\subsection{Contraction of the era amplitude}
\label{sec:era-output}

At their endpoints, the core passages and reflections do not
increase either small wall. The terminal connection supplies a stronger
gain: their sum at the next entrance is bounded by a power
greater than one of their previous sum. This gain will
allow the phase and derivative errors to be summed over
successive eras. To obtain it uniformly near a terminal
phase of zero, we retain the radius in the strong-wall
equation during the incoming passage.

\begin{lemma}
\label{lem:incoming-refined-decay}
Under the hypotheses of Proposition~\ref{prop:incoming-passage},
the same first arrival satisfies
\begin{equation}
 \mathcal P_U(y)
 \leq u\exp\!\left(\frac{2w+3u^2}{k}\right)
                  v^{1+99k/200}.
 \label{eq:incoming-refined-output}
\end{equation}
In particular, if $k\geq19/10$, then
\begin{equation}
 \mathcal P_U(y)\leq e^{1/1000}u v^{19/10},
 \qquad \mathcal P_W(y)\leq wv.
 \label{eq:incoming-terminal-powers}
\end{equation}
\end{lemma}

\begin{proof}
Use the increasing clock $s=\log(V/v)$ on the passage,
and write
\[
 \begin{gathered}
 \rho=H(M-1,V),\qquad c=U(V+W-U),\qquad q=M-1-c,\\
 a=\tfrac12\bigl(1-M^2-(V-W)^2+U^2\bigr).
 \end{gathered}
\]
Thus $dU/ds=(a/q)U$. The region
\eqref{eq:incoming-region} and its clock bounds hold up
to the actual first arrival. The identity
\begin{equation}
 \frac aq=-1-\frac{\rho^2}{2(M-1)}
 +\frac{2VW-W^2+U^2-\bigl(2+\rho^2/(M-1)\bigr)c}{2q}
 \label{eq:incoming-arc-rate-identity}
\end{equation}
keeps the full radius in the negative term. Since
$0<M-1\leq\rho$ and $\rho\geq99k/100$,
\[
 \frac{\rho^2}{2(M-1)}\geq\frac\rho2\geq\frac{99k}{200}.
\]
For the remainder, the bounds
$M-1\geq4k/5$ and $\rho\leq101k/100$ give
\[
 \frac{\rho^2}{M-1}\leq\frac{32}{25}k,
 \qquad 3+\frac{\rho^2}{M-1}\leq4q.
\]
The second inequality uses $q\geq3k/4$ and $k\geq7/4$.
Together with $c\geq-U^2$, these estimates bound the
last term of~\eqref{eq:incoming-arc-rate-identity} by
\[
 \frac{VW}{q}+2U^2\leq\frac{2W}{k}+2U^2.
\]
Consequently
\begin{equation*}
 \frac aq\leq-\left(1+\frac{99k}{200}\right)
                       +\frac{2W}{k}+2U^2.
\end{equation*}

The previously established decay~\eqref{eq:incoming-decay}
controls the two error integrals without a factor of the
waiting length:
\begin{equation*}
 \int_0^s W(\ell)\dd\ell\leq w,
 \qquad
 \int_0^s U(\ell)^2\dd\ell\leq\frac{3u^2}{2k}.
\end{equation*}
Indeed, it is enough to use $W(\ell)\leq we^{-\ell}$
and $U(\ell)^2\leq u^2e^{-2k\ell/3}$.
Integrating the scalar equation for $U$ now gives, for
every $0\leq s\leq\log(1/v)$,
\[
 U(s)\leq u\exp\!\left(\frac{2w+3u^2}{k}\right)
                    e^{-(1+99k/200)s}.
\]
This also holds for $u=0$, by invariance of that wall.
At the first arrival $s=\log(1/v)$, this is
\eqref{eq:incoming-refined-output}.

