\subsection{Radial phase during the initial decay}
\label{sec:small-section-radial}

The radial phase changes less than the full transverse vector.
The rotation visible in
\eqref{eq:small-section-perturbed-equations} makes no
contribution to the Euclidean radius when the strong wall
vanishes. The transported quadratic form retains this
cancellation at positive strong wall. Its drift has an
additional small factor, which removes the logarithmic
loss from the phase estimate and its initial derivative.

For $Z=(s,n_2,n_3)$, write
\[
 v=n_2-n_3,\qquad w=n_2+n_3,\qquad
 \varrho(Z)=\sqrt{s^2+v^2},\qquad
 \mathcal J(n,Z)=\mathcal Q(n;s,v).
\]
The function $\varrho$ is the Euclidean radius in the shear
and wall-difference coordinates. We continue to use maximum
norms for $Z$, for $(n,Z)$, and for their derivatives.

\subsubsection{The vanishing drift coefficients}

Use the coefficients $A,C,K$ of
\eqref{eq:small-section-coefficients} to define
\begin{equation}
 \begin{aligned}
 g(n)&=(A^2,\,2AK,\,K^2+C^2),\\
 k(n)&=(0,\,AK,\,K^2+C^2),\\
 o(n)&=(AK,\,K^2+C^2-A^2,\,-AK).
 \end{aligned}
 \label{eq:radial-coefficient-vectors}
\end{equation}
The coefficients on the right are evaluated at $n$.
Thus $\mathcal J(n,Z)=g(n)\cdot(s^2,sv,v^2)$.
On the negative constraint branch put
\[
 \Delta_Q=v^2-2nw,\qquad
 \Delta_z=\sqrt{\mathcal D_n(Z)}-z(n).
\]
These are the differences in
\eqref{eq:small-section-differences}. Define
\begin{equation}
 \begin{split}
 m(n,Z)={}&n(\Delta_Q-2\Delta_z)g'(n)
               +2\Delta_Qg(n)+2\Delta_zk(n)
               +2\sqrt3\,w\,o(n),\\
 \mathcal L(n,Z)={}&m(n,Z)\cdot(s^2,sv,v^2).
 \end{split}
 \label{eq:radial-drift-coefficients}
\end{equation}
Proposition~\ref{prop:small-section-quadratic-drift} gives
the exact identity
\begin{equation}
 \frac{\dd}{\dd t}\mathcal J(N_1(t),\Pi x(t))
       =\mathcal L(N_1(t),\Pi x(t))
 \label{eq:radial-drift-identity}
\end{equation}
along the constrained flow wherever $p<0$ and $N_1^2<1$.

All coefficient functions in
\eqref{eq:radial-drift-coefficients} are smooth on a
neighborhood of the signed box
$|n|,\|Z\|\leq1/16$. Two identities determine their size:
\begin{equation}
 m(n,0)=0,\qquad Dm(0,0)=0.
 \label{eq:radial-drift-vanishing}
\end{equation}
The first follows from $\Delta_Q=\Delta_z=w=0$ at $Z=0$.
For the second, both differences have zero value and zero
first derivative at the origin, and $o(0)=0$.
In particular, the term linear in $w$ also has zero
first derivative there.

There is a constant $B\geq1$ such that on this box
\begin{equation}
 \begin{split}
 \|m(n,Z)\|&\leq B(|n|+\|Z\|)\|Z\|,\\
 \|Dm(n,Z)\|&\leq B(|n|+\|Z\|),\\
 \|\partial_n m(n,Z)\|&\leq B\|Z\|.
 \end{split}
 \label{eq:radial-coefficient-bounds}
\end{equation}
Indeed, a compact bound on the Hessian makes $Dm$
Lipschitz on the box. Comparing $Dm(n,Z)$ with $Dm(0,0)$
gives the second inequality. Integrating this bound on the
segment from $(n,0)$ to $(n,Z)$ gives the first, by
\eqref{eq:radial-drift-vanishing}. Finally,
$\partial_n m(n,0)=0$; comparing the derivatives at
$(n,0)$ and $(n,Z)$ gives the third.

