\subsection{Variation of the core return}
\label{sec:core-variation}

The passage estimates control the position of an orbit. To control the
measure carried by a family of orbits, we also need the derivative of
its phase at the exit. Two features of the core equations make this
possible. The strong transverse direction remains strongly contracting
throughout the passage, and the coefficients that change the radius
retain the small transverse amplitudes. We compare orbits at the same
angle $z$ and then account for the motion of the two section crossings.

Write the return from Proposition~\ref{prop:large-core-passage} as
\[
 \mathcal R(\kappa,u,w)=(R_*,U_*,W_*).
\]
Since $V=1$ at both crossings, the transverse coordinate $P=VW$
agrees with $W$ there. Thus both the entrance and the exit are
parametrized by the coordinates $(R,U,P)$.

\begin{proposition}[Derivative of the exit radius]
\label{prop:large-core-radius-derivative}
The first return time and $\mathcal R$ are locally $C^1$ at every
entrance with $\kappa\geq6$ and $0\leq u,w\leq\varepsilon_0/\kappa$.
There is an absolute constant $C_*$ such that, for every
$v=(v_\kappa,v_u,v_w)\in\R^3$,
\begin{equation}
 |D\mathcal R_R(\kappa,u,w)v-v_\kappa|
 \leq C_*\left\{\frac{|v_u|}{\kappa}
       +\frac{u+w}{\kappa}
          \bigl(|v_\kappa|+\kappa|v_u|+|v_w|\bigr)\right\}.
 \label{eq:large-core-radius-derivative}
\end{equation}
The assertion includes entrances at which either transverse wall is zero.
\end{proposition}

The dependence on the direction $v$ is useful. Variation of the strong
wall contributes at order $|v_u|/\kappa$, while all other contributions
contain a transverse amplitude. We first establish this estimate along
the passage, with the angle held fixed.

\subsubsection{Variation at a fixed angle}

Let $Y=(R,U,P)$ and denote the right-hand side of
\eqref{eq:core-angle-system} by $F(z,Y)$. Along a reference orbit,
let $\eta=\delta Y$ and $\zeta=\delta z$ be variations at fixed
original time. Set
\begin{equation}
 \xi=\eta-\zeta F(z,Y).
 \label{eq:fixed-angle-variation}
\end{equation}
This is the variation after the change in the angle has been removed.
It satisfies
\begin{equation}
 \frac{d\xi}{dz}=J(z,Y)\xi,\qquad J=D_YF.
 \label{eq:core-variational-equation}
\end{equation}
Indeed, with $q=dz/dt=ARd>0$, the original-time equations are
$dY/dt=qF$ and $dz/dt=q$. Their variations give
\[
 \frac{d\eta}{dt}=\delta q\,F+q(F_z\zeta+J\eta),\qquad
 \frac{d\zeta}{dt}=\delta q,\qquad
 \frac{dF}{dt}=q(F_z+JF).
\]
Substitution into~\eqref{eq:fixed-angle-variation} yields
$d\xi/dt=qJ\xi$, proving~\eqref{eq:core-variational-equation}.
The same calculation shows that adding a multiple of the flow vector
to a variation leaves $\xi$ unchanged. This will allow us to recover
the derivative at the moving exit.

For a fixed entrance radius $\kappa$, use the weighted variation
\begin{equation}
 X=S_\kappa\xi=(\xi_R,\kappa\xi_U,\xi_P),\qquad
 B=S_\kappa J S_\kappa^{-1}.
 \label{eq:core-weighted-variation}
\end{equation}
Here $\kappa$ is the radius of the reference entrance and is held
constant in the linear equation $X'=BX$. This weighting balances
the influence of the strong wall on the radius against its contraction.
All vector norms in this subsection are supremum norms.

