\subsection{The physical phase at the terminal return}
\label{sec:terminal-physical-phase}

We now compare the single-wall phase with the phase of the
actual next core. There are two corrections. The first restores
the weak stable wall and the active wall at the cyclic entrance.
The second follows that entrance through its incoming passage.
Both retain wall factors that compensate for the singular
derivatives of the cyclic coordinates.

Write the ordinary and stable radial entrance coordinates as
\[
 Z_h=(m_e,u_e,v_e,w_e),\qquad
 \widehat Z_h=(r_h,u_e,v_e,w_e),
\]
and let $\kappa_h$ be the radius at the actual first arrival of
Proposition~\ref{prop:terminal-first-arrival}. Define
\begin{equation}
 Q_h=\frac2{\kappa_h},\qquad
 \Xi_h=\frac2{r_h}-\psi_h,\qquad
 \Psi_h=\frac2{\kappa_h}-\frac2{r_h}.
 \label{eq:terminal-two-phase-corrections}
\end{equation}
Thus
\begin{equation}
 Q_h=\psi_h+\Xi_h+\Psi_h
     =\frac2{r_h}+\Psi_h.
 \label{eq:terminal-physical-phase-decompositions}
\end{equation}
The first decomposition preserves the reference radial
derivative $1/2$. The second will give a uniform bound on the
two wall derivatives.

\begin{proposition}[The terminal phase]
\label{prop:terminal-physical-phase}
There is $0<\delta\leq1/40000$ such that, for each
$0<\eta_0\leq\delta$ and integer $J\geq4$, there are
$C\geq1$ and $\varepsilon>0$ with the following property.
Suppose
\begin{equation}
 \eta_0\leq\eta\leq\delta,\quad 0<\alpha\leq1,\quad
 0<b\leq\varepsilon\alpha^{2J},\quad
 u,w>0,\quad u+w\leq b.
 \label{eq:terminal-physical-phase-domain}
\end{equation}
At $Y_0=(2+2\alpha,u,w)$ and fixed $h=\eta\alpha^J$, the
physical phase $Q_h$ is locally $C^2$ and satisfies
\begin{align}
 |Q_h(Y_0)-\alpha|+|\partial_RQ_h(Y_0)-1/2|&\leq Cb,
       \label{eq:terminal-physical-phase-radial}\\
 |\partial_uQ_h(Y_0)|+|\partial_wQ_h(Y_0)|&\leq C.
       \label{eq:terminal-physical-phase-walls}
\end{align}
The constants are uniform over the indicated height interval.
\end{proposition}

\subsubsection{Wall factors at the cyclic entrance}

The ordinary derivative bound for $Z_h$ loses the amplitude
of an individual wall. To keep that amplitude, use the
outgoing multipliers before applying the cyclic map.

\begin{lemma}
\label{lem:terminal-cyclic-wall-factors}
Fix $0<\eta_0\leq1/1000$ and $J\geq2$. There are
$C\geq1$ and $\varepsilon>0$ such that, for
$\eta_0\leq\eta\leq1/1000$, $0<\alpha\leq1$ and
$|u|+|w|\leq b\leq\varepsilon\alpha^{2J}$ with $b>0$,
the entrance at fixed $h=\eta\alpha^J$ has the form
\begin{equation}
 u_e=\frac{4h}{\mathcal H(y_h)},\qquad
 v_e=w\mathcal B_W,\qquad w_e=u\mathcal B_U.
 \label{eq:terminal-cyclic-wall-factors}
\end{equation}
The factors $\mathcal B_U,\mathcal B_W$ are positive and
locally $C^2$, including at signed or zero old walls, and
\begin{align}
 \mathcal B_j&\leq C/\alpha^2,&
 \mathcal B_j^{-1}&\leq C\alpha^2,
       \label{eq:terminal-cyclic-factor-size}\\
 \|D\log\mathcal B_j\|&\leq C/\alpha,&
 \|D\log u_e\|&\leq C/\alpha.
       \label{eq:terminal-cyclic-factor-derivative}
\end{align}
In particular,
\begin{equation}
 |\partial_R z_i|\leq C|z_i|/\alpha,
 \qquad z=Z_h(Y_0),\quad i\in\{u,v,w\}.
 \label{eq:terminal-cyclic-radial-walls}
\end{equation}
\end{lemma}

