\subsection{Phase after the terminal connection}
\label{sec:terminal-phase-value}

A terminal exit of radius $R_0=2+2\alpha$ leads to a new core
whose radius is close to $2/\alpha$. The cyclic entrance already
has this scale, but its ordinary radius still depends on the
auxiliary height. The stable phase removes that dependence
exactly on the reference orbit. Comparing at the same height
then gives the phase error for a perturbed orbit.

We first record the resulting value and wall estimates. Retain
the actual terminal arrival $\mathcal E(h,Y)$ from
\eqref{eq:terminal-height-maps}, and write
\[
 Y=(2+2\alpha,u,w),\qquad h=\eta\alpha^J,\qquad
 y_+=\mathcal E(h,Y),\qquad \kappa=H(M_+-1,1).
\]

\begin{proposition}[The next core and its phase]
\label{prop:terminal-phase-value}
Fix $0<\varepsilon_{\rm in}\leq10^{-4}$ and integers
$q\geq16$, $J\geq q+2$. Choose
\begin{equation}
 0<\eta\leq
 \min\left(\frac1{100},
        \frac{\varepsilon_{\rm in}}{4(11/5)^q}\right).
 \label{eq:terminal-value-height}
\end{equation}
There are $C\geq1$ and $\varepsilon>0$ such that, whenever
\begin{equation}
 0<\alpha\leq1,\qquad
 0<u,w,\qquad u+w\leq b\leq\varepsilon\alpha^{2J},
 \label{eq:terminal-value-domain}
\end{equation}
the actual first arrival of
Proposition~\ref{prop:terminal-first-arrival} satisfies
\begin{align}
 \kappa&\geq\frac1\alpha,&
 \left|\kappa-\frac2\alpha\right|
     &\leq\frac{Cb}{\alpha^2},&
 \left|\frac2\kappa-\alpha\right|&\leq Cb,
 \label{eq:terminal-value-bounds}\\
 0<U_+&\leq\frac{\varepsilon_{\rm in}}{\kappa^{16}},&
 0<W_+&\leq\frac{\varepsilon_{\rm in}}{\kappa^{16}}.
 \label{eq:terminal-core-wall-budget}
\end{align}
If $Cb<1-\alpha$, then $2/\kappa<1$ and $\kappa>2$.
\end{proposition}

The last condition ensures that the arrival lies in the core
domain for the next era. The wall estimates will carry a
radius-dependent smallness condition through successive eras.

\subsubsection{The reference phase}

On the boundary with only the strong wall present, the stable
system~\eqref{eq:stable-boundary-field} reduces to
\[
 \dot m=mu^2,\qquad
 \dot u=\frac{1-m^2+u^2}{2}u,\qquad w=0.
\]
Its conserved quantity is
\begin{equation}
 I(m,u)=\frac{1+m^2+u^2}{m},\qquad
 \lambda_+(I)=\frac{I+\sqrt{I^2-4}}2.
 \label{eq:terminal-strong-invariant}
\end{equation}

\begin{lemma}[The strong-wall boundary limit]
\label{lem:terminal-strong-limit}
For $m\geq2$ and $|u|\leq1/200$,
\begin{equation}
 P(m,u,0)=\lambda_+(I(m,u)).
 \label{eq:terminal-strong-projection}
\end{equation}
For $0<\alpha\leq1$ and $0\leq v\leq\alpha^2/400$, set
\[
 R_0=2+2\alpha,\qquad
 y_0=\bigl(1-\sqrt{R_0^2-v^2},0,v,0\bigr),\qquad
 z_0=\mathsf C_p y_0.
\]
Then
\begin{equation}
 P((z_0)_M,(z_0)_U,0)=1+\frac2\alpha,\qquad
 \widehat\rho(z_0)=\frac2\alpha.
 \label{eq:terminal-reference-radius}
\end{equation}
\end{lemma}

\begin{proof}
The derivative of $I$ along the displayed two-dimensional
system vanishes:
\[
 \left(1-\frac{1+u^2}{m^2}\right)mu^2
 +\frac{2u}{m}\frac{1-m^2+u^2}{2}u=0.
\]
The same identity holds in original time, since the time change
multiplies both components by the same positive factor.
Proposition~\ref{prop:stable-phase-limit} keeps $m(t)>1$
and gives $(m(t),u(t))\to(P,0)$ with $P>1$. Passing to the
limit in $I$ yields
\[
 P^2-I(m,u)P+1=0.
\]
Moreover,
$I(m,u)-2=((m-1)^2+u^2)/m>0$. The two roots are positive,
their product is one, and exactly one exceeds one. This proves
\eqref{eq:terminal-strong-projection}.

