\subsection{Uniform core returns} \label{sec:core-uniform} The large-radius estimates control cores arbitrarily close to a Taub point. The remaining radii form a compact interval, where the exact single-wall orbits provide a family of transverse crossings. Their wall multipliers are strictly less than one. We first preserve these properties in a common neighborhood, and then combine the two ranges in one return estimate. \subsubsection{The single-wall family and its neighborhood} For $\kappa>1$, put $h=\operatorname{arcosh}(\kappa)/\kappa$. With projective time centered at the maximum of $V$, the single-wall orbit from Section~\ref{sec:core-passage} is \begin{equation} (M,U,V,W)(\tau) =\left(1-\kappa\tanh(\kappa\tau),\,0,\, \frac{\kappa}{\cosh(\kappa\tau)},\,0\right). \label{eq:reference-core-pulse} \end{equation} Its two crossings of $V=1$ are at $\tau=-h$ and $\tau=h$, and $V>1$ between them. The projective derivatives of $V$ there are $\sqrt{\kappa^2-1}$ and $-\sqrt{\kappa^2-1}$. Along this orbit, \[ D=\kappa^2+4-2\kappa\tanh(\kappa\tau)>0, \qquad \frac{dt}{d\tau}=\frac D6. \] Thus these are also transverse crossings for the original flow. The original time elapsed between them is \begin{equation} T_0(\kappa) =\frac{(\kappa^2+4)\operatorname{arcosh}\kappa}{3\kappa}. \label{eq:reference-core-time} \end{equation} Indeed, a primitive of $D/6$ is $(\kappa^2+4)\tau/6-\log\cosh(\kappa\tau)/3$, and the logarithmic terms cancel at $\pm h$. For an entrance $Y=(\kappa,u,w)$, use the same embedding \[ (M,U,V,W)=(1+\sqrt{\kappa^2-1},u,1,w) \] as before. Denote its first positive return time to $V=1$ by $T(Y)$, and its returned radius and walls by $\mathcal R(Y)=(R_*,U_*,W_*)$. Equivalently, $T(Y)$ is the infimum of the positive times in the maximal flow domain at which $V=1$; in all domains considered below this infimum is attained. \begin{lemma}[A common neighborhood of the bounded cores] \label{lem:compact-core-regularity} There are $\delta>0$ and $B>0$ such that, for \begin{equation} 2\leq\kappa\leq6,\qquad |u|,|w|\leq\delta, \label{eq:compact-core-box} \end{equation} the first return exists, the projective chart remains regular on $[0,T(Y)]$, and \[ V(t)>1\quad(01$ and $Z=(\kappa,0,0)$. The explicit orbit \eqref{eq:reference-core-pulse} lies in the projective chart throughout its finite crossing segment. Smooth dependence of the original flow and the implicit function theorem at its downward crossing give a local $C^2$ exit time near $Z$. This exit is the first return. At the entrance, transversality makes $t=0$ the only nearby crossing for all nearby initial points of the section. At the exit, transversality makes the constructed event the only nearby crossing. On a closed interval between these two event neighborhoods, the reference orbit has $V>1$, with a positive minimum gap from the section. Continuity preserves this gap for nearby entrances. These three intervals exclude an earlier return. The sign of $V-1$ is positive throughout the intervening open interval by continuity, and the exit derivative remains negative. Compactness of the reference segment also preserves the flow domain and $A>0$ up to the moving exit. Consequently $T$ agrees locally with the implicit exit time, and $\mathcal R$ is $C^2$ by composition with the flow and the radius coordinates. These local functions agree wherever their neighborhoods overlap, since each is the first positive return. The segment \[ \{(\kappa,0,0):2\leq\kappa\leq6\} \] is compact. A closed neighborhood of sufficiently small positive width lies in the union of the regular neighborhoods just constructed. The first and second derivatives of $T$ and $\mathcal R$ are continuous there and hence bounded. Choose a smaller box \eqref{eq:compact-core-box} inside this neighborhood. This proves both the uniform bounds and the asserted regularity around the box. \end{proof} The transverse derivatives on the reference segment can be computed without solving the full variational equation. The equations for the two transverse components of a variation are \[ \dot\xi_U= \left(-\frac{\kappa^2}{2}+\kappa\tanh(\kappa\tau)\right)\xi_U, \qquad \dot\xi_W=(-2+\kappa\tanh(\kappa\tau))\xi_W. \] For a coefficient $-r+\kappa\tanh(\kappa\tau)$, a positive fundamental solution is $e^{-r\tau}\cosh(\kappa\tau)$. Its ratio between $h$ and $-h$ is $e^{-2rh}$. A change of time in a variation adds a multiple of the vector field; on the single-wall orbit that field has zero $U$ and $W$ components. The exit-time correction has the same property. Therefore the actual return derivative satisfies \begin{align} D\mathcal R_U(Z)v&=m_U(\kappa)v_u, &m_U(\kappa)&=e^{-\kappa\operatorname{arcosh}\kappa},\notag\\ D\mathcal R_W(Z)v&=m_W(\kappa)v_w, &m_W(\kappa)&=e^{-4\operatorname{arcosh}(\kappa)/\kappa}. \label{eq:compact-reference-multipliers} \end{align} Both multipliers belong to $(0,1)$. Moreover, \begin{equation} \mathcal R(\kappa,0,0)=(\kappa,0,0),\qquad D\mathcal R(\kappa,0,0)e_R=e_R, \quad e_R=(1,0,0). \label{eq:compact-reference-radial} \end{equation} The second identity follows by differentiating the first along the reference segment. The invariant hyperplanes $u=0$ and $w=0$ give $\mathcal R_U(\kappa,0,w)=0$ and $\mathcal R_W(\kappa,u,0)=0$. On the box of Lemma~\ref{lem:compact-core-regularity}, define \begin{equation} \begin{split} \mu_U(\kappa,u,w)&=\int_0^1 \partial_u\mathcal R_U(\kappa,su,w)\dd s,\\ \mu_W(\kappa,u,w)&=\int_0^1 \partial_w\mathcal R_W(\kappa,u,sw)\dd s. \end{split} \label{eq:compact-wall-factors} \end{equation} The coordinate segments in these integrals remain in the box. The fundamental theorem of calculus and differentiation under the integral give $C^1$ factors satisfying \begin{equation} U_*=u\mu_U(Y),\qquad W_*=w\mu_W(Y). \label{eq:compact-wall-factorization} \end{equation} This representation includes zero and signed wall coordinates. The identities hold locally around each point of the box, since its coordinate segments remain in a neighborhood of regular returns. At $Z=(\kappa,0,0)$ the factors equal the multipliers in~\eqref{eq:compact-reference-multipliers}. Let $q_0$ be the maximum of $m_U$ and $m_W$ over $[2,6]$. It satisfies $00$ such that \begin{equation} 0<\mu_U,\mu_W\leq q<1,\qquad \|D\log\mu_U\|,\|D\log\mu_W\|\leq L. \label{eq:compact-factor-bounds} \end{equation} For the derivative bounds, first choose a closed neighborhood on which the factors remain positive and $C^1$, and then bound the continuous derivatives of their logarithms there. Thus the same neighborhood supports both the factorization and the bounds. We also need the phase errors to vanish with the walls. Set $Z=(\kappa,0,0)$. Apply the mean-value bound on the convex box to $\mathcal R$ and $D\mathcal R$, using the first and second derivative bounds of Lemma~\ref{lem:compact-core-regularity}. After enlarging $B$, \begin{equation} \begin{split} \|\mathcal R(Y)-Z\|&\leq B(|u|+|w|),\\ \|D\mathcal R(Y)-D\mathcal R(Z)\|&\leq B(|u|+|w|), \qquad \|D\mathcal R(Y)\|\leq B. \end{split} \label{eq:compact-linear-errors} \end{equation} Now suppose $u,w\geq0$, $u+w\leq1$, and $v=(1,a,b)$ belongs to the cone~\eqref{eq:core-radial-cone}. Since $\|v-e_R\|\leq\sqrt{u+w}$, the radial identity \eqref{eq:compact-reference-radial} implies \begin{align*} |D\mathcal R_R(Y)v-1| &\leq |D\mathcal R_R(Y)e_R-1| +|D\mathcal R_R(Y)(v-e_R)|\\ &\leq B(u+w)+B\sqrt{u+w} \leq2B\sqrt{u+w}. \end{align*} Using $\kappa\leq6$ in this inequality and \eqref{eq:compact-linear-errors}, we obtain \begin{equation} |R_*-\kappa|\leq C_b\frac{u+w}{\kappa},\qquad |D\mathcal R_R(Y)v-1|\leq C_b\frac{\sqrt{u+w}}{\kappa}, \quad C_b=12B. \label{eq:compact-phase-errors} \end{equation} \begin{lemma}[Strict cone preservation for bounded radii] \label{lem:compact-core-cone} There are $\delta_c>0$ and $\sigma\in(0,1)$ such that, if $2\leq\kappa\leq6$, $0\leq u,w\leq\delta_c$, and $v=(1,a,b)\in\mathcal C(Y)$, then $d=D\mathcal R(Y)v$ satisfies \begin{equation} d_R>0,\qquad |d_U|\leq\sigma\sqrt{U_*}\,d_R,\qquad |d_W|\leq\sigma\sqrt{W_*}\,d_R. \label{eq:compact-cone-margin} \end{equation} \end{lemma} \begin{proof} Put $s=\sqrt q$ and $e=(1-s)/4$, so $00. \] For either wall write $j$ for its amplitude, $a_0$ for the corresponding component of $v$, and $\mu$ for its factor. Put $\ell=D\log\mu(Y)v$. The