\subsection{Nonlinear transport from the small section}
\label{sec:small-section-nonlinear}

The strong wall decays exponentially along the reference orbit.
In logarithmic strong-wall coordinates, the derivative of the
vector field near this orbit is controlled by the decaying wall
and by the size of the three transverse coordinates.
Its time integral can therefore remain small on an
interval whose length grows logarithmically as the initial
transverse size tends to zero. We use this observation to compare
the nonlinear flow with the transfer map $\mathcal T(h)$.
All finite-dimensional spaces in this subsection carry the maximum
norm and its induced operator norms.

\subsubsection{Coordinates and integrated growth}

For $X=(n,s,n_2,n_3)$, put
\[
 \zeta(X)=\sqrt{1-s^2-Q(n,n_2,n_3)}.
\]
The following map records the logarithmic rate of the strong wall
and the three transverse velocities:
\begin{equation}
 \mathcal V(X)=
 \begin{pmatrix}
 Q-1-2\zeta\\
 Qs+\sqrt3(n-n_2-n_3)(n_2-n_3)\\
 n_2(Q-1+\zeta+\sqrt3s)\\
 n_3(Q-1+\zeta-\sqrt3s)
 \end{pmatrix}.
 \label{eq:small-section-rate-map}
\end{equation}
Here $Q=Q(n,n_2,n_3)$ and $\zeta=\zeta(X)$.
The map is smooth on a neighborhood of the signed box
$\|X\|\leq1/16$, by \eqref{eq:small-section-box}, and
\[
 \mathcal V(0)=(-3,0,0,0),\qquad D\mathcal V(0)=0.
\]
The second identity follows directly from
\eqref{eq:small-section-rate-map}: $Q$ and $1-\zeta$ have
vanishing first derivatives at zero, the second row has no
linear term, and each of the last two rows is a coordinate
times a function that vanishes at zero. Compactness and the
mean value theorem now give a constant $B\geq1$ such that
\begin{equation}
 \|D\mathcal V(X)\|\leq B\|X\|,\qquad
 \|D^2\mathcal V(X)\|\leq B
 \quad(\|X\|\leq1/16).
 \label{eq:small-section-compact-derivatives}
\end{equation}

Use the logarithmic coordinate $\ell=\log n$ when $n>0$, and set
\[
 E(\ell,s,n_2,n_3)=(e^\ell,s,n_2,n_3),\qquad
 \mathcal W=\mathcal V\circ E.
\]
The corresponding domain
\[
 \mathcal D=\{(\ell,s,n_2,n_3):
       \ell\leq\log(1/16),\
       |s|,|n_2|,|n_3|\leq1/16\}
\]
is convex. On this domain $\|DE\|\leq1$ and $\|D^2E\|\leq1$.
The chain rule and
\eqref{eq:small-section-compact-derivatives} therefore give a
constant $C\geq32$ such that
\begin{align}
 \|D\mathcal W(y)\|&\leq C\|E(y)\|,
       &\|D^2\mathcal W(y)\|&\leq C,
       \label{eq:small-section-log-derivatives}\\
 \|D\mathcal W(y)-D\mathcal W(\widetilde y)\|
       &\leq C\|y-\widetilde y\|
       &&(y,\widetilde y\in\mathcal D).
       \label{eq:small-section-log-lipschitz}
\end{align}
Choose $C$ also to dominate $B$, and fix it throughout the
section. The second line follows by applying the mean value
theorem to $D\mathcal W$ along the line segment in $\mathcal D$.
The unbounded range of $\ell$ causes no loss in these estimates.

We will repeatedly use the following norm comparison. If
$\Gamma(0)=0$, $\Gamma(t)\geq0$, $\varepsilon\geq0$, and
differentiable functions satisfy
\[
 \|f'(t)\|\leq\Gamma'(t)\|f(t)\|+\varepsilon
 \quad(0\leq t\leq T),
\]
then
\begin{equation}
 \|f(t)\|\leq
       \bigl(\|f(0)\|+\varepsilon t\bigr)e^{\Gamma(t)}.
 \label{eq:small-section-norm-comparison}
\end{equation}
Indeed, for $\delta>0$ compare $\|f(t)\|$ with
\[
 B_\delta(t)=
   \bigl(\|f(0)\|+\delta+\varepsilon t\bigr)
                 e^{\Gamma(t)+\delta t}.
\]
Its initial value is strictly larger, and at any contact
with $\|f(t)\|$ it satisfies
$B_\delta'(t)>\Gamma'(t)B_\delta(t)+\varepsilon$.
The norm comparison principle gives $\|f(t)\|\leq B_\delta(t)$;
letting $\delta$ decrease to zero proves
\eqref{eq:small-section-norm-comparison}. This argument uses
the bound on $\|f'\|$ and remains valid at zeros of $f$.