\begin{lemma}[Coefficients of the variational equation]
\label{lem:core-variational-coefficients}
Throughout the region~\eqref{eq:core-closed-region}, put
$m=\varepsilon_0/\kappa$. The radius row satisfies
\begin{equation}
 |B_{RR}|\leq8m,\qquad
 \left|B_{RU}+\frac V\kappa\right|\leq\frac{8m}\kappa,
 \qquad |B_{RP}|\leq\frac{8m}{V}.
 \label{eq:core-radius-row-rough}
\end{equation}
The transverse rows satisfy
\begin{align}
 B_{UU}&\leq-\frac\kappa4,&
 |B_{UR}|&\leq2\kappa U,& |B_{UP}|&\leq4U,\notag\\
 B_{PP}&\leq-\frac1\kappa,&
 |B_{PR}|&\leq\frac{8P}{\kappa^2},&
 |B_{PU}|&\leq\frac{8P}{\kappa}.
 \label{eq:core-transverse-rows}
\end{align}
The following refinement keeps the amplitudes in the radius row:
\begin{align}
 |B_{RR}|&\leq8\left(U+\frac{W^2}{\kappa^2}\right),\notag\\
 \left|B_{RU}+\frac V\kappa\right|
   &\leq\frac{8(U+W)}\kappa,\qquad
 |B_{RP}|\leq\frac8V\left(U+\frac W\kappa\right).
 \label{eq:core-radius-row-amplitudes}
\end{align}
\end{lemma}

\begin{proof}
We give the coefficient calculations, since the negative diagonal
terms and the powers of the small walls will both be used.
At a fixed angle put $s=\operatorname{sech}z$, $c=-\tanh z$ and
$\delta=U-W$. Then $V=Rs$, $W=P/V$, and
\[
 a=-\frac R2-c+sW+\frac{U^2-W^2}{2R},\qquad
 g=-C+\frac{c\delta^2}{R},\qquad
 F=\left(\frac gd,\frac{Ua}d,-\frac{2P(1+C)}{Rd}\right).
\]
In particular $W_R=-W/R$ and $W_P=1/V$. Differentiation gives
\begin{align*}
 C_R&=U(s-W/R),& C_U&=V-2U+W,& C_P&=U/V,\\
 d_R&=2\delta(U-2W)/R^3,& d_U&=-2\delta/R^2,&
 d_P&=2\delta/(VR^2),\\
 a_R&=-\tfrac12-sW/R+(3W^2-U^2)/(2R^2),&
 a_U&=U/R,& a_P&=(1-W/V)/R.
\end{align*}
For the radius numerator,
\begin{align*}
 g_R&=-U(s-W/R)+c\delta(3W-U)/R^2,\\
 g_U&=-V+2U-W+2c\delta/R,\qquad
 g_P=-U/V-2c\delta/(RV).
\end{align*}
We use $R\geq5$, $1\leq V\leq R$, $0\leq U,W\leq m$,
$Rm\leq1/1000$, and~\eqref{eq:core-elementary-bounds}.
These give
\begin{gather}
 0\leq C\leq\frac1{100},\quad |a|\leq R,\quad
 |g|\leq2Rm,\quad |C_R|\leq U,\quad |C_U|\leq2R,\notag\\
 |d_R|\leq\frac m{R^2},\quad |d_U|\leq\frac{2m}{R^2},\quad
 |d_P|\leq\frac{2m}{VR^2},\quad
 |a_R|\leq\frac34,\quad |a_P|\leq\frac1R.
 \label{eq:core-auxiliary-bounds}
\end{gather}
For example $|\delta|\leq m$, $|U-2W|\leq2m$ and
$4m/R\leq1$ give the first bound for the derivatives of $d$.
The remaining bounds follow directly from the displayed derivatives.
In applying the quotient rule we shall use
\begin{equation}
 \left|\frac{x}{d}-\frac{y}{d^2}\right|
 \leq\frac32|x|+2|y|,
 \qquad d\geq\frac{999}{1000}.
 \label{eq:core-quotient-estimate}
\end{equation}