factorization gives the returned component as $\mu(a_0+j\ell)$, with $|a_0|\leq\sqrt j$. Moreover, \[ |\ell|\leq L,\qquad \sqrt j\,|\ell|\leq e, \qquad \sqrt\mu\leq s<1. \] The product estimate~\eqref{eq:core-wall-cone-inequality} bounds this component by $\sqrt{j\mu}(s+e)$. Finally, \[ s+e=1-3e\leq(1-e)(1-2e)\leq(1-e)d_R. \] Taking $\sigma=1-e$ proves the lemma, including at a zero wall. \end{proof} \subsubsection{One estimate for all radii} The compact and large-radius constructions identify the same return. Indeed, both give the first positive crossing of $V=1$ along the original flow. Their local exit functions therefore agree on a neighborhood of every entrance where both apply. Their derivatives agree there as well, including at the common endpoint $\kappa=6$. \begin{proposition}[Uniform core estimates] \label{prop:uniform-core-estimates} There are constants $\varepsilon\in(0,10^{-4}]$, $C_0>0$ and $c_0\in(0,1/2]$ such that, for \begin{equation} \kappa\geq2,\qquad 0\leq u,w\leq\frac\varepsilon\kappa, \label{eq:uniform-core-domain} \end{equation} the actual first return $T$ and $\mathcal R$ are locally $C^1$. The flow and the projective chart remain regular throughout the crossing, with $V>1$ in its interior and $dV/dt<0$ at its exit. Writing $b=u+w$, one has \begin{equation} |R_*-\kappa|\leq C_0\frac b\kappa,\qquad 0\leq U_*\leq e^{-c_0/\kappa}u,\qquad 0\leq W_*\leq e^{-c_0/\kappa}w. \label{eq:uniform-core-value} \end{equation} An output wall is positive exactly when its input wall is positive. For every $v\in\mathcal C(Y)$ with $v_\kappa=1$, put $d=D\mathcal R(Y)v$. Then \begin{gather} \frac34\leq d_R\leq\frac54,\qquad |d_R-1|\leq C_0\frac{\sqrt b}{\kappa},\qquad |\log d_R|\leq C_0\frac{\sqrt b}{\kappa}, \label{eq:uniform-core-phase}\\ |d_U|\leq\sqrt{U_*}\,d_R,\qquad |d_W|\leq\sqrt{W_*}\,d_R. \label{eq:uniform-core-cone} \end{gather} \end{proposition} \begin{proof} Take $\varepsilon$ smaller than every wall threshold used in the compact construction and in Proposition~\ref{prop:large-core-cone}. For $2\leq\kappa\leq6$, the bounds $u,w\leq\varepsilon/\kappa$ then place the entrance in each compact box. For $\kappa\geq6$, the large-radius results apply directly. The preceding uniqueness argument supplies one locally $C^1$ map and one locally $C^1$ return time on the resulting domain. For the wall contraction, let \[ c_0=\min\{1/2,-2\log q\}>0. \] Since $\log q<0$ and $\kappa\geq2$, $\kappa\log q\leq2\log q\leq-c_0$, whence $q\leq e^{-c_0/\kappa}$. This proves the compact part of the wall estimate by~\eqref{eq:compact-wall-factorization}. In the large-radius range the weak multiplier is at most $e^{-1/(2\kappa)}$, while $e^{-\kappa/40}\leq e^{-1/(2\kappa)}$ for $\kappa\geq6$. Thus~\eqref{eq:core-multiplier-contraction} gives the same bound. The positive factors in both constructions also give the assertion about positivity. In the large-radius value estimate, $w^2\leq w$ reduces Proposition~\ref{prop:large-core-passage} to $|R_*-\kappa|\leq16b/\kappa$. Combine this with \eqref{eq:compact-phase-errors} and Corollary~\ref{cor:large-core-phase-direction}. There is one $C>0$ such that both the value error and the ordinary radial derivative error have the bounds $Cb/\kappa$ and $C\sqrt b/\kappa$, respectively, in the full range. The two cone results give \eqref{eq:uniform-core-cone} on this same domain. Finally decrease $\varepsilon$ so that $\varepsilon\leq(4C)^{-2}$. Since $b\leq2\varepsilon/\kappa \leq\varepsilon$, the radial derivative error is at most $1/4$. Hence $3/4\leq d_R\leq5/4$, and the logarithmic inequality used in Corollary~\ref{cor:large-core-phase-direction} gives $|\log d_R|\leq2|d_R-1|$. Taking $C_0=2C$ proves all the estimates. \end{proof} Both the phase error and its logarithmic derivative can now be compared with the wall loss. For any finite chain whose connecting passages do not increase the wall sum, Lemma~\ref{lem:amplitude-loss} applies with $c=c_0$, and with $p=1$ or $p=1/2$ for the two errors. To use this comparison for a complete era, we next determine how the exact connecting passages change the radius, the two walls and the transverse directions.