\subsection{From the remaining core to the next entrance}
\label{sec:partial-core-connection}

After the downward core crossing, the growing wall determines
the next entrance. The estimates for a long connection apply
once its parameters are expressed in terms of the actual
core exit. We first make this identification, then follow
the radius and its derivative through the connection.
The resulting phase differs from the fractional-linear
expression associated with a radius decrement of two by
an error that retains the initial wall amplitudes.

Let $y_0$ satisfy \eqref{eq:partial-core-initial-state}, and
write
\[
 k=r(y_0),\quad U=U_0,\quad \Pin=V_0W_0,\qquad
 (f,u,w)=\mathcal R_\downarrow(y_0).
\]
The connection parameters are
\begin{equation}
 \pi=(\mu,u,w),\qquad
 \mu=\sqrt{f^2-1}-1,\qquad H(\mu+1,1)=f,
 \label{eq:partial-connection-parameters}
\end{equation}
where $H(a,b)=\sqrt{a^2+b^2}$.
Denote by $T_c(\pi)$ the first $W=1$ time of the
connection in Section~\ref{sec:connection-passage}.
At that event, write the state in the original frame as
$(-m_*,U_*,V_*,1)$. Its core coordinates in the reflected
frame are
\begin{equation}
 \Psi(y_0)=(g,\widehat u,\widehat w)
              =(H(m_*-1,1),U_*,V_*).
 \label{eq:partial-connection-output}
\end{equation}

\begin{proposition}
\label{prop:partial-connection-actual}
Suppose that $Z\geq0$, $2\cosh Z\leq k$,
$z(y_0)\leq Z$, and
\[
 0<\eta\leq\varepsilon_0,\qquad
 0<U,\Pin\leq\eta/k.
\]
Then the connection is defined, and
\begin{equation}
 4\leq\mu,\qquad k/2\leq\mu<k,\qquad
 0<u\leq\frac{\eta}{\mu},\qquad0<w\leq\eta.
 \label{eq:partial-connection-domain}
\end{equation}
More precisely, $u\leq Ue^{-k/12}$ and $w\leq \Pin$.
The parameter map $\pi$, the total time
$\tau_\downarrow+T_c\circ\pi$, and $\Psi$ are locally
$C^2$. With the embeddings and reflection of
Section~\ref{sec:era-accumulation},
\begin{equation}
 \iota_+(\Psi(y_0))
   =\mathsf S\Phi_{\tau_\downarrow(y_0)+T_c(\pi(y_0))}(y_0).
 \label{eq:partial-connection-orbit}
\end{equation}
The connection ends at the first $W=1$ event after the
downward core exit, and $g\geq k/4$.
\end{proposition}

\begin{proof}
At the downward crossing, $V=1$ and $M<1$, so the
actual state is
\[
 (1-\sqrt{f^2-1},u,1,w)=(-\mu,u,1,w).
\]
The core energy bound gives
$|(\mu+1)^2+1-k^2|\leq1/8$. Together with $k\geq6$
and $\mu+1>0$, this implies
$4\leq\mu$, $k/2\leq\mu<k$. For the first two
inequalities, an assumed upper bound
$\mu<4$ or $\mu<k/2$ contradicts the lower energy bound.
For the third, $\mu\geq k$ contradicts the upper bound.

Positivity of the scalar wall multipliers and
Lemma~\ref{lem:partial-core-wall-contraction} give
\[
 0<u\leq Ue^{-k/12}\leq\eta/k\leq\eta/\mu,
 \qquad 0<w\leq \Pin\leq\eta/k\leq\eta.
\]
Thus Proposition~\ref{prop:long-connection-arrival}
applies to the actual exit state. The flow composition
identity gives \eqref{eq:partial-connection-orbit}.
The connection band has $|m_*-\mu|\leq1/2$; hence
$m_*-1\geq\mu-3/2\geq\mu/2\geq k/4$.
It follows that $g=H(m_*-1,1)\geq k/4$ and that
the reflected state belongs to the positive entrance
branch $\iota_+$.