\begin{proof}
Let $B_j$ denote the outgoing wall multiplier evaluated at
the arrival at $V=h$. The exact cyclic formulas give
$\mathcal B_j=4B_j/\mathcal H(y_h)$ and
\eqref{eq:terminal-cyclic-wall-factors}.
These factors are $C^2$ in the initial data. Indeed, if
$N_j'=\gamma_jN_j$ in original normalized time, the
projective multiplier has the expression
\[
 B_j=\exp\left(\log A(0)-\log A(S_h)
                    +\int_0^{S_h}\gamma_j(\phi_t x(Y))\dd t\right).
\]
The event time and flow are $C^2$ on a neighborhood of this
finite segment, the endpoint values of $A$ are positive, and
the moving integral is $C^2$. One may see the last assertion
by writing it as $S_h\int_0^1\gamma_j(\phi_{sS_h}x(Y))\dd s$.

On the same neighborhood,
$B_j=\exp\ell_j(Y,\log(1/h))$, where $\ell_j$ is
defined in~\eqref{eq:terminal-wall-rate-integrals}.
Equation~\eqref{eq:terminal-fine-multiplier-bounds} and
$\alpha\log(1/h)\leq\log(1/\eta_0)+J/e$ imply
\[
 C_0^{-1}\leq B_j\leq C_0,\qquad
 \|D\log B_j\|\leq C_0/\alpha.
\]
Combine these with
$3.99\alpha^2<\mathcal H<4.01\alpha^2$ and
\eqref{eq:terminal-cyclic-log-denominator}. The identities
\[
 D\log\mathcal B_j=D\log B_j-D\log\mathcal H,
 \qquad D\log u_e=-D\log\mathcal H
\]
prove~\eqref{eq:terminal-cyclic-factor-size}--\eqref{eq:terminal-cyclic-factor-derivative}.
In the second identity $h$ is fixed.
Finally, differentiating~\eqref{eq:terminal-cyclic-wall-factors}
in $R$ gives~\eqref{eq:terminal-cyclic-radial-walls}; the
old wall coordinates are constant in this direction.
\end{proof}

\subsubsection{Restoring the weak wall and the active wall}

First consider an arbitrary entrance $y=(m,a,v,c)$ with
$m\geq2$ and $|a|,|c|\leq10^{-4}$. Put
\begin{equation}
 \begin{aligned}
 \chi&=P(m,a,c)-1,& q&=P(m,a,0)-1,& r&=H(\chi,v),\\
 E_0&=P(m,a,c)-P(m,a,0),&
 E&=|c|(|a|+|c|)+v^2.
 \end{aligned}
 \label{eq:terminal-stable-correction-data}
\end{equation}
Here $q$ is the single-wall radius. The decomposition
\begin{equation}
 r-q=E_0+\frac{v^2}{r+\chi}
 \label{eq:terminal-stable-correction-split}
\end{equation}
separates the removed weak wall from the active-wall square.

\begin{lemma}
\label{lem:terminal-stable-correction-direction}
Suppose $\alpha>0$, $m+1\geq1/\alpha$, and a direction
$d=(d_m,d_a,d_v,d_c)$ satisfies, for some $D\geq0$,
\begin{equation}
 \|d\|\leq D/\alpha^2,\qquad
 |d_a|\leq D|a|/\alpha,\quad
 |d_v|\leq D|v|/\alpha,\quad
 |d_c|\leq D|c|/\alpha.
 \label{eq:terminal-stable-correction-direction}
\end{equation}
There is a universal constant $C$ such that
\begin{align}
 |r-q|&\leq C\alpha E,& |D(r-q)[d]|&\leq CDE,
       \label{eq:terminal-stable-radius-correction}\\
 \left|\frac2r-\frac2q\right|&\leq C\alpha^3E,&
 \left|D\left(\frac2r-\frac2q\right)[d]\right|
       &\leq CD\alpha^2E.
       \label{eq:terminal-stable-reciprocal-correction}
\end{align}
These are ordinary derivatives of locally $C^2$ functions.
\end{lemma}