For the rough radius bounds, the numerator derivatives obey
\[
 |g_R|\leq2m,\qquad |g_U+V|\leq3m,\qquad |Vg_P|\leq2m.
\]
Also $V(1-d)\leq m$, and each of
$|gd_R|$, $|gd_U|$, $|Vgd_P|$ is at most $m/10$.
The radius row consists of
\[
 B_{RR}=\frac{g_R}{d}-\frac{gd_R}{d^2},\qquad
 B_{RU}=\frac1\kappa\left(\frac{g_U}{d}-\frac{gd_U}{d^2}\right),
 \qquad B_{RP}=\frac{g_P}{d}-\frac{gd_P}{d^2}.
\]
For its middle entry, combine $g_U+Vd=g_U+V-V(1-d)$ before
using~\eqref{eq:core-quotient-estimate}. This proves
\eqref{eq:core-radius-row-rough}.

The two diagonal entries have the exact forms
\begin{align}
 B_{UU}&=
   a\,\frac{R^2+U^2-W^2}{R^2d^2}+\frac{U^2}{Rd},\notag\\
 B_{PP}&=
   -\frac{2(1+C)}R\,
       \frac{R^2-U^2+W^2}{R^2d^2}-\frac{2UW}{Rd}.
 \label{eq:core-signed-diagonals}
\end{align}
Both fractions with denominator $R^2d^2$ are at least $99/100$.
Since $a\leq-3\kappa/10$, $U^2/(Rd)\leq1/100$, and
$R\leq101\kappa/100$, the asserted negative diagonal bounds follow.
For the off-diagonal entries of the strong row, the quotient rule gives
\[
 B_{UR}=\kappa U\left(\frac{a_R}{d}-\frac{ad_R}{d^2}\right),
 \qquad
 B_{UP}=\kappa U\left(\frac{a_P}{d}-\frac{ad_P}{d^2}\right).
\]
The two parenthesized expressions have absolute values at most $2$
and $2/R$, respectively, by~\eqref{eq:core-auxiliary-bounds} and
\eqref{eq:core-quotient-estimate}. Use $\kappa/R\leq3/2$.
For the weak row, write
\begin{align*}
 B_{PR}&=\frac P{R^2}\left\{
       \frac{2(1+C)-2RC_R}{d}
       +\frac{2(1+C)Rd_R}{d^2}\right\},\\
 B_{PU}&=\frac P\kappa\left\{
       -\frac{2C_U}{Rd}+\frac{2(1+C)d_U}{Rd^2}\right\}.
\end{align*}
The numerator bounds for the two terms in braces are respectively
$(3,1/10)$ and $(4,1/10)$. Applying
\eqref{eq:core-quotient-estimate} and
$\kappa^2/R^2\leq3/2$ proves~\eqref{eq:core-transverse-rows}.

To retain the amplitudes in the radius row, use
\[
 |\delta|\,|U-2W|\leq4mU+2W^2,\qquad
 |\delta|\,|3W-U|\leq5mU+3W^2.
\]
These inequalities follow by replacing the absolute differences by
$U+W$, $U+2W$ and $U+3W$, and then using $U^2,UW\leq mU$.
They give the sharper bounds
\begin{align*}
 |g_R|&\leq2U+3W^2/R^2,&
 |g_U+Vd|&\leq4(U+W),&
 |Vg_P|&\leq2(U+W/R),\\
 |gd_R|&\leq(U+W^2/R^2)/10,&
 |gd_U|&\leq(U+W/R)/10,&
 |Vgd_P|&\leq(U+W/R)/10.
\end{align*}
For the middle numerator, $\delta^2\leq m(U+W)$ controls
$V(1-d)$. For the products with derivatives of $d$, multiply the
two preceding quadratic bounds by $|g|\leq2Rm$; in particular
\[
 |gd_R|\leq\frac{16m^2}{R^2}U+
                         8m\frac{W^2}{R^2},\qquad
 |gd_U|\leq\frac{4m}{R}U+4m\frac WR.
\]
Their coefficients are at most $1/10$, and $Vd_P=-d_U$.
One last application of~\eqref{eq:core-quotient-estimate}, with
$\kappa/R,\kappa^2/R^2\leq3/2$, proves
\eqref{eq:core-radius-row-amplitudes}.
\end{proof}

