinking $V_1$ if necessary. [/step]

[step:Replace the scalar equation microlocally by a diagonal first-order Cauchy system] Define the unrescaled Cauchy vector map \begin{align*} \mathcal C:C^\infty(I_1\times Y)\to C^\infty(I_1\times Y)^m \end{align*} by \begin{align*} \mathcal C(u):=\bigl(u,\partial_tu,\dots,\partial_t^{m-1}u\bigr). \end{align*} The proper support of $P$ and $\Lambda$ lets all microlocal cutoffs be chosen properly supported on the second-countable Hausdorff manifold $I_1\times Y$. Let $v:=\mathcal C(u)$. After applying a microlocal elliptic parametrix for $a_m$ and using the identities $\partial_t v_j=v_{j+1}$ for $0\le j\le m-2$, the scalar equation $Pu=f$ is represented, modulo smoothing errors on the chosen conic set, by a first-order system \begin{align*} (D_t-A(t,y,D_y))v=F. \end{align*} Here $A(t,y,D_y)$ is an $m\times m$ properly supported tangential pseudodifferential matrix of order $1$, $F$ is the forcing vector with only the final component containing the microlocally normalized forcing, and the principal symbol $a_1(t,y,\eta)$ of $A$ has characteristic polynomial \begin{align*} \det(\tau I_m-a_1(t,y,\eta))=\prod_{\ell=1}^m(\tau-\lambda_\ell(t,y,\eta)). \end{align*} Because the roots are real, smooth, homogeneous of degree $1$, and pairwise distinct, the eigenprojectors of $a_1$ are smooth order-zero homogeneous matrix symbols on the shrunken conic set. Quantizing these eigenprojectors gives an elliptic order-zero matrix pseudodifferential operator $M$ such that \begin{align*} M^{-1}(D_t-A)M=D_t-\operatorname{diag}(B_1,\dots,B_m)+L_0 \end{align*} microlocally, where each $B_\ell\in\Psi^1_{\mathrm{prop}}(Y)$ has principal symbol $\lambda_\ell$ and $L_0$ is an order-zero matrix operator absorbed into the transport equations. The rescaled companion map $U_\Lambda$ from the statement is related to this unrescaled system by the elliptic diagonal tangential operator with entries $\Lambda^{m-1-j}$ on the $j$th component, so using $\mathcal C$ here is exactly the unrescaled Cauchy convention used for the data $g_j$. The diagonalizing matrix at $t=0$ for the unrescaled trace vector is the microlocal Vandermonde matrix $\mathcal V:V_1\to \operatorname{GL}(m,\mathbb C)$ defined by \begin{align*} \mathcal V(y,\eta)_{j\ell}:=\bigl(i\lambda_\ell(0,y,\eta)\bigr)^j. \end{align*} Here $0\le j\le m-1$ and $1\le \ell\le m$. Its determinant is \begin{align*} \det \mathcal V(y,\eta)=i^{m(m-1)/2}\prod_{1\le k<\ell\le m}\bigl(\lambda_\ell(0,y,\eta)-\lambda_k(0,y,\eta)\bigr), \end{align*} which is elliptic on $V_1$ because the roots are pairwise distinct. Hence $\mathcal V^{-1}$ is a matrix of order-zero tangential pseudodifferential symbols on $V_1$. [/step]

[step:Construct the branch evolution Fourier integral operators]For each branch $\ell$, let \begin{align*} \Phi_\ell^t:V_1\to T^*Y\setminus 0 \end{align*} denote the following projected Hamilton flow. Starting from $(y,\eta)\in V_1$, lift it to the characteristic point $(0,y;\lambda_\ell(0,y,\eta),\eta)$ on $q_\ell^{-1}(0)$, flow by the Hamilton vector field $H_{q_\ell}$ until the time coordinate equals $t$, and then project the resulting point $(t,y_t;\lambda_\ell(t,y_t,\eta_t),\eta_t)$ to $(y_t,\eta_t)\in T^*Y\setminus 0$. The hypotheses of the Hamilton-Jacobi phase theorem [citetheorem:TEMP-55] are satisfied: $Y$ is smooth, $\lambda_\ell$ is smooth, real-valued, and positively homogeneous of degree $1$ in $\eta$, and the shrinking of $I_1$ and $V_1$ keeps the flow inside one properly supported conic coordinate neighbourhood. Hence the flow graph is parametrized by a homogeneous nondegenerate Hamilton-Jacobi phase function. The hypotheses of the transport solvability theorem [citetheorem:TEMP-56] are also satisfied: the phase is nondegenerate, the transport vector field is smooth on the conic support, and the initial hypersurface $t=0$ is noncharacteristic for $q_\ell$ because $\partial_\tau q_\ell=1$. Solving the transport equations to all homogeneous orders and applying asymptotic summation therefore gives a properly supported Fourier integral operator \begin{align*} S_\ell(t,0):\mathcal D'(Y)\to\mathcal D'(Y) \end{align*} depending smoothly on $t\in I_1$, with canonical relation equal to the graph of $\Phi_\ell^t$. In the diagonalized first-order system, the $\ell$th diagonal equation has principal symbol $q_\ell$ in the $D_t$ convention, and the phase and amplitude are chosen so that this diagonal equation is solved modulo $\Psi^{-\infty}$ on the branch flow-out. The initial normalization is \begin{align*} S_\ell(0,0)-\operatorname{id}_Y\in\Psi^{-\infty} \end{align*} microlocally on $V_1$.[/step]

