\subsection{From the small section to core coordinates}
\label{sec:small-section-projection}

After the initial decay, the shear still has size $a$, whereas
the denominator $1+p$ of the core coordinates has size $a^2$.
The core radius is consequently large. Its reciprocal is the
quantity that remains comparable to the radial phase of the
preceding subsection. To make this comparison also at the level
of derivatives, we retain the small factors in the weak walls
and then rescale the coordinate change by $a$.

Throughout this subsection, finite-dimensional norms are maximum
norms. We enlarge the constant $C\geq32$ from
Sections~\ref{sec:small-section-nonlinear}
and~\ref{sec:small-section-radial} when necessary, and thereafter
keep it fixed.

\subsubsection{Weak-wall factors}

For $X=(n,s,w_2,w_3)$, let
\[
 g_\pm(X)=Q(n,w_2,w_3)-1+\zeta(X)\pm\sqrt3s,
 \qquad \zeta(X)=\sqrt{1-s^2-Q(n,w_2,w_3)}.
\]
These are the two weak-wall rates. They are smooth on a
neighborhood of $\|X\|\leq1/16$ and vanish at zero.
Compact derivative bounds, the mean value theorem, and
$\|DE\|\leq1$ in logarithmic coordinates give
\begin{equation}
 \max\{|g_+(X)|,|g_-(X)|\}\leq C\|X\|,
 \qquad
 \|D(g_\pm\circ E)(y)\|\leq C
 \quad(\|E(y)\|\leq1/16).
 \label{eq:projection-weak-rates}
\end{equation}

\begin{lemma}
\label{lem:small-section-weak-factors}
Suppose $h>0$ and \eqref{eq:small-section-smallness} holds.
For $Z=(s,n_2,n_3)$ with $\|Z\|<b/2$ and $i=2,3$,
there are positive $C^2$ functions
$\mu_i(Z)$, defined on ordinary neighborhoods of these points,
such that
\begin{equation}
 \begin{split}
 N_i(\phi_T\Xi_h(Z))&=n_i\mu_i(Z),\\
 |\log\mu_i(Z)|&\leq\frac14,\qquad
 \frac12<\mu_i(Z)<2,\\
 \|D\log\mu_i(Z)\|&\leq2CT,\qquad
 \|D\mu_i(Z)\|\leq4CT.
 \end{split}
 \label{eq:projection-weak-multipliers}
\end{equation}
For a direction $v=(v_s,v_2,v_3)$ with $v_i=0$,
\begin{equation}
 \bigl|D_Z[N_i(\phi_T\Xi_h(Z))]v\bigr|
       \leq8CT\,|N_i(\phi_T\Xi_h(Z))|\,\|v\|.
 \label{eq:projection-weak-direction}
\end{equation}
\end{lemma}

\begin{proof}
Use the actual logarithmic-coordinate flow $U_{h,t}$ and put
\[
 L_i(Z)=\int_0^T g_i(E(U_{h,t}(Z)))\dd t,
 \qquad \mu_i(Z)=e^{L_i(Z)},
 \qquad g_2=g_+,\quad g_3=g_-.
\]
The scalar wall equation gives the first identity in
\eqref{eq:projection-weak-multipliers}, including at zero or
signed initial walls. The orbit estimate and
\eqref{eq:projection-weak-rates} imply
\[
 |L_i(Z)|\leq C\int_0^T(he^{-2t}+b)\dd t
       \leq\frac{Ch}{2}+CbT\leq\frac14.
\]
Since $e^{1/4}<2$, this gives both multiplier bounds.

The actual flow is jointly $C^2$ on a neighborhood of the
compact time segment. Differentiating its rate integral at
fixed $T$, and using $\|D_ZU_{h,t}\|\leq2$, yields
\[
 \|DL_i(Z)\|\leq\int_0^T 2C\dd t=2CT.
\]
The same regularity permits a second differentiation under
the finite integral, so $L_i$ and $\mu_i$ are $C^2$.
The chain rule gives $\|D\mu_i\|\leq4CT$.
Finally, when $v_i=0$, differentiation of the wall
identity leaves only $n_iD\mu_i\,v$.
Its absolute value is at most $4CT|n_i|\|v\|$.
The lower bound $\mu_i>1/2$ gives
$|n_i|\leq2|N_i(\phi_T\Xi_h(Z))|$ and proves
\eqref{eq:projection-weak-direction}.
\end{proof}

