\subsection{Outgoing and incoming passages} \label{sec:compact-connection-passage} At bounded exit radii, the outgoing part of a connection belongs to a compact family of finite orbit segments. Its endpoint can be chosen where the departing wall has a fixed small height $\eta$. After reflection, the next wall is still small and must grow to height one. In the projective clock, this second passage takes a time of order $\log(1/v)$, where $v$ is the initial height of the growing wall. We first estimate that passage, retaining the decay accumulated during the wait, and then join it to the finite outgoing segment. \subsubsection{Arrival from a small active wall} Use the projective coordinates in the incoming frame, so that $V$ is the growing wall. Write the initial point as $y=(m,u,v,w)$ and put \[ k=H(m-1,v),\qquad \varepsilon_0=10^{-4}. \] The radius $k$ labels this incoming passage. A dot denotes projective time and $t$ denotes the original normalized time. \begin{proposition}[Incoming passage] \label{prop:incoming-passage} Suppose \begin{equation} k\geq\frac74,\qquad m\geq1,\qquad 0\leq u,w\leq\varepsilon_0,\qquad01$. The arrival time and state $\mathcal P(y)$ are locally $C^2$ in all four ordinary initial coordinates, including at $u=0$ or $w=0$. Set $L=\log(1/v)$ and use $s=\log(V/v)$ along the passage. For $0\leq s\leq L$, \begin{align} 0\leq U(s)&\leq u e^{-(1+k/3)s},& 0\leq W(s)&\leq w e^{-(1+1/k)s}, \label{eq:incoming-decay}\\ |\rho(s)-k|&\leq\frac{12}{k} \bigl[u(v+w)+u^2+w^2\bigr]<\frac{k}{200},& \rho&=H(M-1,V). \label{eq:incoming-drift} \end{align} The durations in the two time variables satisfy \begin{equation} T_{\rm in}(y)\leq\frac{2(k+1)^2}{k}L, \qquad \frac{L}{2k}\leq\tau_{\rm in}\leq\frac{2L}{k}. \label{eq:incoming-time} \end{equation} At arrival the wall bounds become \begin{equation} \mathcal P_U(y)\leq u v^{1+k/3},\qquad \mathcal P_W(y)\leq w v^{1+1/k}. \label{eq:incoming-output} \end{equation} Each output wall is positive exactly when its input wall is positive. If in addition $k\leq4$, then \begin{equation} \mathcal P_U(y)\leq u v^{19/12},\qquad \mathcal P_W(y)\leq w v^{13/9}. \label{eq:incoming-compact-powers} \end{equation} \end{proposition} \begin{proof} We first work on any initial segment in the closed region \begin{equation} M\geq1,\qquad |\rho-k|\leq\frac{k}{100},\qquad 0\leq U,W\leq\varepsilon_0,\qquad0\leq V\leq1. \label{eq:incoming-region} \end{equation} Along the actual orbit, $V$ remains positive by its multiplicative equation and the assumption $v>0$. The identity $\rho^2=(M-1)^2+V^2$, together with $k\geq7/4$, gives \begin{equation} \frac45k\leq M-1\leq\frac{101}{100}k. \label{eq:incoming-phase-band} \end{equation} For the lower bound, square both sides and use $(99k/100)^2-1\geq(4k/5)^2$; the upper bound follows from $M-1\leq\rho$. Recall $C=U(V+W-U)$ and put \[ q=M-1-C,\qquad a=\tfrac12\bigl(1-M^2-(V-W)^2+U^2\bigr). \] Since $-U^2\leq C\leq2U$, the numerical margins in \eqref{eq:incoming-phase-band} imply \begin{equation} \frac34k\leq q\leq\frac{21}{20}k. \label{eq:incoming-clock-speed} \end{equation} In particular, $ds/d\tau=q>0$. The two decreasing walls obey \begin{equation} \frac a q\leq-\left(1+\frac k3\right),\qquad \frac{-M-1-C}{q}\leq-\left(1+\frac1k\right). \label{eq:incoming-clock-rates} \end{equation} Here the identities that retain the needed signs are \begin{align*} \frac a q &=-\frac{M+1}{2} +\frac{-(V-W)^2+U^2-(M+1)C}{2q},\\ \frac{-M-1-C}{q}&=-1-\frac{2(1+C)}q. \end{align*} For the first remainder, $C\geq-U^2$ and $M+2\leq4q$ give an upper bound $2U^2$. Thus the first rate is at most $-1-2k/5+2\varepsilon_0^2\leq-1-k/3$. For the second rate, $2(1+C)/q\geq2(1-\varepsilon_0^2)/((21/20)k)\geq1/k$. Integrating the