\begin{proof}
Write $\ell=m+1$ and $s=|a|+|c|$.
Integrate the derivatives of $P$ along $(m,a,tc)$,
$0\leq t\leq1$. Proposition~\ref{prop:stable-refined-derivatives}
gives
\begin{align}
 |E_0|&\leq C|c|s/\ell,&
 |(E_0)_m|&\leq C|c|s/\ell^2,\notag\\
 |(E_0)_a|&\leq C|c|/\ell,&
 |(E_0)_c|&\leq Cs/\ell.
 \label{eq:terminal-removed-wall-derivatives}
\end{align}
For the middle two bounds, apply the fundamental theorem of
calculus to $P_m$ and $P_a$ along this segment, using the
mixed and stable Hessian estimates. The last bound is the
first derivative of $P$ itself. All points of the segment
have total wall amplitude at most $s$.

The bound~\eqref{eq:stable-phase-tail} gives
$|P-m|\leq1/100$ at these inputs, and hence
\begin{equation}
 \chi,q\geq\ell/4\geq1/(4\alpha),\qquad
 r\geq\chi>0,
 \label{eq:terminal-stable-correction-lower}
\end{equation}
since $m\geq2$. Equation~\eqref{eq:terminal-stable-correction-split}
now proves the first value estimate. To differentiate it,
use $r^2=\chi^2+v^2$. After subtracting $D\chi[d]$ from
$Dr[d]$, the exact formula is
\begin{equation}
 D(r-q)[d]=DE_0[d]+\frac{v d_v}{r}
          -\frac{v^2DP[d]}{r(r+\chi)}.
 \label{eq:terminal-stable-correction-differential}
\end{equation}
The four bounds in~\eqref{eq:terminal-removed-wall-derivatives},
together with~\eqref{eq:terminal-stable-correction-direction}
and $\ell^{-1}\leq\alpha$, give
$|DE_0[d]|\leq3CD|c|s$.
The uniform first derivative of $P$ gives
$|DP[d]|\leq C D/\alpha^2$. Since
$r^{-1},(r+\chi)^{-1}\leq4\alpha$, the last two terms in
\eqref{eq:terminal-stable-correction-differential} are bounded
by $4Dv^2$ and $16CDv^2$. This proves the derivative
estimate in~\eqref{eq:terminal-stable-radius-correction}.

For completeness, the reciprocal difference of two positive
functions $f,g$ satisfies, with $e=f-g$,
\begin{equation*}
 \frac2f-\frac2g=-\frac{2e}{gf},\qquad
 D\left(\frac2f-\frac2g\right)[d]
   =-\frac{2De[d]}{f^2}
     +\frac{2e(f+g)Dg[d]}{g^2f^2}.
\end{equation*}
If $\lambda>0$, $f,g\geq1/\lambda$ and
$|Dg[d]|\leq D_0/\lambda^2$, then
\begin{equation}
 \left|\frac2f-\frac2g\right|\leq2\lambda^2|e|,
 \qquad
 \left|D\left(\frac2f-\frac2g\right)[d]\right|
       \leq2\lambda^2|De[d]|+4D_0\lambda|e|.
 \label{eq:terminal-reciprocal-difference-bounds}
\end{equation}
Indeed, $(f+g)/(g^2f^2)=1/(g^2f)+1/(gf^2)\leq2\lambda^3$.
Apply this with $f=r$, $g=q$, $\lambda=4\alpha$ and a constant
$D_0$ bounded by a universal multiple of $D$.
The latter bound follows from the uniform derivative of
$P(m,a,0)$ and $\|d\|\leq D/\alpha^2$.
Equation~\eqref{eq:terminal-stable-radius-correction} gives
\eqref{eq:terminal-stable-reciprocal-correction}.
Positivity in~\eqref{eq:terminal-stable-correction-lower}
and the regularity of $P$ also prove the stated $C^2$ property.
\end{proof}

Apply the lemma to the cyclic entrance. Fix
$0<\eta_0\leq1/40000$, $J\geq2$, and
$\eta\in[\eta_0,1/40000]$. For sufficiently small
$b\leq\varepsilon\alpha^{2J}$, including signed old walls,
the geometry gives
\[
 m_e>2,\qquad m_e+1\geq1/\alpha,\qquad
 |u_e|,|w_e|\leq10^{-4}.
\]
Proposition~\ref{prop:terminal-entrance-derivatives} and
Lemma~\ref{lem:terminal-cyclic-wall-factors} show that
$d=\partial_RZ_h$ satisfies
\eqref{eq:terminal-stable-correction-direction} with a
constant $D$ depending only on $\eta_0,J$.
The identity $2/q=\vartheta(m_e,u_e)$ holds on an ordinary
neighborhood by Lemma~\ref{lem:terminal-strong-limit}.
The chain rule therefore identifies the composed reciprocal
correction with $\Xi_h$, including its derivative.