On the incoming domain,
$(2w+3u^2)/k\leq1/1000$. If $k\geq19/10$, then
$1+99k/200\geq19/10$. Since $0<v<1$, the first bound
in~\eqref{eq:incoming-terminal-powers} follows.
The second follows from
$\mathcal P_W(y)\leq wv^{1+1/k}\leq wv$.
\end{proof}

Let $\mathcal Y$ be the actual terminal output at the
natural height, as in~\eqref{eq:terminal-natural-bounded-phase}.
The cyclic change of axes amplifies two incoming walls
by at most $\alpha^{-2}$. The following estimate shows
how the incoming decay compensates for this amplification
under the same smallness conditions.

\begin{proposition}
\label{prop:terminal-amplitude}
Fix $0<\eta\leq1/40000$ and an integer $J\geq20$.
There are $C\geq1$ and $0<\varepsilon\leq1$ such that,
for $0<\alpha\leq1$, $u,w>0$, and
\[
 u+w\leq b\leq\varepsilon\alpha^{2J},\qquad
 Y=(2+2\alpha,u,w),
\]
the two small walls at the next entrance satisfy
\begin{equation}
 \mathcal Y_U(Y)>0,\qquad \mathcal Y_W(Y)>0,\qquad
 \mathcal Y_U(Y)+\mathcal Y_W(Y)\leq Cb^{19/10}.
 \label{eq:terminal-amplitude}
\end{equation}
\end{proposition}

\begin{proof}
For this fixed $\eta$, choose a common amplitude threshold for the cyclic
geometry, the wall factors, and the first incoming
arrival. At $h=\eta\alpha^J$, let
$y_e=(m_e,u_e,v_e,w_e)$ be the cyclic entrance.
Lemma~\ref{lem:terminal-cycle-entrance},
Proposition~\ref{prop:terminal-first-arrival}, and
Lemma~\ref{lem:terminal-cyclic-wall-factors} give, with
$q=0$ and $\varepsilon_{\rm in}=10^{-4}$ in the
first-arrival proposition,
\begin{gather*}
 k=H(m_e-1,v_e)>\frac{19}{10\alpha}\geq\frac{19}{10},
 \qquad m_e>2,\qquad0<v_e<\tfrac12,\\
 0<u_e,w_e\leq10^{-4},\qquad
 u_e\leq2\eta\alpha^{J-2},\qquad
 v_e\leq\frac{Aw}{\alpha^2},\quad
 w_e\leq\frac{Au}{\alpha^2}
\end{gather*}
for a constant $A\geq1$.
Lemma~\ref{lem:incoming-refined-decay} therefore applies to
this actual entrance. At its first arrival,
\begin{equation}
 \mathcal Y_U(Y)\leq e^{1/1000}u_e v_e^{19/10},
 \qquad \mathcal Y_W(Y)\leq w_ev_e.
 \label{eq:terminal-amplitude-products}
\end{equation}
Both walls are positive by their multiplicative equations.
The state in~\eqref{eq:terminal-amplitude-products} is
$\mathcal Y(Y)$ because the natural height at this $Y$
is exactly $\eta\alpha^J$.

Shrink the threshold so that $\varepsilon\leq1$.
Since $J\geq20$, $0<\alpha\leq1$, and $2\eta\leq1$,
\begin{equation*}
 u_e\leq\alpha^4,\qquad b\leq\alpha^{40},\qquad
 b^{1/10}\leq\alpha^4.
\end{equation*}
It follows that
\begin{align*}
 u_e v_e^{19/10}
 &\leq A^{19/10}w^{19/10}\alpha^{1/5}
       \leq A^2b^{19/10},\\
 w_ev_e
 &\leq\frac{A^2uw}{\alpha^4}
       \leq\frac{A^2b^2}{\alpha^4}
       \leq A^2b^{19/10}.
\end{align*}
Insert these bounds into~\eqref{eq:terminal-amplitude-products}
and take $C=(e^{1/1000}+1)A^2$.
\end{proof}

Return to the era maps~\eqref{eq:era-phase-maps}, and
denote their two small output walls and radius by
\begin{equation}
 U_m^+=(\mathcal Y_m)_U,\qquad
 W_m^+=(\mathcal Y_m)_W,\qquad
 \kappa_m^+=H((\mathcal Y_m)_M-1,(\mathcal Y_m)_V).
 \label{eq:era-output-coordinates}
\end{equation}
All are functions of the initial core coordinates $X$.
On the era domain, $\kappa_m^+=2/P_m$.