The signed coordinates used for ordinary differentiation also
have forward solutions. To see this, the multiplicative wall
equations in the proof of Lemma~\ref{lem:forward} give
\[
 |N_i(t)|\leq |N_i(0)|e^t
\]
for any solution on $G=1$, with no sign restriction on the
walls. If $|N_i|\leq M$, the identity
$Q=2\sum_iN_i^2-(\sum_iN_i)^2$ and the constraint give
$p^2+s^2\leq1+9M^2$.
Thus the full state has norm at most $1+3M$.
The same continuation argument proves forward existence,
and every initially positive wall remains positive.

For $h,b>0$, define
\begin{equation}
 \Gamma(t)=C\left(\frac h2(1-e^{-2t})+bt\right),
 \qquad
 \Gamma'(t)=C(he^{-2t}+b).
 \label{eq:small-section-growth-integral}
\end{equation}
The conditions used below are
\begin{equation}
 Ch\leq\frac14,\qquad
 0<b\leq\frac1{16},\qquad
 T\geq0,\qquad CbT\leq\frac18.
 \label{eq:small-section-smallness}
\end{equation}
They imply $h\leq1/128$ and
$0\leq\Gamma(t)\leq1/4$ for $0\leq t\leq T$.

\begin{proposition}
\label{prop:small-section-orbit-control}
Suppose $h>0$ and \eqref{eq:small-section-smallness} holds.
Let $x=(p,s,N_1,N_2,N_3)$ satisfy $G(x)=1$, $p<0$,
$0<N_1\leq h$, and $\|\Pi x\|\leq b/2$.
For $0\leq t\leq T$, the actual orbit satisfies
\begin{equation}
 \begin{split}
 0<N_1(\phi_t x)&\leq he^{-2t},
       \qquad p(\phi_t x)\leq-\frac{15}{16},\\
 \|\Pi\phi_t x\|&\leq
       \|\Pi x\|e^{\Gamma(t)}<b.
 \end{split}
 \label{eq:small-section-orbit-control}
\end{equation}
In particular these conclusions hold for $x=\Xi_h(Z)$ with
$\|Z\|\leq b/2$.
\end{proposition}

\begin{proof}
First consider a time interval on which
\begin{equation}
 p\leq-\frac12,\qquad N_1\leq2h,\qquad
 \|\Pi\phi_t x\|\leq b.
 \label{eq:small-section-control-region}
\end{equation}
Positivity of $N_1$ and $2h\leq1/16$ place the four physical
coordinates in the signed box. The constraint selects its
negative root, so $p\leq-15/16$, and
\eqref{eq:small-section-transverse} gives
$N_1'/N_1\leq-2$. Consequently $N_1(t)\leq he^{-2t}$.

Write $Z(t)=\Pi\phi_t x$ and $n=N_1(t)$.
The last three rows of $\mathcal V(n,0)$ vanish.
Along the segment from $(n,0)$ to $(n,Z(t))$,
\eqref{eq:small-section-compact-derivatives} gives
\[
 \|Z'(t)\|\leq C\max(n,\|Z(t)\|)\|Z(t)\|
      \leq C(n+\|Z(t)\|)\|Z(t)\|.
\]
Thus $\|Z'(t)\|\leq\Gamma'(t)\|Z(t)\|$ on the interval
under consideration. Equation~\eqref{eq:small-section-norm-comparison}
with $\varepsilon=0$ yields the last bound in
\eqref{eq:small-section-orbit-control}, including its strict
inequality because
\[
 \|Z(0)\|e^{\Gamma(t)}\leq\frac b2 e^{1/4}<b.
\]

Initially all three inequalities in
\eqref{eq:small-section-control-region} are strict: the
constraint and the box estimate give $p\leq-15/16$,
while $N_1\leq h<2h$ and $\|Z(0)\|\leq b/2<b$.
On every closed interval where they hold, the preceding
estimates improve them strictly. Continuity therefore
extends such an interval past its right endpoint whenever
that endpoint is smaller than $T$. Taking the supremum of
these intervals proves
\eqref{eq:small-section-control-region} throughout $[0,T]$,
and hence \eqref{eq:small-section-orbit-control}.
The chart $\Xi_h(Z)$ satisfies the required initial conditions
by \eqref{eq:small-section-box}.
\end{proof}