Fix the shear families \eqref{eq:radial-shear-family}, and write
\begin{equation}
 \begin{split}
 T&=\frac q2\log(1/a),\qquad
 x(\rho)=\phi_T\Xi_h(Z_\varsigma(\rho)),\\
 x(\rho)&=(p(\rho),s(\rho),n(\rho),w_2(\rho),w_3(\rho)),\\
 X(\rho)&=(n(\rho),s(\rho),w_2(\rho),w_3(\rho)),
       \qquad a\leq\rho\leq2a.
 \end{split}
 \label{eq:projection-cutoff-state}
\end{equation}
All derivatives in $\rho$ below keep $a,h,q,n_2,n_3$ fixed;
in particular they keep $T$ fixed.
For an arbitrary four-coordinate state $X=(n,s,w_2,w_3)$,
define
\begin{equation}
 \begin{split}
 d(X)&=s^2+Q(n,w_2,w_3)
      =s^2+n^2+(w_2-w_3)^2-2n(w_2+w_3),\\
 \mathcal H(X)&=\frac{d(X)}{1+\sqrt{1-d(X)}}.
 \end{split}
 \label{eq:projection-height}
\end{equation}
When $d(X)<1$, rationalization gives
$\mathcal H(X)=1-\sqrt{1-d(X)}$.
Consequently, on the negative constraint branch,
\[
 \mathcal H(X)=1+p.
\]

\begin{proposition}
\label{prop:small-section-core-coordinates}
There is $h_*>0$ such that, for every $0<h\leq h_*$ and
$q>2$, the following holds for all sufficiently small $a>0$,
uniformly over \eqref{eq:radial-shear-family}.
The state \eqref{eq:projection-cutoff-state} exists, has ordinary
derivatives in $\rho$, and satisfies
\begin{equation}
 \begin{split}
 p&\leq-\frac{15}{16},\qquad
       0<n\leq ha^q,\qquad \|X\|\leq8a,\qquad \|X'\|\leq2,\\
 \frac{3a}{4}&\leq\varsigma s\leq\frac{9a}{4},
       \qquad |n'|\leq2n,\\
 \frac{n_i}{2}&\leq w_i\leq2n_i,\qquad
       |w_i'|\leq8CTw_i\leq\frac{8Cw_i}{a}
       \quad(i=2,3).
 \end{split}
 \label{eq:projection-cutoff-bounds}
\end{equation}
Its height $A=1+p$ is positive, and
\begin{equation}
 \frac{a^2}{4}\leq A\leq6a^2,\qquad |A'|\leq512a.
 \label{eq:projection-height-bounds}
\end{equation}
The actual core coordinates
\[
 Y(\rho)=(M,U,V,W)=\frac{\sqrt3}{A}(s,n,w_2,w_3)
\]
have ordinary derivatives and obey
\begin{equation}
 \begin{aligned}
 \frac1{8a}\leq\varsigma M&\leq\frac{18}{a},
       &0<U&\leq8ha^{q-2},\\
 \frac{n_2}{12a^2}\leq V&\leq\frac{16n_2}{a^2},
       &\frac{n_3}{12a^2}\leq W&\leq\frac{16n_3}{a^2},\\
 0\leq V+W&\leq\frac{64\varepsilon}{a},
       &0\leq VW&\leq\frac{256n_2n_3}{a^4}.
 \end{aligned}
 \label{eq:projection-coordinate-bounds}
\end{equation}
\end{proposition}

\begin{proof}
Choose $h_*$ below the threshold of
Theorem~\ref{thm:small-section-logarithmic-transport} and so
that $Ch_*\leq1/4$ and $h_*\leq1/64$.
Set $b=8a$. By decreasing the threshold for $a$, arrange
\begin{equation*}
 0<a\leq1,\qquad 8a\leq1/16,\qquad
 C(8a)T\leq1/8,\qquad aT\leq1.
\end{equation*}
The initial family has norm at most $2a<b/2$.
Proposition~\ref{prop:small-section-orbit-control} gives the
claims about $p,n$ and the center coordinates, since
$e^{-2T}=a^q$.
The physical-state derivative bound
\eqref{eq:radial-state-derivative} gives $\|X'\|\leq2$
and $|n'|\leq2n$. Lemma~\ref{lem:small-section-weak-factors}
gives the two weak-wall bounds, including when an initial
wall is zero; the last inequality uses $aT\leq1$.