For the cyclic reference point, put
$g_0=\sqrt{R_0^2-v^2}$ and
\[
 \mathcal H_0=(2-g_0)^2+v^2=R_0^2+4-4g_0.
\]
The cyclic formula gives
\begin{equation*}
 m_0=\frac{R_0^2-4}{\mathcal H_0},\qquad
 u_0=\frac{4v}{\mathcal H_0},\qquad v_0=w_0=0.
\end{equation*}
Since $v\leq\alpha/100$, the endpoint algebra in the proof of
Lemma~\ref{lem:terminal-cycle-entrance} gives
\[
 1.99\alpha<g_0-2<2.01\alpha,\qquad
 3.99\alpha^2<\mathcal H_0<4.01\alpha^2.
\]
It follows that $m_0-1>1.9/\alpha$ and
$0\leq u_0\leq2v/\alpha^2\leq1/200$.
Thus~\eqref{eq:terminal-strong-projection} applies.

The identity $g_0^2+v^2=R_0^2$ gives
\[
 \mathcal H_0^2+(R_0^2-4)^2+16v^2
     =\bigl(2(R_0^2-4)+16\bigr)\mathcal H_0.
\]
Substitution into~\eqref{eq:terminal-strong-invariant} therefore
gives
\begin{equation}
 I(m_0,u_0)=2+\frac{16}{R_0^2-4},\qquad
 \lambda_+(I(m_0,u_0))=\frac{R_0+2}{R_0-2}
                     =1+\frac2\alpha.
 \label{eq:terminal-reference-invariant}
\end{equation}
For the second equality, the displayed fraction exceeds one and
satisfies the defining quadratic, so it is the upper root.
Since $v_0=0$, the stable radial phase is $P(m_0,u_0,0)-1$,
which proves~\eqref{eq:terminal-reference-radius}.
\end{proof}

On the actual outgoing orbit with $u=w=0$, the radius is constant:
the radial equation in~\eqref{eq:terminal-outgoing-field} has
zero right-hand side. Thus its state at $V=h$ is exactly the
point $y_0$ above with $v=h$. For $J\geq2$ and
$\eta\leq1/400$, the reference stable radius is consequently
$2/\alpha$ at every chosen height $h=\eta\alpha^J$.

We shall use the stable phase on the signed strip
\begin{equation}
 \mathcal S=\{(m,u,w):m\geq2,\ |u|,|w|\leq1/400\}.
 \label{eq:terminal-stable-strip}
\end{equation}
There is a constant $L\geq1$ such that
\begin{equation}
 \|DP\|\leq L,\qquad
 |P(X)-P(X_0)|\leq L\|X-X_0\|,\qquad X,X_0\in\mathcal S.
 \label{eq:terminal-stable-lipschitz}
\end{equation}
Here and below the finite-dimensional norms are maximum norms.
Indeed, \eqref{eq:stable-phase-differential} bounds $DP$
independently of $m$ whenever $|u|+|w|<1/100$. Integrate this
bound along the line segment joining $X_0$ to $X$, which stays
in $\mathcal S$. This argument also includes its zero-wall faces.

\begin{lemma}[Stable radius at the cyclic entrance]
\label{lem:terminal-stable-comparison}
Suppose
\begin{equation}
 \begin{gathered}
  0<\alpha\leq1,\qquad 0\leq E\leq\alpha^2/10000,\qquad
  |\rho-(2+2\alpha)|\leq E,\\
  |U|+|W|\leq E,\qquad 0\leq v\leq\alpha^2/1000.
 \end{gathered}
 \label{eq:terminal-comparison-domain}
\end{equation}
For $y=(1-\sqrt{\rho^2-v^2},U,v,W)$,
\begin{equation}
 \left|\widehat\rho(\mathsf C_p y)-\frac2\alpha\right|
       \leq\frac{42LE}{\alpha^2}.
 \label{eq:terminal-stable-comparison}
\end{equation}
In particular, for fixed $0<\eta\leq1/1000$ and $J\geq2$,
there are $C\geq1$ and $\varepsilon>0$ such that
\begin{equation}
 \left|\widehat\rho(\mathsf C_p y_h(Y))-\frac2\alpha\right|
       \leq\frac{Cb}{\alpha^2}
 \label{eq:terminal-actual-stable-comparison}
\end{equation}
whenever $0<\alpha\leq1$, $0<b\leq\varepsilon\alpha^{2J}$,
$|u|+|w|\leq b$, $Y=(2+2\alpha,u,w)$ and $h=\eta\alpha^J$.
\end{lemma}

