\subsection{Transverse contraction and invariant directions}
\label{sec:core-multipliers}

The weak transverse wall contracts by a factor approaching one as
$\kappa\to\infty$. Its derivative must therefore be controlled on
the same scale as this small loss. The variation of the strong wall
$U$ enters the weak-wall equation. Integrating the decay of this
variation supplies an additional inverse power of the radius in
the weak multiplier estimate.

\subsubsection{Positive multipliers and their derivatives}

For the angle system~\eqref{eq:core-angle-system}, define
\begin{equation}
 L_U=\frac ad,\qquad L_P=-\frac{2(1+C)}{Rd},\qquad
 a=-\frac R2+\tanh z+\frac W{\cosh z}
                         +\frac{U^2-W^2}{2R}.
 \label{eq:core-logarithmic-rates}
\end{equation}
Thus $U'=L_U\,U$ and $P'=L_P\,P$, where primes denote angle derivatives.
For an entrance $Y_0=(\kappa,u,w)$, let $z_-(Y_0)$ and $z_+(Y_0)$
be its entrance and first-exit angles. Write
$Y_*=\mathcal R(Y_0)$ and set
\begin{equation}
 \lambda_j(Y_0)=\exp\left(
      \int_{z_-(Y_0)}^{z_+(Y_0)}L_j(z,Y(z;Y_0))\dd z\right),
 \qquad j\in\{U,P\}.
 \label{eq:core-positive-multipliers}
\end{equation}
The scalar equations give
\begin{equation}
 U_*=u\lambda_U,\qquad W_*=w\lambda_P.
 \label{eq:core-multiplier-formula}
\end{equation}
The second identity uses $P=W$ at both crossings. The factors are
strictly positive, including at entrances with a zero wall.

These factors and their logarithms are locally $C^1$. To see this
directly in the original time, write the integral in
\eqref{eq:core-positive-multipliers} as
\[
 \int_0^{t_*(Y_0)} q(t;Y_0)L_j(z(t;Y_0),Y(t;Y_0))\dd t,
 \qquad q=\frac{dz}{dt}=ARd.
\]
The integrand is jointly $C^1$ on a neighborhood of the compact
crossing segment, and $t_*$ is locally $C^1$ by
Proposition~\ref{prop:large-core-radius-derivative}. Differentiation
of this moving-endpoint integral proves the claim. Only $R$, $V$ and
$d$ occur as denominators in the rates, and all three stay positive.

The proof of~\eqref{eq:core-wall-rates} bounds the coefficients
themselves, even when the corresponding wall is zero. In projective
time they are at most $-\kappa^2/10$ and $-2$.
Integrating these inequalities and using
$\tau_*\geq1/(4\kappa)$ gives
\begin{equation}
 0<\lambda_U\leq e^{-\kappa/40},\qquad
 0<\lambda_P\leq e^{-1/(2\kappa)}.
 \label{eq:core-multiplier-contraction}
\end{equation}

\begin{proposition}[Logarithmic derivatives of the wall multipliers]
\label{prop:core-log-multiplier-derivatives}
For $\kappa\geq6$ and $0\leq u,w\leq\varepsilon_0/\kappa$, put
$Q_0=|v_\kappa|+\kappa|v_u|+|v_w|$. Then
\begin{align}
 |D\log\lambda_U(Y_0)v|
   &\leq A_U\log(4\kappa)Q_0,\notag\\
 |D\log\lambda_P(Y_0)v|
   &\leq A_P\frac{\log(4\kappa)}{\kappa^2}Q_0,
 \label{eq:core-log-multiplier-derivatives}
\end{align}
where $A_U=20K+4$, $A_P=118K+8$, and $K$ is the constant in
\eqref{eq:core-variation-growth}.
\end{proposition}

\begin{proof}
At fixed angle, the two rates satisfy the following bounds on the
core region. We use them first to estimate the multiplier derivatives
and verify their coefficients at the end of the proof:
\begin{gather}
 |L_U|\leq2\kappa,\qquad |L_P|\leq4/\kappa,\notag\\
 \begin{array}{c|ccc}
       & |\partial_R| & |\partial_U| & |\partial_P|\\ \hline
 L_U   & 2 & 4\varepsilon_0/\kappa^2 & 4/\kappa\\
 L_P   & 8/\kappa^2 & 8 & 8\varepsilon_0/\kappa^2
 \end{array}
 \label{eq:core-log-rate-bounds}
\end{gather}
The entries in the array are upper bounds for the indicated derivatives.

