\subsection{Phase through an era}
\label{sec:era-phase}

The terminal connection closes the sequence constructed in
Proposition~\ref{prop:core-itinerary}. Its bounded phase
can be compared directly with the fractional part of the
initial half-radius. The comparison remains uniform in the
number of intervening reflections.

Fix $\eta,J$ in the terminal construction. Let $Q$, $T$,
and $\mathcal Y$ denote its bounded phase, elapsed time,
and four-coordinate output state at the natural auxiliary
height, as in~\eqref{eq:terminal-natural-bounded-phase}.
For a prescribed number $m$ of reflections, define
\begin{equation}
 \mathcal H_m=F\circ G^m,\qquad
 P_m=Q\circ\mathcal H_m,\qquad
 \mathcal Y_m=\mathcal Y\circ\mathcal H_m.
 \label{eq:era-phase-maps}
\end{equation}
Thus $\mathcal H_m$ is the last downward core state, and
$P_m$ is twice the reciprocal radius at the following
upward entrance. The time $T\circ\mathcal H_m$ measures
only the terminal connection; each preceding core and
reflection has its own positive elapsed time.

\begin{theorem}[The bounded era phase]
\label{thm:era-bounded-phase}
There is $0<\delta\leq1/40000$ such that, for each
$0<\eta\leq\delta$ and integer $J\geq4$, there are
$0<\varepsilon\leq10^{-4}$ and $C\geq1$ with the
following property. Suppose $m\in\N$, $0<\beta<1/2$, and
$X=(r,u,w)$ satisfy
\begin{gather}
 0<u,w\leq\varepsilon/r,\qquad
 b_0=u+w\leq\varepsilon(\beta/2)^{2J},\notag\\
 m+1+\beta\leq r/2\leq m+2-\beta,\qquad
 Cb_0<\beta/4.
 \label{eq:era-phase-domain}
\end{gather}
The $m+1$ cores and $m$ reflections exist, and their last
core has radius $2<R_m<4$. The subsequent terminal
connection has a first positive upward arrival at the new
$V=1$ section. The maps $\mathcal H_m,P_m,\mathcal Y_m$
and $T\circ\mathcal H_m$ are locally $C^1$ at $X$.
Writing $f=r/2-(m+1)$, one has
\begin{equation}
 \beta/2<P_m(X)<1-\beta/2,\qquad
 |P_m(X)-f|\leq Cb_0.
 \label{eq:era-phase-value}
\end{equation}
The next core radius is $\kappa_+=2/P_m(X)>2$, and
$\lfloor r/2\rfloor=m+1$. For every normalized radial
cone direction $v$ at $X$,
\begin{equation}
 DP_m(X)[v]>0,\qquad
 |\log(2DP_m(X)[v])|\leq C\sqrt{b_0}.
 \label{eq:era-phase-differential}
\end{equation}
\end{theorem}

\begin{proof}
Let $\varepsilon_k,K$ be the constants of
Proposition~\ref{prop:core-itinerary}, and let
$\varepsilon_t,A$ be those of
Corollary~\ref{cor:terminal-bounded-phase-direction}, with
$A\geq1$. Choose
\[
 \varepsilon=\min(\varepsilon_k,\varepsilon_t),\qquad
 C=K+A+1.
\]
The hypotheses imply $Kb_0<\beta/2$, so the core itinerary
is defined. Set
\[
 Z=\mathcal H_m(X)=(R,U,W),\qquad
 b=U+W,\qquad \alpha=(R-2)/2.
\]
Its terminal estimates give
\begin{equation}
 0<b\leq b_0,\qquad
 3\beta/4<\alpha<1-3\beta/4.
 \label{eq:era-terminal-input}
\end{equation}
In particular,
\[
 b\leq\varepsilon(\beta/2)^{2J}
            \leq\varepsilon_t\alpha^{2J}.
\]
At the last downward crossing, $R>2$ and
$0<U,W\leq b_0\leq10^{-4}$. The branch identification in
the proof of Lemma~\ref{lem:core-reflection-step} shows that
the actual state there is $\iota_-(Z)$.
The terminal passage therefore applies at this endpoint. It gives
the stated first arrival, its upward crossing, and locally
$C^2$ time and state as functions of $Z$.
Composing with the $C^1$ map $\mathcal H_m$ gives all
the regularity assertions at $X$.