[guided]Fix one branch $\ell$. The diagonalized first-order system has an $\ell$th scalar diagonal equation whose principal symbol in the $D_t=-i\partial_t$ convention is \begin{align*} q_\ell(t,y;\tau,\eta)=\tau-\lambda_\ell(t,y,\eta). \end{align*} The sign convention matters: an oscillatory factor with phase $\phi_\ell(t,y,\theta)-z\cdot\theta$ is annihilated at principal level by $D_t-\lambda_\ell(t,y,D_y)$ precisely when the phase satisfies the Hamilton-Jacobi equation \begin{align*} \partial_t\phi_\ell(t,y,\theta)-\lambda_\ell(t,y,\partial_y\phi_\ell(t,y,\theta))=0 \end{align*} with initial condition $\phi_\ell(0,y,\theta)=y\cdot\theta$ in local coordinates. Theorem [citetheorem:TEMP-55] applies because $\lambda_\ell$ is smooth, real-valued, and positively homogeneous of degree $1$ in the cotangent variable, and because $I_1$ and $V_1$ have been shrunk so that the Hamilton flow remains in the chosen conic coordinate neighbourhood. It supplies a homogeneous nondegenerate phase parametrizing the graph of the branch flow $\Phi_\ell^t$. With this phase fixed, choose a classical amplitude $a_\ell(t,y,z,\theta)$ of order $0$ supported by the same proper microlocal cutoffs. Applying the $\ell$th diagonal first-order operator to the corresponding oscillatory integral gives an asymptotic expansion in homogeneous degrees of $\theta$. The degree-one term is zero by the Hamilton-Jacobi equation. At each lower degree one obtains a transport equation along the Hamilton vector field of $q_\ell$, with initial normalization chosen so that the leading amplitude at $t=0$ is $1$ on $V_1$. The solvability theorem for transport equations [citetheorem:TEMP-56] applies because the phase is the Hamilton-Jacobi phase just constructed, the transport vector field is smooth on the conic support, and the initial hypersurface $t=0$ is noncharacteristic for the branch equation. Solving the transport equations to all homogeneous orders and applying classical asymptotic summation gives a properly supported Fourier integral operator \begin{align*} S_\ell(t,0):\mathcal D'(Y)\to\mathcal D'(Y) \end{align*} whose canonical relation is the graph of $\Phi_\ell^t$. The constructed amplitude makes the $\ell$th diagonal equation hold modulo $\Psi^{-\infty}$ microlocally on the branch flow-out, and the initial amplitude normalization gives \begin{align*} S_\ell(0,0)-\operatorname{id}_Y\in\Psi^{-\infty} \end{align*} microlocally on $V_1$. Thus $S_\ell(t,0)$ is the microlocal branch evolution from Cauchy data at $t=0$.[/guided]

[step:Choose initial amplitudes by inverting the microlocal Vandermonde system] Let \begin{align*} A_{\ell j}\in\Psi^0_{\mathrm{prop}}(Y), \qquad 1\le \ell\le m,\quad 0\le j\le m-1, \end{align*} be tangential pseudodifferential operators whose principal symbol matrix is $\mathcal V^{-1}$ on $V_1$ for the unrescaled trace system. If one uses the rescaled trace variables $U_\Lambda$, the corresponding inverse matrix is obtained by composing the $j$th trace component with the elliptic factor $\Lambda^{-(m-1-j)}$; the present theorem uses the unrescaled data $g_j$. Define \begin{align*} E_jg:=\sum_{\ell=1}^m S_\ell(t,0)A_{\ell j}g \end{align*} for $g\in\mathcal D'(Y)$. Since $S_\ell(0,0)=\operatorname{id}_Y$ modulo smoothing and since applying $\partial_t^k$ to the branch phase at $t=0$ contributes the principal factor $(i\lambda_\ell(0,y,\eta))^k$, the Cauchy trace matrix of the family $\{S_\ell(t,0)A_{\ell j}\}_{\ell,j}$ is \begin{align*} \mathcal V\mathcal V^{-1}=I_m \end{align*} modulo lower-order errors. Solving successively for the lower-order amplitudes removes these lower-order errors to all orders. Hence \begin{align*} \partial_t^k E_jg|_{t=0}-\delta_{kj}g\in C^\infty \end{align*} microlocally on $V_1$ for all $0\le k,j\le m-1$. [/step]