\begin{proof}
Set $R_0=2+2\alpha$, $g=\sqrt{\rho^2-v^2}$ and
$g_0=\sqrt{R_0^2-v^2}$. Denote the two cyclic denominators
by $\mathcal H$ and $\mathcal H_0$, comparing $y$ with the
zero-wall point $y_0$ at this same $v$. The elementary endpoint
bounds from the proof of
Lemma~\ref{lem:terminal-cycle-entrance} give
\[
 \mathcal H,\mathcal H_0\geq3.9\alpha^2,\qquad
 0<g-2<3\alpha,\qquad 0<g_0-2<3\alpha.
\]
Rationalizing the square roots gives
\[
 (g-g_0)(g+g_0)=(\rho-R_0)(\rho+R_0).
\]
Here $g+g_0\geq3$ and $\rho+R_0\leq9$, so
$|g-g_0|\leq3E$. Also,
\[
 \mathcal H-\mathcal H_0
  =(g-g_0)(g+g_0-4)+(U-W)^2-2v(U+W).
\]
The first term is at most $18\alpha E$ in absolute value.
The remaining terms are at most $E^2+2vE\leq\alpha E$.
Consequently
\begin{equation*}
 |g-g_0|\leq3E,\qquad
 |\mathcal H-\mathcal H_0|\leq20\alpha E.
\end{equation*}

To compare a cyclic quotient, use
\begin{equation*}
 \frac n{\mathcal H}-\frac{n_0}{\mathcal H_0}
 =\frac{n-n_0}{\mathcal H}
  +\frac{n_0(\mathcal H_0-\mathcal H)}
          {\mathcal H\mathcal H_0}.
\end{equation*}
If $|n-n_0|\leq12E$ and $|n_0|\leq12\alpha$, the two terms
are at most $4E/\alpha^2$ and $30E/\alpha^2$, respectively.
Thus their sum is at most $40E/\alpha^2$.
This applies to $n=4(g-2)$, $n_0=4(g_0-2)$ and to
$n=n_0=4v$. Writing $z=\mathsf C_p y=(m_e,u_e,v_e,w_e)$,
we obtain
\begin{equation*}
 |m_e-m_0|,\ |u_e-u_0|\leq40E/\alpha^2,\qquad
 |v_e|,\ |w_e|\leq2E/\alpha^2.
\end{equation*}

Both $(m_e,u_e,w_e)$ and $(m_0,u_0,0)$ lie in
$\mathcal S$: the shear coordinates exceed two, while
\[
 |u_e|,\ |u_0|\leq2v/\alpha^2\leq1/400,\qquad
 |w_e|\leq2E/\alpha^2\leq1/400.
\]
Equations~\eqref{eq:terminal-stable-lipschitz} and
\eqref{eq:terminal-reference-radius} therefore give
\[
 |P(m_e,u_e,w_e)-(1+2/\alpha)|
       \leq40LE/\alpha^2.
\]
The norm inequality
$|H(a,c)-d|\leq|a-d|+|c|$ for $d\geq0$, applied with
$a=P(m_e,u_e,w_e)-1$ and $d=2/\alpha$, proves
\eqref{eq:terminal-stable-comparison}, since $L\geq1$.

For the actual endpoint, the outgoing estimates give
$|\rho-R_0|\leq C_*b$ and $|U|+|W|\leq C_*b$.
Choose $\varepsilon$ so that
$C_*\varepsilon\leq1/10000$. Then
$E=C_*b\leq\alpha^2/10000$, while
$v=h=\eta\alpha^J\leq\alpha^2/1000$.
Equation~\eqref{eq:terminal-stable-comparison} applies, proving
the last assertion with $C=42LC_*$.
\end{proof}

\subsubsection{The actual next core}

\begin{proof}[Proof of Proposition~\ref{prop:terminal-phase-value}]
Proposition~\ref{prop:terminal-first-arrival} supplies an
admissible incoming endpoint
\[
 z=\mathsf C_p y_h(Y)=(m_e,u_e,v_e,w_e).
\]
Since $\eta\leq\varepsilon_{\rm in}/4\leq1/40000$,
Lemma~\ref{lem:terminal-stable-comparison} applies there.
Put $k=H(m_e-1,v_e)$ and $r=\widehat\rho(z)$.