We may choose the same $B$ so that
\begin{equation}
 \begin{split}
 \|D\mathcal J(n,Z)\|&\leq B\|Z\|,\\
 \|g(n)-g(0)\|&\leq B|n|,\qquad
 \|g'(n)\|\leq B.
 \end{split}
 \label{eq:radial-gram-bounds}
\end{equation}
For the first inequality, $D\mathcal J(n,0)=0$ because
$\mathcal J$ is quadratic in $s,v$. A compact Hessian
bound and the same segment comparison apply. The other
two inequalities follow from a compact derivative bound
for $g$, with $g(0)=(1,0,1)$.

For a coefficient vector $a$, the quadratic polynomial
$a_1s^2+a_2sv+a_3v^2$ has absolute value at most
$2\|a\|(s^2+v^2)$. Its differential, when
$|s|\leq r$, $|v|\leq2r$, $|\delta s|\leq d$, and
$|\delta v|\leq2d$, has absolute value at most
\[
 12\|\delta a\|r^2+24\|a\|rd.
\]
These bounds follow by expanding the three monomials and
using $|sv|\leq s^2+v^2$. Apply them to $a=m(n,Z)$,
and use \eqref{eq:small-section-quadratic-lower}.
For $0\leq n\leq1/16$ and $\|Z\|\leq1/16$ we obtain
\begin{equation}
 \begin{split}
 |\mathcal L(n,Z)|
     &\leq C(n\|Z\|+\|Z\|^2)\mathcal J(n,Z),\\
 \|D_Z\mathcal L(n,Z)\|
     &\leq C(n\|Z\|^2+\|Z\|^3),\\
 |\partial_n\mathcal L(n,Z)|
     &\leq C\|Z\|^3,
 \end{split}
 \label{eq:radial-drift-bounds}
\end{equation}
with $C=36B$. The last line uses the third bound in
\eqref{eq:radial-coefficient-bounds}, keeping the strong-wall
direction separate from the transverse directions.

We also need to compare the transported and Euclidean
radii. The bounds \eqref{eq:small-section-coefficient-bounds}
give $\|g(n)-g(0)\|\leq4n$ for $0\leq n\leq1/16$.
Consequently
\begin{equation}
 \begin{split}
 |\mathcal J(n,Z)-\varrho(Z)^2|
      &\leq8n\varrho(Z)^2,\\
 \tfrac12\varrho(Z)^2
      &\leq\mathcal J(n,Z)\leq2\varrho(Z)^2.
 \end{split}
 \label{eq:radial-quadratic-comparison}
\end{equation}
The lower bound is \eqref{eq:small-section-quadratic-lower}.
For $\varrho(Z)>0$, rationalizing the two square roots gives
\[
 |\sqrt{\mathcal J(n,Z)}-\varrho(Z)|
      \leq8n\varrho(Z).
\]
When $\varrho(Z)=0$, both sides vanish.
If $\|Z\|\leq b$, then $|s|\leq b$, $|v|\leq2b$,
and $\varrho(Z)\leq3b$. Hence
\begin{equation}
 \sqrt{\mathcal J(n,Z)}\leq4b,\qquad
 |\sqrt{\mathcal J(n,Z)}-\varrho(Z)|\leq24nb.
 \label{eq:radial-endpoint-value}
\end{equation}

\subsubsection{The accumulated radial displacement}

Choose $C\geq32$ large enough for
\eqref{eq:radial-drift-bounds},
\eqref{eq:radial-gram-bounds}, and the estimates of
Section~\ref{sec:small-section-nonlinear}, and fix this
constant. We impose
\begin{equation}
 h>0,\qquad Ch\leq\frac14,\qquad
 0<b\leq\frac1{16},\qquad
 T\geq0,\qquad CbT\leq\frac18.
 \label{eq:radial-smallness}
\end{equation}
For $\|Z\|\leq b/2$, write
\begin{equation}
 \begin{split}
 \nu_t(Z)&=N_1(\phi_t\Xi_h(Z)),\qquad
 Z_t=H_{h,t}(Z),\\
 J_t(Z)&=\mathcal J(\nu_t(Z),Z_t),\qquad
 r_t(Z)=\sqrt{J_t(Z)}.
 \end{split}
 \label{eq:radial-observables}
\end{equation}
The height $h$ is fixed in this notation.
In particular, $J_0(Z)=\mathcal J(h,Z)$ and
$r_0(Z)/\sqrt3=\Phi_h(Z)$.