\subsubsection{Ordinary initial derivatives}

For $y=(\ell,s,n_2,n_3)$, let
\[
 L(p,s,N_1,N_2,N_3)=(\log N_1,s,N_2,N_3),\qquad
 \widehat\Xi(y)=(-\zeta(E(y)),s,e^\ell,n_2,n_3).
\]
On the open domain where the initial radicand is positive,
the actual flow exists, and its endpoint has $p<0$ and $N_1>0$,
define
\[
 \Psi_t(y)=L\bigl(\phi_t\widehat\Xi(y)\bigr).
\]
The flow, the chart, and $L$ are jointly $C^2$ there.
The constraint and \eqref{eq:normalized-flow} give the local
identity
\begin{equation}
 \partial_t\Psi_t(y)=\mathcal W(\Psi_t(y)),
 \qquad \Psi_0(y)=y.
 \label{eq:small-section-mixed-flow}
\end{equation}
Because this identity holds on an open neighborhood,
differentiation in $y$ gives the ordinary variational equation
\begin{equation}
 J_v'(t)=D\mathcal W(\Psi_t(y))J_v(t),
 \qquad
 J_v(t)=D_y\Psi_t(y)v,\qquad J_v(0)=v.
 \label{eq:small-section-mixed-variation}
\end{equation}

For initial data on the fixed section, write
\begin{equation}
 U_{h,t}(Z)=\Psi_t(\log h,Z),\qquad
 H_{h,t}(Z)=\Pi\phi_t\Xi_h(Z).
 \label{eq:small-section-nonlinear-maps}
\end{equation}
The last three components of $U_{h,t}$ are $H_{h,t}$.
Their initial values are $(\log h,Z)$ and $Z$ respectively.

\begin{proposition}
\label{prop:small-section-variation-control}
Under \eqref{eq:small-section-smallness}, with $h>0$,
the maps in \eqref{eq:small-section-nonlinear-maps} are
differentiable at every $\|Z\|\leq b/2$ for
$0\leq t\leq T$. For $\|Z\|,\|W\|\leq b/2$,
\begin{align}
 \|D U_{h,t}(Z)\|&\leq e^{\Gamma(t)}\leq e^{1/4}<2,
       \label{eq:small-section-variation-size}\\
 \|U_{h,t}(Z)-U_{h,t}(W)\|
       &\leq e^{1/4}\|Z-W\|,
       \label{eq:small-section-flow-comparison}\\
 \|D U_{h,t}(Z)-D U_{h,t}(W)\|
       &\leq8Ct\|Z-W\|.
       \label{eq:small-section-variation-comparison}
\end{align}
The derivative comparison
\eqref{eq:small-section-variation-comparison} also holds
with $H_{h,t}$ in place of $U_{h,t}$.
\end{proposition}

\begin{proof}
Proposition~\ref{prop:small-section-orbit-control} places each
orbit in the open regularity domain of $\Psi$ and gives
\[
 \|E(U_{h,t}(Z))\|\leq he^{-2t}+b,\qquad
 U_{h,t}(Z)\in\mathcal D.
\]
It follows from \eqref{eq:small-section-log-derivatives}
that the coefficient in
\eqref{eq:small-section-mixed-variation} has norm at most
$\Gamma'(t)$. The inclusion $v\mapsto(0,v)$ has norm one,
so \eqref{eq:small-section-norm-comparison} proves
\eqref{eq:small-section-variation-size}. The mean value
theorem on the convex ball $\|Z\|\leq b/2$ gives
\eqref{eq:small-section-flow-comparison}. Ordinary
differentiability at its boundary follows from the same
open regularity domain.

For the final estimate, fix $v\in\R^3$ and put
\[
 J_Z(t)=D U_{h,t}(Z)v,\quad
 J_W(t)=D U_{h,t}(W)v,\quad
 A_Z(t)=D\mathcal W(U_{h,t}(Z)).
\]
Define $A_W$ in the same way. The difference of the two
variational equations is
\[
 (J_Z-J_W)'=A_Z(J_Z-J_W)+(A_Z-A_W)J_W.
\]
Equations \eqref{eq:small-section-log-lipschitz},
\eqref{eq:small-section-flow-comparison}, and
\eqref{eq:small-section-variation-size} imply
\[
 \|A_Z-A_W\|\leq2C\|Z-W\|,\qquad
 \|J_W(t)\|\leq2\|v\|.
\]
The difference starts at zero. Applying
\eqref{eq:small-section-norm-comparison} with
$\varepsilon=4C\|Z-W\|\|v\|$ therefore gives
\[
 \|J_Z(t)-J_W(t)\|
   \leq4Ct\,e^{\Gamma(t)}\|Z-W\|\|v\|
   \leq8Ct\|Z-W\|\|v\|.
\]
Taking the operator norm proves
\eqref{eq:small-section-variation-comparison}.
Projection onto the last three coordinates has norm one,
which gives the assertion for $H_{h,t}$.
\end{proof}

