\subsection{The terminal cone}
\label{sec:terminal-cone}

The terminal passage sends a radius close to two to a large
radius in the next core. Its radial derivative has size
$\alpha^{-2}$. To compare the transverse derivatives with this
scale, we retain the two logarithmic wall responses separately
and multiply each by the square root of its own output wall.
The decay during the incoming passage then controls both slopes,
even when the two input walls have very different sizes.

For the fixed-height passage, write
\[
 \mathcal R_h(Y)=(\kappa_h(Y),U_+(Y),W_+(Y)),\qquad
 Y=(R,u,w),
\]
where $U_+,W_+$ are the wall components of
$\mathcal E(h,Y)$. All derivatives with a subscript $h$ below
hold this height fixed. The radial cone at $Y$ is
\begin{equation}
 \mathcal K_Y=\{d:\ |d_u|\leq\sqrt u\,|d_R|,
                         \ |d_w|\leq\sqrt w\,|d_R|\}.
 \label{eq:terminal-radial-cone}
\end{equation}
Every nonzero vector in it can be divided by $d_R$ to give
a direction $d=(1,d_u,d_w)$ with
$|d_u|\leq\sqrt u$, $|d_w|\leq\sqrt w$.

\begin{theorem}[Terminal cone preservation]
\label{thm:terminal-cone}
There is $0<\delta\leq1/40000$ such that, for every
$0<\eta_0\leq\delta$ and integer $J\geq5$, there are
$C\geq1$ and $\varepsilon>0$ with the following property.
Suppose
\begin{equation}
 \eta_0\leq\eta\leq\delta,\qquad 0<\alpha\leq1,\qquad
 u,w>0,\qquad u+w\leq b\leq\varepsilon\alpha^{2J},
 \quad b>0.
 \label{eq:terminal-cone-domain}
\end{equation}
At $Y=(2+2\alpha,u,w)$ and $h=\eta\alpha^J$, the actual
return $\mathcal R_h$ is locally $C^2$ and has positive
output walls. For every normalized direction in
\eqref{eq:terminal-radial-cone}, put $p=D\kappa_h(Y)[d]$.
Then
\begin{equation}
 \begin{gathered}
 p<0,\qquad |p|\geq\frac{3}{4\alpha^2},\qquad
 |\alpha^2p+1|\leq C\sqrt b,\\
 \max\left\{
   \frac{|DU_+[d]|}{\sqrt{U_+}\,|p|},
   \frac{|DW_+[d]|}{\sqrt{W_+}\,|p|}
       \right\}\leq Cb^{7/24}<\frac12.
 \end{gathered}
 \label{eq:terminal-radial-cone-bounds}
\end{equation}
Thus $D\mathcal R_h$ maps every nonzero vector of
$\mathcal K_Y$ strictly inside the radial cone at its output,
with the radial orientation reversed.
\end{theorem}

We prove the radial bound first, then the two wall bounds.
Throughout the proof, constants may depend on $\eta_0,J$
but are independent of $\alpha,\eta,u,w$. Each smallness
condition is imposed by reducing the same positive
$\varepsilon$.

\subsubsection{Radius and wall responses}

The inversion $\kappa_h=2/Q_h$ converts the bounded phase
estimates into ordinary radius derivatives.

\begin{lemma}
\label{lem:terminal-radius-columns}
On the domain~\eqref{eq:terminal-cone-domain} with $J\geq4$,
after choosing $\delta$ and $\varepsilon$ small enough,
\begin{align}
 \left|\kappa_h-\frac2\alpha\right|
       &\leq\frac{Cb}{\alpha^2},&
 \left|(\kappa_h)_R+\frac1{\alpha^2}\right|
       &\leq\frac{Cb}{\alpha^3},\notag\\
 |(\kappa_h)_u|+|(\kappa_h)_w|
       &\leq\frac C{\alpha^2}.
 \label{eq:terminal-radius-columns}
\end{align}
The radius is positive and locally $C^2$.
\end{lemma}

\begin{proof}
Let $A\geq1$ be a common constant in
Proposition~\ref{prop:terminal-physical-phase}. Since
$\alpha^{2J}\leq\alpha$, we can require $Ab\leq\alpha/4$.
Then
\[
 \alpha/2\leq Q_h\leq2\alpha,\qquad
 D\kappa_h=-\frac{2DQ_h}{Q_h^2}.
\]
The value and wall estimates follow from
$|Q_h-\alpha|\leq Ab$ and $Q_h^{-2}\leq4/\alpha^2$.
For the radial column, subtract its reference value before
estimating:
\begin{equation*}
 -\frac{2(Q_h)_R}{Q_h^2}+\frac1{\alpha^2}
 =-\frac{2((Q_h)_R-1/2)}{Q_h^2}
   +\frac{(Q_h-\alpha)(Q_h+\alpha)}{\alpha^2Q_h^2}.
\end{equation*}
The two terms are bounded by $8Ab/\alpha^2$ and
$12Ab/\alpha^3$, respectively. Since $\alpha\leq1$,
their sum is at most $20Ab/\alpha^3$.
The value constant is $4A$ and the sum of the wall constants
is $8A$, so a single enlarged $C$ gives all three bounds.
Positivity and local regularity follow from those of $Q_h$.
\end{proof}