The terminal phase estimate reads
\begin{equation}
 |P_m(X)-\alpha|\leq Ab\leq Ab_0.
 \label{eq:era-terminal-phase-error}
\end{equation}
As $Ab_0\leq Cb_0<\beta/4$, combining this with
\eqref{eq:era-terminal-input} gives the interval in
\eqref{eq:era-phase-value}. Also,
\[
 |\alpha-f|
   =\tfrac12|R-(r-2m)|\leq\tfrac K2 b_0.
\]
Adding this estimate to~\eqref{eq:era-terminal-phase-error}
gives $|P_m(X)-f|\leq(A+K/2)b_0\leq Cb_0$.
The output radius is positive and satisfies $P_m=2/\kappa_+$.
Since $0<P_m<1$, it follows that $\kappa_+>2$.
The floor identity comes from
Proposition~\ref{prop:core-itinerary}.

For the derivative, put $d=D\mathcal H_m(X)[v]$ and
$s=d_r$. By~\eqref{eq:core-itinerary-differential},
$s>0$, $|\log s|\leq K\sqrt{b_0}$, and
$\widehat d=d/s$ lies in the normalized cone at $Z$.
The terminal bounded phase satisfies
\[
 DQ(Z)[\widehat d]>0,\qquad
 |\log(2DQ(Z)[\widehat d])|\leq A\sqrt b.
\]
The chain rule and linearity give
\begin{equation*}
 2DP_m(X)[v]=s\,2DQ(Z)[\widehat d]>0.
\end{equation*}
Both factors are positive. Taking logarithms and using
$b\leq b_0$ yields
\[
 |\log(2DP_m(X)[v])|
 \leq K\sqrt{b_0}+A\sqrt b
 \leq C\sqrt{b_0},
\]
as required.
\end{proof}

The constructed segment contains $m+1$ successive cores and
ends at the entrance after its first terminal connection.
When its initial entrance also follows a terminal connection,
it is a complete era in Definition~\ref{def:era}, of length
$\lfloor r/2\rfloor$. The theorem identifies its output
phase and its derivative from the actual crossings.

The differential estimate is adapted to curves transverse
to the small wall directions. Let $U,W$ be $C^1$ near a
radius $r$ and consider the graph
\[
 \gamma(s)=(s,U(s),W(s)),\qquad
 |U'(r)|\leq\sqrt{U(r)},\quad
 |W'(r)|\leq\sqrt{W(r)}.
\]
Its tangent at $r$ is a normalized radial cone direction.

\begin{corollary}
\label{cor:era-graph-inverse}
Suppose $X=\gamma(r)$ satisfies
\eqref{eq:era-phase-domain}, with $u=U(r)$, $w=W(r)$.
Then $\varphi=P_m\circ\gamma$ is $C^1$ near $r$, has
$\varphi'(r)>0$, and admits a $C^1$ local inverse $g$ at
$\varphi(r)$. At the corresponding points,
\begin{equation}
 g(\varphi(r))=r,\qquad
 g'(\varphi(r))=\frac1{\varphi'(r)}>0,
 \label{eq:era-graph-inverse-derivative}
\end{equation}
and
\begin{equation}
 |\log(2\varphi'(r))|\leq C\sqrt{U(r)+W(r)},\qquad
 |\log(g'(\varphi(r))/2)|\leq C\sqrt{U(r)+W(r)}.
 \label{eq:era-graph-inverse-distortion}
\end{equation}
The identities $g\circ\varphi=\mathrm{id}$ and
$\varphi\circ g=\mathrm{id}$ hold on suitable open
neighborhoods of $r$ and $\varphi(r)$, respectively.
\end{corollary}

\begin{proof}
Differentiating the graph gives
\[
 \varphi'(r)=DP_m(X)[(1,U'(r),W'(r))].
\]
Theorem~\ref{thm:era-bounded-phase} supplies its positivity
and the first logarithmic bound. The one-dimensional inverse
function theorem gives the local inverse and
\eqref{eq:era-graph-inverse-derivative}. Finally,
\[
 g'(\varphi(r))/2=(2\varphi'(r))^{-1},
\]
so the two logarithmic errors have equal absolute value.
\end{proof}

The local inverse is therefore controlled by the same initial
amplitude as the forward phase, independently of the era
length. To obtain the interval branches used in the measure
argument, we next need the output wall estimates and a common
phase interval on which these local inverses join.