\begin{theorem}[Amplitude and cone contraction through an era]
\label{thm:era-output}
The constants in Theorem~\ref{thm:era-bounded-phase} can
be chosen so that, for each $J\geq20$, the same domain
\eqref{eq:era-phase-domain} also has the following
properties. The maps $\mathcal Y_m$ and $\kappa_m^+$
are locally $C^1$, their small output walls are positive,
and
\begin{equation}
 U_m^+(X)+W_m^+(X)\leq Cb_0^{19/10},\qquad
 Cb_0^{7/24}<\tfrac12.
 \label{eq:era-output-amplitude}
\end{equation}
For every normalized radial cone direction $v$ at $X$,
put $p=D\kappa_m^+(X)[v]$. Then
\begin{equation}
 \begin{gathered}
 p<0,\\
 \max\left\{
  \frac{|DU_m^+(X)[v]|}{\sqrt{U_m^+(X)}\,|p|},
  \frac{|DW_m^+(X)[v]|}{\sqrt{W_m^+(X)}\,|p|}
 \right\}\leq Cb_0^{7/24}.
 \end{gathered}
 \label{eq:era-output-cone}
\end{equation}
The constants are independent of $m$ and of the ratio
of the two input walls.
\end{theorem}

\begin{proof}
Intersect the allowed ranges of $\eta$ in the bounded
phase and terminal cone estimates. For fixed $\eta,J$,
let $K$ be the core itinerary constant, $A$ the constant
of Corollary~\ref{cor:terminal-natural-cone}, and $B$
the constant of Proposition~\ref{prop:terminal-amplitude}.
Choose $C\geq K+A+B$ also large enough for
Theorem~\ref{thm:era-bounded-phase}. Choose $\varepsilon$
below all the corresponding amplitude thresholds and
so small that
\begin{equation}
 C\varepsilon^{7/24}<\tfrac12.
 \label{eq:era-output-threshold}
\end{equation}

For $Z=\mathcal H_m(X)=(R,U,W)$, set $b=U+W$ and
$\alpha=(R-2)/2$. The core itinerary and the domain give
\[
 0<b\leq b_0,\qquad 3\beta/4<\alpha<1-3\beta/4,
 \qquad b\leq\varepsilon(\beta/2)^{2J}
                         \leq\varepsilon\alpha^{2J}.
\]
Thus both terminal estimates apply at $Z$.
The terminal amplitude is at most $Bb^{19/10}$, which
is at most $Cb_0^{19/10}$. The actual terminal state
and radius are locally $C^2$ at $Z$, so their compositions
with $\mathcal H_m$ are locally $C^1$ at $X$.
Positivity is inherited from the terminal return.
Also $b_0\leq\varepsilon$, and hence
\eqref{eq:era-output-threshold} gives the second
inequality in~\eqref{eq:era-output-amplitude}.

To compare the cones, put $d=D\mathcal H_m(X)[v]$.
The core itinerary gives $s=d_r>0$ and a normalized
cone vector $\widehat d=d/s$ at $Z$. If
$p_t=D\kappa(Z)[\widehat d]$, then the terminal cone
estimate gives $p_t<0$ and
\[
 \frac{|D\mathcal Y_U(Z)[\widehat d]|}
      {\sqrt{\mathcal Y_U(Z)}\,|p_t|}
 \leq Ab^{7/24},\qquad
 \frac{|D\mathcal Y_W(Z)[\widehat d]|}
      {\sqrt{\mathcal Y_W(Z)}\,|p_t|}
 \leq Ab^{7/24}.
\]
By the chain rule, $p=sp_t<0$ and each output wall
derivative is $s$ times its terminal derivative in
direction $\widehat d$. This positive factor cancels
from each ratio. Finally $Ab^{7/24}\leq Cb_0^{7/24}$,
which proves~\eqref{eq:era-output-cone}.
\end{proof}

In particular, along a radial graph satisfying the input
cone bound, the output radius has negative derivative.
When the image is parametrized locally by that radius,
its two wall slopes satisfy the same square-root bounds
with a factor at most $Cb_0^{7/24}<1/2$. Thus the
class of graphs used to control the phase is preserved
with a strict margin at the next entrance.