For regularity, $f>1$ makes the change
\eqref{eq:partial-connection-parameters} smooth.
Proposition~\ref{prop:partial-core-smooth-exit}
and Proposition~\ref{prop:long-connection-arrival}
give the $C^2$ time and state maps. Their composition
proves the claim, including at an incoming boundary
point with $V_0=1$.
\end{proof}

Consider a differentiable family $Y(\rho)$ satisfying
the hypotheses of
Proposition~\ref{prop:partial-core-radial-family} at
the parameter under consideration. Use $U,\Pin,k,z$ for
its initial core quantities and
$(f,u,w)=\mathcal R_\downarrow\circ Y$ for the
outgoing quantities. Suppose also that $U,\Pin>0$ and
$z\leq Z$, $2\cosh Z\leq k$ for some $Z\geq0$.
Write $(g,\widehat u,\widehat w)=\Psi\circ Y$.
All primes below are ordinary derivatives in $\rho$,
with the scale $a$ fixed.

The connection error from
\eqref{eq:connection-phase-error-definition}, evaluated
on this curve, is
\begin{equation}
 \Delta=g-f+2.
 \label{eq:partial-connection-error}
\end{equation}
This equality holds on a neighborhood of the base
parameter. Indeed, \eqref{eq:partial-connection-parameters}
there identifies the connection's entrance radius with
$f$, so it can also be differentiated.

\begin{proposition}
\label{prop:partial-connection-phase}
For each $K\geq1$ there is $C_K>0$, depending only
on $K$, such that
\begin{equation}
 |\Delta|\leq C_Ka(U+\Pin^2),\qquad
 |\Delta'|\leq C_K(U+\Pin^2).
 \label{eq:partial-connection-error-bounds}
\end{equation}
Put $E_K=16K+4C_pK^2+C_K$. Then $g\geq k/4$ and
\begin{equation}
 |g-(k-2)|\leq E_K(U+a\Pin^2),\qquad
 |g'-k'|\leq E_K\frac{U+a\Pin}{a^2}.
 \label{eq:partial-connection-radius-bounds}
\end{equation}
The reciprocal phase error
\begin{equation}
 \mathcal E=\frac2g-\frac{2/k}{1-2/k}
 \label{eq:partial-connection-reciprocal-error}
\end{equation}
is differentiable and satisfies
\begin{align}
 |\mathcal E|&\leq8E_KK^2a^2(U+a\Pin^2),\notag\\
 |\mathcal E'|&\leq(8E_KK^2+32E_KK^4)(U+a\Pin).
 \label{eq:partial-connection-reciprocal-bounds}
\end{align}
\end{proposition}

\begin{proof}
Proposition~\ref{prop:partial-connection-actual},
with $\eta=\varepsilon_0$, supplies the connection
and its regularity. In particular $g\geq k/4$.
The preceding radius and multiplier estimates give
\begin{gather*}
 |f'|\leq F_K/a^2,\qquad F_K=K+8C_pK^2,\\
 0<u\leq Ue^{-k/12},\qquad0<w\leq \Pin,\\
 |u'|\leq u\left(\frac Ka+\frac{A_m\log(4k)}{a^2}\right),
 \qquad
 |w'|\leq w\left(\frac Ka+B_m\log(4k)\right).
\end{gather*}
The first estimate follows from
\eqref{eq:partial-core-family-radius-error} and
$U+a\Pin\leq2$. The constants $A_m,B_m$ are those in
\eqref{eq:partial-core-radial-multiplier-constants}.

The exponential decay also removes the extra inverse
scale in the strong derivative. Since $1/a\leq Kk$,
\[
 \frac{e^{-k/12}\log(4k)}a
 \leq K\bigl(e^{-k/24}\log(4k)\bigr)
          \bigl(ke^{-k/24}\bigr)\leq576K.
\]
The two factors in parentheses are each at most $24$.
Using $\log(4k)\leq k\leq K/a$ for the weak wall, we obtain
\begin{equation*}
 a|u'|\leq A_cU,\qquad a|w'|\leq B_c\Pin,\qquad
 A_c=K(1+576A_m),\quad B_c=K(1+B_m).
\end{equation*}