Put $N_0=|u|(h+b)+w^2$. The wall factors give
\[
 |u_e|\leq2h/\alpha^2,\qquad
 |v_e|\leq C|w|/\alpha^2,\qquad
 |w_e|\leq C|u|/\alpha^2.
\]
Consequently
\[
 |w_e|(|u_e|+|w_e|)+v_e^2\leq C N_0/\alpha^4.
\]
Equation~\eqref{eq:terminal-stable-reciprocal-correction}
now gives the useful amplitude estimates
\begin{equation}
 |\Xi_h|\leq C N_0/\alpha,\qquad
 |\partial_R\Xi_h|\leq C N_0/\alpha^2.
 \label{eq:terminal-stable-correction-old-walls}
\end{equation}
After reducing $\varepsilon$ to at most one, $b,h\leq\alpha^2$.
Thus $N_0\leq3b\alpha^2$, and
\begin{equation}
 |\Xi_h|+|\partial_R\Xi_h|\leq Cb.
 \label{eq:terminal-stable-correction-linear}
\end{equation}

\subsubsection{The actual incoming correction}

Return now to positive old walls. Let $\delta_{\rm in}$ be
the entrance threshold in Lemma~\ref{lem:terminal-incoming-phase}
and choose $\delta=\min(\delta_{\rm in}/4,1/40000)$.
By~\eqref{eq:terminal-incoming-wall-order}, that lemma applies
uniformly to all cyclic entrances with $\eta\leq\delta$,
after reducing the old input threshold.
In particular, $0<w_e\leq u_e\leq1$ and $0<v_e<1$.
At fixed height write
\begin{equation}
 \Theta_h=\kappa_h-r_h=F(\widehat Z_h)-r_h.
 \label{eq:terminal-incoming-radius-error}
\end{equation}
The equality holds on an initial neighborhood: the local
inverse of the stable radial coordinate takes
$\widehat Z_h$ back to the actual cyclic entrance, and
$F$ is its physical incoming arrival radius.

\begin{lemma}
\label{lem:terminal-incoming-reciprocal-error}
Fix $0<\eta_0\leq\delta$ and $J\geq2$.
For the positive inputs above, after choosing a sufficiently
small uniform $\varepsilon$, the functions $\Theta_h$ and
$\Psi_h$ are locally $C^2$ and
\begin{align}
 |\Theta_h|&\leq C\alpha u_ev_e,&
 |\partial_R\Theta_h|+|\partial_u\Theta_h|
      &\leq C u_ev_e/\alpha^2,\notag\\
 |\partial_w\Theta_h|&\leq C u_e/\alpha^2,
       \label{eq:terminal-incoming-radius-error-bounds}\\
 r_h,\kappa_h&\geq1/\alpha,&
 |\Psi_h|&\leq C\alpha^3u_ev_e,\notag\\
 |\partial_R\Psi_h|+|\partial_u\Psi_h|&\leq C u_ev_e,&
 |\partial_w\Psi_h|&\leq C u_e.
       \label{eq:terminal-incoming-reciprocal-bounds}
\end{align}
If $J\geq4$, these imply
\begin{align}
 |\Psi_h|&\leq Cw\alpha^{J-1},\notag\\
 |\partial_R\Psi_h|+|\partial_u\Psi_h|
     &\leq Cw\alpha^{J-4},&
 |\partial_w\Psi_h|&\leq C\alpha^{J-2}.
 \label{eq:terminal-incoming-reciprocal-powers}
\end{align}
\end{lemma}

