[proofplan] We work microlocally on conic neighbourhoods where $U$ is elliptic and $Q$ is a two-sided microlocal inverse. The canonical relations of $U$, $P$, and $Q$ compose as graph, diagonal, inverse graph, and this composition is exactly the diagonal relation in $T^*Y \setminus 0$; the Fourier integral operator composition theorem therefore places $QPU$ in the pseudodifferential calculus on $Y$. The principal symbol calculation is the leading-order part of the same composition formula: $U$ transports the input half-density along $\kappa$, $P$ multiplies by $p$ at the transported covector, and $Q$ transports back by the inverse elliptic symbol. The inverse factors cancel because $Q$ is a microlocal inverse, leaving $p \circ \kappa$. [/proofplan] [step:Fix the microlocal neighbourhoods and graph convention] Let $V_Y \subset \Omega_Y$ be an open conic set whose closure is contained in the region under consideration, and put \begin{align*} V_X := \kappa(V_Y) \subset \Omega_X. \end{align*} All equalities below are understood microlocally over $V_Y$ and $V_X$. We use the twisted canonical relation convention stated in the theorem: $U$ has relation \begin{align*} C_\kappa = \{(\kappa(\rho),\rho) : \rho \in \Omega_Y\} \subset (T^*X \setminus 0) \times (T^*Y \setminus 0), \end{align*} where $\rho$ denotes a point of $T^*Y \setminus 0$. The microlocal inverse $Q$ has relation \begin{align*} C_{\kappa^{-1}} = \{(\rho,\kappa(\rho)) : \rho \in \Omega_Y\} \subset (T^*Y \setminus 0) \times (T^*X \setminus 0). \end{align*} Since $P \in \Psi^r(X)$, its canonical relation is the diagonal \begin{align*} \Delta_X := \{(\zeta,\zeta) : \zeta \in T^*X \setminus 0\} \subset (T^*X \setminus 0) \times (T^*X \setminus 0). \end{align*} [/step] [step:Compose the canonical relations to obtain the diagonal on $T^*Y$] The canonical relation of $QPU$ is contained in the composed relation \begin{align*} C_{\kappa^{-1}} \circ \Delta_X \circ C_\kappa. \end{align*} We compute this composition explicitly. Let $\rho \in V_Y$. First $C_\kappa$ sends $\rho$ to $\kappa(\rho) \in V_X$. The diagonal relation $\Delta_X$ keeps this covector fixed. Finally $C_{\kappa^{-1}}$ sends $\kappa(\rho)$ back to $\rho$. Hence every point produced by the composition has the form $(\rho,\rho)$ with $\rho \in V_Y$. Conversely, if $\rho \in V_Y$, then the chain \begin{align*} \rho \xrightarrow{C_\kappa} \kappa(\rho) \xrightarrow{\Delta_X} \kappa(\rho) \xrightarrow{C_{\kappa^{-1}}} \rho \end{align*} shows that $(\rho,\rho)$ belongs to the composed relation. Therefore \begin{align*} C_{\kappa^{-1}} \circ \Delta_X \circ C_\kappa = \Delta_Y \end{align*} microlocally over $V_Y$, where \begin{align*} \Delta_Y := \{(\rho,\rho) : \rho \in T^*Y \setminus 0\} \end{align*} is the diagonal canonical relation on $T^*Y \setminus 0$. [guided] The purpose of this step is to determine what kind of operator $QPU$ can be before computing its symbol. The operator $U$ moves singularities from $Y$ to $X$ according to $\kappa$, the pseudodifferential operator $P$ does not move singularities on $X$ because its relation is diagonal, and $Q$ moves singularities back from $X$ to $Y$ according to $\kappa^{-1}$. Let $\rho \in V_Y$ be a covector on $Y$. By our convention for the graph relation, \begin{align*} C_\kappa = \{(\kappa(\rho),\rho) : \rho \in \Omega_Y\}, \end{align*} so $U$ carries $\rho$ to $\kappa(\rho)$. Since $P$ is pseudodifferential, its relation is \begin{align*} \Delta_X = \{(\zeta,\zeta) : \zeta \in T^*X \setminus 0\}, \end{align*} which means that $P$ preserves the covector $\kappa(\rho)$. Finally, since $Q$ quantises $\kappa^{-1}$, its relation is \begin{align*} C_{\kappa^{-1}} = \{(\rho,\kappa(\rho)) : \rho \in \Omega_Y\}, \end{align*} so $Q$ carries $\kappa(\rho)$ back to $\rho$. Thus the only possible input-output pair in $T^*Y \times T^*Y$ is $(\rho,\rho)$. This proves \begin{align*} C_{\kappa^{-1}} \circ \Delta_X \circ C_\kappa \subset \Delta_Y \end{align*} microlocally over $V_Y$. The reverse inclusion follows from the same displayed chain: for each $\rho \in V_Y$, the intermediate covector $\kappa(\rho) \in V_X$ provides an actual composable path through $C_\kappa$, $\Delta_X$, and $C_{\kappa^{-1}}$. Hence \begin{align*} C_{\kappa^{-1}} \circ \Delta_X \circ C_\kappa = \Delta_Y. \end{align*} This is the geometric core of [Egorov's theorem](/theorems/896): conjugation by a canonical graph turns the diagonal relation on $X$ into the diagonal relation on $Y$. [/guided] [/step] [step:Apply the Fourier integral composition theorem to identify the operator class] We now invoke the standard clean graph-composition theorem for Fourier integral operators (citing a result not yet in the wiki: Composition theorem for Fourier integral operators). Its hypotheses are satisfied microlocally because $C_\kappa$ and $C_{\kappa^{-1}}$ are canonical graphs and the intermediate diagonal relation $\Delta_X$ composes transversely with graphs of diffeomorphisms. The orders add as \begin{align*} 0 + r + 0 = r. \end{align*} The preceding step shows that the resulting canonical