\subsection{Radial directions through the incoming passage}
\label{sec:entrance-radial-direction}

An arbitrarily small active wall can produce a long wait before
the core. The wall derivatives grow with this waiting time, but
the walls themselves decay. Keeping half of that decay gives
uniform estimates for the square-root slopes used by the core
and connection maps. The core radius has a simpler response:
its derivative changes by one power of the small scale less
than its leading radial derivative.

For $y=(m,u,v,w)$ write
\[
 k=r(y)=H(m-1,v),\qquad b=u+w,\qquad
 J=uv+v^2b^2,\qquad L=\log(1/v).
\]
Let $\mathcal P(y)$ be the actual first upward arrival,
and denote its core radius and two walls by
\[
 \kappa(y)=r(\mathcal P(y)),\qquad
 U_+(y)=\mathcal P_U(y),\qquad W_+(y)=\mathcal P_W(y).
\]
The stable radius used in
Sections~\ref{sec:entrance-phase-transport}--\ref{sec:entrance-physical-radius}
will be denoted here by
\[
 r_s(y)=H(P(m,u,w)-1,v).
\]
This notation distinguishes it from the input core radius $r$.

\begin{proposition}
\label{prop:entrance-ordinary-direction}
There are universal constants $C>0$ and
$0<\varepsilon_1\leq\varepsilon_0$ with the following
property. Suppose
\[
 k\geq7/4,\qquad m\geq1,\qquad
 0<u,w\leq\varepsilon_1,\qquad0<v<1.
\]
For an ordinary initial direction
$d=(d_m,d_u,d_v,d_w)$ and a number $D_0\geq0$, assume
\begin{equation}
 |d_m|\leq D_0k^2,\qquad
 |d_u|\leq D_0ku,\qquad |d_v|\leq D_0kv,\qquad
 |d_w|\leq D_0kw.
 \label{eq:entrance-relative-direction}
\end{equation}
Then
\begin{align}
 |\kappa(y)-k|&\leq C\frac{b^2+J}{k},
 \label{eq:entrance-ordinary-radius-value}\\
 |D\kappa(y)d-Dr(y)d|&\leq CD_0(b^2+kJ),
 \label{eq:entrance-ordinary-radius-direction}\\
 |DU_+(y)d|&\leq CD_0k^2\sqrt{uv}\sqrt{U_+(y)},\notag\\
 |DW_+(y)d|&\leq CD_0k\sqrt{wv}\sqrt{W_+(y)}.
 \label{eq:entrance-ordinary-wall-direction}
\end{align}
The derivatives are those of the actual arrival maps
on their ordinary open domains.
\end{proposition}

\begin{proof}
We first compare the two input radii, then use the
arrival columns already proved in the stable radial
coordinates. Put
\[
 c=m-1,\qquad \chi=P(m,u,w)-1,\qquad s=r_s(y).
\]
The incoming radius band gives $m>2$ and
$c,\chi>0$. Choose $A>0$ for the projection estimates
\eqref{eq:entrance-projection-scales}. They imply
\[
 |\chi-c|\leq Ab^2/k,\qquad |s-k|\leq Ab^2/k.
\]
Since $s,k\geq k/2$ and $c/k\leq1$, the identities
\[
 \frac{\chi}{s}-\frac ck
   =\frac{\chi-c}{s}-\frac ck\,\frac{s-k}{s},
 \qquad
 \frac1s-\frac1k=-\frac{s-k}{sk}
\]
give
\begin{equation*}
 \left|\frac{\chi}{s}-\frac ck\right|
       \leq\frac{4Ab^2}{k^2},\qquad
 \left|\frac1s-\frac1k\right|
       \leq\frac{4Ab^2}{k^3}.
\end{equation*}
The same projection bounds, applied to
\eqref{eq:entrance-relative-direction}, give
\[
 |DP(y)d-d_m|
 \leq \frac{Ab^2}{k^2}|d_m|
       +\frac{Ab}{k}(|d_u|+|d_w|)
 \leq2AD_0b^2.
\]
The exact differential comparison is
\begin{equation*}
 \begin{split}
 Dr_s(y)d-Dr(y)d
   ={}&\frac{\chi}{s}\bigl(DP(y)d-d_m\bigr)\\
     &+\left(\frac{\chi}{s}-\frac ck\right)d_m
       +v\left(\frac1s-\frac1k\right)d_v.
 \end{split}
\end{equation*}
Here $\chi/s\leq1$ and $v\leq k$. Substitution yields
\begin{equation}
 |Dr_s(y)d-Dr(y)d|\leq10AD_0b^2,\qquad
 |Dr_s(y)d|\leq(10A+2)D_0k^2.
 \label{eq:entrance-input-radius-response}
\end{equation}
For the second inequality, use
$Dr(y)d=(cd_m+vd_v)/k$, so that
$|Dr(y)d|\leq2D_0k^2$, and $b^2\leq1\leq k^2$.

