\subsection{A common interval for the era branches}
\label{sec:era-intervals}

The local inverse in Corollary~\ref{cor:era-graph-inverse}
becomes an interval branch once the initial radius runs
across a prescribed era. The phase error controls its
two endpoints, while the positive derivative ensures
uniqueness throughout. These facts give a common output
interval for every radial graph in the same amplitude
range.

\begin{theorem}[The interval inverse]
\label{thm:era-interval-inverse}
Choose $\eta,J,\varepsilon,C$ as in
Theorem~\ref{thm:era-bounded-phase}. Let $m\in\N$,
$0<\beta<1/4$, and set
\begin{equation}
 a_-=2(m+1+\beta),\qquad
 a_+=2(m+2-\beta),\qquad I_m=[a_-,a_+].
 \label{eq:era-branch-radius-interval}
\end{equation}
Suppose $B>0$ satisfies
\begin{equation}
 Ba_+\leq\varepsilon,\qquad
 B\leq\varepsilon(\beta/2)^{2J},\qquad
 CB<\beta/4.
 \label{eq:era-interval-budget}
\end{equation}
Let $U,W$ be $C^1$ on a neighborhood of $I_m$ and
assume, for every $r\in I_m$,
\begin{equation}
 U(r),W(r)>0,\qquad U(r)+W(r)\leq B,\qquad
 |U'(r)|\leq\sqrt{U(r)},\quad
 |W'(r)|\leq\sqrt{W(r)}.
 \label{eq:era-interval-graphs}
\end{equation}
Write $\gamma(r)=(r,U(r),W(r))$ and
$\varphi=P_m\circ\gamma$. Then $\varphi$ is strictly
increasing on $I_m$, with
\begin{equation}
 \varphi(a_-)<2\beta,\qquad
 1-2\beta<\varphi(a_+).
 \label{eq:era-interval-endpoints}
\end{equation}
Its inverse $g$ on $\varphi(I_m)$ satisfies
$g(\varphi(r))=r$ for every $r\in I_m$.
For every $y\in[2\beta,1-2\beta]$, one has
\begin{equation}
 a_-<g(y)<a_+,\qquad \varphi(g(y))=y,
 \label{eq:era-interval-inverse-identities}
\end{equation}
and $g(y)$ is the unique radius in $I_m$ with this phase.
The inverse is $C^1$ near each such $y$, including the
endpoints of the retained interval, and
\begin{gather}
 g'(y)=\frac1{\varphi'(g(y))}>0,\notag\\
 \left|\frac{g(y)}2-(m+1+y)\right|\leq CB,
 \qquad \left|\log\frac{g'(y)}2\right|\leq C\sqrt B.
 \label{eq:era-interval-inverse-estimates}
\end{gather}
\end{theorem}

\begin{proof}
For $r\in I_m$, one has $0<r\leq a_+$ and hence
$Br\leq Ba_+\leq\varepsilon$. In particular,
\[
 0<U(r),W(r)\leq B\leq\varepsilon/r.
\]
The other conditions of~\eqref{eq:era-phase-domain}
follow from~\eqref{eq:era-interval-budget} and
\eqref{eq:era-interval-graphs}. Thus the actual era
phase is defined along the whole graph. By
Theorem~\ref{thm:era-bounded-phase} and the chain rule,
it is locally $C^1$ at every $r\in I_m$ and satisfies
\begin{gather}
 \left|\varphi(r)-\left(\frac r2-(m+1)\right)\right|
       \leq CB,\notag\\
 \varphi'(r)>0,\qquad
 |\log(2\varphi'(r))|\leq C\sqrt B.
 \label{eq:era-interval-pointwise}
\end{gather}

The mean value theorem makes $\varphi$ strictly increasing
on $I_m$. Continuity gives
\[
 \varphi(I_m)=[\varphi(a_-),\varphi(a_+)],
\]
so there is a unique inverse on this image. At the two
endpoints, the first bound
in~\eqref{eq:era-interval-pointwise} gives
\[
 \varphi(a_-)\leq\beta+CB<\tfrac54\beta<2\beta,
 \qquad
 \varphi(a_+)\geq1-\beta-CB
                    >1-\tfrac54\beta>1-2\beta.
\]
This proves~\eqref{eq:era-interval-endpoints} and places
the entire closed interval $[2\beta,1-2\beta]$ strictly
inside the open image
$(\varphi(a_-),\varphi(a_+))$.
Strict monotonicity then gives
\eqref{eq:era-interval-inverse-identities} and uniqueness.

For the ordinary differentiability assertion, fix such
a $y$ and put $r=g(y)$. Since $r\in(a_-,a_+)$ and
$\varphi'(r)>0$, the inverse function theorem supplies
a $C^1$ local inverse $j$ near $y$. By continuity,
shrink its neighborhood so that its values remain in
$(a_-,a_+)$ and its arguments remain in
$(\varphi(a_-),\varphi(a_+))$. Both $j$ and $g$ then
invert $\varphi$ inside $I_m$, where it is injective.
Hence $j=g$ on this ordinary open neighborhood of $y$,
and $g'(y)=1/\varphi'(g(y))>0$.
This argument also applies when $y=2\beta$ or
$y=1-2\beta$ because both endpoints lie in the open image.

Finally apply~\eqref{eq:era-interval-pointwise} at $r=g(y)$.
The value estimate follows from $\varphi(g(y))=y$ and
\[
 \frac{g(y)}2-(m+1+y)
   =-\left[y-\left(\frac{g(y)}2-(m+1)\right)\right].
\]
For the derivative, the reciprocal identity gives
\[
 \log\frac{g'(y)}2=-\log(2\varphi'(g(y))),
\]
which proves the last bound.
\end{proof}

The retained interval depends only on $\beta$.
The input radius interval depends on the era length,
but the inverse error and logarithmic derivative bounds
depend only on the common amplitude $B$.
Together with Theorem~\ref{thm:era-output}, this prepares
the choice of a phase margin and an amplitude scale
that can be propagated from one era to the next.