\begin{proof}
For an input direction $d=(d_R,d_u,d_w)$,
differentiate the three wall factors:
\begin{equation*}
 \begin{aligned}
 Du_e[d]&=u_eD\log u_e[d],\\
 Dv_e[d]&=\mathcal B_Wd_w+v_eD\log\mathcal B_W[d],\\
 Dw_e[d]&=\mathcal B_Ud_u+w_eD\log\mathcal B_U[d].
 \end{aligned}
\end{equation*}
The chain rule in~\eqref{eq:terminal-incoming-radius-error}
therefore gives
\begin{equation*}
 \begin{aligned}
 D\Theta_h[d]={}&(F_r-1)Dr_h[d]+F_u u_eD\log u_e[d]\\
   &+F_v v_eD\log\mathcal B_W[d]
      +F_w w_eD\log\mathcal B_U[d]\\
   &+F_v\mathcal B_Wd_w+F_w\mathcal B_Ud_u.
 \end{aligned}
\end{equation*}
All derivatives of $F$ are evaluated at $\widehat Z_h$.
For a coordinate unit vector $d$, the first four terms
are bounded in absolute value by
\[
 C u_ev_e(\alpha^{-2}+1+\alpha^{-1}+w_e)
      \leq4C u_ev_e/\alpha^2.
\]
Here we used~\eqref{eq:terminal-incoming-phase-bounds},
$\|D\widehat Z_h\|\leq C/\alpha^2$, and
\eqref{eq:terminal-cyclic-factor-derivative}.
The last two terms retain different amplitudes:
\begin{equation*}
 |F_v\mathcal B_W|\leq C u_e/\alpha^2,
 \qquad |F_w\mathcal B_U|\leq C u_ev_e/\alpha.
\end{equation*}
For $d=e_R$ neither occurs; for $e_u$ only the second
occurs; for $e_w$ only the first occurs. This proves the
derivative bounds for $\Theta_h$. Its value bound is the
first estimate of Lemma~\ref{lem:terminal-incoming-phase}.
Local $C^2$ regularity follows from the same composition.

The lower bound for $r_h$ follows from
\eqref{eq:terminal-cycle-end-radius} and
$|P(m_e,u_e,w_e)-m_e|\leq1/100$:
\[
 r_h\geq P(m_e,u_e,w_e)-1
       >\frac{1.9}{\alpha}-\frac1{100}
       \geq\frac3{2\alpha}.
\]
Since $u_e\leq1$ and $v_e\leq Cb/\alpha^2$, the value
bound for $\Theta_h$ is at most $C'b/\alpha$.
Decreasing $\varepsilon$ so that $C'b\leq1/2$ gives
$\kappa_h=r_h+\Theta_h\geq1/\alpha$.
In particular, all reciprocal functions are locally $C^2$.

For each input coordinate $j$, the derivative of $r_h$
is at most $D_0/\alpha^2$. Apply
\eqref{eq:terminal-reciprocal-difference-bounds} with
$f=\kappa_h$, $g=r_h$ and $\lambda=\alpha$:
\[
 |\Psi_h|\leq2\alpha^2|\Theta_h|,\qquad
 |\partial_j\Psi_h|
      \leq2\alpha^2|\partial_j\Theta_h|
                      +4D_0\alpha|\Theta_h|.
\]
Equation~\eqref{eq:terminal-incoming-radius-error-bounds},
$\alpha\leq1$ and $v_e\leq1$ prove
\eqref{eq:terminal-incoming-reciprocal-bounds}.
Finally
\[
 u_e\leq2\eta\alpha^{J-2}\leq\alpha^{J-2},\qquad
 v_e\leq Cw/\alpha^2
\]
give $u_ev_e\leq Cw\alpha^{J-4}$. Substitution yields
\eqref{eq:terminal-incoming-reciprocal-powers}.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:terminal-physical-phase}]
Choose a common input threshold for
Proposition~\ref{prop:terminal-single-composition},
\eqref{eq:terminal-stable-correction-linear}, and
Lemma~\ref{lem:terminal-incoming-reciprocal-error}.
The choice of $\delta$ also places every endpoint in the
incoming domain of Proposition~\ref{prop:terminal-first-arrival}
with $q=0$, so all three terms in
\eqref{eq:terminal-physical-phase-decompositions} refer to the
same physical first arrival.
Since $J\geq4$, $w\leq b$ and $0<\alpha\leq1$,
\eqref{eq:terminal-incoming-reciprocal-powers} gives
\[
 |\Psi_h|+|\partial_R\Psi_h|\leq Cb.
\]
Adding this to~\eqref{eq:terminal-stable-correction-linear}
and~\eqref{eq:terminal-single-composition-bounds} proves
\eqref{eq:terminal-physical-phase-radial}.