In the local input coordinates $(r_s,u,v,w)$, let
$F$ be the physical arrival radius of
\eqref{eq:entrance-physical-return}. Its derivative
columns, with the waiting factors bounded by
\eqref{eq:entrance-terminal-clock}, imply
\begin{equation*}
 \begin{split}
 |F-s|&\leq CJ/k,\qquad |F_s-1|\leq CJ/k^2,\\
 |F_u|&\leq C(v+v^2b)/k,\qquad
 |F_v|\leq C(u+vb^2),\\
 |F_w|&\leq C(uv+v^2b)/k.
 \end{split}
\end{equation*}
These are direct weakenings of
Proposition~\ref{prop:entrance-physical-bounds}, after
a universal enlargement of $C$.
The transformed direction is
$e=(Dr_s(y)d,d_u,d_v,d_w)$. Put
$D_1=(10A+2)D_0$. All its components satisfy
\eqref{eq:entrance-relative-direction} with $D_1$ and
with the first component interpreted as radial.
Expanding $DF\,e-e_s$ by its four columns gives
\[
 |DF\,e-e_s|
 \leq CD_1\{J+u(v+v^2b)+kJ+w(uv+v^2b)\}
 \leq4CD_1kJ.
\]
The last inequality uses $u(v+v^2b)\leq J$ and
$w(uv+v^2b)\leq J$, together with $k\geq1$.
The local inverse chart identifies $DF\,e$ with
$D\kappa(y)d$. Combining this estimate with
\eqref{eq:entrance-input-radius-response} proves
\eqref{eq:entrance-ordinary-radius-direction}.
The value estimate follows in the same way from
$|\kappa-r_s|\leq CJ/k$ and $|r_s-r|\leq Ab^2/k$.

For the walls, use the logarithmic arrival columns
\eqref{eq:entrance-three-wall-columns} and
\eqref{eq:entrance-active-wall-columns}. In the same
radial coordinates they give
\begin{align*}
 |D\ell_U\,e|
 &\leq C L|e_s|+(1/u+C)|e_u|
                    +(Ck/v)|e_v|+C|e_w|,\\
 |D\ell_W\,e|
 &\leq (CL/k^2)|e_s|+(C/k^2)|e_u|
                    +(C/v)|e_v|+(1/w+C/k^2)|e_w|.
\end{align*}
The relative wall bounds cancel the inverse input
amplitudes. Using $u,w\leq1$ and $k\geq1$ gives,
after another universal enlargement of $C$,
\begin{equation}
 \begin{split}
 |DU_+(y)d|&\leq CD_0k^2(1+L)U_+(y),\\
 |DW_+(y)d|&\leq CD_0(k+L)W_+(y).
 \end{split}
 \label{eq:entrance-ordinary-log-wall-response}
\end{equation}
The actual decay at arrival is
\[
 U_+\leq uv e^{-kL/3},\qquad
 W_+\leq wv e^{-L/k}.
\]
Taking square roots, while leaving one square-root
factor of each output wall unchanged in
\eqref{eq:entrance-ordinary-log-wall-response},
reduces the remaining time factors to
\begin{equation*}
 (1+L)e^{-kL/6}\leq6,\qquad
 (k+L)e^{-L/(2k)}\leq2k.
\end{equation*}
Indeed, both follow from $(1+x)e^{-x}\leq1$ for
$x\geq0$, with $x=kL/6$ and $x=L/(2k)$ respectively.
This proves \eqref{eq:entrance-ordinary-wall-direction}.
\end{proof}