Let $\xi$ be the variation at fixed angle corresponding to $v$, and
put $X=S_\kappa\xi$, $x=X_U$ and $Q=2KQ_0$. The preceding subsection
gives $\|X\|_\infty\leq Q$ throughout the passage. Consequently
\begin{equation}
 |D_YL_U\xi|\leq3Q,\qquad
 |D_YL_P\xi|\leq\frac{9Q}{\kappa^2}
                            +\frac8\kappa|x|.
 \label{eq:core-log-rate-variation}
\end{equation}
The strong component is kept separately in the second estimate.
By~\eqref{eq:core-strong-variation-integrals},
\[
 \frac8\kappa\int_{z_-}^{z_+}|x|\dd z
 \leq\frac{32|x(z_-)|+384uQ}{\kappa^2}
 \leq\frac{33Q}{\kappa^2}.
\]
The last step uses $|x(z_-)|\leq Q$ and $u\leq1/60000$.
Since $z_+-z_-\leq2\log(4\kappa)$, integration of
\eqref{eq:core-log-rate-variation} gives
\begin{equation}
 |I_U|\leq6\log(4\kappa)Q,\qquad
 |I_P|\leq51\frac{\log(4\kappa)}{\kappa^2}Q,
 \quad
 I_j=\int_{z_-}^{z_+}D_YL_j\xi\dd z.
 \label{eq:core-log-integral-bounds}
\end{equation}
Here $1\leq\log(4\kappa)$ absorbs the constant term in the weak
estimate.

It remains to include the moving angles. The exact derivative formula is
\begin{equation}
 D\log\lambda_j(Y_0)v
 =I_j+L_j(z_+,Y_*)\,Dz_+(Y_0)v
          -L_j(z_-,Y_0)\,Dz_-(Y_0)v.
 \label{eq:core-moving-multiplier-derivative}
\end{equation}
One can verify this formula before changing time. Let
$\zeta=\delta z$ and $\eta=\delta Y$ be variations at fixed original
time, so that $\xi=\eta-\zeta F$. The chain rule yields
\[
 \delta(qL_j)=qD_YL_j\xi+\frac{d}{dt}(L_j\zeta).
\]
Integrate this identity from $0$ to $t_*$. The derivative of the
upper time endpoint contributes $q_*L_j(z_+,Y_*)Dt_*v$;
together with $L_j(z_+,Y_*)\zeta(t_*)$ it is exactly
$L_j(z_+,Y_*)Dz_+v$. The lower boundary term is $-L_j(z_-,Y_0)Dz_-v$.
This proves~\eqref{eq:core-moving-multiplier-derivative}.

The entrance angle derivative and the exit tangency calculation give
\begin{equation}
 |Dz_-v|\leq\frac{2Q_0}{\kappa},\qquad
 |Dz_+v|\leq\frac{2Q}{\kappa}.
 \label{eq:core-moving-angle-bounds}
\end{equation}
For the second bound, the notation in~\eqref{eq:core-exit-projection}
gives $Dz_+v=-\xi_R(z_+)/(r c_*+f_*)$.
Since $r c_*=M_*-1\leq-3\kappa/4$ and $f_*\leq0$, its denominator
has absolute value at least $3\kappa/4$.
Thus the two boundary contributions have total absolute value at most
$4Q+4Q_0$ for $L_U$, and $(8Q+8Q_0)/\kappa^2$ for $L_P$.
Combine these bounds with~\eqref{eq:core-log-integral-bounds},
use $1\leq\log(4\kappa)$, and substitute $Q=2KQ_0$.
The resulting constants are $20K+4$ and $118K+8$.

