\subsection{Core coordinates and the first return} \label{sec:core-passage} An angular coordinate turns the core passage into an interval on which the transverse decay can be integrated explicitly. We use it to prove finite arrival at the outgoing section and to compare the radius drift with the loss in the small walls. All estimates refer to the exact equations~\eqref{eq:projective-flow}. \subsubsection{Coordinates and estimates along the core} For $V>0$, set \begin{equation} R=\sqrt{(M-1)^2+V^2},\qquad z=\operatorname{arsinh}\frac{1-M}{V},\qquad P=VW. \label{eq:core-angle-coordinates} \end{equation} Then \begin{equation} M=1-R\tanh z,\qquad V=\frac R{\cosh z},\qquad W=\frac PV. \label{eq:core-angle-inverse} \end{equation} The product $P$ will decrease when $P\geq0$ and $C=U(V+W-U)\geq0$. Define \[ d=1-\frac{(U-W)^2}{R^2}. \] Differentiation with respect to the projective time $\tau$ gives \begin{equation} \dot R=-CR-\tanh z\,(U-W)^2, \qquad \dot z=Rd,\qquad \dot P=-2(1+C)P. \label{eq:core-angle-clock} \end{equation} In particular $z$ is a time coordinate wherever $d>0$. In that coordinate the equations for $(R,U,P)$ become \begin{align} \frac{dR}{dz} &=\frac{-C-\tanh z\,(U-W)^2/R}{d},\notag\\ \frac{dU}{dz} &=\frac{U}{d}\left(-\frac R2+\tanh z+\frac W{\cosh z} +\frac{U^2-W^2}{2R}\right),\notag\\ \frac{dP}{dz}&=-\frac{2(1+C)P}{Rd}. \label{eq:core-angle-system} \end{align} To verify~\eqref{eq:core-angle-clock}, differentiate $R^2$ using \eqref{eq:energy-identities}. For $z$, direct differentiation yields \[ \dot z=\frac{(M-1)\dot V-V\dot M}{RV} =R-\frac{(U-W)^2}{R}. \] The equation for $P$ is already in~\eqref{eq:energy-identities}. Dividing by $Rd$ and substituting~\eqref{eq:core-angle-inverse} gives \eqref{eq:core-angle-system}. On the single active wall $U=W=0$, the radius is constant, say $R=\kappa$, and $z$ increases at speed $\kappa$ in projective time. The entrance and exit at $V=1$ occur at $z=-\operatorname{arcosh}\kappa$ and $z=\operatorname{arcosh}\kappa$. Thus the active wall follows the explicit profile $V=\kappa/\cosh z$. The same coordinate will control the nearby nonlinear passage without making its duration a multiplier in every error estimate. Fix $\kappa\geq6$, and put $\varepsilon_0=10^{-4}$. Consider the closed region \begin{equation} |R^2-\kappa^2|\leq\frac18,\qquad V\geq1, \qquad 0\leq U\leq\frac{\varepsilon_0}{\kappa}, \qquad 0\leq P\leq\frac{\varepsilon_0}{\kappa}. \label{eq:core-closed-region} \end{equation} Since $W=P/V$, both small walls are at most $\varepsilon_0/\kappa\leq1/60000$. In this region, \begin{equation} \frac{99}{100}\kappa\leq R\leq\frac{101}{100}\kappa, \qquad \frac{999}{1000}\leq d\leq1, \qquad C\geq0. \label{eq:core-elementary-bounds} \end{equation} The radius bounds follow by squaring, and the bounds for $d$ follow from $|U-W|\leq\varepsilon_0/\kappa$. Also $V+W-U>0$, which gives the sign of $C$. Two further estimates keep the coordinates and their clock regular. The denominator $D$ from~\eqref{eq:projective-inverse} satisfies \begin{equation} 2\leq D\leq2\kappa^2,\qquad \frac3{\kappa^2}\leq A=\frac6D\leq3, \qquad \frac{dz}{dt}=ARd\geq\frac1\kappa. \label{eq:core-chart-clock} \end{equation} For the lower bound, complete the square: \[ D=3+M^2+(V-U-W)^2-4UW\geq2. \] For the upper bound, use \[ D=R^2+2(M-1)+4+(U-W)^2-2V(U+W) \] together with $|M-1|\leq R$ and~\eqref{eq:core-elementary-bounds}. The bounds for $A$ and $ARd$ follow immediately. Furthermore $\cosh z=R/V\leq101\kappa/100$, so \begin{equation} |z|\leq\log(4\kappa). \label{eq:core-angle-range} \end{equation} Here we used $e^{|z|}\leq2\cosh z$. We call $U$ the strong transverse wall because of its rapid contraction; $V$ is the active wall crossing the core. The strong wall $U$ and the product $P$ decrease