Return to the positive shear family
\eqref{eq:positive-shear-joint-family}. In this subsection
we assume \eqref{eq:positive-shear-common-domain} with
$n_3>0$. Write
\[
 Y(\rho)=(M,U,V,W),\qquad k(\rho)=r(Y(\rho)),
 \qquad Z(\rho)=\mathcal P(Y(\rho)),
\]
and
\[
 \kappa(\rho)=r(Z(\rho)),\qquad
 U_+(\rho)=Z_U(\rho),\qquad W_+(\rho)=Z_W(\rho).
\]
The incoming maps in these expressions are used at
parameters with $V(\rho)<1$. Derivatives keep
$a,h,q,n_2,n_3$ fixed.

\begin{lemma}
\label{lem:small-section-entrance-directions}
There are $h_*>0$ and $D_*>0$, chosen before $h,q,\ell$,
such that the following holds. For every
$0<h\leq h_*$, $q>3$, $\ell>4$ and
$0<\delta\leq\varepsilon_0$, all sufficiently small
$a>0$ satisfy, at every parameter with $V<1$,
\begin{equation}
 \begin{split}
 &0<U,W\leq\delta,\qquad
       U+VW\leq\frac{\delta}{2k},\\
 &|M'|\leq D_*k^2,\qquad
       |U'|\leq D_*kU,\qquad
       |V'|\leq D_*kV,\qquad |W'|\leq D_*kW.
 \end{split}
 \label{eq:small-section-entrance-directions}
\end{equation}
The radius bounds $1/(16a)\leq k\leq32/a$ and $k\geq8$
also hold, and $Y$ is ordinarily differentiable.
\end{lemma}

\begin{proof}
Use Lemma~\ref{lem:small-section-entrance-input} with
$\eta=\delta$, and Proposition~\ref{prop:small-section-coordinate-response}.
Their common small-$h$ threshold is independent of
$q,\ell,\delta$. The first lemma gives
$U+VW\leq\delta/(2k)$ and hence $U\leq\delta$.
Require also $a\leq\min\{1,\delta/16\}$. Since
$n_3\leq a^\ell$ and $\ell>4$,
\[
 W\leq16n_3/a^2\leq16a^2\leq16a\leq\delta.
\]
Both walls are positive by their multiplicative
equations and $h,n_3>0$.
The lower radius bound gives $1/a\leq16k$.
Thus the relative derivative estimates
\eqref{eq:partial-core-coordinate-response} imply
\eqref{eq:small-section-entrance-directions} with
$D_*=256K_0$.
\end{proof}

\begin{proposition}
\label{prop:small-section-entrance-radial-speed}
There are constants $h_*>0$ and $B>0$, chosen before
$h,q,\ell$, such that for every $0<h\leq h_*$,
$q>3$, $\ell>4$, and all sufficiently small $a>0$,
the preceding family satisfies, wherever $V<1$,
\begin{equation}
 |\kappa-k|\leq Ba,\qquad
 |\kappa'-k'|\leq B/a,\qquad
 \frac1{8192a^2}\leq-\kappa'\leq\frac{1537}{a^2}.
 \label{eq:small-section-entrance-radial-speed}
\end{equation}
Moreover, for any prescribed $\theta>0$ the scale
threshold can be decreased so that
\begin{equation}
 |U_+'|\leq\theta\sqrt{U_+}\,|\kappa'|,\qquad
 |W_+'|\leq\theta\sqrt{W_+}\,|\kappa'|.
 \label{eq:small-section-entrance-narrow-cone}
\end{equation}
All these are ordinary derivatives, and $U_+,W_+>0$.
\end{proposition}

\begin{proof}
Take $\delta\leq\varepsilon_1$ in
Lemma~\ref{lem:small-section-entrance-directions}.
Proposition~\ref{prop:entrance-ordinary-direction}
applies to $d=Y'(\rho)$ with $D_0=D_*$.
Here $b=U+W\leq1$ and $J=UV+V^2b^2\leq2$.
Using the two radius bounds gives
\[
 |\kappa-k|\leq3C/k\leq48Ca,\qquad
 |\kappa'-k'|\leq CD_*(1+2k)
                         \leq96CD_*/a.
\]
Choose $B=48C+96CD_*+1$. The original radial speed
is bounded by \eqref{eq:partial-core-radius-response}.
If $a\leq1/(8192B)$, then
$B/a\leq1/(8192a^2)$.
Subtracting this error from the lower bound for
$-k'$, and adding it to the upper bound, proves
\eqref{eq:small-section-entrance-radial-speed}.