The reference orbit has zero transverse coordinates, so
$H_{h,t}(0)=0$. Its initial derivative is precisely the
map $Y_h(t)$ of \eqref{eq:small-section-flow-differential}.
Define the remainder relative to the limiting transfer by
\begin{equation}
 R_{h,t}(Z)=H_{h,t}(Z)-\mathcal T(h)Z.
 \label{eq:small-section-remainder}
\end{equation}

\begin{proposition}
\label{prop:small-section-nonlinear-remainder}
For $h>0$ satisfying \eqref{eq:small-section-smallness},
$0\leq t\leq T$, and $\|Z\|\leq b/2$, the remainder is
differentiable at $Z$ and satisfies
\begin{align}
 \|R_{h,t}(Z)\|
      &\leq4Ct\|Z\|^2+6he^{-2t}\|Z\|,
      \label{eq:small-section-remainder-value}\\
 \|D R_{h,t}(Z)\|
      &\leq8Ct\|Z\|+6he^{-2t}.
      \label{eq:small-section-remainder-derivative}
\end{align}
\end{proposition}

\begin{proof}
The derivative splits as
\[
 D R_{h,t}(Z)=
   D H_{h,t}(Z)-D H_{h,t}(0)
      +Y_h(t)-\mathcal T(h).
\]
Proposition~\ref{prop:small-section-variation-control}
and \eqref{eq:small-section-reference-tail} prove
\eqref{eq:small-section-remainder-derivative}.
To recover the value, set $g(\theta)=R_{h,t}(\theta Z)$
for $0\leq\theta\leq1$. Then $g(0)=0$ and
\[
 \|g'(\theta)\|
   \leq8Ct\theta\|Z\|^2+6he^{-2t}\|Z\|.
\]
Norm comparison with the scalar polynomial
$4Ct\theta^2\|Z\|^2+6he^{-2t}\theta\|Z\|$
at $\theta=1$ proves \eqref{eq:small-section-remainder-value}.
\end{proof}

\begin{theorem}[Transport at logarithmic time]
\label{thm:small-section-logarithmic-transport}
There exists $h_*>0$ such that, for every $0<h\leq h_*$,
$C_0>0$, and $q>2$, there are $K>0$ and $a_*>0$
with the following property. If $0<a\leq a_*$ and
$\|Z\|\leq C_0a$, put
\[
 T=\frac q2\log(1/a).
\]
The exact flow through $\Xi_h(Z)$ exists at time $T$, and
\begin{equation}
 0<N_1(\phi_T\Xi_h(Z))\leq ha^q,\qquad
 H_{h,T}(Z)=\mathcal T(h)Z+R_{h,T}(Z).
 \label{eq:small-section-logarithmic-transfer}
\end{equation}
The remainder has an ordinary derivative in $Z$ and
\begin{equation}
 \begin{split}
 \|R_{h,T}(Z)\|&\leq Ka^2(1+|\log a|),\\
 \|D R_{h,T}(Z)\|&\leq Ka(1+|\log a|).
 \end{split}
 \label{eq:small-section-logarithmic-remainder}
\end{equation}
Here the derivative is taken with $h$ and $T$ fixed.
\end{theorem}

\begin{proof}
Take $h_*=1/(4C)$ and set $b=2C_0a$.
Since $a|\log a|\to0$ as $a\downarrow0$, there is an
$a_*\leq1$ such that
\[
 2C_0a\leq\frac1{16},\qquad
 C(2C_0a)\frac q2\log(1/a)\leq\frac18
 \quad(0<a\leq a_*).
\]
Thus \eqref{eq:small-section-smallness} holds.
Proposition~\ref{prop:small-section-orbit-control} gives
the wall bound because $e^{-2T}=a^q$.
Proposition~\ref{prop:small-section-nonlinear-remainder}
gives the more explicit estimates
\begin{align*}
 \|R_{h,T}(Z)\|
     &\leq4CC_0^2a^2T+6hC_0a^{q+1},\\
 \|D R_{h,T}(Z)\|
     &\leq8CC_0aT+6ha^q.
\end{align*}
For $a\leq1$ and $q>2$, one has
$a^{q+1}\leq a^2$ and $a^q\leq a$.
Hence \eqref{eq:small-section-logarithmic-remainder}
holds with
\[
 K=\max\{2CC_0^2q+6hC_0,\ 4CC_0q+6h\}>0.
\]
\end{proof}

The theorem controls the full transverse displacement and
its ordinary derivative while the strong wall decreases
from $h$ to at most $ha^q$. For the radial phase, a finer
estimate comes from the cancellation in the transported
quadratic form \eqref{eq:small-section-quadratic-drift}.
We turn next to that scalar estimate before following the
orbit to a core entrance.