The incoming phase bound~\eqref{eq:incoming-physical-phase-value}
implies
\begin{equation}
 |\kappa-r|\leq Dv_e
 \label{eq:terminal-active-value-bound}
\end{equation}
with a universal $D>0$. Indeed, $u_e+w_e\leq1$,
$u_e\leq1$, $0<v_e<1$, and $k\geq1$, so
\[
 u_ev_e+v_e^2(u_e+w_e)^2\leq2v_e.
\]
Since $v_e\leq2C_*b/\alpha^2$, adding
\eqref{eq:terminal-active-value-bound} to
\eqref{eq:terminal-actual-stable-comparison} gives
$|\kappa-2/\alpha|\leq Cb/\alpha^2$ after increasing $C$.

Reduce $\varepsilon$ so that $C\varepsilon\leq1$.
Then $Cb\leq\alpha$, and the lower bound in the last estimate
gives $\kappa\geq1/\alpha>0$. The identity
\begin{equation*}
 \frac2\kappa-\alpha
   =-\frac{\alpha}{\kappa}\left(\kappa-\frac2\alpha\right)
\end{equation*}
and $\alpha/\kappa\leq\alpha^2$ prove the bounded phase
estimate in~\eqref{eq:terminal-value-bounds}.
If $Cb<1-\alpha$, its upper bound gives $2/\kappa<1$,
hence $\kappa>2$.

For the walls, the incoming radius control and decay give
\begin{equation}
 \frac{99}{100}k\leq\kappa\leq\frac{101}{100}k,\qquad
 0<U_+\leq u_e,\qquad0<W_+\leq w_e.
 \label{eq:terminal-incoming-power-transfer}
\end{equation}
The weighted entrance estimates
\eqref{eq:terminal-entrance-smallness} state
$u_e k^q,w_e k^q\leq\varepsilon_{\rm in}/2$.
As $k\geq1$ and $q\geq16$,
\[
 \kappa^{16}
 \leq(101/100)^{16}k^{16}
 \leq2k^{16}\leq2k^q.
\]
Multiplying the two wall bounds in
\eqref{eq:terminal-incoming-power-transfer} by this inequality
proves~\eqref{eq:terminal-core-wall-budget}.
Positivity follows from the positive wall multipliers along
the same finite orbit.
\end{proof}

The phase comparison above uses finite differences at a common
height. For the differential analysis we also need the ordinary
cyclic coordinates and their stable radial correction. Define
\begin{equation}
 \begin{aligned}
  Z_h(Y)&=\mathsf C_p y_h(Y)=(m_e,u_e,v_e,w_e),\\
  \widehat Z_h(Y)&=
       \bigl(\widehat\rho(Z_h(Y)),u_e,v_e,w_e\bigr).
 \end{aligned}
 \label{eq:terminal-cyclic-entrance-maps}
\end{equation}
In all derivatives of these maps, $h$ is held fixed.

\begin{proposition}[Cyclic entrance derivatives]
\label{prop:terminal-entrance-derivatives}
Fix $0<\eta_0\leq1/1000$ and $J\geq2$.
There are $C\geq1$ and $\varepsilon>0$ such that, for
\begin{equation}
 \begin{gathered}
  \eta_0\leq\eta\leq1/1000,\qquad0<\alpha\leq1,\\
  0<b\leq\varepsilon\alpha^{2J},\qquad |u|+|w|\leq b,
 \end{gathered}
 \label{eq:terminal-cyclic-derivative-domain}
\end{equation}
the maps $Z_h$ and $\widehat Z_h$ are $C^2$ near
$Y=(2+2\alpha,u,w)$, where $h=\eta\alpha^J$. At this point,
\begin{align}
 \|DZ_h\|,\ \|D\widehat Z_h\|&\leq C/\alpha^2,
 \label{eq:terminal-cyclic-derivative-bounds}\\
 \|D\log(\mathcal H\circ y_h)\|&\leq C/\alpha.
 \label{eq:terminal-cyclic-log-denominator}
\end{align}
The estimates include signed and zero old walls.
\end{proposition}

\begin{proof}
At fixed height, reconstruct the full outgoing state from
the radius map $E_h$ of~\eqref{eq:terminal-outgoing-radius-map}:
\[
 (\rho,U,W)\longmapsto
       (1-\sqrt{\rho^2-h^2},U,h,W).
\]
For $\rho\geq7/4$ and $|h|\leq1$, the norm of this derivative
is at most two, since $\rho/\sqrt{\rho^2-h^2}\leq2$.
The reconstruction agrees with $y_h$ on an initial
neighborhood: $M_h<1$ and the first-arrival identity persist
there. Proposition~\ref{prop:terminal-outgoing-derivatives}
therefore gives
\begin{equation}
 \|Dy_h(Y)\|\leq2C_0
 \label{eq:terminal-full-tail-derivative}
\end{equation}
with one $C_0$ for the whole height interval.