The negative entries in~\eqref{eq:core-signed-diagonals} prevent a
large growth factor. More precisely, the sum of the diagonal entry
and the absolute values of the off-diagonal entries in each row obeys
\begin{align}
 B_{RR}+|B_{RU}|+|B_{RP}|&\leq2\operatorname{sech}z+24m,\notag\\
 |B_{UR}|+B_{UU}+|B_{UP}|&\leq0,\notag\\
 |B_{PR}|+|B_{PU}|+B_{PP}&\leq0.
 \label{eq:core-signed-row-bounds}
\end{align}
For the first row use $V/\kappa\leq2\operatorname{sech}z$ and
$V\geq1$. The other two follow from $U,P\leq m$ and
\eqref{eq:core-transverse-rows}.

\begin{lemma}[Uniform growth of the weighted variation]
\label{lem:core-variation-growth}
On every interval $[\alpha,\beta]$ in the core region,
every solution of~\eqref{eq:core-variational-equation} satisfies
\begin{equation}
 \|S_\kappa\xi(\beta)\|_\infty
 \leq K\|S_\kappa\xi(\alpha)\|_\infty,
 \qquad K=\exp(2\pi+48\varepsilon_0).
 \label{eq:core-variation-growth}
\end{equation}
\end{lemma}

\begin{proof}
If the signed row sums of a matrix $B$ are at most $\rho$, then for
sufficiently small $h\geq0$,
\[
 \|X+hBX\|_\infty\leq(1+h\rho)\|X\|_\infty.
\]
Indeed $1+hB_{ii}\geq0$ for every $i$, so the diagonal term can be
estimated with its sign intact. Applied to $X'=BX$, this inequality
and the first-order remainder give the upper right slope bound
$D^+\|X\|_\infty\leq\rho\|X\|_\infty$.
Multiplication by $\exp(-\int\rho)$ and scalar comparison imply
\[
 \|X(\beta)\|_\infty
 \leq\|X(\alpha)\|_\infty
       \exp\left(\int_\alpha^\beta\rho(z)\dd z\right).
\]
Take $\rho(z)=2\operatorname{sech}z+24m$ in
\eqref{eq:core-signed-row-bounds}. A primitive is
\[
 H(z)=2\arctan(\sinh z)+24mz.
\]
By~\eqref{eq:core-angle-range},
\[
 H(\beta)-H(\alpha)
 \leq2\pi+48m\log(4\kappa)
 \leq2\pi+48\varepsilon_0.
\]
The last inequality uses $\log(4\kappa)\leq\kappa$ for
$\kappa\geq6$, which follows from
$e^\kappa\geq1+\kappa+\kappa^2/2\geq4\kappa$.
\end{proof}

\subsubsection{Integrated decay and the radius component}

The uniform bound controls the entire variation. The radius estimate
requires more information about the strong component, whose forcing
decays together with $U$. Let $\alpha$ be the entrance angle, so that
$\cosh\alpha=\kappa$, and suppose $\|X(z)\|_\infty\leq Q$
on an initial segment $[\alpha,\beta]$. Write $x=X_U$ and
$x_0=x(\alpha)$. By~\eqref{eq:core-transverse-rows},
\[
 x'=B_{UU}x+h,\qquad B_{UU}\leq-\kappa/4,\qquad
 |h|\leq3\kappa UQ.
\]
Multiply this equation by $e^{\kappa(z-\alpha)/4}$.
The new damping coefficient is $B_{UU}+\kappa/4\leq0$, and the
absolute value of its forcing is at most $3\kappa uQ$, since
$U(z)\leq u e^{-\kappa(z-\alpha)/4}$.
The scalar comparison argument used above therefore gives
\begin{equation}
 |x(z)|\leq e^{-\kappa(z-\alpha)/4}
       \bigl(|x_0|+3\kappa uQ(z-\alpha)\bigr).
 \label{eq:core-strong-variation}
\end{equation}
There is no division by $U$ in this estimate.

