\subsection{Phase and amplitude after the delayed return} \label{sec:small-section-delayed-output} The long decay interval has brought the second weak-wall ordering to a complete core entrance. We now compare the next entrance with the original small-section data. The phase comparison uses the bounded radius displacement through one core and connection. For the amplitudes, the product of the two weak walls is the useful quantity: it is nonincreasing throughout the slow interval and controls both walls at the next entrance. Use the actual return $G=(g,\widehat u,\widehat w)$ and the upward radius $\kappa$ from the preceding subsection. Put \begin{equation} \xi=\frac2g,\qquad \Phi=\frac2\kappa,\qquad F(\rho)=\Phi_h(\rho,\zeta,\nu). \label{eq:delayed-output-phases} \end{equation} Throughout this subsection $q>3$, $D>4$, $P>6D+2$, and the labels satisfy~\eqref{eq:delayed-core-labels}. Derivatives are ordinary derivatives in $\rho$, with all other parameters fixed. All scale thresholds are uniform over these labels. \begin{proposition} \label{prop:delayed-output-phase} There is $h_*>0$ such that, for every $03$ and $D>4$, one can choose $C=C_{h,q,D}>0$ before $P$ with the following property. For every $P>6D+2$ and all sufficiently small $a>0$, the actual return phase satisfies \begin{equation} |\xi-F|\leq Ca^2,\qquad |\xi'-F'|\leq Ca, \qquad \frac1{8\sqrt3}\leq\xi'\leq C. \label{eq:delayed-output-phase} \end{equation} It is jointly $C^2$ in $(\zeta,\nu,\rho)$ near each input. \end{proposition} \begin{proof} Take the common thresholds for the delayed core and connection. Their ordinary regularity was proved in Proposition~\ref{prop:delayed-return-connection}. At the upward arrival, $\kappa\geq6$ and $U_++W_+\leq\varepsilon_0/\kappa$. The full-core and connection estimates used in \eqref{eq:entrance-complete-radius-drop} therefore apply here as well. With $b=U_++W_+^2$, they give \begin{equation} |g-(\kappa-2)|\leq\frac{64b}{\kappa},\qquad 1\leq\kappa-g\leq3,\qquad \frac\kappa2\leq g\leq\kappa. \label{eq:delayed-output-radius-drop} \end{equation} The input smallness makes $64b/\kappa\leq1$. The two derivative ratios in Proposition~\ref{prop:delayed-return-directions} give \begin{align*} |g'-\kappa'| &\leq \frac{|f'|}{4\mu}+\frac{|\kappa'|}{32\kappa}\\ &\leq\left(\frac58+\frac1{32}\right) \frac{|\kappa'|}{\kappa} \leq\frac{|\kappa'|}{\kappa}. \end{align*} Here $|f'|\leq(5/4)|\kappa'|$ and $\mu\geq\kappa/2$. Combining this with~\eqref{eq:delayed-entrance-speed} yields \begin{gather*} \kappa,g\geq\frac1{96a},\qquad |g-\kappa|\leq3, \qquad |\kappa'|\leq\frac{24576}{a^2},\\ |g'-\kappa'|\leq\frac{1179648}{a}. \end{gather*} Apply the reciprocal identities \eqref{eq:return-phase-radius-identities}, with $\kappa$ in place of $k$. The common lower radius bound gives \[ \frac1{g\kappa}\leq96^2a^2,\qquad \frac{g+\kappa}{g^2\kappa^2}\leq2\cdot96^3a^3. \] Consequently \begin{equation} |\xi-\Phi|\leq2\cdot96^2\cdot3a^2\leq Aa^2, \qquad |\xi'-\Phi'|\leq Aa, \label{eq:delayed-output-reciprocal-error} \end{equation} where the absolute constant \[ A=2\cdot96^2\cdot1179648 +4\cdot24576\cdot96^3\cdot3 \] is independent of all parameters. Let $K=K_{h,q,D}$ be the comparison constant in Theorem~\ref{thm:delayed-entrance-phase}. Adding its value and derivative errors to~\eqref{eq:delayed-output-reciprocal-error} gives the first two claims with constant $A+K$. Decrease the scale threshold so that \[ Aa\leq\frac1{8\sqrt3},\qquad Aa\leq1. \] The bounds $1/(4\sqrt3)\leq\Phi'\leq2/\sqrt3+1$ then give \[ \frac1{8\sqrt3}\leq\xi'\leq\frac2{\sqrt3}+2. \] Thus $C=A+K+2/\sqrt3+2$ suffices, and is chosen before $P$. Joint regularity follows from the ordinary return already constructed. \end{proof} For $0\leq t\leq T$, write \[ Y_t=(M_t,U_t,V_t,W_t) =\operatorname{proj}(\phi_{t_0+t}x_0),\qquad A_t=1+\Sigma_+(\phi_{t_0+t}x_0). \] The height is positive on this interval. In the original normalized time, the product identity \eqref{eq:energy-identities} reads \begin{equation} \frac{d}{dt}(V_tW_t) =-2A_t\bigl[1+U_t(V_t+W_t-U_t)\bigr]V_tW_t. \label{eq:delayed-output-product-equation} \end{equation} The slow-decay bounds give \[ 0\leq U_t\leq20ha^{q-2}\leq1 \] for a sufficiently small $a$, uniformly in $t$. Since $V_t,W_t\geq0$, \[ 1+U_t(V_t+W_t-U_t)\geq1-U_t^2\geq0. \] It follows that $V_tW_t$ is nonincreasing. At $t=0$, the fast-segment coordinate estimate \eqref{eq:projection-coordinate-bounds} gives \begin{equation} 0\leq V_TW_T\leq V_0W_0 \leq\frac{256\zeta\nu}{a^4}. \label{eq:delayed-output-product-bound} \end{equation} This argument also applies when either initial weak wall is zero, since it uses the product equation rather than logarithms. \begin{proposition} \label{prop:delayed-output-amplitude} For a sufficiently small universal $h_*>0$, each $00,\qquad \widehat u+\widehat w\leq\frac{512\zeta\nu}{a^4}. \label{eq:delayed-output-amplitude} \end{equation} These are the coordinates of the actual reflected entrance at $t_{\rm next}>t_0+T$ in \eqref{eq:delayed-return-original-orbit}. \end{proposition} \begin{proof} Use the amplitude threshold $\eta=\varepsilon_0$ in Proposition~\ref{prop:delayed-return-connection}. The incoming passage and the core give, successively, \[ W_+\leq V_TW_T,\qquad w\leq W_+. \] The connection estimate \eqref{eq:connection-output-weak-bound} then yields \[ \widehat u+\widehat w\leq2w\leq2V_TW_T \leq\frac{512\zeta\nu}{a^4}. \] Both walls are positive along these passages. The radius bounds follow from \eqref{eq:delayed-output-radius-drop} and $1/(48a)\leq\kappa\leq128/a$. Finally, the downward arrival and the connection each take positive time after $t_0+T$; their orbit identity was proved in Proposition~\ref{prop:delayed-return-connection}. \end{proof} The delayed return thus has the same phase accuracy and the same product-dependent amplitude bound as the earlier weak-wall ordering. Together with its square-root tangent cone, these estimates provide the local data for placing both families on admissible statistical graphs.