Differentiate $H(\mu+1,1)=f$:
\[
 \chi\mu'=f',\qquad
 \chi=\frac{\mu+1}{H(\mu+1,1)}\geq\frac12.
\]
Thus $|\mu'|\leq2F_K/a^2$.
Also $\mu\geq k/2$ and $k\geq2/(Ka)$ imply
$1/\mu\leq Ka$.
The value bound \eqref{eq:connection-radius-decrement}
therefore yields
\[
 |\Delta|\leq24Ka(U+\Pin^2).
\]
Apply the three ordinary derivative bounds in
\eqref{eq:connection-phase-error-derivatives} to
$(\mu',u',w')$. They give
\begin{align*}
 |\Delta'|
 &\leq50000\left(
   \frac{u+w^2}{\mu^2}|\mu'|
      +\frac{|u'|}{\mu}
      +\frac{u+w}{\mu}|w'|\right)\\
 &\leq50000\bigl(2K^2F_K+KA_c+KB_c\bigr)(U+\Pin^2).
\end{align*}
For the last term, use $(U+\Pin)\Pin\leq U+\Pin^2$, since
$\Pin\leq1$. We may consequently choose
\[
 C_K=1+24K+50000(2K^2F_K+KA_c+KB_c).
\]
This proves \eqref{eq:partial-connection-error-bounds}.

Now use the local identity $g=f-2+\Delta$.
Adding the radius errors, with
$a(U+\Pin^2)\leq U+a\Pin^2$, gives the value estimate in
\eqref{eq:partial-connection-radius-bounds}.
For the derivative, use
$a^2(U+\Pin^2)\leq U+a\Pin$ and
\eqref{eq:partial-core-family-radius-error}.
The single constant $E_K$ bounds both sums.

Finally set $b=k-2$ and $e=g-b$. Since $k\geq6$,
the fractional expression in
\eqref{eq:partial-connection-reciprocal-error} equals
$2/b$ on a neighborhood of the base point. Moreover,
\[
 g,b\geq\frac{k}{4}\geq\frac1{2Ka}.
\]
The reciprocal identities give
\[
 \mathcal E=-\frac{2e}{gb},\qquad
 \mathcal E'=-\frac{2e'}{g^2}
              +\frac{2e(g+b)k'}{g^2b^2}.
\]
They imply
\[
 |\mathcal E|\leq8K^2a^2|e|,\qquad
 |\mathcal E'|\leq8K^2a^2|e'|+32K^4a|e|.
\]
Insert \eqref{eq:partial-connection-radius-bounds}
and use $a(U+a\Pin^2)\leq U+a\Pin$. This proves
\eqref{eq:partial-connection-reciprocal-bounds}.
\end{proof}

We next apply the cone estimate to the same family.
For the positive-shear inputs from the small section,
the outgoing cone can be made arbitrarily narrow before
the connection is taken.

\begin{lemma}
\label{lem:small-section-core-cone-decay}
After decreasing $h_*$ if necessary, for each
$0<h\leq h_*$ there is $C_h>0$, chosen before
$q>3$ and $\ell>4$, with the following property.
For every $\delta>0$ and sufficiently small $a>0$,
the inputs of
Corollary~\ref{cor:small-section-partial-core-cone}
satisfy
\begin{equation}
 E_a=a^{(q-2)/2}+\sqrt{n_3/a},\qquad C_hE_a\leq\delta,
 \label{eq:small-section-core-cone-rate}
\end{equation}
and
\begin{equation}
 |u'|\leq C_hE_a\sqrt u\,|f'|,\qquad
 |w'|\leq C_hE_a\sqrt w\,|f'|.
 \label{eq:small-section-core-cone-decay}
\end{equation}
Here $(f,u,w)=\mathcal R_\downarrow\circ Y$ and $f'<0$.
\end{lemma}

\begin{proof}
The initial core amplitudes obey
$U\leq32h a^{q-2}$ and $\Pin\leq2048n_3/a^3$.
Thus
\[
 \sqrt U\leq\sqrt{32h}\,a^{(q-2)/2},\qquad
 a\sqrt \Pin\leq\sqrt{2048}\sqrt{n_3/a}.
\]
Use \eqref{eq:partial-core-radial-cone} with $D=4096$.
Its two coefficients are bounded by $C_hE_a$ if
\[
 C_h=1+8192(K+24A_m)\sqrt{32h}
                         +8192K(1+B_m)\sqrt{2048}.
\]
Finally $n_3\leq a^\ell$ implies
$\sqrt{n_3/a}\leq a^{(\ell-1)/2}$. Both exponents
are positive, so the further scale restriction
$C_hE_a\leq\delta$ is uniform in the permitted $n_3$.
The sign of $f'$ follows from
\eqref{eq:partial-core-cone-radial-speed}.
\end{proof}

