\subsection{The connection phase at bounded radii}
\label{sec:compact-connection-phase}

The finite outgoing phase has the first-order expansion
$R-2-2u/R$. The incoming estimates now show that this expansion
also describes the radius at the next core entrance. The remaining
error contains the old weak wall as a factor, because that wall
becomes the growing wall of the incoming passage.

For the actual reflection of
Proposition~\ref{prop:compact-reflection-passage}, denote the
next core radius by $R_+(R,u,w)$. Choose the intermediate height
$\eta$ sufficiently small and then hold it fixed. Its reflected
entrance, in radial coordinates, is
\begin{equation}
 z_\eta(R,u,w)
     =\bigl(F_\eta(R,u,w),\,uB_U(R,u,w),\,
                         wB_W(R,u,w),\,\eta\bigr).
 \label{eq:compact-radial-entrance}
\end{equation}
Here $F_\eta$ is the finite outgoing phase and $B_U,B_W$ are the
positive factors from Lemma~\ref{lem:compact-outgoing-tail}.

\begin{lemma}[Error through the incoming segment]
\label{lem:compact-incoming-error}
There exists $\delta>0$ such that, for every fixed
$0<\eta\leq\delta$, there are $\varepsilon>0$ and $C_\eta>0$
with the following property. For
\[
 31/8\leq R\leq6,\qquad 0<u,w\leq\varepsilon,
\]
the actual reflection is defined and the function
$E_\eta=R_+-F_\eta$ is locally $C^2$. It satisfies
\begin{align}
 |E_\eta|+|\partial_RE_\eta|
     &\leq C_\eta w(u+w),\notag\\
 |\partial_uE_\eta|&\leq C_\eta w,\qquad
 |\partial_wE_\eta|\leq C_\eta(u+w).
 \label{eq:compact-incoming-error}
\end{align}
\end{lemma}

\begin{proof}
Take $\delta$ at most the universal stable-wall threshold in
Proposition~\ref{prop:entrance-physical-bounds}.
For fixed $\eta$, compactness of the finite outgoing family gives
a common signed neighborhood on which $F_\eta,B_U,B_W$ are
$C^2$, the two factors are positive, and
\[
 \|DF_\eta\|,\quad B_U,\quad B_W,\quad
 \|DB_U\|,\quad\|DB_W\|\leq A_\eta
\]
for some $A_\eta\geq1$. The bounds follow from the finite phase
regularity and~\eqref{eq:compact-tail-factor-bounds}.
Reduce the input threshold so that $u+w\leq1$,
$uB_U\leq\delta$, and $wB_W\leq1/2$.
The reflected entrance has $k_\eta\geq7/4$ and first coordinate
greater than one. The physical incoming estimates therefore apply.

At any given input choose its local incoming radial chart, and
write $F$ for the physical arrival radius in that chart.
The composition of actual first-arrival maps holds on a
neighborhood, so
\begin{equation*}
 R_+=F\circ z_\eta,\qquad
 E_\eta=F\circ z_\eta-F_\eta
\end{equation*}
there. This also proves the $C^2$ assertion.
In~\eqref{eq:entrance-compact-phase-bounds}, substitute the
incoming amplitudes
$\widetilde u=uB_U$, $\widetilde v=wB_W$ and
$\widetilde w=\eta$. With $b=u+w$, the result is
\begin{align*}
 |E_\eta|+|F_r-1|&\leq B A_\eta^2 wb,\\
 |F_u|&\leq B A_\eta w,\qquad
 |F_v|\leq B A_\eta b.
\end{align*}
The derivatives of $F$ here are evaluated at $z_\eta$.

For $h=(h_R,h_u,h_w)$, the chain rule gives the exact identity
\begin{equation*}
\begin{split}
 DE_\eta[h]={}&(F_r-1)DF_\eta[h]\\
 &+F_u\bigl(B_Uh_u+u\,DB_U[h]\bigr)\\
 &+F_v\bigl(B_Wh_w+w\,DB_W[h]\bigr).
\end{split}
\end{equation*}
The fourth coordinate of $z_\eta$ is constant.
Each of the three terms containing a derivative of
$F_\eta,B_U,B_W$ is bounded, for any unit coordinate direction,
by a constant depending on $\eta$ times $wb$.
For example, the latter two are at most
$BA_\eta^2wu$ and $BA_\eta^2bw$.
In the radial direction these are all the terms.
In the $u$ direction there is in addition $F_uB_U$, bounded
by $BA_\eta^2w$; in the $w$ direction there is
$F_vB_W$, bounded by $BA_\eta^2b$.
Use $b,w\leq1$ and enlarge the common constant to obtain
\eqref{eq:compact-incoming-error}.
\end{proof}

\begin{proposition}[Expansion of the actual connection phase]
\label{prop:compact-connection-phase}
There are $\varepsilon>0$ and $C>0$ such that, for
$31/8\leq R\leq6$ and $0<u,w\leq\varepsilon$, the same actual
reflection is locally $C^2$ and its next core radius satisfies
\begin{align}
 \left|R_+(R,u,w)-\left(R-2-\frac{2u}{R}\right)\right|
     &\leq C(u+w)^2,\label{eq:compact-phase-expansion}\\
 |\partial_RR_+-1|&\leq C(u+w^2),
 \label{eq:compact-phase-radial}\\
 \left|\partial_uR_++\frac2R\right|+|\partial_wR_+|
     &\leq C(u+w).
 \label{eq:compact-phase-walls}
\end{align}
\end{proposition}

\begin{proof}
Fix one height $\eta$ allowed by
Lemma~\ref{lem:compact-incoming-error}, and take a common
input threshold with Proposition~\ref{prop:finite-phase-expansion}.
The identity $R_+=F_\eta+E_\eta$ and $w(u+w)\leq(u+w)^2$
give~\eqref{eq:compact-phase-expansion}.
For the radial derivative, use the stronger finite estimate
$|\partial_RF_\eta-1|\leq C_\eta(u+w^2)$ and observe that
\[
 w(u+w)=uw+w^2\leq u+w^2.
\]
This proves~\eqref{eq:compact-phase-radial}.
The two wall derivatives follow by adding
\eqref{eq:finite-phase-wall-bound} to the corresponding bounds
in~\eqref{eq:compact-incoming-error}.
The constant is fixed once $\eta$ has been chosen.
\end{proof}

Thus the first-order strong-wall response survives the entire
connection. The expansion has no linear weak-wall term, and the
radial derivative retains the more precise scale $u+w^2$.
These distinctions will matter when the phase estimate is combined
with the contraction of the two output walls.