relation is $\Delta_Y$. By the standard identification of Fourier integral operators associated with the diagonal relation with pseudodifferential operators (citing a result not yet in the wiki: Pseudodifferential operators as Fourier integral operators associated to the diagonal), it follows that \begin{align*} QPU \in \Psi^r(Y) \end{align*} microlocally on $V_Y$. [/step] [step:Compute the leading symbol by composing the elliptic symbols] Let \begin{align*} u: V_Y \to \mathcal{L}_U \end{align*} denote the principal symbol of $U$ along $C_\kappa$, viewed as a non-vanishing section of the relevant half-density and Maslov line bundle $\mathcal{L}_U$. Let \begin{align*} q: V_X \to \mathcal{L}_Q \end{align*} denote the principal symbol of $Q$ along $C_{\kappa^{-1}}$. Since $Q$ is a microlocal inverse for $U$, the leading symbol of $QU$ is the identity symbol on $V_Y$. Equivalently, under the natural contraction of the half-density and Maslov factors along the composed graph, \begin{align*} q(\kappa(\rho))\,u(\rho) = 1 \end{align*} for every $\rho \in V_Y$. We apply the principal symbol composition formula for Fourier integral operators (citing a result not yet in the wiki: Principal symbol calculus for Fourier integral operator composition). At the covector $\rho \in V_Y$, the factor $U$ contributes $u(\rho)$, the pseudodifferential operator $P$ contributes multiplication by $p(\kappa(\rho))$, and the factor $Q$ contributes $q(\kappa(\rho))$. Therefore the principal symbol of $QPU$ at $\rho$ is \begin{align*} \sigma_r(QPU)(\rho) = q(\kappa(\rho))\,p(\kappa(\rho))\,u(\rho). \end{align*} Since $p(\kappa(\rho))$ is scalar, it commutes with the line-bundle contractions, and the inverse-symbol identity gives \begin{align*} \sigma_r(QPU)(\rho) = p(\kappa(\rho)). \end{align*} Thus \begin{align*} \sigma_r(QPU) = p \circ \kappa \end{align*} on $V_Y$. [guided] The symbol calculation mirrors the geometric calculation, but now we track the leading amplitudes. Let \begin{align*} u: V_Y \to \mathcal{L}_U \end{align*} be the principal symbol of $U$ along the graph of $\kappa$. The target $\mathcal{L}_U$ is the half-density and Maslov line attached to the Fourier integral operator; the important point is that ellipticity means $u(\rho)$ is non-zero for each $\rho \in V_Y$. Let \begin{align*} q: V_X \to \mathcal{L}_Q \end{align*} be the corresponding principal symbol of $Q$ along the graph of $\kappa^{-1}$. Because $Q$ is a microlocal inverse for $U$, the composition $QU$ is microlocally the identity on $V_Y$. Applying the leading part of the Fourier integral symbol composition formula to $QU$, the symbol of $QU$ at $\rho$ is the contracted product of the symbol of $U$ at $\rho$ and the symbol of $Q$ at the transported covector $\kappa(\rho)$. Since the identity operator has principal symbol $1$, this gives \begin{align*} q(\kappa(\rho))\,u(\rho) = 1. \end{align*} Now insert $P$ between $Q$ and $U$. The pseudodifferential operator $P$ has principal symbol $p$, and because $U$ first transports $\rho$ to $\kappa(\rho)$ in $T^*X$, the symbol of $P$ is evaluated at $\kappa(\rho)$. Hence the principal symbol composition formula gives \begin{align*} \sigma_r(QPU)(\rho) = q(\kappa(\rho))\,p(\kappa(\rho))\,u(\rho). \end{align*} The factor $p(\kappa(\rho))$ is scalar multiplication on the same one-dimensional symbol line, so we may write \begin{align*} \sigma_r(QPU)(\rho) = p(\kappa(\rho))\,q(\kappa(\rho))\,u(\rho). \end{align*} Using the inverse-symbol identity for $Q$ and $U$, \begin{align*} q(\kappa(\rho))\,u(\rho) = 1, \end{align*} we obtain \begin{align*} \sigma_r(QPU)(\rho) = p(\kappa(\rho)). \end{align*} Since this holds for every $\rho \in V_Y$, the principal symbol is exactly \begin{align*} \sigma_r(QPU) = p \circ \kappa. \end{align*} This is where the chosen graph convention matters: because $U$ carries covectors from $Y$ to $X$ by $\kappa$, the symbol of $P$ is pulled back from $T^*X$ to $T^*Y$ by composition with $\kappa$, not with $\kappa^{-1}$. [/guided] [/step] [step:Record the lower-order terms from the full symbolic expansion] The same symbolic composition theorem has an asymptotic version for classical symbols. After fixing a quantisation convention, half-density normalisation, and representative amplitudes for $U$, $P$, and $Q$, the full symbol of $QPU$ is obtained by composing the full symbol expansions of these three operators. Its leading term is the already computed expression $p \circ \kappa$. Each lower-order term is a universal finite expression, at that order, involving derivatives of the lower-order amplitudes of $U$ and $Q$, derivatives of the full symbol of $P$, and the phase and quantisation convention used to realise the canonical graph. Hence conjugation by the elliptic Fourier integral operator transports the full principal symbol sequence through $\kappa$, with the stated convention-dependent lower-order corrections. Since $V_Y$ was an arbitrary conic neighbourhood in the prescribed microlocal region, the conclusion holds microlocally on the whole region under consideration. [/step]