At the endpoint, the uniform geometry gives
\begin{equation*}
 \mathcal H\geq3.9\alpha^2,\qquad
 |M+1|\leq3\alpha,\qquad |U|+|V|+|W|\leq3\alpha.
\end{equation*}
Write the four cyclic components as $c_i+N_i/\mathcal H$,
where $c=(1,0,0,0)$ and
\[
 N=\bigl(-4(M+1),4V,4W,4U\bigr).
\]
For each ordinary coordinate $j$,
\[
 |N_i|\leq12\alpha,\qquad
 |\partial_jN_i|\leq4,\qquad
 |\partial_j\mathcal H|\leq6\alpha.
\]
The last bound follows directly from
\[
 d\mathcal H
  =2(M+1)dM+2(U-V-W)dU
      +2(V-U-W)dV+2(W-U-V)dW.
\]
Quotient differentiation yields
\[
 |\partial_j(\mathsf C_p)_i|
 \leq\frac4{3.9\alpha^2}
       +\frac{72}{3.9^2\alpha^2}
 \leq\frac6{\alpha^2}.
\]
Summing the four entries in each row, and differentiating the
logarithm of the positive denominator, gives
\begin{equation*}
 \|D\mathsf C_p\|\leq24/\alpha^2,\qquad
 \|D\log\mathcal H\|\leq24\alpha/\mathcal H\leq7/\alpha.
\end{equation*}
Together with~\eqref{eq:terminal-full-tail-derivative}, these
prove the bound for $Z_h$ and
\eqref{eq:terminal-cyclic-log-denominator}.

It remains to apply the stable radial coordinate change.
On $\mathcal S$, \eqref{eq:stable-phase-tail} gives
$|P-m|\leq1/100$, hence $\chi=P-1>0$. For arbitrary signed $v$,
put $r=H(\chi,v)$. Then
\begin{equation*}
 dr=\frac{\chi\,dP+v\,dv}{r},\qquad
 \chi/r\leq1,\qquad |v|/r\leq1.
\end{equation*}
Equation~\eqref{eq:terminal-stable-lipschitz} bounds the
derivative of $(m,u,v,w)\mapsto(r,u,v,w)$ by $L+1$.
This map is locally $C^2$ because $P$ is $C^2$ and $r>0$.

At the actual cyclic endpoint, the estimates of
Lemma~\ref{lem:terminal-height-uniform} give
$m_e>2$, $|u_e|\leq2\eta\leq1/400$ and
$|w_e|\leq2C_*b/\alpha^2\leq1/400$, after decreasing
$\varepsilon$. Thus the signed radial bound applies.
Composition proves~\eqref{eq:terminal-cyclic-derivative-bounds}
for $\widehat Z_h$ and its $C^2$ regularity.
\end{proof}

One consequence is useful when varying the phase while keeping
a numerical height fixed. For $J\geq2$, choose $\eta>0$ with
$2\eta\leq1/1000$ and set $h=\eta\alpha_0^J$, where
$\alpha_0>0$. If
\begin{equation}
 0<\alpha\leq1,\qquad
 \left|\log\frac{\alpha}{\alpha_0}\right|<\frac{\log2}{J},
 \qquad
 0<b\leq\varepsilon\alpha_0^{2J+1},\qquad |u|+|w|\leq b,
 \label{eq:terminal-fixed-height-phase-interval}
\end{equation}
then the conclusions of
Proposition~\ref{prop:terminal-entrance-derivatives} hold at
$Y=(2+2\alpha,u,w)$ with the same fixed $h$, after choosing
$C,\varepsilon$ depending only on $\eta,J$.
Indeed, exponentiating the logarithmic inequality gives
\[
 \frac{\alpha_0^J}{2}<\alpha^J<2\alpha_0^J,\qquad
 \frac\eta2<\eta_{\rm eff}:=\frac{\eta\alpha_0^J}{\alpha^J}
                 <2\eta.
\]
Also $\alpha_0\leq2\alpha$, so
$\alpha_0^{2J+1}\leq8\alpha^{2J+1}\leq8\alpha^{2J}$.
Apply the proposition with lower coefficient $\eta/2$ and
effective coefficient $\eta_{\rm eff}$, reducing its input
threshold by a factor of eight. The identity
$h=\eta_{\rm eff}\alpha^J$ leaves the numerical height unchanged.

These coordinate estimates account for the singular change of
scale at the terminal cycle. To estimate the bounded phase
derivative close to its reference value $1/2$, we next retain
the cancellation in~\eqref{eq:terminal-reference-invariant}
before taking norms.