Two integrals of~\eqref{eq:core-strong-variation} will be used:
\begin{align}
 \int_\alpha^\beta\frac{|x(z)|}{\cosh z}\dd z
   &\leq\frac{12|x_0|+432uQ}{\kappa^2},\notag\\
 \int_\alpha^\beta|x(z)|\dd z
   &\leq\frac{4|x_0|+48uQ}{\kappa}.
 \label{eq:core-strong-variation-integrals}
\end{align}
For the first, use
$\operatorname{sech}(\alpha+s)\leq e^s/\kappa$ and
$\kappa/4-1\geq\kappa/12$. Both estimates then follow from
\[
 \int_0^L e^{-cs}(A+Bs)\dd s\leq\frac Ac+\frac B{c^2}
 \qquad(A,B,L\geq0,\ c>0).
\]
The same endpoint estimate for $\cosh^2$ used in
Lemma~\ref{lem:core-radius-drift} gives, with $\gamma=(101/99)^2$,
\begin{align}
 \int_\alpha^\beta U\dd z&\leq\frac{4u}{\kappa},&
 \int_\alpha^\beta\frac1{V^2}\dd z&\leq2\gamma,\notag\\
 \int_\alpha^\beta\frac W{\kappa V}\dd z
   &\leq\frac{2\gamma w}{\kappa},&
 \int_\alpha^\beta W^2\dd z&\leq2\gamma w^2.
 \label{eq:core-amplitude-integrals}
\end{align}
Here $P(z)\leq w$, $R\geq99\kappa/100$, and the two endpoint
values of $\cosh z$ are at most $101\kappa/100$. These facts prove
the inverse-wall integral; the last two bounds follow from $W=P/V$.

\begin{lemma}[Drift of a radial variation]
\label{lem:core-radial-variation}
Under the preceding assumptions,
\begin{equation}
 |\xi_R(\beta)-\xi_R(\alpha)|
 \leq256\left\{\frac{|x_0|}{\kappa^2}
                         +\frac{(u+w)Q}{\kappa}\right\}.
 \label{eq:core-radial-variation}
\end{equation}
\end{lemma}

\begin{proof}
The refined radius row~\eqref{eq:core-radius-row-amplitudes} yields
\begin{equation}
 |\xi_R'|
 \leq\left(16U+\frac{8W^2}{\kappa^2}
                         +\frac{8W}{\kappa V}\right)Q
      +\frac{2|x|}{\cosh z}+\frac{|x|}{\kappa}.
 \label{eq:core-radial-variation-integrand}
\end{equation}
Indeed $V/\kappa\leq2\operatorname{sech}z$, $1/V\leq1$, and
$8(U+W)\leq1$. Integrating with
\eqref{eq:core-strong-variation-integrals} and
\eqref{eq:core-amplitude-integrals} bounds the result by
\[
 \frac{28|x_0|}{\kappa^2}
 +\left(\frac{64}{\kappa}+\frac{912}{\kappa^2}\right)uQ
 +\frac{16\gamma w^2Q}{\kappa^2}
 +\frac{16\gamma wQ}{\kappa}.
\]
Use $\kappa\geq6$, $w\leq\kappa$ and $\gamma<2$ to obtain
\eqref{eq:core-radial-variation}.
\end{proof}