\begin{proposition}
\label{prop:radial-value-control}
Under \eqref{eq:radial-smallness}, for $\|Z\|\leq b/2$
and $0\leq t\leq T$,
\begin{align}
 \tfrac12J_0(Z)&\leq J_t(Z)\leq2J_0(Z),
       \label{eq:radial-nondegeneration}\\
 |r_t(Z)-r_0(Z)|
       &\leq Chb^2+2Cb^3t,
       \label{eq:radial-accumulated-value}\\
 |\varrho(Z_t)-r_0(Z)|
       &\leq Chb^2+2Cb^3t+24hbe^{-2t}.
       \label{eq:radial-full-value}
\end{align}
The value estimates include $\varrho(Z)=0$.
\end{proposition}

\begin{proof}
Proposition~\ref{prop:small-section-orbit-control} gives
$0<\nu_t\leq he^{-2t}$ and $\|Z_t\|\leq b$.
By \eqref{eq:radial-drift-identity} and
\eqref{eq:radial-drift-bounds},
\begin{equation}
 |J_t'|\leq
 C(\nu_t\|Z_t\|+\|Z_t\|^2)J_t
 \leq C(he^{-2t}+b)J_t.
 \label{eq:radial-relative-drift}
\end{equation}
In the last step we used $\|Z_t\|\leq b\leq1$.
Multiplying the two differential inequalities by
$e^{-\Gamma(t)}$ and $e^{\Gamma(t)}$, respectively, where
$\Gamma$ is \eqref{eq:small-section-growth-integral}, gives
\[
 J_0e^{-\Gamma(t)}\leq J_t\leq J_0e^{\Gamma(t)}.
\]
Since $\Gamma(t)\leq1/4$ and $e^{1/4}<2$, this proves
\eqref{eq:radial-nondegeneration} without dividing by $J_t$.

If $J_0=0$, the upper bound forces $J_t=0$ throughout,
and \eqref{eq:radial-accumulated-value} follows.
Otherwise the lower bound keeps $r_t>0$, and
\eqref{eq:radial-relative-drift}, before its last
simplification, yields
\[
 |r_t'|=\frac{|J_t'|}{2r_t}
   \leq\frac C2(\nu_tb+b^2)r_t
   \leq2C(\nu_tb^2+b^3).
\]
Here $r_t\leq4b$ follows from
\eqref{eq:radial-endpoint-value}. Comparing with the
primitive
$Chb^2(1-e^{-2t})+2Cb^3t$ proves
\eqref{eq:radial-accumulated-value}.
Finally, \eqref{eq:radial-endpoint-value} at the endpoint
gives $|r_t-\varrho(Z_t)|\leq24hbe^{-2t}$.
The triangle inequality proves \eqref{eq:radial-full-value}.
\end{proof}

\subsubsection{Initial derivatives on a radial annulus}

For differentiation of the radius, assume in addition that
\begin{equation}
 \varrho(Z)\geq cb,\qquad c>0.
 \label{eq:radial-annulus}
\end{equation}
Equations \eqref{eq:radial-quadratic-comparison} and
\eqref{eq:radial-nondegeneration} give
\begin{equation}
 r_t(Z)\geq\frac{cb}{2},\qquad
 \varrho(Z_t)\geq\frac{cb}{4}.
 \label{eq:radial-positive-radii}
\end{equation}
These positive lower bounds provide ordinary neighborhoods
on which both square roots are smooth.