\emph{Verification of the coefficient bounds.}
Put $m=\varepsilon_0/\kappa$. In terms of the current radius $R$,
the bounds underlying~\eqref{eq:core-log-rate-bounds} are
\[
 \begin{array}{c|ccc}
       & |\partial_R| & |\partial_U| & |\partial_P|\\ \hline
 L_U   & 2 & 7m/(2R) & 2/R\\
 L_P   & 5/R^2 & 6 & 7m/(2R).
 \end{array}
\]
The quotient rule gives $(L_U)_i=a_i/d-ad_i/d^2$. For the weak rate, set
$n=-2(1+C)/R$. Then $(L_P)_i=n_i/d-nd_i/d^2$, with
\[
 n_R=\frac{2(1+C)}{R^2}-\frac{2C_R}{R},\qquad
 n_U=-\frac{2C_U}{R},\qquad n_P=-\frac{2C_P}{R}.
\]
The auxiliary bounds~\eqref{eq:core-auxiliary-bounds} for
$a$, $C$, $d$ and their derivatives give
\begin{align*}
 (|a_R|,|a_U|,|a_P|)&\leq(3/4,m/R,1/R),\\
 (|ad_R|,|ad_U|,|ad_P|)&\leq(m/R,2m/R,2m/R),\\
 (|n_R|,|n_U|,|n_P|)&\leq(3/R^2,4,2m/R),\\
 (|nd_R|,|nd_U|,|nd_P|)&\leq(1/R^2,1,m/R),
\end{align*}
where vector inequalities are componentwise. Applying
\[
 \left|\frac{x}{d}-\frac{y}{d^2}\right|
       \leq\frac{11}{10}(|x|+|y|)
       \qquad(d\geq999/1000)
\]
proves the radius-dependent bounds. Converting them using
$\kappa/R\leq10/9$ and $\kappa^2/R^2\leq3/2$ proves
\eqref{eq:core-log-rate-bounds}. The rate bounds themselves follow
from $|a|\leq R$ and $0\leq C\leq1/100$.
\end{proof}

\subsubsection{An invariant cone}

For $Y_0=(\kappa,u,w)$ define the forward radial cone
\begin{equation}
 \mathcal C(Y_0)=\left\{v:\ v_\kappa\geq0,
       \ |v_u|\leq\sqrt u\,v_\kappa,
       \ |v_w|\leq\sqrt w\,v_\kappa\right\}.
 \label{eq:core-radial-cone}
\end{equation}
Every nonzero vector in this cone has $v_\kappa>0$. We may therefore
normalize it by $v_\kappa=1$, as in
\eqref{eq:core-square-root-directions}. At the exit the cone is defined
in the same way using $(R_*,U_*,W_*)$.

\begin{proposition}[Preservation with a quantitative margin]
\label{prop:large-core-cone}
There is an absolute $\varepsilon_c\in(0,\varepsilon_0]$ such that
for $\kappa\geq6$ and $0\leq u,w\leq\varepsilon_c/\kappa$,
the derivative of the core return maps $\mathcal C(Y_0)$ into
$\mathcal C(\mathcal R(Y_0))$. More precisely, if
$v\in\mathcal C(Y_0)$ and $v_\kappa=1$, write
$d=D\mathcal R(Y_0)v=(d_R,d_U,d_W)$. Then
\begin{align}
 d_R&>0,& |d_R-1|&\leq\frac1{32\kappa},\notag\\
 |d_U|&\leq\left(1-\frac1{16\kappa}\right)\sqrt{U_*}\,d_R,&
 |d_W|&\leq\left(1-\frac1{16\kappa}\right)\sqrt{W_*}\,d_R.
 \label{eq:large-core-cone-margin}
\end{align}
\end{proposition}