For a normalized cone direction, these columns give
\begin{align}
 |\alpha^2D\kappa_h[d]+1|
 &\leq\alpha^2|(\kappa_h)_R+\alpha^{-2}|
       +\alpha^2\bigl(|(\kappa_h)_u|+|(\kappa_h)_w|\bigr)
          \max(|d_u|,|d_w|)\notag\\
 &\leq Cb/\alpha+C\sqrt b
 \leq2C\sqrt b.
 \label{eq:terminal-radius-cone-error}
\end{align}
Here $b\leq\alpha^2$, after taking $\varepsilon\leq1$,
implies $b/\alpha\leq\sqrt b$. Requiring
$2C\sqrt b\leq1/4$ proves the negative sign and the lower
bound in~\eqref{eq:terminal-radial-cone-bounds}.

At the cyclic entrance use the notation
\[
 Z_h=(m_e,u_e,v_e,w_e),\qquad
 \widehat Z_h=(r_h,u_e,v_e,w_e),\qquad
 L=\log(1/v_e).
\]
The geometric bounds and the wall factors give
\begin{equation}
 \frac1\alpha\leq k=H(m_e-1,v_e)\leq\frac3\alpha,
 \qquad 0<v_e<1,
 \label{eq:terminal-log-entrance-domain}
\end{equation}
with $u_e,w_e$ in the domain of
Propositions~\ref{prop:entrance-three-columns}
and~\ref{prop:entrance-active-column}. This follows by taking
$\eta$ below one quarter of their wall threshold and then
using Proposition~\ref{prop:terminal-first-arrival} with $q=0$.

\begin{lemma}
\label{lem:terminal-log-wall-directions}
For $J\geq2$, uniformly on a domain of the form
\eqref{eq:terminal-cone-domain}, the output walls are positive
and locally $C^2$. In every normalized cone direction,
\begin{align}
 |D\log U_+[d]|&\leq
 C\left(\frac{1+L}{\alpha^2}
               +\frac1{\alpha\sqrt w}\right),\notag\\
 |D\log W_+[d]|&\leq
 C\left(L+\frac1\alpha+\frac1{\sqrt w}
                          +\frac1{\sqrt u}\right).
 \label{eq:terminal-output-log-directions}
\end{align}
\end{lemma}

\begin{proof}
The two passive coordinates at the entrance retain the
input wall as a factor:
\[
 v_e=w\mathcal B_W,\qquad w_e=u\mathcal B_U.
\]
By Lemma~\ref{lem:terminal-cyclic-wall-factors}, the factors
are positive and locally $C^2$, and their logarithmic
derivatives are bounded by $C/\alpha$. The same bound
holds for $D\log u_e$.
Since $b\leq1$, a normalized cone direction has maximum
norm at most one. Proposition~\ref{prop:terminal-entrance-derivatives}
and logarithmic differentiation therefore give
\begin{align*}
 |Dr_h[d]|&\leq C/\alpha^2,&
 |D\log u_e[d]|&\leq C/\alpha,\\
 |D\log v_e[d]|&\leq C/\alpha+1/\sqrt w,&
 |D\log w_e[d]|&\leq C/\alpha+1/\sqrt u.
\end{align*}
For example, $D\log v_e[d]=d_w/w+D\log\mathcal B_W[d]$.

Put $a=u_e$, $v=v_e$, $c=w_e$, and
$e=D\widehat Z_h[d]$. With a common $B\geq1$, the bounds
just proved read
\begin{align*}
 |e_r|&\leq B/\alpha^2,&
 |e_a|&\leq aB/\alpha,\\
 |e_v|&\leq v(B/\alpha+s),&
 |e_c|&\leq c(B/\alpha+t),
 \qquad s=1/\sqrt w,\quad t=1/\sqrt u.
\end{align*}
One further product is useful in the strong output row:
\begin{equation}
 ct=\mathcal B_U\sqrt u\leq B/\alpha^2.
 \label{eq:terminal-weak-factor-product}
\end{equation}
This uses $u\leq1$ and the upper factor bound. It is uniform
in the ratio $u/w$.