\begin{theorem}
\label{thm:small-section-connection}
There are $h_*>0$ and constants $K\geq1$, $C_K>0$
such that the following holds for $0<h\leq h_*$,
$q>3$, $\ell>4$ and sufficiently small $a>0$.
Take the positive-shear family
\eqref{eq:radial-shear-family}, now with
$0<n_3\leq a^\ell$, and its actual state
$Y(\rho)$ at $T=(q/2)\log(1/a)$.
At every $\rho\in[a,2a]$ with $V(Y(\rho))\geq1$,
the next entrance
$G(\rho)=\Psi(Y(\rho))=(g,\widehat u,\widehat w)$
is on the original orbit, at time
\begin{equation}
 T+\tau_\downarrow(Y(\rho))+T_c(\pi(\rho)),
 \label{eq:small-section-connection-time}
\end{equation}
with the reflected frame.
It satisfies the phase estimates
\eqref{eq:partial-connection-error-bounds}--%
\eqref{eq:partial-connection-reciprocal-bounds}.

The maps $f$ and $G$ are differentiable, $g>0$,
$f',g'<0$, and
\begin{equation}
 \left|\frac{g'}{f'}-1\right|\leq\frac1{4\mu},\qquad
 \frac34\leq\frac{g'}{f'}\leq\frac54.
 \label{eq:small-section-connection-radial-ratio}
\end{equation}
The two walls are positive, and
\begin{equation}
 |\widehat u'|\leq\sqrt{\widehat u}\,|g'|,\qquad
 |\widehat w'|\leq\sqrt{\widehat w}\,|g'|.
 \label{eq:small-section-connection-cone}
\end{equation}
Both reciprocal phase derivatives are positive, and
\begin{equation}
 \frac{(2/g)'}{(2/f)'}
             =\left(\frac fg\right)^2\frac{g'}{f'}>0.
 \label{eq:small-section-connection-phase-ratio}
\end{equation}
\end{theorem}

\begin{proof}
Choose the common small-$h$ threshold for the preceding
small-section estimates. Their $K$ is chosen before
$h,q,\ell$, and
Proposition~\ref{prop:partial-connection-phase} chooses
$C_K$ from $K$ alone.
The actual strong wall is positive because $h>0$.
The sign preservation of $N_3$ and the positive
projective height give $W(Y(\rho))>0$; hence $\Pin>0$.
The other hypotheses of the phase proposition follow
from Corollary~\ref{cor:small-section-partial-core-response}
and $z\leq0$. Take $Z=0$.

For the cone, use the threshold $\varepsilon_\ell$
in \eqref{eq:long-connection-cone-threshold}.
Proposition~\ref{prop:small-section-core-data}, with
entry bound $\eta=\varepsilon_\ell$, and
Proposition~\ref{prop:partial-connection-actual} give
\[
 0<u\leq\varepsilon_\ell/\mu,\qquad
 0<w\leq\varepsilon_\ell.
\]
Apply Lemma~\ref{lem:small-section-core-cone-decay}
with $\delta=1$. The input curve then satisfies
$|u'|\leq\sqrt u\,|f'|$ and
$|w'|\leq\sqrt w\,|f'|$.
Differentiating \eqref{eq:partial-connection-parameters}
shows that its connection tangent is
$v=(\mu',u',w')$ with $\chi\mu'=f'$.
Since $f'<0$, the vector $v/f'$ has unit input
radial derivative and lies in the cone of
Proposition~\ref{prop:long-connection-cone}.

Let $d$ be its image under the connection differential.
Linearity gives $G'=f'd$. That proposition yields
$d_r\geq3/4$ and $|d_r-1|\leq1/(4\mu)$, which prove
\eqref{eq:small-section-connection-radial-ratio}
and $g'<0$. Its transverse bounds imply
\eqref{eq:small-section-connection-cone}.
Positive wall multipliers along both passages give
$\widehat u,\widehat w>0$.
The chain rule for reciprocal functions proves
\eqref{eq:small-section-connection-phase-ratio}.

Finally, the downward event is the first $V=1$
crossing strictly after $T$ by
Corollary~\ref{cor:small-section-partial-core-cone}.
The following connection ends at its first $W=1$
crossing. Applying \eqref{eq:partial-connection-orbit}
after the fixed initial time $T$ gives
\eqref{eq:small-section-connection-time}.
All derivatives refer to these actual event maps,
including when $V(Y(\rho))=1$.
\end{proof}

The same original curve has now reached an upward
core entrance with a controlled phase and a tangent
in the radial cone. The next step is to obtain such an entrance
also when the fixed initial decay ends outside the
core, giving a common section on which to apply the estimates.