\begin{proof}
Write $v=(1,a,b)$, $Q_0=1+\kappa|a|+|b|$ and
$\ell_j=D\log\lambda_j(Y_0)v$. Differentiating
\eqref{eq:core-multiplier-formula} gives
\begin{equation}
 d_U=\lambda_U(a+u\ell_U),\qquad
 d_W=\lambda_P(b+w\ell_P).
 \label{eq:core-transverse-product-derivative}
\end{equation}
If $0\leq u,w\leq\varepsilon/\kappa$ and $\varepsilon\leq1$,
the bounds on $a,b$ imply
\begin{equation}
 \sqrt u\,Q_0\leq3\sqrt\varepsilon,\qquad
 \sqrt w\,Q_0\leq3\sqrt\varepsilon,\qquad
 \sqrt u+(u+w)Q_0\leq7\sqrt\varepsilon.
 \label{eq:core-cone-size}
\end{equation}
For example,
$\sqrt u\,Q_0\leq\sqrt u+\kappa u+\sqrt{uw}$,
where each term is at most $\sqrt\varepsilon$.
For the weak component use
$\kappa\sqrt{uw}\leq\varepsilon$ and $w\leq\varepsilon$.
Multiplying these first two estimates by $\sqrt u$ and $\sqrt w$
proves the last one, since $\varepsilon\leq\sqrt\varepsilon$.

Choose $\varepsilon_c>0$, at most $\varepsilon_0$, so small that
\begin{equation*}
 7C_*\sqrt{\varepsilon_c}\leq\frac1{32},\qquad
 38400A_U\sqrt{\varepsilon_c}\leq\frac1{32},\qquad
 3A_P\sqrt{\varepsilon_c}\leq\frac1{32}.
\end{equation*}
All constants here are independent of $\kappa$.
Set $e=1/(32\kappa)$ and $L=\log(4\kappa)$.
Proposition~\ref{prop:large-core-radius-derivative} and
\eqref{eq:core-cone-size} imply $|d_R-1|\leq e$, so
$d_R\geq1-e>0$.

The square roots of both multipliers are bounded by $1-4e$.
Indeed,
\begin{equation*}
 \sqrt{\lambda_U}\leq e^{-\kappa/80}
       \leq e^{-1/(4\kappa)},\qquad
 \sqrt{\lambda_P}\leq e^{-1/(4\kappa)}\leq1-4e.
\end{equation*}
The last inequality follows from
$e^{-x}\leq(1+x)^{-1}\leq1-x/2$ for $0\leq x\leq1$.
The strong multiplier also satisfies
\begin{equation}
 \kappa^2e^{-\kappa/80}\leq12800,
 \qquad e^{-\kappa/80}L\leq\frac{12800}{\kappa}.
 \label{eq:core-strong-exponential-bound}
\end{equation}
The first estimate follows from $e^x\geq x^2/2$ with $x=\kappa/80$;
the second uses $L\leq\kappa$.

Proposition~\ref{prop:core-log-multiplier-derivatives} and
\eqref{eq:core-cone-size} now give
\begin{align*}
 \sqrt{\lambda_U}\sqrt u\,|\ell_U|
  &\leq3A_U\sqrt{\varepsilon_c}\,e^{-\kappa/80}L
    \leq\frac{38400A_U\sqrt{\varepsilon_c}}\kappa\leq e,\\
 \sqrt{\lambda_P}\sqrt w\,|\ell_P|
  &\leq\frac{3A_PL\sqrt{\varepsilon_c}}{\kappa^2}
    \leq\frac{3A_P\sqrt{\varepsilon_c}}\kappa\leq e.
\end{align*}
For either wall, the product derivative in
\eqref{eq:core-transverse-product-derivative} has the bound
\begin{equation}
 |\lambda(a_0+j\ell)|
 \leq\sqrt{j\lambda}
       \left(\sqrt\lambda+\sqrt\lambda\sqrt j\,|\ell|\right)
 \quad\text{if } |a_0|\leq\sqrt j, j\geq0.
 \label{eq:core-wall-cone-inequality}
\end{equation}
This follows by the triangle inequality and
$j=(\sqrt j)^2$; it includes $j=0$.
The parenthesis is at most $1-3e$. Finally,
\[
 1-3e\leq(1-2e)(1-e)\leq(1-2e)d_R.
\]
Use $1-2e=1-1/(16\kappa)$ and
\eqref{eq:core-multiplier-formula} to obtain
\eqref{eq:large-core-cone-margin}. Homogeneity gives the assertion
for every vector of the cone.
\end{proof}

The margin in~\eqref{eq:large-core-cone-margin} tends to zero with
$1/\kappa$, just as the weak-wall loss does. It is nevertheless
uniform relative to that loss: the same entrance threshold works for
arbitrarily large radii.