The local inverse of the entrance radial coordinates identifies
the terminal output with the incoming return on a neighborhood
of $Y$. Thus its logarithmic derivative is the incoming
logarithmic differential applied to $e$. Using
\eqref{eq:entrance-three-wall-columns}
and~\eqref{eq:entrance-active-wall-columns}, with a common
constant $A\geq1$, its two rows satisfy
\begin{align*}
 |D\log U_+[d]|\leq{}&
 AL|e_r|+(1/a+A)|e_a|
                +(Ak/v)|e_v|+A|e_c|,\\
 |D\log W_+[d]|\leq{}&
 (AL/k^2)|e_r|+(A/k^2)|e_a|
                +(A/v)|e_v|+(1/c+A/k^2)|e_c|.
\end{align*}
In the first row, the four terms are bounded by
\[
 \frac{ABL}{\alpha^2},\qquad
 \frac{2AB}{\alpha^2},\qquad
 \frac{3AB}{\alpha^2}+\frac{3As}{\alpha},\qquad
 \frac{2AB}{\alpha^2},
\]
respectively. Here $a,c\leq1$, $k\leq3/\alpha$, and
\eqref{eq:terminal-weak-factor-product} were used.
In the second row, $k^{-2}\leq\alpha^2$ gives the bounds
\[
 ABL,\qquad AB/\alpha,\qquad
 AB/\alpha+As,\qquad 2AB/\alpha+2At.
\]
Adding each row proves~\eqref{eq:terminal-output-log-directions}.
The regularity and positivity are those of the actual passage
and its multiplicative wall equations.
\end{proof}

The strong output row has no $u^{-1/2}$ term. Its dependence
on that input is multiplied by $w_e=u\mathcal B_U$ before
the estimate is taken. Keeping this factor is what allows
the strong and weak walls to be treated independently.

\subsubsection{Decay absorbs the logarithmic singularities}

The incoming wall estimates in Proposition~\ref{prop:incoming-passage}
and $k\geq7/4$ imply
\begin{equation}
 \sqrt{U_+}\leq\sqrt{u_e}\,v_e^{19/24},\qquad
 \sqrt{W_+}\leq\sqrt{w_e}\sqrt{v_e}.
 \label{eq:terminal-output-square-roots}
\end{equation}
Indeed $1+k/3\geq19/12$ and $1+1/k\geq1$.
The entrance estimates also give, with $A\geq1$,
\begin{align}
 \sqrt{u_e}&\leq\alpha^{(J-2)/2},&
 v_e&\leq Aw/\alpha^2,\notag\\
 \sqrt{v_e}&\leq A\sqrt w/\alpha,&
 \sqrt{w_e}&\leq A\sqrt u/\alpha.
 \label{eq:terminal-entrance-square-roots}
\end{align}
For the first inequality, use $u_e\leq2\eta\alpha^{J-2}$
and $2\eta\leq1$. The others follow directly from the
cyclic wall factors, enlarging $A$ to cover their square roots.

We first record the powers left by the strong output. For
$0<v\leq1$,
\begin{equation}
 v^{19/24}(1+|\log v|)\leq24v^{3/4}.
 \label{eq:terminal-log-power}
\end{equation}
To see this, put $t=v^{1/24}$. The elementary inequality
$1-1/t\leq\log t$ implies $t(1-\log t)\leq1$;
hence $t(1-24\log t)\leq24$.
Also, if $v\leq Aw/\alpha^2$ and $0\leq q\leq1$, then
\begin{equation}
 v^q\leq A\alpha^{-2q}w^q,
 \label{eq:terminal-wall-power}
\end{equation}
because $A^q\leq A$. With $\beta=(J-2)/2$, these two
inequalities yield
\begin{align}
 &\alpha^\beta\left[
 v^{19/24}(1+|\log v|)
       +\frac{\alpha v^{19/24}}{\sqrt w}\right]\notag\\
 &\hspace{12mm}\leq24A\left[
   \alpha^{(J-5)/2}w^{3/4}
       +\alpha^{J/2-19/12}w^{7/24}\right].
 \label{eq:terminal-strong-powers}
\end{align}
The second exponent of $w$ is $19/24-1/2=7/24$.

For the weak output, the four terms can be bounded without
comparing $u$ and $w$. From
\eqref{eq:terminal-entrance-square-roots}, $v_e\leq1$, and
$\sqrt{v_e}|\log v_e|\leq2$, we obtain
\begin{align}
 \alpha^2\sqrt{w_e}\sqrt{v_e}\,L
     &\leq2A\sqrt u,&
 \alpha\sqrt{w_e}\sqrt{v_e}
     &\leq A\sqrt u,\notag\\
 \frac{\alpha^2\sqrt{w_e}\sqrt{v_e}}{\sqrt w}
     &\leq A^2\sqrt u,&
 \frac{\alpha^2\sqrt{w_e}\sqrt{v_e}}{\sqrt u}
     &\leq A^2\sqrt w.
 \label{eq:terminal-weak-products}
\end{align}
Their sum is bounded by $4A^2(\sqrt u+\sqrt w)$.