\subsubsection{The two moving crossings}

We now identify the derivatives just estimated with derivatives of
the actual first return. The vector field and coordinate changes are
smooth on a neighborhood of the compact crossing segment. At the
exit $dV/dt<0$, so the implicit function theorem gives a locally $C^1$
crossing time near $t_*$. This crossing remains the first positive
return. To see this, choose small intervals about $0$ and $t_*$ on
which the corresponding crossings are unique by transversality.
On the intervening compact interval the reference solution has
$V>1$, uniformly separated from $1$. Continuous dependence preserves
this separation for nearby entrance data. At the entrance the
crossing time stays exactly zero because the incoming section is
$V=1$. Thus no earlier positive crossing can appear.
The same compactness argument preserves the flow domain and $A>0$
up to the moving return time. It also supplies a local $C^1$
extension when an entrance wall is zero.

For the entrance parametrization~\eqref{eq:large-core-entrance},
$Y=(\kappa,u,w)$ and $z=\alpha=-\operatorname{arcosh}\kappa$.
Consequently the initial value of the variation at fixed angle is
\begin{equation}
 \xi(\alpha)=v+\frac{v_\kappa}{\sqrt{\kappa^2-1}}
                               F(\alpha,\kappa,u,w).
 \label{eq:core-incoming-variation}
\end{equation}
At the entrance, the equations imply
\[
 |F_R|\leq2(u+w^2/\kappa),\qquad
 |F_U|\leq2\kappa u,\qquad |F_P|\leq4w/\kappa.
\]
The first bound follows from~\eqref{eq:core-radius-integrand} with
$V=1$; the other two follow from $|a|\leq\kappa$,
$C\leq1/100$ and $d\geq999/1000$.
Since $\sqrt{\kappa^2-1}\geq\kappa/2$,
\begin{align}
 |\xi_R(\alpha)-v_\kappa|
   &\leq4(u/\kappa+w^2/\kappa^2)|v_\kappa|,\notag\\
 |\kappa\xi_U(\alpha)|
   &\leq\kappa|v_u|+4\kappa u|v_\kappa|,\notag\\
 |\xi_P(\alpha)-v_w|
   &\leq8w|v_\kappa|/\kappa^2.
 \label{eq:core-incoming-bounds}
\end{align}
In particular, if
\[
 Q_0=|v_\kappa|+\kappa|v_u|+|v_w|,
\]
then $\|S_\kappa\xi(\alpha)\|_\infty\leq2Q_0$.
Lemma~\ref{lem:core-variation-growth} therefore gives the uniform
bound $Q=2KQ_0$ for use in~\eqref{eq:core-radial-variation}.

At the exit put $r=R_*$, $c_*=-\tanh z_*$ and
$f_*=F_R(z_*,Y_*)$. A variation tangent to $V=1$ satisfies
\[
 \delta R+r c_*\delta z=0,
 \qquad \xi_R=\delta R-f_*\delta z.
\]
The first identity is the differential of $V=R/\cosh z$ on that
section. Eliminating $\delta z$ gives
\begin{equation}
 D\mathcal R_Rv=\sigma\,\xi_R(z_*),\qquad
 \sigma=\frac{c_*}{c_*+f_*/r}.
 \label{eq:core-exit-projection}
\end{equation}
The change in the exit time adds a multiple of the flow vector and
hence does not change the projected variation in
\eqref{eq:fixed-angle-variation}.

At this crossing $c_*\leq-1/2$, $C\geq0$ and $\tanh z_*>0$.
Thus $f_*\leq0$, and
\[
 |f_*|\leq2(U_*+W_*^2/r),\qquad
 0<\sigma\leq1,\qquad
 |\sigma-1|\leq\frac{2|f_*|}{r}
                    \leq8(u/\kappa+w^2/\kappa^2).
\]
The last estimate uses $U_*\leq u$, $W_*\leq w$ and
$r\geq99\kappa/100$. The exit projection therefore preserves the
amplitude dependence of the interior estimate.