For the wall derivatives, use the second decomposition in
\eqref{eq:terminal-physical-phase-decompositions}.
As $r_h\geq1/\alpha$ and $\|Dr_h\|\leq C_0/\alpha^2$,
\[
 \left\|D\left(\frac2{r_h}\right)\right\|
       \leq2C_0.
\]
The two wall columns of $\Psi_h$ are bounded by
\eqref{eq:terminal-incoming-reciprocal-powers}, reducing
$\varepsilon$ to at most one if necessary. Their sum proves
\eqref{eq:terminal-physical-phase-walls}.
The local $C^2$ assertions proved above give the regularity
of $Q_h$.
\end{proof}

The auxiliary height may now follow the phase parameter.
For fixed $\eta,J$, set
\begin{equation}
 a(Y)=\eta\left(\frac{R-2}{2}\right)^J,\qquad
 Q(Y)=Q_{a(Y)}(Y).
 \label{eq:terminal-natural-bounded-phase}
\end{equation}
Define the corresponding time and state by
$T(Y)=\mathcal T(a(Y),Y)$ and
$\mathcal Y(Y)=\mathcal E(a(Y),Y)$, using
\eqref{eq:terminal-height-maps}.

\begin{corollary}
\label{cor:terminal-bounded-phase-direction}
For $0<\eta\leq\delta$ and $J\geq4$, there are
$C\geq1$ and $\varepsilon>0$ such that all positive inputs
with $0<\alpha\leq1$ and
$u+w\leq b\leq\varepsilon\alpha^{2J}$, $b>0$, have the
following properties at $Y_0=(2+2\alpha,u,w)$.
The functions $T,\mathcal Y,Q$ are locally $C^2$;
$T$ is the actual first positive arrival at the new $V=1$,
its crossing is upward, and $\mathcal Y$ is the state there.
Moreover,
\begin{equation}
 |Q(Y_0)-\alpha|\leq Cb.
 \label{eq:terminal-natural-phase-value}
\end{equation}
For every direction $d=(1,d_u,d_w)$ with
$|d_u|\leq\sqrt u$, $|d_w|\leq\sqrt w$,
\begin{equation}
 DQ(Y_0)[d]\geq\frac14,\qquad
 |2DQ(Y_0)[d]-1|\leq C\sqrt b,\qquad
 |\log(2DQ(Y_0)[d])|\leq C\sqrt b.
 \label{eq:terminal-natural-phase-direction}
\end{equation}
\end{corollary}

\begin{proof}
The height function $a$ is continuous and $a(Y_0)=\eta\alpha^J$.
Lemma~\ref{lem:terminal-height-freezing} identifies the time
and state with their fixed-height versions on an ordinary
neighborhood of $Y_0$. The same identity holds for their
radius and its reciprocal, which are positive there.
It proves the first-arrival statements, regularity, and
equality of the derivatives of $Q$ and $Q_h$ at this input.

Let $A$ be a common constant in
\eqref{eq:terminal-physical-phase-radial}--\eqref{eq:terminal-physical-phase-walls}.
For $b\leq1$, the directional derivative obeys
\[
 |2DQ[d]-1|
  \leq2Ab+2A\max(|d_u|,|d_w|)
  \leq4A\sqrt b.
\]
Reduce $\varepsilon$ further so that $4A\sqrt b\leq1/4$.
Then $2DQ[d]\geq1/2$, hence $DQ[d]\geq1/4$.
The elementary bound $|\log t|\leq2|t-1|$ for $t\geq1/2$
gives the logarithmic estimate. Increasing $C$ also includes
the value estimate inherited from
Proposition~\ref{prop:terminal-physical-phase}.
\end{proof}

The bounded phase therefore has a positive derivative on the
same square-root directions used in the core and reflection
estimates. To carry these directions through the whole
terminal return, we next estimate the two wall components of
its derivative.