\begin{proposition}
\label{prop:radial-derivative-control}
For every $c>0$ there is $K_c>0$ such that, under
\eqref{eq:radial-smallness} and
\eqref{eq:radial-annulus}, with $\|Z\|\leq b/2$ and
$0\leq t\leq T$,
\begin{align}
 \|D(r_t-r_0)(Z)\|&\leq K_c(hb+b^2t),
       \label{eq:radial-accumulated-derivative}\\
 \|D(r_t-\varrho\circ H_{h,t})(Z)\|
       &\leq K_c\nu_t(Z).
       \label{eq:radial-endpoint-derivative}
\end{align}
All derivatives are ordinary initial derivatives at fixed
$h,t$. The constant $K_c$ is independent of $h,b,T$.
\end{proposition}

\begin{proof}
First recover the strong-wall derivative from its
logarithm. Proposition~\ref{prop:small-section-variation-control}
gives $\|D U_{h,t}(Z)\|\leq2$, and $\log\nu_t$ is its
first component. Since $\nu_t>0$,
\begin{equation}
 \|D\nu_t(Z)\|\leq2\nu_t(Z),\qquad
 \|D H_{h,t}(Z)\|\leq2,\qquad
 \|D(\nu_t,H_{h,t})(Z)\|\leq2.
 \label{eq:radial-state-derivative}
\end{equation}
The last bound uses $\nu_t\leq1$. In the rest of the
proof $D$ denotes differentiation in the initial variable.
Put
\[
 F_t(Z)=\mathcal L(\nu_t(Z),H_{h,t}(Z)),
 \qquad V_t(Z)=\frac{F_t(Z)}{2r_t(Z)}.
\]
The chain rule, \eqref{eq:radial-gram-bounds}, and the
three estimates in \eqref{eq:radial-drift-bounds} give
\begin{equation}
 \begin{split}
 \|D J_t\|&\leq2Cb,\\
 |F_t|&\leq C(\nu_tb+b^2)J_t,\\
 \|D F_t\|&\leq4C(\nu_tb^2+b^3).
 \end{split}
 \label{eq:radial-composed-derivatives}
\end{equation}
For the last inequality the strong-wall contribution is
at most $2C\nu_tb^3$, whereas the transverse contribution
is at most $2C(\nu_tb^2+b^3)$. Since $b\leq1$, their sum
has the stated bound.

Using \eqref{eq:radial-positive-radii}, differentiation of
the square root and of its time rate gives
\begin{equation*}
 D r_t=\frac{D J_t}{2r_t},\qquad
 D V_t=\frac{D F_t}{2r_t}
             -\frac{F_t}{2J_t}D r_t.
\end{equation*}
It follows that, for a constant depending only on $c$,
\begin{equation*}
 \|D r_t\|\leq K_c,\qquad
 \|D V_t\|\leq K_c(\nu_tb+b^2).
\end{equation*}
The actual flow and the positive square root defining
$r_t$ are jointly $C^2$ on an open neighborhood of each
point under consideration. The local time identity
$\partial_t r_t=V_t$ therefore implies, for every fixed
initial direction $u$,
\[
 \frac{\dd}{\dd t}\bigl(D r_t(Z)u\bigr)=D V_t(Z)u.
\]
Compare its absolute derivative with
$K_c\|u\|(hbe^{-2t}+b^2)$.
The primitive
$K_c\|u\|\bigl(hb(1-e^{-2t})/2+b^2t\bigr)$
gives \eqref{eq:radial-accumulated-derivative} after taking
the operator norm.