[step:Build the zero-Cauchy inhomogeneous parametrix from the branch Green kernel] Let $(t,y)$ denote the target spacetime variables and let $(\sigma,z)$ denote the source spacetime variables. The diagonal first-order system and the branch propagators $S_\ell(t,\sigma)$ give a properly supported distribution kernel \begin{align*} K_R\in\mathcal D'(I_1\times Y\times I_1\times Y) \end{align*} for the zero-Cauchy Green operator, after inserting fixed microlocal cutoffs supported in $W$. This is the retarded inhomogeneous operator obtained by solving the diagonal first-order system branch by branch with the propagators $S_\ell(t,\sigma)$, including the order-zero coupling $L_0$ by the same transport recursion and a Volterra Neumann series in the Cauchy-retarded time variable. Proper support is preserved because all spatial cutoffs in the kernels are properly supported and the source time $\sigma$ is restricted to the compact interval between $0$ and $t$ after shrinking $I_1$. Define \begin{align*} R:\mathcal D'(I_1\times Y)\to\mathcal D'(I_1\times Y) \end{align*} by its properly supported Schwartz kernel $K_R$; equivalently, when $f$ is smooth this operator is the Duhamel integral over $0\le \sigma\le t$ for $t\ge 0$ and over $t\le \sigma\le 0$ for $t\le 0$, with the lower-order coupling terms of the diagonalized first-order system included in the transport equations and removed to all orders by the Volterra construction. This kernel definition is meaningful for arbitrary $f\in\mathcal D'(I_1\times Y)$ and does not require pointwise time slices $f(\sigma)$. Its microlocal support is contained in the Cauchy-retarded region where $\sigma$ lies between $0$ and $t$. Applying the diagonal first-order operator to the Duhamel kernel differentiates the retarded integral in $t$ and produces the identity source term from the endpoint $\sigma=t$; the branch equations and transport equations cancel the remaining oscillatory terms to infinite order. Transferring the resulting diagonal Green operator back through the elliptic diagonalization and the unrescaled companion map $\mathcal C$ gives \begin{align*} PRf-f\in C^\infty \end{align*} microlocally on $W$, and also gives the zero Cauchy traces \begin{align*} \partial_t^jRf|_{t=0}\in C^\infty \end{align*} microlocally on $V_1$ for $0\le j\le m-1$. [/step]

[step:Assemble the solution and identify its canonical relations] For microlocalized data $g_0,\dots,g_{m-1}$ and forcing term $f$, define \begin{align*} u:=\sum_{j=0}^{m-1}E_jg_j+Rf. \end{align*} The diagonal first-order Cauchy construction, transferred back through the elliptic diagonalization and the unrescaled companion transformation $\mathcal C$, gives \begin{align*} P\sum_{j=0}^{m-1}E_jg_j\in C^\infty \end{align*} microlocally on $W$, while the zero-Cauchy Green construction gives $PRf-f\in C^\infty$ microlocally on $W$. Therefore \begin{align*} Pu-f\in C^\infty \end{align*} microlocally on $W$. The Cauchy matching from the Vandermonde construction and the vanishing of the retarded Cauchy traces give \begin{align*} \partial_t^k u|_{t=0}-g_k\in C^\infty \end{align*} microlocally on $V_1$ for $0\le k\le m-1$. Since every $S_\ell$ is associated with the graph of the Hamilton flow of $q_\ell$, each $E_j$ is a finite sum of Fourier integral operators associated with the corresponding branch canonical relations, and the same is true for the kernel of $R$ with the additional retarded relation $\sigma\le t$. [/step]

