\subsection{Initial data and the remaining core passage} \label{sec:small-section-partial-core} We first control the core coordinates and their variation along the shear family, then follow the state to its outgoing section. Selecting the exit by its downward crossing makes the return smooth across initial states on the incoming boundary. \subsubsection{Entrance data and the incoming passage} For the cutoff state in \eqref{eq:projection-cutoff-state}, write $Y(\rho)=(M,U,V,W)$ and $P(\rho)=V(\rho)W(\rho)$. The following bounds hold for both shear signs. As in the preceding subsection, $\rho$ varies in $[a,2a]$ while the time $T=(q/2)\log(1/a)$ is fixed. \begin{proposition} \label{prop:small-section-coordinate-response} There are constants $h_*>0$ and $K_0>0$ such that, for every $02$, all sufficiently small $a>0$ satisfy the following estimates, uniformly over \eqref{eq:radial-shear-family}: \begin{equation} \begin{split} |M'|&\leq\frac{K_0}{a^2},\\ |U'|&\leq\frac{K_0U}{a},\qquad |V'|\leq\frac{K_0V}{a},\qquad |W'|\leq\frac{K_0W}{a},\\ |P'|&\leq\frac{K_0P}{a}. \end{split} \label{eq:partial-core-coordinate-response} \end{equation} Each positive wall has logarithmic derivative bounded in absolute value by $K_0/a$. If $n_2,n_3>0$, then \begin{equation} |(\log U)'|+|(\log V)'|+|(\log W)'| \leq\frac{K_0}{a}. \label{eq:partial-core-log-response} \end{equation} If $n_2=0$, then $V=V'=0$; if $n_3=0$, then $W=W'=0$. In either case $P=P'=0$. \end{proposition} \begin{proof} The height estimates give \[ |A'|\leq512a\leq\frac{2048A}{a}. \] For nonnegative $f$, positive $g$, and a constant $c\geq0$, the quotient rule shows that \[ |f'|\leq K_f f/a,\quad |g'|\leq K_g g/a \quad\Longrightarrow\quad \left|\left(\frac{cf}{g}\right)'\right| \leq(K_f+K_g)\frac{cf/g}{a}. \] This calculation remains valid when $f=0$. By \eqref{eq:projection-cutoff-bounds}, each physical wall $n,w_2,w_3$ has relative derivative bounded by $8C/a$. Applying the quotient estimate with $g=A$, $c=\sqrt3$, and \[ B=8C+2048 \] gives the three wall inequalities with $B$ in place of $K_0$. The product rule then gives $|P'|\leq2BP/a$. Division by a positive wall gives its logarithmic derivative, so the sum of the three logarithmic bounds is at most $3B/a$. For the shear, \eqref{eq:projection-cutoff-bounds} gives $|s'|\leq2$ and $\varsigma s\geq3a/4$. Consequently \[ |(\varsigma s)'|\leq\frac{4\varsigma s}{a}. \] The same quotient estimate, now followed by $\varsigma M\leq18/a$, yields \[ |M'|\leq(4+2048)\frac{\varsigma M}{a} \leq\frac{18(4+2048)}{a^2}. \] Thus $K_0=4B+40000$ suffices for all the estimates. It is chosen before $h$ and $q$. The vanishing assertions follow first from the value bounds in \eqref{eq:projection-coordinate-bounds}, and then from the corresponding relative derivative bounds. \end{proof} From now on take the positive shear sign $\varsigma=1$, and define \[ k(\rho)=\sqrt{(M(\rho)-1)^2+V(\rho)^2}. \] When $V(\rho)>0$, also define \[ z(\rho)=\operatorname{arsinh} \frac{1-M(\rho)}{V(\rho)}. \] The symbol $z$ here is the actual core angle \eqref{eq:core-angle-coordinates}. \begin{proposition} \label{prop:small-section-core-data} There are constants $h_*>0$ and $K_1>0$ such that for every $02$, all sufficiently small $a>0$ have the following properties. For the positive-shear family \eqref{eq:radial-shear-family} with $n_3\leq a^\ell$, \begin{equation} \begin{split} M&>1,\qquad k\geq6,\qquad \frac1{16a}\leq k\leq\frac{32}{a},\\ \frac1{4096a^2}&\leq-k'\leq\frac{1536}{a^2}. \end{split} \label{eq:partial-core-radius-response} \end{equation} The radius has an ordinary derivative. Wherever $V>0$, the angle also has an ordinary derivative and \begin{equation} |z'|\leq\frac{K_1}{a}. \label{eq:partial-core-angle-response} \end{equation} If $V\geq1$, then \begin{equation} -\operatorname{arcosh}k\leq z\leq0. \label{eq:partial-core-incoming-angle} \end{equation} For $q>3$, $\ell>4$, and any $0<\eta\leq10^{-4}$, one can further decrease the scale threshold so that \begin{equation} U+P\leq\frac{\eta}{k}. \label{eq:partial-core-entry-smallness} \end{equation} In particular, when $V\geq1$, the actual state belongs to \eqref{eq:core-closed-region} with $\kappa=k$. \end{proposition} \begin{proof} Take the common threshold of the coordinate and phase estimates from the preceding subsection, and require $a\leq1/128$. Equation~\eqref{eq:projection-coordinate-bounds} gives \[ \frac1{8a}\leq M\leq\frac{18}{a}, \qquad 0\leq V\leq V+W\leq\frac1a. \] Thus $M>1$, and \[ k\geq M-1\geq\frac1{16a}\geq6,\qquad k\leq M+V\leq\frac{32}{a}. \] If $V\geq1$, the identity $\cosh z=k/V$ implies $\cosh|z|\leq k$. Since $M>1$, we have $z\leq0$, which proves \eqref{eq:partial-core-incoming-angle}. For the radial derivative use the actual phase $f=2/k$. The radius bounds give $a/16\leq f\leq32a$, and Theorem~\ref{thm:small-section-projected-phase} gives $1/8\leq f'\leq3$. Hence $k=2/f$ is ordinarily differentiable and \[ -k'=\frac{2f'}{f^2},\qquad \frac1{4096a^2}\leq\frac{2f'}{f^2} \leq\frac{1536}{a^2}. \] These numerical bounds are independent of the phase comparison constant; its dependence on $h$ only changes the permitted threshold for $a$. For $V>0$, direct differentiation gives \begin{equation*} z'=\frac{(M-1)V'-VM'}{kV}. \end{equation*} Use $|M-1|\leq k$, $|V'|\leq K_0V/a$, and $|M'|\leq K_0/a^2\leq16K_0k/a$. The factor $V$ cancels in both terms, proving \eqref{eq:partial-core-angle-response} with $K_1=17K_0$. No lower bound on a positive $V$ is needed in this estimate. It remains to obtain \eqref{eq:partial-core-entry-smallness}. The coordinate bounds and $n_2\leq a$ imply \[ 03$ and $\ell>4$. Choose the scale threshold so that the right-hand side is at most $\eta$. At $V\geq1$ the initial radius equals $k$, while $U,P\leq\eta/k\leq10^{-4}/k$. These are precisely the remaining inequalities of \eqref{eq:core-closed-region}. \end{proof} \subsubsection{Radius drift from an arbitrary initial angle} The initial angle need not be the incoming endpoint $-\operatorname{arcosh}k$. In particular the initial active wall can be comparable to $k$. The following version of the radius estimate keeps this possibility. \begin{lemma} \label{lem:partial-core-radius-drift} Let a solution of \eqref{eq:core-angle-system} remain in \eqref{eq:core-closed-region}, with $\kappa=k\geq6$, for $z_0\leq z\leq z_1$. Suppose $R(z_0)=k$, and put $u_0=U(z_0)$ and $P_0=P(z_0)$. Then \begin{equation} |R(z_1)-k|\leq16\left(u_0+\frac{P_0^2}{k}\right), \qquad |R(z_1)^2-k^2|<\frac1{16}. \label{eq:partial-core-radius-drift} \end{equation} \end{lemma} \begin{proof} The contraction estimates \eqref{eq:core-wall-rates} give \[ U(z)\leq u_0e^{-k(z-z_0)/4},\qquad P(z)\leq P_0. \] Thus \[ W(z)\leq P_0,\qquad W(z)\leq\frac{P_0\cosh z}{R(z)},\qquad V(z)\leq\frac{11k}{10}. \] The four integral estimates in the proof of Lemma~\ref{lem:core-radius-drift} now apply with $V/k$ in place of $V$. Indeed, $V/k\leq11/10\leq(11/10)e^{z-z_0}$. Multiplying the resulting first integral by $k$ gives \begin{equation*} \begin{aligned} \int_{z_0}^{z_1}UV\dd z&\leq14u_0, &\int_{z_0}^{z_1}UW\dd z&\leq\frac{4u_0P_0}{k},\\ \int_{z_0}^{z_1}\frac{U^2}{R}\dd z &\leq\frac{3u_0^2}{k^2}, &\int_{z_0}^{z_1}\frac{W^2}{R}\dd