For a family almost tangent to the radial direction, the following
refinement retains its actual slope and measures the slope generated
from an initially radial family.

\begin{corollary}
\label{cor:large-core-narrow-cone}
Under the entrance assumptions of Proposition~\ref{prop:large-core-cone},
let $0\leq\theta\leq1$ and
\[
 v=(1,a,b),\qquad |a|\leq\theta\sqrt u,
                       \qquad |b|\leq\theta\sqrt w.
\]
Put
\[
 S=\frac{\sqrt{u+w}\log(4\kappa)}{\kappa^2},\qquad
 C_2=2(1+12800A_U+2A_P),\qquad d=D\mathcal R(Y_0)v.
\]
Then $d_R\geq3/4$, the radial estimate
\eqref{eq:core-phase-direction-error} holds, and
\begin{equation}
 |d_U|\leq C_2(\theta+S)\sqrt{U_*}\,d_R,\qquad
 |d_W|\leq C_2(\theta+S)\sqrt{W_*}\,d_R.
 \label{eq:large-core-narrow-cone}
\end{equation}
\end{corollary}

\begin{proof}
The direction belongs to the cone, so the radial conclusions follow
from Proposition~\ref{prop:large-core-cone} and
Corollary~\ref{cor:large-core-phase-direction}.
Write $\varepsilon=\varepsilon_c$, $L=\log(4\kappa)$,
$E=e^{-\kappa/80}$ and $Q_0=1+\kappa|a|+|b|$.
Keeping $\theta$ in the calculation of~\eqref{eq:core-cone-size}
gives
\begin{equation*}
 \sqrt u\,Q_0\leq\sqrt u+2\varepsilon\theta,\qquad
 \sqrt w\,Q_0\leq\sqrt w+2\varepsilon\theta.
\end{equation*}
The logarithmic derivative estimates therefore imply
\[
 \sqrt u\,|\ell_U|\leq A_UL(\sqrt u+2\varepsilon\theta),
 \qquad
 \sqrt w\,|\ell_P|\leq\frac{A_PL}{\kappa^2}
                                    (\sqrt w+2\varepsilon\theta).
\]
By~\eqref{eq:core-strong-exponential-bound},
\begin{align*}
 EL\sqrt u&\leq12800S,&
 2\varepsilon EL&\leq12800,\\
 L\sqrt w/\kappa^2&\leq S,&
 2\varepsilon L/\kappa^2&\leq2.
\end{align*}
For the second estimate in the first row, use
$2\varepsilon L\leq2\kappa\leq\kappa^2$.
Together with $\sqrt{\lambda_U}\leq E\leq1$ and
$\sqrt{\lambda_P}\leq1$, these bounds yield
\begin{align*}
 \sqrt{\lambda_U}\theta+
       \sqrt{\lambda_U}\sqrt u\,|\ell_U|
   &\leq(1+12800A_U)\theta+12800A_US,\\
 \sqrt{\lambda_P}\theta+
       \sqrt{\lambda_P}\sqrt w\,|\ell_P|
   &\leq(1+2A_P)\theta+A_PS.
\end{align*}
Both right-hand sides are at most
$(1+12800A_U+2A_P)(\theta+S)$.
The product estimate~\eqref{eq:core-wall-cone-inequality}, with its
first $\sqrt\lambda$ term replaced by $\theta\sqrt\lambda$, then
bounds $|d_U|$ and $|d_W|$ by this factor times $\sqrt{U_*}$ and
$\sqrt{W_*}$, respectively. Since $2d_R\geq1$, this is
\eqref{eq:large-core-narrow-cone}.
\end{proof}

For positive walls, division by $\sqrt{U_*}d_R$ and
$\sqrt{W_*}d_R$ expresses~\eqref{eq:large-core-narrow-cone}
as a bound on the normalized slopes of the image family. In
particular a radial tangent, for which $\theta=0$, acquires a slope
of order $\sqrt{u+w}\log(4\kappa)/\kappa^2$.