We now combine these powers with the logarithmic derivatives.
For $J\geq4$, the radial lower bound already proved gives
$\alpha^2|p|\geq3/4$. For either positive output wall $x$,
\begin{equation}
 \frac{|Dx[d]|}{\sqrt x\,|p|}
   =\frac{\alpha^2\sqrt x\,|D\log x[d]|}
          {\alpha^2|p|}
   \leq2\alpha^2\sqrt x\,|D\log x[d]|.
 \label{eq:terminal-slope-conversion}
\end{equation}
The first row of~\eqref{eq:terminal-output-log-directions},
multiplied by $\alpha^2\sqrt{U_+}$ and estimated with
\eqref{eq:terminal-output-square-roots}
and~\eqref{eq:terminal-strong-powers}, therefore gives
\begin{equation}
 \frac{|DU_+[d]|}{\sqrt{U_+}\,|p|}
 \leq C\left[
   \alpha^{(J-5)/2}w^{3/4}
       +\alpha^{J/2-19/12}w^{7/24}\right].
 \label{eq:terminal-strong-slope}
\end{equation}
The second row, together with
\eqref{eq:terminal-output-square-roots}
and~\eqref{eq:terminal-weak-products}, gives
\begin{equation}
 \frac{|DW_+[d]|}{\sqrt{W_+}\,|p|}
       \leq C(\sqrt u+\sqrt w).
 \label{eq:terminal-weak-slope}
\end{equation}

\begin{proof}[Completion of the proof of Theorem~\ref{thm:terminal-cone}]
Choose $\delta$ to satisfy the phase, entrance-column, and
wall-factor thresholds above. For $J\geq5$, the powers
of $\alpha$ in~\eqref{eq:terminal-strong-slope} are
nonnegative. As $u,w\leq b\leq1$,
\[
 w^{3/4}+w^{7/24}\leq2b^{7/24},\qquad
 \sqrt u+\sqrt w\leq2b^{7/24}.
\]
After enlarging $C$, both slopes are at most $Cb^{7/24}$.
Reduce $\varepsilon$ once more so that
$C\varepsilon^{7/24}<1/2$ and the radial error in
\eqref{eq:terminal-radius-cone-error} is at most $1/4$.
All thresholds are independent of the individual walls.
The actual-return regularity and positivity were established
in the preceding lemmas. Finally, the inequalities are
homogeneous in the direction, so the normalized conclusion
extends to every nonzero vector of $\mathcal K_Y$.
\end{proof}

The same cone estimate holds for the actual return when the
auxiliary height follows the input phase. With $a(Y)$ and
$\mathcal Y(Y)$ as in~\eqref{eq:terminal-natural-bounded-phase},
set
\[
 \kappa(Y)=H(\mathcal Y_M(Y)-1,\mathcal Y_V(Y)),\qquad
 \mathcal R(Y)=(\kappa(Y),\mathcal Y_U(Y),\mathcal Y_W(Y)).
\]

\begin{corollary}
\label{cor:terminal-natural-cone}
For fixed $0<\eta\leq\delta$ and $J\geq5$, there are
$C\geq1$ and $\varepsilon>0$ such that, whenever
$0<\alpha\leq1$ and $u,w>0$ satisfy
$u+w\leq b\leq\varepsilon\alpha^{2J}$, $b>0$,
the map $\mathcal R$ is locally $C^2$ at
$Y=(2+2\alpha,u,w)$. Its differential satisfies all the
cone estimates of Theorem~\ref{thm:terminal-cone} at that point.
\end{corollary}

\begin{proof}
Apply Theorem~\ref{thm:terminal-cone} with $\eta_0=\eta$.
After intersecting its amplitude threshold with that of
Lemma~\ref{lem:terminal-height-freezing}, the latter identifies
$\mathcal E(a(Z),Z)$ and $\mathcal E(h,Z)$ on a neighborhood
of $Y$, where $h=a(Y)=\eta\alpha^J$.
Consequently $\mathcal R$ and $\mathcal R_h$ agree there.
Their values and ordinary differentials agree, giving all
the stated estimates.
\end{proof}

The terminal passage therefore preserves the radial cones
used through the preceding core passages and reflections.
Its reversal of the radius is accompanied by a positive
bounded-phase derivative, as in
Corollary~\ref{cor:terminal-bounded-phase-direction}.
These two descriptions provide the radial orientation and
the transverse control needed to compose a complete era.