z &\leq\frac{4P_0^2}{k}. \end{aligned} \end{equation*} The last estimate uses only the endpoint bounds $\cosh z_0,\cosh z_1\leq11k/10$, which still follow from $V\geq1$. Integrating \eqref{eq:core-radius-integrand}, we obtain \[ |R(z_1)-k| \leq \frac{14u_0+4u_0P_0/k+3u_0^2/k^2+4P_0^2/k} {999/1000} \leq16(u_0+P_0^2/k). \] For the last inequality it suffices to use $4u_0P_0/k\leq u_0$ and $3u_0^2/k^2\leq u_0/2$. Multiplication by $R(z_1)+k\leq201k/100$ then gives \[ |R(z_1)^2-k^2| \leq\frac{804}{25}(ku_0+P_0^2)<\frac1{16}, \] because $ku_0\leq10^{-4}$ and $P_0\leq1/60000$. \end{proof} \subsubsection{The first outgoing event} We formulate the passage for a general initial core state. This also identifies the event reached by the small-section family. \begin{theorem} \label{thm:partial-core-first-exit} Let $y_0=(M_0,U_0,V_0,W_0)$ satisfy \begin{equation} \begin{split} k&=\sqrt{(M_0-1)^2+V_0^2}\geq6,\\ V_0&>1\quad\hbox{or}\quad(V_0=1\ \hbox{and}\ M_0\geq1),\\ 0\leq U_0&\leq10^{-4}/k,\qquad 0\leq P_0:=V_0W_0\leq10^{-4}/k. \end{split} \label{eq:partial-core-initial-state} \end{equation} Lift $y_0$ by \eqref{eq:projective-inverse} and follow the normalized flow. Write its core coordinates as \[ \beta(t)=(M(t),U(t),V(t),W(t)). \] There is a first positive time $t_*$ with $V(t_*)=1$. For $01$, and \begin{equation} M(t_*)<1,\qquad z(t_*)>0,\qquad \frac{\dd V}{\dd t}(t_*)\leq-\frac{A(t_*)k}{2}<0, \qquad t_*\leq2k\log(4k). \label{eq:partial-core-exit-event} \end{equation} Throughout $0\leq t\leq t_*$, the projective chart is regular, \eqref{eq:core-closed-region} holds, and \begin{equation} \begin{split} A(t)&\geq\frac3{2k^2},\\ |R(t)-k|&\leq16\left(U_0+\frac{P_0^2}{k}\right), \qquad |R(t)^2-k^2|<\frac1{16}. \end{split} \label{eq:partial-core-prefix-estimates} \end{equation} \end{theorem} \begin{proof} The initial state lies in \eqref{eq:core-closed-region}, so \eqref{eq:core-chart-clock} makes its inverse chart regular. The lift has $G=1$ and nonnegative walls, and hence exists for every forward normalized time. If $V_0>1$, the active-wall inequality is initially strict. If $V_0=1$, the radius condition and $M_0\geq1$ place the state on the incoming branch, where $dV/dt\geq A(0)k/2>0$. As in the proof of Proposition~\ref{prop:large-core-passage}, continue the actual orbit under the closed conditions \eqref{eq:core-closed-region} and $A\geq3/(2k^2)$. The wall equations preserve their zero faces and point inward at the positive upper faces. On each closed time segment the quantities $A,V,d$ are positive. They remain so on an open interval containing that segment, where the strictly increasing angle gives a legitimate change of time. Lemma~\ref{lem:partial-core-radius-drift}, applied on its angle image, improves the energy bound to $1/16$. Equation~\eqref{eq:core-chart-clock} improves the height bound to $3/k^2$. Thus the closed region continues past every finite endpoint at which $V>1$. It cannot contain the forward orbit indefinitely. On any such segment, \[ \frac{t}{k}\leq z(t)-z(0)\leq2\log(4k), \] by \eqref{eq:core-chart-clock} and \eqref{eq:core-angle-range}. The first boundary encountered is consequently $V=1$, at some $t_*\leq2k\log(4k)$. At any return approached from $V\geq1$, the left derivative of $V$ is nonpositive. The two crossing alternatives on $V=1$, proved in Proposition~\ref{prop:large-core-passage}, then give $M<1$ and $dV/dt\leq-Ak/2$. In particular an interior touch would immediately leave $V\geq1$, so no earlier return occurs. The definition of $z$ gives $z(t_*)>0$. Applying the radius lemma to every initial segment proves \eqref{eq:partial-core-prefix-estimates}. \end{proof} For a projective state $y$ near an initial state in \eqref{eq:partial-core-initial-state}, let $\beta_y(t)$ denote the core coordinates of the normalized flow starting at its inverse image. Define $\tau_\downarrow(y)$ to be the first positive time such that \[ V_y(\tau_\downarrow(y))=1,\qquad \partial_tV_y(\tau_\downarrow(y))<0, \] whenever that time exists. Write its output in radius and wall coordinates as \begin{equation} \mathcal R_\downarrow(y)= (R,U,P)\bigl(\beta_y(\tau_\downarrow(y))\bigr). \label{eq:partial-core-oriented-return} \end{equation} Since $V=1$ at this event, its third component is the outgoing weak wall $W$. \begin{proposition} \label{prop:partial-core-smooth-exit} At every state satisfying \eqref{eq:partial-core-initial-state}, $\tau_\downarrow$ and $\mathcal R_\downarrow$ are defined and $C^2$ on an ordinary open neighborhood in $\R^4$. At the stated initial state, $\tau_\downarrow=t_*$ from Theorem~\ref{thm:partial-core-first-exit}. The nearby flow segments up to their moving exit times remain in the projective chart. If $V_0>1$, the same event is the first positive crossing of $V=1$ throughout a sufficiently small neighborhood. \end{proposition} \begin{proof} The normalized flow and coordinate changes are jointly $C^2$ on a neighborhood of the compact orbit segment. At $t_*$ the event function has a strictly negative time derivative. The implicit function theorem gives a $C^2$ time $\tau(y)$ and a unique nearby crossing. Its orientation remains downward by continuity. To prove that this branch is the first downward crossing, consider the whole interval $[0,t_*]$. Near $t_*$ local uniqueness excludes a second crossing. At each earlier positive time, $V>1$, so continuity excludes a nearby hit of $V=1$. At time zero there are two possibilities. If $V_0>1$, the same avoidance applies. If $V_0=1$, the initial crossing speed is strictly positive and remains positive on a neighborhood of that point in state and time. Any nearby crossing there is therefore upward. These neighborhoods cover the compact segment. Taking a finite subcover and using the continuity of $\tau$ gives one initial neighborhood on which there is no downward crossing in $00$ on the entire moving interval $[0,\tau(y)]$. Thus $\tau=\tau_\downarrow$ on this neighborhood. When $V_0>1$, the argument excludes every earlier crossing, proving the last assertion. The endpoint depends $C^2$ on $y$. At the endpoint $V=1>0$, so the radius and product coordinates in \eqref{eq:partial-core-oriented-return} are smooth. Their composition proves the asserted regularity of $\mathcal R_\downarrow$. \end{proof} We can now apply these results to the positive-shear family. Fix $q>3$, $\ell>4$, $0<\eta\leq10^{-4}$, and $n_3\leq a^\ell$, and choose the common scale threshold in Proposition~\ref{prop:small-section-core-data}. Whenever its actual state at time $T$ has $V\geq1$, it satisfies \eqref{eq:partial-core-initial-state}. Its first outgoing time, measured from the original small section, is \[ t_{\mathrm{out}}(\rho) =T+\tau_\downarrow(Y(\rho))>T. \] The group property of the normalized flow identifies this concatenation with the same original orbit. The time and endpoint are $C^2$ functions of the initial coordinates near each such point, with $T$ fixed. The bounds \eqref{eq:partial-core-exit-event} and \eqref{eq:partial-core-prefix-estimates} apply after translating time by $T$. Together with the relative wall, radius, and angle derivatives, this supplies the initial data for estimating the variation of the remaining core passage.