scalar inequalities proves \eqref{eq:incoming-decay} on every such segment. A zero wall remains zero by its multiplicative equation, so these estimates also hold on the boundary. The radial equation, obtained from~\eqref{eq:energy-identities}, is \begin{equation*} \dot\rho=-C\rho+\frac{M-1}{\rho}(U-W)^2. \end{equation*} In the present region, $\rho/q\leq2$, $0\leq(M-1)/\rho\leq1$, and $1/q\leq2/k$. Also $|C|\leq U(V+W)+U^2$ and $(U-W)^2\leq U^2+W^2$. Consequently \begin{equation*} \left|\frac{d\rho}{ds}\right| \leq4\left[U(V+W)+U^2+\frac{W^2}{k}\right]. \end{equation*} The time-dependent terms in this bound are integrable uniformly in the length of the passage. Indeed, $V=ve^s$ and \eqref{eq:incoming-decay} give \[ U(V+W)+U^2+\frac{W^2}{k} \leq\bigl[u(v+w)+u^2\bigr]e^{-ks/3} +\frac{w^2}{k}e^{-2s}. \] Integration from zero to any current clock value yields \begin{equation} |\rho(s)-k| \leq\frac{12}{k}\bigl[u(v+w)+u^2\bigr]+ \frac{2w^2}{k} \leq\frac{12}{k}\bigl[u(v+w)+u^2+w^2\bigr]. \label{eq:incoming-integrated-drift} \end{equation} The bracket is at most $\varepsilon_0+3\varepsilon_0^2$. Since $12(\varepsilon_0+3\varepsilon_0^2)<(7/4)^2/200$, this improves the radius band strictly to~\eqref{eq:incoming-drift}. We next show that the actual solution reaches the section while these estimates hold. On~\eqref{eq:incoming-region}, \begin{equation} 00$. For the upper bound, drop the negative cross terms in $D$, use $M\leq2(k+1)$ and $U,V,W\leq1$, and obtain $D\leq4(k+1)^2+6\leq6(k+1)^2$. Thus the projective chart is regular initially. The normalized solution exists for all positive time by the proof of Lemma~\ref{lem:forward}, which also applies when a wall is zero. Continue the initial segment in the closed region with the additional condition $A\geq1/[2(k+1)^2]$. At an endpoint before $V=1$, \eqref{eq:incoming-integrated-drift} puts the radius strictly inside its allowed band, and~\eqref{eq:incoming-phase-band} puts $M$ strictly above one. The upper faces $U=\varepsilon_0$ and $W=\varepsilon_0$ point inward by~\eqref{eq:incoming-clock-rates}; zero walls are invariant. Finally~\eqref{eq:incoming-chart-bound} improves the additional chart condition. Continuity and these strict margins extend the segment whenever $V<1$. An indefinitely long segment with $V<1$ is impossible, because \begin{equation*} \frac{ds}{dt}=Aq\geq\frac{k}{2(k+1)^2}. \end{equation*} The clock starts at zero and is less than $L$ before arrival. This proves existence of a first arrival and the original-time bound in~\eqref{eq:incoming-time}. Strict positivity of $AqV$ proves monotonicity and transversality. Integrating $d\tau=ds/q$ and using~\eqref{eq:incoming-clock-speed} gives the projective-time bounds. Substituting $s=L$ into \eqref{eq:incoming-decay} proves~\eqref{eq:incoming-output}. The multiplicative equations give positivity of every wall that was initially positive. For local regularity, fix one such finite passage. Its flow domain, positive chart and positive derivative $dV/dt$ persist on the compact segment for nearby initial points and nearby endpoint times. The implicit function theorem gives a local $C^2$ solution of the arrival equation. Since initially $V<1$ and its derivative remains positive up to that event, this solution is still the first arrival. Composition with the flow gives the $C^2$ arrival state. This argument takes place in an ordinary open neighborhood in $\R^4$, so it also covers zero initial $U$ or $W$. Finally suppose $k\leq4$. Then $q\leq21/5$, and \[ \frac{2(1+C)}q\geq\frac{2(1-\varepsilon_0^2)}{21/5} \geq\frac49. \] The weak-wall rate in~\eqref{eq:incoming-clock-rates} improves to $-13/9$, while the strong-wall rate is at most $-19/12$ because $k\geq7/4$. Integrating