[step:Verify the Sobolev mapping exponents] For each branch, the canonical relation of $S_\ell(t,0)$ is a local canonical graph from $T^*Y\setminus 0$ to $T^*(I_1\times Y)\setminus 0$. The standard Sobolev mapping theorem for properly supported Fourier integral operators associated with canonical graphs applies to these branch operators and to the retarded spacetime branch kernels. The unrescaled datum $\partial_t^ju|_{t=0}$ is converted to a branch amplitude by an operator of tangential order $j$ and then lifted from the hypersurface $Y$ to spacetime, producing the additional trace shift $1/2$; hence the Cauchy solution operator has order $j+1/2$ relative to the target Sobolev index. The scalar equation has order $m$, so the zero-Cauchy inhomogeneous Green operator gains $m$ derivatives relative to the source Sobolev index. Therefore an $H^s_{\mathrm{loc}}(I_1\times Y)$ solution requires the $j$th unrescaled Cauchy datum to lie in \begin{align*} H^{s-j-1/2}_{\mathrm{loc}}(Y) \end{align*} and the forcing term to lie in \begin{align*} H^{s-m}_{\mathrm{loc}}(I_1\times Y). \end{align*} Thus the parametrix defines the continuous microlocal map \begin{align*} \prod_{j=0}^{m-1}H^{s-j-1/2}_{\mathrm{loc}}(Y)\times H^{s-m}_{\mathrm{loc}}(I_1\times Y)\to H^s_{\mathrm{loc}}(I_1\times Y) \end{align*} over $V_1$ and $W$. If the datum is $Q_j\partial_t^ju|_{t=0}$ with $Q_j\in\Psi^{r_j}_{\mathrm{prop}}(Y)$ elliptic, microlocal elliptic regularity gives a parametrix $Q_j^{-1}\in\Psi^{-r_j}_{\mathrm{prop}}(Y)$ on $V_1$. Applying this parametrix shifts the required Sobolev exponent by $-r_j$, giving \begin{align*} s-j-\frac{1}{2}-r_j. \end{align*} [/step]

[step:Remove residual errors and prove uniqueness modulo smooth terms] The transport equations determine the amplitudes to every homogeneous order, and classical asymptotic summation for the branch Fourier integral operators produces full classical amplitudes with smoothing residual errors. More precisely, the error operator for the coupled Cauchy problem has a properly supported smooth Schwartz kernel microlocally on the chosen cutoffs and vanishes to infinite order at the initial trace operators. Composing this smoothing error with the already constructed properly supported Cauchy-retarded parametrix gives another smoothing operator with the same microlocal Cauchy-retarded support property; iterating the correction gives a locally finite Neumann series in smoothing kernels. Hence the correction changes $E_j$ and $R$ only by smoothing operators, preserves the Cauchy matching modulo $C^\infty$, and does not change the branch canonical relations or the retarded support modulo smoothing terms. It remains to prove uniqueness. Suppose $u_1,u_2\in\mathcal D'(I_1\times Y)$ satisfy the same microlocal Cauchy problem modulo $C^\infty$ on $V_1$ and $W$, and set \begin{align*} w:=u_1-u_2. \end{align*} Then \begin{align*} Pw\in C^\infty \end{align*} microlocally on $W$, and every Cauchy trace $\partial_t^jw|_{t=0}$ is smooth microlocally on $V_1$. Applying the branch diagonalization reduces this to homogeneous first-order real principal type equations with smooth initial data. The hypotheses of propagation of microlocal smoothness on the characteristic sheets for strictly hyperbolic scalar operators [citetheorem:TEMP-36] hold because, on $W$, the scalar principal symbol has real smooth pairwise distinct characteristic roots, the initial hypersurface is noncharacteristic, and $Pw$ and the Cauchy traces of $w$ are microlocally smooth on the relevant incoming sheet. Therefore [citetheorem:TEMP-36] gives \begin{align*} w\in C^\infty \end{align*} microlocally on $W$. Hence the parametrix solution is unique modulo smooth terms on $W$. Applying this conclusion to the difference of two parametrices constructed with the same cutoffs shows that their difference maps every admissible datum to a smooth function microlocally on $W$. To pass from this mapping statement to the kernel statement, test the difference operator on compactly supported oscillatory distributions microlocalized at an arbitrary input covector in the cutoff region. If the primed wave front relation of the difference kernel met the relevant input-output relation over $V\times W$, the wave front set mapping theorem for properly supported Fourier integral kernels [citetheorem:TEMP-5] would produce a nonsmooth output for some such microlocal input. Since uniqueness gives smooth output for every microlocal input, the primed wave front relation is empty over the cutoff relation. Thus the difference has a smooth Schwartz kernel microlocally on $W$, which is exactly that the two parametrices differ by a smoothing operator microlocally on $W$. [/step]