It remains to compare the two endpoint radii at the
derivative level. Define the quadratic defect
\[
 \mathcal E(n,Z)=\mathcal J(n,Z)-\varrho(Z)^2
       =(g(n)-g(0))\cdot(s^2,sv,v^2).
\]
The quadratic differential estimate used above and
\eqref{eq:radial-gram-bounds} give, for a direction
$(\eta,W)$,
\[
 |D\mathcal E(n,Z)(\eta,W)|
   \leq12C|\eta|\|Z\|^2+24C|n|\|Z\|\|W\|.
\]
Let $S_t(Z)=\varrho(H_{h,t}(Z))^2$.
Composing with the actual state and using
\eqref{eq:radial-state-derivative} yields
\begin{equation}
 \|D(J_t-S_t)\|
    \leq24C\nu_tb^2+48C\nu_tb
    \leq72C\nu_tb.
 \label{eq:radial-defect-derivative}
\end{equation}
Both $J_t$ and $S_t$ are positive by
\eqref{eq:radial-positive-radii}. The exact differential
identity
\begin{equation}
 D(r_t-\sqrt{S_t})
   =\frac{D(J_t-S_t)}{2\sqrt{S_t}}
       +\frac{\sqrt{S_t}-r_t}{2r_t\sqrt{S_t}}D J_t
 \label{eq:radial-root-difference-derivative}
\end{equation}
preserves the small difference of the two radii.
Use \eqref{eq:radial-defect-derivative},
$|r_t-\sqrt{S_t}|\leq24\nu_tb$ from
\eqref{eq:radial-endpoint-value}, and
$\|D J_t\|\leq2Cb$ from
\eqref{eq:radial-composed-derivatives}.
Each term on the right of
\eqref{eq:radial-root-difference-derivative} is bounded
by a constant depending on $c$ times $\nu_t$.
This proves \eqref{eq:radial-endpoint-derivative}, after
enlarging $K_c$ to cover both conclusions.
\end{proof}

Define the phase error
\begin{equation}
 E_{h,t}(Z)=\frac{\varrho(H_{h,t}(Z))}{\sqrt3}-\Phi_h(Z)
          =\frac{\varrho(Z_t)-r_0(Z)}{\sqrt3}.
 \label{eq:radial-phase-error}
\end{equation}
Combining the two parts of
Proposition~\ref{prop:radial-derivative-control}, and using
$\nu_t\leq he^{-2t}$, gives
\begin{equation}
 \|D E_{h,t}(Z)\|
       \leq K_c(hb+b^2t+he^{-2t})
 \label{eq:radial-full-derivative}
\end{equation}
on the annulus \eqref{eq:radial-annulus}. The value estimate
comes from \eqref{eq:radial-full-value}; division by
$\sqrt3>1$ only decreases its right-hand side.

\begin{theorem}[Radial phase approximation]
\label{thm:small-section-radial-approximation}
There exists $h_*>0$ such that for every $0<h\leq h_*$ and
$c_0,C_0>0$ there is a constant $K>0$ with the following
property. For each $q>2$ there is $a_*>0$ such that, if
\[
 0<a\leq a_*,\qquad
 \|Z\|\leq C_0a,\qquad \varrho(Z)\geq c_0a,\qquad
 T=\frac q2\log(1/a),
\]
then the actual flow through $\Xi_h(Z)$ exists at $T$,
$E_{h,T}$ has an ordinary derivative at $Z$, and
\begin{equation}
 |E_{h,T}(Z)|\leq Ka^2,\qquad
 \|D E_{h,T}(Z)\|\leq Ka.
 \label{eq:radial-logarithmic-approximation}
\end{equation}
The derivative is taken with $h,T$ fixed.
The constant $K$ can be chosen before $q$.
\end{theorem}