For the wall slopes, the radius upper bound and the
new lower speed give
\[
 k^2\leq\frac{1024}{a^2}
              \leq2^{23}|\kappa'|.
\]
Let $E=2^{23}CD_*$ and choose
\[
 \delta=\min\{\varepsilon_1,(\theta/E)^2\}.
\]
After decreasing $a$ as required by the preceding
lemma, both $\sqrt{UV}$ and $\sqrt{WV}$ are at most
$\sqrt\delta$, because $V<1$.
Use \eqref{eq:entrance-ordinary-wall-direction} and
$k\leq k^2$ to obtain
\[
 |U_+'|\leq E\sqrt\delta\,\sqrt{U_+}\,|\kappa'|,
 \qquad
 |W_+'|\leq E\sqrt\delta\,\sqrt{W_+}\,|\kappa'|.
\]
Since $E\sqrt\delta\leq\theta$, these are
\eqref{eq:small-section-entrance-narrow-cone}.
Differentiability and positivity follow from the
ordinary arrival map and its positive wall multipliers.
\end{proof}

The first downward event from $Y$ follows a full
core passage from $Z$. To transfer derivatives as well
as states, this identification is needed on a
neighborhood of the parameter.

Put $X=(\kappa,U_+,W_+)$ and let $\mathcal R$ denote
the full core return in radius and wall coordinates.
Near a parameter with $V<1$, the upward arrival
satisfies $Z_V=1$ and $Z_M>1$, so
$Z=\iota_+(X)$. The full core time and the downward
time from this state agree at the reference point.
Both are continuous nearby and solve the same
transverse arrival equation near that time. The local
uniqueness from the implicit function theorem makes
them equal on a neighborhood. Combining this fact
with \eqref{eq:entrance-downward-identity} gives
\begin{equation}
 \mathcal R_\downarrow(Y(\rho))
                   =\mathcal R(X(\rho))
 \quad\hbox{on a neighborhood of the reference parameter}.
 \label{eq:entrance-full-core-identity}
\end{equation}
Consequently their ordinary derivatives agree there.

\begin{proposition}
\label{prop:entrance-core-connection-cone}
For the positive-wall family above, after a common
decrease of the scale thresholds, write
\[
 (f,u,w)=\mathcal R_\downarrow(Y),\qquad
 G=\Psi(Y)=(g,\widehat u,\widehat w),\qquad
 \mu=\sqrt{f^2-1}-1.
\]
At every parameter with $V<1$,
$\kappa',f',g'<0$, and
\begin{align}
 \left|\frac{f'}{\kappa'}-1\right|
       &\leq\frac1{32\kappa},&
 \frac34\leq\frac{f'}{\kappa'}&\leq\frac54,
 \label{eq:entrance-full-core-radial-ratio}\\
 |u'|&\leq\left(1-\frac1{16\kappa}\right)\sqrt u\,|f'|,&
 |w'|&\leq\left(1-\frac1{16\kappa}\right)\sqrt w\,|f'|,
 \label{eq:entrance-full-core-cone}\\
 \left|\frac{g'}{f'}-1\right|
       &\leq\frac1{4\mu},&
 \frac34\leq\frac{g'}{f'}&\leq\frac54,
 \label{eq:entrance-connection-radial-ratio}\\
 |\widehat u'|&\leq\sqrt{\widehat u}\,|g'|,&
 |\widehat w'|&\leq\sqrt{\widehat w}\,|g'|.
 \label{eq:entrance-connection-cone}
\end{align}
Both reciprocal phase derivatives are positive, and
\begin{equation}
 \frac{(2/g)'}{(2/f)'}
                  =\left(\frac fg\right)^2\frac{g'}{f'}.
 \label{eq:entrance-connection-reciprocal-ratio}
\end{equation}
\end{proposition}