to $s=L$ proves \eqref{eq:incoming-compact-powers}. \end{proof} \subsubsection{A finite outgoing segment} Return now to the frame of the preceding core. Its downward exit is \begin{equation} y_-(R,u,w)=(1-\sqrt{R^2-1},u,1,w),\qquad \frac{31}{8}\leq R\leq6. \label{eq:compact-outgoing-entry} \end{equation} The interval extends below the reflection threshold $R=4$, providing a neighborhood in which the returning maps can be differentiated. Fix a height $0<\eta\leq1/8$. Let $S_\eta$ be the first positive time at which the old wall $V$ reaches $\eta$, and let $y_\eta=(M_\eta,U_\eta,\eta,W_\eta)$ denote the state there. The axis reflection acts in these coordinates by \[ \mathsf S(M,U,V,W)=(-M,U,W,V). \] Thus $\mathsf S y_\eta$ is an entrance for Proposition~\ref{prop:incoming-passage} once its walls and radius lie in the stated range. \begin{lemma}[The outgoing segment and its reflected entrance] \label{lem:compact-outgoing-tail} For each fixed $\eta\in(0,1/8]$ there are $\delta>0$ and $B>0$ such that, on \begin{equation} \frac{31}{8}\leq R\leq6,\qquad |u|,|w|\leq\delta, \label{eq:compact-tail-box} \end{equation} the first arrival at $V=\eta$ exists, its whole segment lies in the projective chart, and \[ \frac{dV}{dt}<0,\qquad |W|<\frac12 \quad\hbox{on }[0,S_\eta]. \] The time and state are $C^2$ on a neighborhood of this box. There are positive $C^2$ factors $B_U,B_W$ such that \begin{equation} \mathsf S y_\eta=(-M_\eta,uB_U,wB_W,\eta). \label{eq:compact-tail-factors} \end{equation} For $j=U,W$ they satisfy \begin{equation} B_j+B_j^{-1}+\|D\log B_j\|+\|D^2\log B_j\|\leq B. \label{eq:compact-tail-factor-bounds} \end{equation} With $k_\eta=H(-M_\eta-1,W_\eta)$, one has \begin{gather} -M_\eta-1\geq\frac74,\qquad \frac74\leq k_\eta\leq4-\frac{\eta^2}{24}<4, \label{eq:compact-tail-radius-range}\\ \left|k_\eta-\left(\sqrt{R^2-\eta^2}-2\right)\right| \leq B(|u|+|w|)\leq\frac{\eta^2}{24}. \label{eq:compact-tail-radius-error} \end{gather} The norms in~\eqref{eq:compact-tail-factor-bounds} are induced by the maximum norm on the three entrance parameters. \end{lemma} \begin{proof} On the single-wall orbit~\eqref{eq:reference-core-pulse}, with $\kappa=R$, this segment runs from $\tau=\operatorname{arcosh}(R)/R$ to $\tau=\operatorname{arcosh}(R/\eta)/R$. Its endpoint is \begin{equation*} y_\eta(R,0,0)=(1-\sqrt{R^2-\eta^2},0,\eta,0). \end{equation*} The reference $V$ is strictly decreasing throughout, its derivative is negative at the endpoint, and $D>0$. The segment is finite in original time as well. Smooth dependence and the implicit function theorem therefore give a $C^2$ first arrival near each $(R,0,0)$. Negativity of $dV/dt$ on the compact segment ensures that the nearby arrival is still the first one. It also persists up to the moving endpoint, as do the chart condition and $|W|<1/2$. The reference parameter interval is compact. As in Lemma~\ref{lem:compact-core-regularity}, take a closed neighborhood inside the open set of these regular first arrivals. The time, state and their first two derivatives are bounded there. Restricting to a smaller box~\eqref{eq:compact-tail-box} and applying the mean-value bound along its wall directions gives \begin{equation} \|y_\eta(R,u,w)-y_\eta(R,0,0)\|\leq B_0(|u|+|w|) \label{eq:compact-tail-state-error} \end{equation} for some $B_0>0$. The positive factors can be defined directly from the normalized wall equations. If $x$ is the normalized initial point associated with~\eqref{eq:compact-outgoing-entry}, set \begin{equation} B_j=\exp\left(\log A(0)-\log A(S_\eta) +\int_0^{S_\eta}\lambda_{i(j)}(\phi_t x)\dd t\right), \qquad i(U)=1,\quad i(W)=3, \label{eq:compact-tail-factor-formula} \end{equation} where the rates $\lambda_i$ are those in the proof of Lemma~\ref{lem:forward}. Multiplying the exact normalized wall formula by $\sqrt3/A(S_\eta)$ gives~\eqref{eq:compact-tail-factors}, including for zero or signed walls. The logarithms of the factors are $C^2$: the finite moving integral may be written as $S_\eta\int_0^1\lambda_{i(j)}(\phi_{aS_\eta}x)\dd a$ and differentiated twice on a neighborhood of the compact segment. The factors are positive, so compactness bounds their inverses and the derivatives of their logarithms. This proves \eqref{eq:compact-tail-factor-bounds} after enlarging $B$. Put $r_0=\sqrt{R^2-\eta^2}-2$. The reference radius has the bounds \begin{equation} \frac{15}{8}-\frac{8\eta^2}{31} \leq r_0\leq4-\frac{\eta^2}{12}. \label{eq:compact-tail-reference-range} \end{equation} Indeed, with $b=\sqrt{R^2-\eta^2}$, \[ R-b=\frac{\eta^2}{R+b},\qquad \frac{\eta^2}{2R}\leq R-b\leq\frac{\eta^2}{R}, \] and $31/8\leq R\leq6$. The state estimate \eqref{eq:compact-tail-state-error} bounds both $|(-M_\eta-1)-r_0|$ and $|W_\eta|$ by $B_0(|u|+|w|)$. For $r_0\geq0$ the norm inequality \[ |H(a,b)-r_0|\leq|a-r_0|+|b| \] therefore gives $|k_\eta-r_0|\leq2B_0(|u|+|w|)$. Take $B\geq2B_0$ and then $\delta\leq\eta^2/(48B)$. Both $k_\eta$ and $-M_\eta-1$ differ from $r_0$ by at most $\eta^2/24$. Equation~\eqref{eq:compact-tail-reference-range} and $\eta\leq1/8$ now give~\eqref{eq:compact-tail-radius-range} and~\eqref{eq:compact-tail-radius-error}. The lower bound on $-M_\eta-1$ also makes $k_\eta$ a $C^2$ function on a neighborhood of the box. \end{proof} \subsubsection{The actual reflection and its wall loss} The two segments now fit on one exact orbit. The lower radius bound in~\eqref{eq:compact-tail-radius-range} permits the incoming passage; the upper bound gives the fixed powers in \eqref{eq:incoming-compact-powers}. \begin{proposition}[Existence and wall decay at bounded radii] \label{prop:compact-reflection-passage} Fix $0<\eta\leq\varepsilon_0$. There are $\varepsilon>0$ and $C_\eta>0$ such that, for \begin{equation} \frac{31}{8}\leq R\leq6,\qquad 01$. Its time and state, and hence its core coordinates $\mathcal F=(H(M_+-1,1),u_+,w_+)$, are locally $C^2$. Moreover, \begin{equation} 01$. The two constituent first arrivals remain valid in full neighborhoods of the given input and of $y_e$. Shrink the input neighborhood so that its reflected endpoints lie in the latter neighborhood. The same first-arrival argument applies throughout this neighborhood, identifying the actual map with \eqref{eq:compact-reflection-composition}. The $C^2$ assertion follows by composition. Taking the core radius is $C^2$ as well, since its final active wall equals one. The compact incoming powers and $B_U,B_W\leq B$ give \[ u_+\leq (uB_U)(wB_W)^{19/12} \leq B^{1+19/12}u w^{19/12},\qquad w_+\leq\eta(wB_W)^{13/9} \leq\eta B^{13/9}w^{13/9}. \] Choose $C_\eta=B^{1+19/12}+\eta B^{13/9}$ and reduce $\varepsilon$ once more so that \[ C_\eta\varepsilon^{19/12}\leq1, \qquad C_\eta\varepsilon^{4/9}\leq1. \] These inequalities prove the two nonincrease statements in \eqref{eq:compact-reflection-wall-powers}. Positivity follows from the positive finite factors and the multiplicative equations during the incoming passage. \end{proof} The powers in~\eqref{eq:compact-reflection-wall-powers} quantify the wall loss accumulated during a possibly long wait at bounded radius. They will also control the transverse derivatives after multiplication by the output walls. The radial estimates require finer information: the incoming drift bound \eqref{eq:incoming-drift} still contains the fixed height $\eta$. We therefore next retain the cancellation between the outgoing segment and the incoming passage in their common radial phase.