throughout any passage contained in \eqref{eq:core-closed-region}. More precisely, \begin{equation} \frac{dU}{dz}\leq-\frac\kappa4 U, \qquad \frac{dP}{dz}\leq0, \qquad \dot U\leq-\frac{\kappa^2}{10}U, \qquad \dot P\leq-2P. \label{eq:core-wall-rates} \end{equation} These inequalities also hold when a wall vanishes. For the first, the logarithmic numerator in~\eqref{eq:core-angle-system} is bounded by \[ -\frac R2+1+W+\frac{U^2}{2R}\leq-\frac{3\kappa}{10}. \] It is negative, and division by $00$, and multiply by $U$ to include $U=0$. The two estimates for $P$ follow from $C\geq0$. \begin{lemma}[Radius drift] \label{lem:core-radius-drift} Suppose a solution of~\eqref{eq:core-angle-system} lies in \eqref{eq:core-closed-region} for $a\leq z\leq b$, and $R(a)=\kappa$, $V(a)=1$. Write $u=U(a)$ and $w=P(a)=W(a)$. Then \begin{equation} |R(b)-\kappa|\leq\frac{16}{\kappa}(u+w^2), \qquad |R(b)^2-\kappa^2|<\frac1{16}. \label{eq:core-radius-drift} \end{equation} \end{lemma} \begin{proof} The first two inequalities in~\eqref{eq:core-wall-rates} imply \[ U(z)\leq u e^{-\kappa(z-a)/4},\qquad P(z)\leq w. \] Since $(\log\operatorname{sech}z)'=-\tanh z\leq1$ and $\cosh a=\kappa$, we also have \[ V(z)\leq\frac{11}{10}e^{z-a},\qquad W(z)\leq w,\qquad W(z)\leq\frac{w\cosh z}{R(z)}. \] The radius equation and $C\geq0$ give \begin{equation} \left|\frac{dR}{dz}\right| \leq\frac1{999/1000} \left(UV+UW+\frac{U^2}{R}+\frac{W^2}{R}\right). \label{eq:core-radius-integrand} \end{equation} Indeed $(U-W)^2\leq U^2+W^2$ and $C=UV+UW-U^2\leq UV+UW$. We estimate the four integrals separately. Using $R\geq9\kappa/10$ and $\kappa/4-1\geq\kappa/12$, \begin{align} \int_a^b UV\dd z&\leq\frac{14u}{\kappa},& \int_a^b UW\dd z&\leq\frac{4uw}{\kappa},\notag\\ \int_a^b\frac{U^2}{R}\dd z&\leq\frac{3u^2}{\kappa^2},& \int_a^b\frac{W^2}{R}\dd z&\leq\frac{4w^2}{\kappa}. \label{eq:core-four-integrals} \end{align} The first three bounds follow by integration of the exponential majorants, with decay rates $\kappa/12$, $\kappa/4$ and $\kappa/2$. For the last one, observe that \[ \int_a^b\cosh^2z\dd z \leq[\sinh z\cosh z]_a^b \leq\cosh^2a+\cosh^2b \leq2(11\kappa/10)^2. \] The first inequality follows by differentiating $\sinh z\cosh z$; the second uses $|\sinh z|\leq\cosh z$. Combining this with $W^2/R\leq w^2\cosh^2z/(9\kappa/10)^3$ proves the fourth estimate. Integrate~\eqref{eq:core-radius-integrand} and use \eqref{eq:core-four-integrals}. Since $u,w\leq1/60000$, \[ \frac{14u/\kappa+4uw/\kappa+3u^2/\kappa^2+4w^2/\kappa} {999/1000} \leq\frac{16}{\kappa}(u+w^2). \] This proves the radius estimate. Multiplying it by $R(b)+\kappa\leq201\kappa/100$ gives \[ |R(b)^2-\kappa^2| \leq\frac{804}{25}(u+w^2)<\frac1{16}, \] which strictly improves the assumed energy bound. \end{proof} The endpoint estimate for $\cosh^2z$ is useful for every initial segment of the crossing. It retains the smallness of the weak wall even when the angle interval is long. \subsubsection{The return and its amplitude loss} We allow $U$ or $W$ to vanish when establishing local estimates. The forward-existence proof of Lemma~\ref{lem:forward} applies unchanged on $G=1$ with nonnegative curvature variables: the exponential formula preserves their signs and all finite-time bounds remain valid. \begin{proposition}[Large-radius core passage] \label{prop:large-core-passage} Let $\kappa\geq6$ and $0\leq u,w\leq\varepsilon_0/\kappa$. The projective initial state \begin{equation} (M,U,V,W)=(1+\sqrt{\kappa^2-1},u,1,w) \label{eq:large-core-entrance} \end{equation} lifts by~\eqref{eq:projective-inverse} to a constrained solution of \eqref{eq:normalized-flow}. It has a first positive time $t_*$ at which $V=1$. The projective chart remains regular on $[0,t_*]$, and $V>1$ on $(0,t_*)$. At the exit, $M<1$ and $dV/dt<0$. The solution lies in~\eqref{eq:core-closed-region} throughout this interval, and its exit values satisfy \begin{equation} |R_* -\kappa|\leq\frac{16}{\kappa}(u+w^2),\qquad U_*\leq e^{-\kappa/40}u,\qquad W_*\leq e^{-1/(2\kappa)}w. \label{eq:large-core-exit} \end{equation} With $\tau(0)=0$ and $\tau_* =\int_0^{t_*}A(t)\dd t$, \begin{equation} \frac1{4\kappa}\leq\tau_*\leq\frac{4\log(4\kappa)}\kappa, \qquad t_*\leq2\kappa\log(4\kappa). \label{eq:large-core-duration} \end{equation} \end{proposition} \begin{proof} The initial state has $R=\kappa$ and $D\geq2$, so the inverse coordinate map is defined and gives $G=1$, $A>0$. On $V=1$, the region~\eqref{eq:core-closed-region} implies \[ |M-1|\geq\frac{3\kappa}{4},\qquad 0\leq C=U(1+W-U)\leq\frac1{1000}. \] Hence the sign of $\dot V=M-1-C$ is positive on the incoming branch $M>1$ and negative on the outgoing branch $M<1$; its absolute value is at least $\kappa/2$. In particular the initial crossing is upward. Follow the normalized solution while it lies in \eqref{eq:core-closed-region}, also imposing $A\geq3/(2\kappa^2)$. These closed conditions hold on an initial interval. The upper faces for $U$ and $P$ point strictly inward by \eqref{eq:core-wall-rates}; their zero faces are invariant. Lemma~\ref{lem:core-radius-drift} improves the energy bound to $1/16$, and~\eqref{eq:core-chart-clock} improves the lower bound for $A$ to $3/\kappa^2$. Consequently a finite endpoint of this interval can only occur at $V=1$. If $V>1$ there, continuity and the inward inequalities extend the interval. The normalized solution itself exists at every positive time, so no other finite endpoint is possible. Such an interval cannot persist indefinitely. Indeed, \eqref{eq:core-chart-clock} and~\eqref{eq:core-angle-range} give, at every time in it, \[ \frac t\kappa\leq z(t)-z(0)\leq2\log(4\kappa). \] There is therefore a finite return to $V=1$ with the stated bound on $t_*$. At any return approached from $V\geq1$ the left derivative of $V$ is nonpositive. The incoming sign alternative is excluded, so this return is on $M<1$ and is strictly downward. An earlier interior touch of $V=1$ would likewise be followed immediately by $V<1$, contradicting the closed-region condition. Thus $t_*$ is the first return and $V>1$ before it. The radius estimate is Lemma~\ref{lem:core-radius-drift} at the exit. To estimate the projective duration, use $\dot z=Rd\geq\kappa/2$ and~\eqref{eq:core-angle-range} for the upper bound. For the lower bound, the phase changes by at least $\kappa$ between the two crossings. The factorization \[ -\dot M=(V+W-U)(V-W+MU) \] gives $V^2/2\leq-\dot M\leq2V^2\leq4\kappa^2$ in the region. To check this, $|M|\leq6\kappa/5$ and the wall bounds place each of the two factors between $9V/10$ and $11V/10$. Integrating $-\dot M\leq4\kappa^2$ yields $\tau_*\geq1/(4\kappa)$. Finally~\eqref{eq:core-wall-rates} gives \[ U_*\leq u e^{-\kappa^2\tau_*/10},\qquad P_*\leq w e^{-2\tau_*}. \] At the exit $P_*=W_*$ because $V_*=1$. Substitution of the lower duration bound proves both contractions in~\eqref{eq:large-core-exit}. \end{proof} Although the contraction of the weak wall in~\eqref{eq:large-core-exit} approaches one as $\kappa\to\infty$, its loss has exactly the scale needed for the radius error. The following elementary estimate makes this relation precise. \begin{lemma}[Amplitude loss] \label{lem:amplitude-loss} Let $c,p>0$, $\kappa\geq2$, and $0\leq b\leq e^{-c/\kappa}a$, where $a\geq0$. Then \begin{equation} \frac{a^p}{\kappa} \leq\frac{1+pc/2}{pc}(a^p-b^p). \label{eq:one-core-loss} \end{equation} Consequently, if nonnegative amplitudes $a_0,\ldots,a_n$ and intermediate values $b_0,\ldots,b_{n-1}$ satisfy \[ a_{j+1}\leq b_j\leq e^{-c/\kappa_j}a_j,\qquad \kappa_j\geq2\quad(0\leq j