To retain the sign and size of the shear, use the full
transverse approximation of
Theorem~\ref{thm:small-section-logarithmic-transport} with
$C_0=2$. Decrease the threshold again so that its remainder
is at most $a/8$.
Also,
\[
 \|(\mathcal T(h)-I)Z_\varsigma(\rho)\|
       \leq3h\|Z_\varsigma(\rho)\|
       \leq6ha\leq a/8.
\]
Thus the center coordinates at time $T$ differ from
$Z_\varsigma(\rho)$ by at most $a/4$. This proves
$3a/4\leq\varsigma s\leq9a/4$.

The remaining assertions concern the coordinate change.
The preceding bounds, $h\leq1$, and $q>2$ give
\[
 0<n\leq a^2,\qquad
 0\leq w_2,w_3\leq4\varepsilon a\leq a/16.
\]
Since $a\leq1/128$, the expression for $d$ in
\eqref{eq:projection-height} satisfies
\begin{align*}
 d(X)&\geq\frac{9a^2}{16}-\frac{a^3}{4}
             \geq\frac{a^2}{2},\\
 d(X)&\leq\frac{81a^2}{16}+a^4+\frac{a^2}{256}
             \leq6a^2\leq\frac12.
\end{align*}
The denominator of $\mathcal H$ lies between one and two.
This proves the two height bounds.
For a tangent vector $v=(v_n,v_s,v_2,v_3)$, an equivalent
expression for the defect gives
\[
 Dd(X)v=
 2s v_s+4(nv_n+w_2v_2+w_3v_3)
       -2(n+w_2+w_3)(v_n+v_2+v_3).
\]
Hence $|Dd(X)v|\leq32\|X\|\|v\|$.
Along our curve,
\[
 A'=\frac{Dd(X)X'}{2\sqrt{1-d(X)}},
 \qquad |A'|\leq32(8a)2=512a.
\]
This is an ordinary derivative: the negative constraint
branch persists in a neighborhood of each initial point,
so the identity $A=\mathcal H(X)$ holds there as well.
The positivity of $A$ also makes $Y$ differentiable in such
a neighborhood.

Finally, divide the shear and wall estimates by the two
height bounds, using $1\leq\sqrt3\leq2$. For example,
\[
 \frac1{8a}
   =\frac{3a/4}{6a^2}
   \leq\frac{\sqrt3\,\varsigma s}{A}
   \leq\frac{2(9a/4)}{a^2/4}
   =\frac{18}{a}.
\]
The same calculation gives the separate bounds on $U,V,W$.
Adding the bounds on $V,W$ and multiplying them proves
the last line of \eqref{eq:projection-coordinate-bounds}.
\end{proof}

\subsubsection{Rescaling the reciprocal radius}

In core coordinates the radius is
\[
 r(Y)=\sqrt{(M-1)^2+V^2},\qquad Y=(M,U,V,W).
\]
Its reciprocal phase in physical coordinates is
\begin{equation}
 \mathcal P(X)=
 \frac{2\mathcal H(X)}
      {\sqrt{(\sqrt3s-\mathcal H(X))^2+3w_2^2}}.
 \label{eq:projection-phase}
\end{equation}
When $X$ belongs to the negative constraint branch and
$\mathcal H(X)>0$, direct substitution in the core
coordinates gives
\begin{equation}
 \frac2{r(Y)}=\mathcal P(X).
 \label{eq:projection-phase-identity}
\end{equation}
We compare this expression with
$\sqrt{s^2+(w_2-w_3)^2}/\sqrt3$.
Both quantities have size $a$. The numerator of $\mathcal P$
has size $a^2$ and its denominator has size $a$;
rescaling makes their relative orders explicit.

