\subsection{The radius at the next core}
\label{sec:entrance-physical-radius}

The stable radial coordinate controls the passage through a long
incoming segment. At the next core, the return map is expressed in
the core radius $\rho=H(M-1,V)$. The difference between these
two radii is quadratic in the output walls. Retaining those walls
in the chain rule makes the conversion uniform in the waiting time.

Use the input coordinates and arrival map of
Section~\ref{sec:entrance-variation}, and define
\begin{equation}
 F(z)=\rho\bigl(\mathcal P(\Psi(z))\bigr),\qquad
 z=(r,u,v,w).
 \label{eq:entrance-physical-return}
\end{equation}
All derivatives below are in these input coordinates, with
$k=H(m-1,v)$ fixed at the reference point.

\begin{proposition}[Physical arrival estimates]
\label{prop:entrance-physical-bounds}
There are universal constants $C>0$ and
$0<\varepsilon\leq10^{-4}$ such that, if
\[
 k\geq7/4,\qquad m\geq1,\qquad
 0<u,w\leq\varepsilon,\qquad 0<v<1,
\]
then $F$ is locally $C^2$. At $z=(r(y),u,v,w)$, put
$L=\log(1/v)$ and $\omega=e^{-2L/k}$. Then
\begin{align}
 |F-r|&\leq\frac Ck\bigl[uv+v^2w^2(1+\omega)\bigr],
 \label{eq:entrance-physical-value}\\
 |\partial_rF-1|
   &\leq\frac C{k^2}
       \bigl[uv+v^2w^2\{1+(1+L/k)\omega\}\bigr],
 \label{eq:entrance-physical-radial}\\
 |\partial_uF|&\leq Cv/k,\qquad
 |\partial_vF|\leq C(u+vw^2),\notag\\
 |\partial_wF|&\leq\frac Ck\bigl[uv+v^2w(1+\omega)\bigr].
 \label{eq:entrance-physical-walls}
\end{align}
\end{proposition}

Write $U_+=\mathcal P_U(y)$, $W_+=\mathcal P_W(y)$ and
$b_+=U_++W_+$. The decay estimates give
\begin{equation}
 U_+\leq uv\,a,\qquad W_+\leq wv\,c,\qquad
 a=e^{-kL/3},\quad c=e^{-L/k},\quad c^2=\omega.
 \label{eq:entrance-terminal-decay}
\end{equation}
Both $a$ and $c$ belong to $(0,1]$. The elementary inequalities
$xe^{-x}\leq1$ and $(1+x)e^{-x}\leq1$, for $x\geq0$, imply
\begin{equation}
 La\leq3/k,\qquad (1+L/k)\omega\leq1.
 \label{eq:entrance-terminal-clock}
\end{equation}
These bounds absorb the waiting length in radial derivatives.
The other columns require the wall amplitudes to remain visible.

\begin{lemma}[Endpoint products]
\label{lem:entrance-terminal-products}
Set $S=v^2w^2\omega$. For the positive inputs of
Proposition~\ref{prop:entrance-physical-bounds},
\begin{align}
 b_+^2&\leq2(uv+S),&
 kLb_+U_+&\leq6uv,\notag\\
 (L/k)b_+W_+&\leq3uv+(L/k)S,&
 b_+U_+&\leq2uv,\notag\\
 b_+W_+&\leq w(uv+v^2w\omega),&
 b_+&\leq2v.
 \label{eq:entrance-terminal-products}
\end{align}
\end{lemma}

\begin{proof}
All initial amplitudes are at most one. Equation
\eqref{eq:entrance-terminal-decay} gives
$U_+^2\leq uv$, $W_+^2\leq S$, and $U_+,W_+\leq v$.
This proves the first and last estimates.
Since $U_+,W_+\leq1$, it also gives
\[
 b_+U_+\leq2uv\,a,\qquad
 b_+W_+\leq uv\,a+S.
\]
Multiplication by $kL$ or $L/k$, followed by
\eqref{eq:entrance-terminal-clock} and $k\geq1$, proves the
two estimates containing $L$. Dropping $a\leq1$ gives
$b_+U_+\leq2uv$.
Finally $W_+\leq w$, so
$U_+W_+\leq uvw$ and
$b_+W_+\leq uvw+S=w(uv+v^2w\omega)$.
\end{proof}

Let $J=r-\rho$ be the correction from
Lemma~\ref{lem:entrance-radius-correction}, and set
\[
 E=J\circ\mathcal P\circ\Psi.
\]
The identity $F=Q-E$ holds on a neighborhood of the input.
For a direction $d$, let
$\xi=D(\mathcal P\circ\Psi)[d]$ be the actual endpoint variation.
The output active wall is identically one, hence $\xi_V=0$.
In endpoint radial coordinates the remaining components of $\xi$
are
\[
 DQ[d],\qquad U_+D\ell_U[d],\qquad W_+D\ell_W[d].
\]
The correction bounds therefore yield
\begin{equation}
 |DE[d]|\leq T\left[
       \frac{b_+^2}{k^2}|DQ[d]|
       +\frac{b_+}{k}
          \bigl(U_+|D\ell_U[d]|+W_+|D\ell_W[d]|\bigr)
                       \right]
 \label{eq:entrance-terminal-correction}
\end{equation}
for a universal $T>0$. The output remains in the incoming region
with reference radius $k$, so this is the same $k$ as in the
input estimates.