\begin{proof}
Choose the core threshold $\varepsilon_c$ from
Proposition~\ref{prop:large-core-cone}.
Use Lemma~\ref{lem:small-section-entrance-directions}
with $\delta=\varepsilon_c$, and
Proposition~\ref{prop:small-section-entrance-radial-speed}
with $\theta=1$.
By Lemma~\ref{lem:entrance-core-smallness}, the upward
arrival has
\[
 \kappa\geq6,\qquad
 0<U_+,W_+\leq\varepsilon_c/\kappa.
\]
Since $\kappa'<0$, the tangent $X'/\kappa'$ has radial
component one and wall components bounded by
$\sqrt{U_+}$ and $\sqrt{W_+}$.
Apply the core cone estimate to this vector.
By linearity and \eqref{eq:entrance-full-core-identity},
the radial component of its image is $f'/\kappa'$.
The estimate \eqref{eq:large-core-cone-margin} gives
\eqref{eq:entrance-full-core-radial-ratio};
in particular $f'<0$.
Multiplying the two wall inequalities by $|\kappa'|$
gives \eqref{eq:entrance-full-core-cone}.

Next choose the long-connection threshold
$\varepsilon_\ell$ of \eqref{eq:long-connection-cone-threshold}.
Apply Lemma~\ref{lem:small-section-entrance-input} with
$\eta=\varepsilon_\ell$. At the ensuing core exit,
Proposition~\ref{prop:partial-connection-actual} gives
\[
 \mu\geq4,\qquad
 0<u\leq\varepsilon_\ell/\mu,\qquad
 0<w\leq\varepsilon_\ell.
\]
The factor in \eqref{eq:entrance-full-core-cone} is
at most one, so the input tangent to this connection
satisfies $|u'|\leq\sqrt u\,|f'|$ and
$|w'|\leq\sqrt w\,|f'|$.
In connection coordinates, set
$\chi=(\mu+1)/H(\mu+1,1)$; differentiation gives
$\chi\mu'=f'$.
Thus $(\mu',u',w')/f'$ has unit physical radial
derivative. Proposition~\ref{prop:long-connection-cone},
followed by multiplication by $f'<0$, proves
\eqref{eq:entrance-connection-radial-ratio} and
\eqref{eq:entrance-connection-cone}, with $g'<0$.
Finally $f,g>0$. Differentiating their reciprocals
gives the positivity and
\eqref{eq:entrance-connection-reciprocal-ratio}.
\end{proof}

\begin{corollary}[The whole positive shear interval]
\label{cor:positive-shear-all-active-cone}
There is $h_*>0$ such that, for every $0<h\leq h_*$,
$q>3$, $\ell>4$, and all sufficiently small $a>0$,
the common return $G=(g,\widehat u,\widehat w)$ of
\eqref{eq:positive-shear-common-connection} satisfies
\begin{equation}
 g'<0,\qquad
 |\widehat u'|\leq\sqrt{\widehat u}\,|g'|,\qquad
 |\widehat w'|\leq\sqrt{\widehat w}\,|g'|,\qquad
 (2/g)'>0
 \label{eq:positive-shear-all-active-cone}
\end{equation}
at every $\rho\in[a,2a]$ under
\eqref{eq:positive-shear-common-domain} with $n_3>0$.
The return and its phase are $C^2$ functions
throughout this interval.
\end{corollary}

\begin{proof}
Take the common thresholds of
Theorem~\ref{thm:small-section-connection} and
Proposition~\ref{prop:entrance-core-connection-cone}.
At a point with $V\geq1$, the former applies; at a
point with $V<1$, the latter applies.
Corollary~\ref{cor:positive-shear-common-connection}
identifies their return with the same $C^2$
map $G$, including at $V=1$.
Thus the derivative bounds hold at every parameter.
The phase is $C^2$ because $g>0$, and its derivative
is $-2g'/g^2>0$.
\end{proof}

The incoming wait has therefore preserved the radial
orientation and supplied the transverse slope control
needed at the next core. The subsequent phase comparison
can now be made along the whole positive shear interval.