\begin{proof}[Proof of Proposition~\ref{prop:large-core-radius-derivative}]
Local $C^1$ regularity and first-return persistence were proved above.
Since $0<\sigma\leq1$, equation~\eqref{eq:core-exit-projection} gives
\begin{align*}
 |D\mathcal R_Rv-v_\kappa|
 &\leq |\xi_R(z_*)-\xi_R(\alpha)|
       +|\xi_R(\alpha)-v_\kappa|
       +|\sigma-1|\,|v_\kappa|\\
 &\leq256\left\{\frac{|\kappa\xi_U(\alpha)|}{\kappa^2}
                    +\frac{(u+w)2KQ_0}{\kappa}\right\}
       +12(u/\kappa+w^2/\kappa^2)|v_\kappa|.
\end{align*}
Use~\eqref{eq:core-incoming-bounds}, $|v_\kappa|\leq Q_0$ and
$w^2/\kappa^2\leq w/\kappa$. The right-hand side is at most
\[
 \frac{256|v_u|}{\kappa}
       +(1040+512K)\frac{(u+w)Q_0}{\kappa}.
\]
Taking $C_*=1040+512K$ proves the proposition.
\end{proof}

The preceding estimate becomes small along families whose tangent
directions satisfy
\begin{equation}
 v_\kappa=1,\qquad |v_u|\leq\sqrt u,\qquad |v_w|\leq\sqrt w.
 \label{eq:core-square-root-directions}
\end{equation}
This choice permits unequal wall sizes and allows either wall to vanish.

\begin{corollary}
\label{cor:large-core-phase-direction}
For every entrance in Proposition~\ref{prop:large-core-radius-derivative}
and every direction satisfying~\eqref{eq:core-square-root-directions},
\begin{equation}
 |D\mathcal R_Rv-1|\leq C_1\frac{\sqrt{u+w}}\kappa,
 \qquad C_1=5C_*.
 \label{eq:core-phase-direction-error}
\end{equation}
There is an absolute $\varepsilon_1\in(0,\varepsilon_0]$ such that
if $0\leq u,w\leq\varepsilon_1/\kappa$, then
\begin{equation}
 \frac34\leq D\mathcal R_Rv\leq\frac54,\qquad
 |\log(D\mathcal R_Rv)|\leq2C_1\frac{\sqrt{u+w}}\kappa.
 \label{eq:core-phase-logarithm}
\end{equation}
\end{corollary}

\begin{proof}
Put $b=u+w$. Then $b\leq1$ and $\kappa b\leq2\varepsilon_0\leq2$.
Hence
\[
 b(1+\kappa|v_u|+|v_w|)\leq4\sqrt b,\qquad
 |v_u|\leq\sqrt b.
\]
Substitution into~\eqref{eq:large-core-radius-derivative} proves
\eqref{eq:core-phase-direction-error}. Choose
$\varepsilon_1\leq\min\{\varepsilon_0,(4C_1)^{-2}\}$.
Since $b\leq\varepsilon_1$, the error in
\eqref{eq:core-phase-direction-error} is at most $1/4$.
Finally, if $|t-1|\leq e\leq1/4$, then $t\geq1/2$ and
$1/t\leq1+2e$. The elementary inequalities
$1-1/t\leq\log t\leq t-1$ give $|\log t|\leq2e$.
Apply this with $t=D\mathcal R_Rv$ and
$e=C_1\sqrt b/\kappa$.
\end{proof}

The two scales now agree with the losses from
Lemma~\ref{lem:amplitude-loss}: phase displacement is bounded by
a constant times $(u+w)/\kappa$, and logarithmic phase distortion
by a constant times $\sqrt{u+w}/\kappa$. Their sums over many cores
can therefore be controlled by the decrease in the corresponding
powers of the transverse amplitude. To apply this observation along
successive returns, the transverse derivative must also preserve the
directions~\eqref{eq:core-square-root-directions} after normalization
by the new radial component. This requires the derivative estimates
for the two wall multipliers.