To remove the scale, let $u=(\lambda,\alpha,\beta)\in\R^3$
and $z=(z_1,z_2)\in\R^2$. On the open set where the
square roots below have positive arguments, define
\begin{equation}
 \begin{split}
 X(u,z)&=(\lambda\alpha,\lambda z_1,\lambda z_2,\lambda\beta),\\
 d_*(u,z)&=z_1^2+Q(\alpha,z_2,\beta),\\
 H_*(u,z)&=\frac{d_*(u,z)}
                  {1+\sqrt{1-\lambda^2d_*(u,z)}},\\
 D_*(u,z)&=(\sqrt3z_1-\lambda H_*(u,z))^2+3z_2^2,\\
 F(u,z)&=\frac{2H_*(u,z)}{\sqrt{D_*(u,z)}}
              -\frac{\sqrt{z_1^2+(z_2-\beta)^2}}{\sqrt3}.
 \end{split}
 \label{eq:projection-rescaling}
\end{equation}
Homogeneity gives $d(X(u,z))=\lambda^2d_*(u,z)$ and
$\mathcal H(X(u,z))=\lambda^2H_*(u,z)$.
Thus, on this domain with $\lambda>0$,
\begin{equation}
 \mathcal P(X(u,z))
       -\frac{\sqrt{(\lambda z_1)^2+
                    (\lambda z_2-\lambda\beta)^2}}{\sqrt3}
       =\lambda F(u,z).
 \label{eq:projection-scaled-error}
\end{equation}

The decisive feature of this expression is its vanishing
on the whole slice $u=0$. For $z\ne0$,
\[
 H_*(0,z)=\frac{z_1^2+z_2^2}{2},\qquad
 D_*(0,z)=3(z_1^2+z_2^2),\qquad F(0,z)=0.
\]
Fix $c>0$ and $M_0\geq c$, and restrict $z$ to the compact set
\[
 \mathcal K=\{z:\|z\|\leq M_0,\ |z_1|\geq c\}.
\]
All radicands and denominators in \eqref{eq:projection-rescaling}
are positive at $(0,z)$ for $z\in\mathcal K$. They remain
positive on a neighborhood of this compact slice.
Choose $\delta>0$ so that a closed neighborhood of radius
$\delta$ lies in this smooth domain, and let $B\geq1$
bound both $\|DF\|$ and $\|D^2F\|$ there.
The mean value theorem along the segment from $(0,z)$ to
$(u,z)$ gives
\begin{equation}
 \begin{split}
 |F(u,z)|&\leq B\|u\|,\qquad \|DF(u,z)\|\leq B,\\
 |DF(u,z)(0,\delta z)|&\leq B\|u\|\|\delta z\|
 \quad(z\in\mathcal K,\ \|u\|\leq\delta).
 \end{split}
 \label{eq:projection-scaled-bounds}
\end{equation}
For the last inequality, differentiate $F(0,z)=0$ to obtain
$DF(0,z)(0,\delta z)=0$, and apply the same mean value
estimate to $DF$. In particular,
\begin{equation}
 |DF(u,z)(\delta u,\delta z)|
       \leq B\|\delta u\|+B\|u\|\|\delta z\|.
 \label{eq:projection-separated-derivative}
\end{equation}
This retains the small factor in the directions tangent
to the limiting slice.

We now impose the additional smallness condition
\begin{equation}
 n_3\leq a^P,\qquad P>2.
 \label{eq:projection-small-weak-wall}
\end{equation}
The scaled normal coordinates then tend to zero, while
the shear remains in a fixed compact annulus.

\begin{theorem}
\label{thm:small-section-projected-phase}
There is $h_*>0$ such that for each $0<h\leq h_*$ one
can choose $K>0$ with the following property.
For every $q>2$ and $P>2$, there is $a_*>0$ such that,
whenever $0<a\leq a_*$,
\eqref{eq:radial-shear-family} and
\eqref{eq:projection-small-weak-wall} hold, the actual
phase
\[
 f_{\mathrm p}(\rho)=\frac2{r(Y(\rho))}
 \quad(a\leq\rho\leq2a)
\]
has an ordinary derivative at every point of the interval and
\begin{equation}
 \begin{split}
 |f_{\mathrm p}(\rho)-\Phi_h(Z_\varsigma(\rho))|
       &\leq Ka^2,\\
 \left|f_{\mathrm p}'(\rho)
       -\frac{\dd}{\dd\rho}\Phi_h(Z_\varsigma(\rho))\right|
       &\leq Ka,\\
 \frac1{4\sqrt3}\leq f_{\mathrm p}'(\rho)
       &\leq\frac2{\sqrt3}+1.
 \end{split}
 \label{eq:projection-phase-comparison}
\end{equation}
In particular the reciprocal core radius is strictly
increasing along both shear families.
\end{theorem}