\begin{proof}
Take $h_*=1/(4C)$, $b=2C_0a$, and $c=c_0/(2C_0)$.
For each fixed $q>2$, the limit $a|\log a|\to0$ allows
us to choose $a_*\leq1$ so that
\begin{equation}
 2C_0a\leq\frac1{16},\qquad
 C(2C_0a)T\leq\frac18,\qquad aT\leq1
 \quad(0<a\leq a_*).
 \label{eq:radial-cutoff-smallness}
\end{equation}
The orbit exists and satisfies all previous estimates.
Since $e^{-2T}=a^q$, its phase value error obeys
\[
 |E_{h,T}(Z)|
   \leq4CC_0^2ha^2+16CC_0^3a^3T
                      +48C_0ha^{q+1}.
\]
The middle term is at most $C_0^2a^2$ by the second
condition in \eqref{eq:radial-cutoff-smallness}, and
$a^{q+1}\leq a^2$. Thus the value constant may be taken
to be
\[
 K_{\mathrm v}=4CC_0^2h+C_0^2+48C_0h.
\]
Equation~\eqref{eq:radial-full-derivative} gives a constant
$K_{\mathrm d}$ depending only on $c_0,C_0$ such that
\[
 \|D E_{h,T}(Z)\|
      \leq K_{\mathrm d}(ha+a^2T+ha^q).
\]
For example,
$K_{\mathrm d}=K_c(2C_0+4C_0^2+1)$ suffices.
Now $aT\leq1$ and $a^q\leq a$, so the last bound is
at most $K_{\mathrm d}(2h+1)a$.
Taking
$K=\max\{K_{\mathrm v},K_{\mathrm d}(2h+1)\}$
proves \eqref{eq:radial-logarithmic-approximation}.
This choice is independent of $q$.
\end{proof}

Return to the curves $Z_\varsigma(\rho)=(\varsigma\rho,n_2,n_3)$
of Section~\ref{sec:small-section-transport}. For
\begin{equation}
 \varsigma\in\{-1,1\},\qquad
 0\leq\varepsilon\leq\frac1{64},\qquad
 0\leq n_2,n_3\leq2\varepsilon a,
 \label{eq:radial-shear-family}
\end{equation}
define the phase after the initial decay by
\[
 f(\rho)=\frac{\varrho(H_{h,T}(Z_\varsigma(\rho)))}{\sqrt3},
 \qquad T=\frac q2\log(1/a).
\]
The time $T$ is fixed along the entire interval $[a,2a]$.

\begin{corollary}
\label{cor:small-section-shear-monotonicity}
For each $0<h\leq h_*$ there is $K>0$ such that, for
every $q>2$, the following holds for all sufficiently
small $a>0$, uniformly over
\eqref{eq:radial-shear-family}. The function $f$ has an
ordinary derivative at every $\rho\in[a,2a]$, and
\begin{equation}
 \begin{split}
 |f(\rho)-\Phi_h(Z_\varsigma(\rho))|&\leq Ka^2,\\
 \left|f'(\rho)
       -\frac{\dd}{\dd\rho}\Phi_h(Z_\varsigma(\rho))\right|
       &\leq Ka,\\
 \frac1{4\sqrt3}\leq f'(\rho)&\leq\frac2{\sqrt3}+1.
 \end{split}
 \label{eq:radial-shear-monotonicity}
\end{equation}
In particular, $f$ is strictly increasing. For
$a\leq x\leq y\leq2a$,
\begin{equation}
 \frac{y-x}{4\sqrt3}
      \leq f(y)-f(x)
      \leq\left(\frac2{\sqrt3}+1\right)(y-x).
 \label{eq:radial-shear-secants}
\end{equation}
\end{corollary}

\begin{proof}
Equation~\eqref{eq:small-section-shear-annulus} gives
$\|Z_\varsigma(\rho)\|\leq2a$ and
$\varrho(Z_\varsigma(\rho))\geq a$.
Apply Theorem~\ref{thm:small-section-radial-approximation}
with $c_0=1$ and $C_0=2$.
Since $Z_\varsigma'=(\varsigma,0,0)$ has maximum norm one,
the chain rule gives the first two inequalities in
\eqref{eq:radial-shear-monotonicity}.
Proposition~\ref{prop:small-section-reference-slope}
bounds the reference derivative between
$1/(2\sqrt3)$ and $2/\sqrt3$.
Reduce the scale threshold so that
$Ka\leq1/(4\sqrt3)\leq1$.
The third line follows, and the mean value theorem gives
strict monotonicity and \eqref{eq:radial-shear-secants}.
\end{proof}

The actual phase after the initial decay thus has a
uniform positive derivative on both shear families.
We next carry its comparison with $\Phi_h$ through the
remaining passage to a core entrance, keeping track of
the arrival time and the entrance coordinates.