\begin{proof}[Proof of Proposition~\ref{prop:entrance-physical-bounds}]
Choose a common constant $A\geq1$ for the phase and logarithmic
wall estimates in Propositions~\ref{prop:entrance-three-columns}
and~\ref{prop:entrance-active-column}. Decrease $\varepsilon$ to
their common threshold. The regularity of the actual arrival and
the positivity of its radius give the stated $C^2$ regularity.
The value estimate is Proposition~\ref{prop:incoming-phase-value}.
It remains to convert the four derivative columns.

For the radial column, put $H=uv+(vw)^2\leq2$. Then
\[
 |Q_r|\leq1+2A,\qquad
 |(\ell_U)_r|\leq AL,\qquad
 |(\ell_W)_r|\leq AL/k^2.
\]
Using Lemma~\ref{lem:entrance-terminal-products} in
\eqref{eq:entrance-terminal-correction}, we obtain
\begin{align*}
 |E_r|
 &\leq\frac T{k^2}
       \left[2(1+2A)(uv+S)+A\{9uv+(L/k)S\}\right]\\
 &\leq\frac{T(2+13A)}{k^2}
                     \bigl[uv+(1+L/k)S\bigr].
\end{align*}
Adding $|Q_r-1|\leq AH/k^2$ proves
\eqref{eq:entrance-physical-radial}.

For the wall columns, define
\[
 J_0=uv+v^2w(1+\omega),\qquad K_0=u+vw^2.
\]
The endpoint products imply
\begin{gather*}
 \frac{b_+^2}{k^2}\leq4,\qquad
 b_+^2\leq4v,\qquad b_+^2\leq2J_0,\\
 \frac{b_+U_+}{u}\leq2v,\qquad
 \frac{b_+W_+}{w}\leq J_0,\qquad
 \frac{b_+U_+}{v}\leq2u,\qquad
 \frac{b_+W_+}{v}\leq K_0.
\end{gather*}
For example, $b_+^2\leq4v^2\leq4v$ follows from
$b_+\leq2v$, and $b_+^2\leq2J_0$ follows from
$w^2\leq w$. The final inequality follows from
$b_+W_+\leq uvw+v^2w^2\omega$ using $w,\omega\leq1$.
The divisions are legitimate because $u,v,w>0$.

The logarithmic estimates, with their singular initial terms
retained, read
\begin{align*}
 |(\ell_U)_u|&\leq1/u+A,&
 |(\ell_W)_u|&\leq A/k^2,\\
 |(\ell_U)_w|&\leq A,&
 |(\ell_W)_w|&\leq1/w+A/k^2,\\
 |(\ell_U)_v|&\leq Ak/v,&
 |(\ell_W)_v|&\leq A/v.
\end{align*}
For the strong column, the second term inside the brackets of
\eqref{eq:entrance-terminal-correction} is at most
\[
 \frac1k\left(\frac{b_+U_+}{u}+A b_+^2\right)
       \leq(2+4A)\frac vk.
\]
For the weak column it is at most
\[
 \frac1k\left(\frac{b_+W_+}{w}+A b_+^2\right)
       \leq(1+2A)\frac{J_0}k.
\]
For the active column it is at most
\[
 A\left(\frac{b_+U_+}{v}+\frac{b_+W_+}{kv}\right)
       \leq A(2u+K_0)\leq3AK_0.
\]
The first term in the correction has the same respective scales:
$|Q_u|\leq Av/k$, $|Q_w|\leq AJ_0/k$ and
$|Q_v|\leq AK_0$, while $b_+^2/k^2\leq4$.
Thus, with $B=T(2+8A)$,
\[
 |E_u|\leq Bv/k,\qquad
 |E_w|\leq BJ_0/k,\qquad
 |E_v|\leq BK_0.
\]
Since $DF=DQ-DE$, addition of the corresponding phase bounds
proves~\eqref{eq:entrance-physical-walls}.
A common enlargement of $C$ includes the value and radial bounds.
\end{proof}

One useful consequence expresses the estimates without a waiting
factor. Since $(1+L/k)\omega\leq1$ and $u+w\leq1$, a universal
constant $B$ gives
\begin{equation}
\begin{split}
 |F-r|+|F_r-1|&\leq Bv(u+v),\\
 |F_u|&\leq Bv,\qquad |F_v|\leq B(u+v).
\end{split}
\label{eq:entrance-compact-phase-bounds}
\end{equation}
Indeed, $uv+v^2w^2\leq v(u+v)$, $k\geq1$ and
$u+vw^2\leq u+v$. These bounds are adapted to composition with
the finite outgoing segment: its new growing wall is proportional
to the old weak wall.