\begin{proof}
Take $h_*$ below the thresholds of
Proposition~\ref{prop:small-section-core-coordinates} and
Corollary~\ref{cor:small-section-shear-monotonicity}, and
also below $1/16$.
For the scaling estimates fix $c=3/4$ and $M_0=8$.
The resulting constants $\delta,B$ are independent of
$h,q,P$.

At the actual state $X(\rho)$, use
\[
 u(\rho)=\left(a,\frac{n(\rho)}a,\frac{w_3(\rho)}a\right),
 \qquad
 z(\rho)=\left(\frac{s(\rho)}a,\frac{w_2(\rho)}a\right).
\]
Then $X(u(\rho),z(\rho))=X(\rho)$ and
$z(\rho)\in\mathcal K$ by
\eqref{eq:projection-cutoff-bounds}.
For $a\leq1$, the assumptions $q,P>2$ give
\[
 n\leq a^2,\qquad 0\leq w_3\leq2n_3\leq2a^2,
 \qquad \|u\|\leq2a.
\]
Choose the scale threshold so that $2a\leq\delta$.
Since $a$ is fixed when differentiating in $\rho$,
\[
 u'=\left(0,\frac{n'}a,\frac{w_3'}a\right),
 \qquad z'=\left(\frac{s'}a,\frac{w_2'}a\right).
\]
The cutoff estimates imply
\begin{equation*}
 \|u'\|\leq16C,\qquad \|z'\|\leq\frac2a.
\end{equation*}
Indeed, $|n'|\leq2n\leq2n/a$ for $a\leq1$, while
$|w_3'|\leq8Cw_3/a$; division by $a$ gives the stated
uniform bound on $u'$.

Let
\[
 e(\rho)=\mathcal P(X(\rho))
       -\frac{\sqrt{s(\rho)^2+(w_2(\rho)-w_3(\rho))^2}}{\sqrt3}.
\]
Equation~\eqref{eq:projection-scaled-error} reads
$e(\rho)=aF(u(\rho),z(\rho))$.
The value and the separated derivative estimates yield
\[
 |e(\rho)|\leq2Ba^2,\qquad
 |e'(\rho)|\leq aB(16C+4).
\]
Thus both errors are bounded by $K_{\mathrm e}a^2$ and
$K_{\mathrm e}a$, respectively, where
$K_{\mathrm e}=B(16C+6)$ is independent of $h,q,P$.

At each initial point the negative constraint branch and
positive height persist on an ordinary neighborhood.
The scaled formula is smooth there as well.
Consequently \eqref{eq:projection-phase-identity} is a
local identity of differentiable functions. It gives
\[
 f_{\mathrm p}(\rho)=e(\rho)
       +\frac{\varrho(H_{h,T}(Z_\varsigma(\rho)))}{\sqrt3},
\]
together with the corresponding derivative identity.
Corollary~\ref{cor:small-section-shear-monotonicity}
compares the second term with $\Phi_h$ with constants
$K_{\mathrm r}a^2$ and $K_{\mathrm r}a$.
Choose $K=K_{\mathrm e}+K_{\mathrm r}$ before choosing
$q,P$. The first two bounds of
\eqref{eq:projection-phase-comparison} follow.

Finally, Proposition~\ref{prop:small-section-reference-slope}
gives the reference derivative between
$1/(2\sqrt3)$ and $2/\sqrt3$.
Decrease $a_*$ so that $Ka\leq1/(4\sqrt3)\leq1$.
This proves the final line, and strict monotonicity follows
from the mean value theorem.
\end{proof}

The reciprocal radius has therefore retained the reference
phase through both the initial evolution and the change to
core coordinates. We next connect the resulting state to
a complete core entrance, using its wall sizes and the
ordinary derivatives established above.