\subsection{Successor graphs}
\label{sec:era-successors}

The image of a retained era is a radial graph over a
closed interval of next radii. To repeat the same
construction, we extend its two small walls to all
radii while preserving their positivity and slope bounds.
The superlinear amplitude estimate leaves enough room
for this extension.

The cone inequality becomes especially simple after
taking square roots: $|U'|\leq\theta\sqrt U$ is equivalent
to $|(\sqrt U)'|\leq\theta/2$ when $U>0$.
This gives an explicit extension with a controlled
amplitude increase.

\begin{lemma}
\label{lem:positive-amplitude-extension}
Let $l<r$ and $0\leq\theta\leq1$. Suppose $u,w$ are
positive $C^1$ functions on $[l,r]$, with continuous
one-sided derivatives at its endpoints, and
\begin{equation}
 u+w\leq A,\qquad |u'|\leq\theta\sqrt u,\qquad
 |w'|\leq\theta\sqrt w.
 \label{eq:amplitude-extension-inputs}
\end{equation}
They admit positive $C^1$ extensions $\widetilde u,
\widetilde w$ to $\R$, agreeing with both values and
derivatives on $[l,r]$, such that
\begin{equation}
 \widetilde u+\widetilde w\leq\frac{25}{16}A,\qquad
 |\widetilde u'|\leq\theta\sqrt{\widetilde u},\qquad
 |\widetilde w'|\leq\theta\sqrt{\widetilde w}.
 \label{eq:amplitude-extension-bounds}
\end{equation}
\end{lemma}

\begin{proof}
Define
\[
 c(t)=\max(1-t,0),\qquad H(t)=\int_0^t c(s)\dd s.
\]
The function $H$ is $C^1$, with $H'=c$ and $H(0)=0$.
For $t\geq0$,
\[
 H(t)=
 \begin{cases}
  t-t^2/2,&0\leq t\leq1,\\
  1/2,&t\geq1,
 \end{cases}
 \qquad 0\leq H(t)\leq\tfrac12,\quad0\leq c(t)\leq1.
\]
Given an endpoint value $s>0$, a derivative $d$, and
an outward orientation $e\in\{-1,1\}$, put
\begin{equation}
 E_{s,d,e}(x)=s+eds\,H(ex/s).
 \label{eq:root-extension-formula}
\end{equation}
Here $x$ is the signed displacement from the endpoint,
so $ex\geq0$ on the exterior half-line. Direct
differentiation gives
\begin{equation}
 E_{s,d,e}(0)=s,\qquad
 E_{s,d,e}'(x)=d\,c(ex/s),\qquad E_{s,d,e}'(0)=d.
 \label{eq:root-extension-derivative}
\end{equation}
If $|d|\leq\theta/2$, then on this half-line
\begin{equation}
 (1-\theta/4)s\leq E_{s,d,e}(x)
       \leq(1+\theta/4)s,\qquad
 |E_{s,d,e}'(x)|\leq\theta/2.
 \label{eq:root-extension-size}
\end{equation}
Indeed, $|edsH(ex/s)|\leq\theta s/4$.
The lower bound is positive because $\theta\leq1$.

Apply this construction to $f=\sqrt u$.
On $[l,r]$ its derivative is the continuous function
$f'=u'/(2\sqrt u)$, whose absolute value is at most
$\theta/2$. Set
\begin{equation}
 \widetilde f(t)=
 \begin{cases}
 E_{f(l),f'(l),-1}(t-l),&t<l,\\
 f(t),&l\leq t\leq r,\\
 E_{f(r),f'(r),1}(t-r),&r<t.
 \end{cases}
 \label{eq:root-curve-extension}
\end{equation}
The values and one-sided derivatives agree at both
joining points by~\eqref{eq:root-extension-derivative}.
The derivative is continuous there, since $c$ is
continuous and $c(0)=1$. Thus $\widetilde f$ is positive
and $C^1$ on $\R$, with $|\widetilde f'|\leq\theta/2$.
Use the same formula for $g=\sqrt w$, obtaining
$\widetilde g$.

Define $\widetilde u=\widetilde f^2$ and
$\widetilde w=\widetilde g^2$. Squaring preserves their
values on $[l,r]$, and
$2f f'=u'$, $2g g'=w'$ there, including at the endpoints.
Everywhere on $\R$,
\[
 |\widetilde u'|
    =2\widetilde f|\widetilde f'|
       \leq\theta\widetilde f
       =\theta\sqrt{\widetilde u},
\]
and the same holds for $\widetilde w$.
On each exterior half-line, the two upper bounds
in~\eqref{eq:root-extension-size} refer to the same
endpoint. Therefore
\[
 \widetilde u+\widetilde w
       \leq(1+\theta/4)^2 A\leq\frac{25}{16}A.
\]
On $[l,r]$ the original bound is stronger.
\end{proof}

Use the input interval $I_m=[a_-,a_+]$ of
\eqref{eq:era-branch-radius-interval}. Along its graph,
write
\begin{equation}
 k(r)=\kappa_m^+(\Gamma(r)),\qquad
 u_0(r)=U_m^+(\Gamma(r)),\qquad
 w_0(r)=W_m^+(\Gamma(r)).
 \label{eq:era-output-curve}
\end{equation}
The next bounded phase is $\varphi_m(r)=2/k(r)$.
The retained phase interval corresponds to
\begin{equation}
 L_\beta=[l_\beta,r_\beta],\qquad
 l_\beta=\frac2{1-2\beta},\qquad r_\beta=\frac1\beta
 \label{eq:era-output-radius-window}
\end{equation}
in the next radius. Let $\xi_m$ be the inverse of $k$
on its full interval image, and reparametrize the two
wall functions there by
\begin{equation}
 u(t)=u_0(\xi_m(t)),\qquad
 w(t)=w_0(\xi_m(t)).
 \label{eq:era-radius-amplitudes}
\end{equation}
The next proposition verifies these definitions on the
whole interval needed for the retained phases.

\begin{proposition}
\label{prop:era-amplitude-extension}
For $J\geq20$, the constants $\varepsilon,C$ in the
interval and output estimates can be chosen so that
every graph satisfying
\eqref{eq:era-interval-budget}--\eqref{eq:era-interval-graphs}
has positive $C^1$ extensions $U^+,W^+$ of the
functions~\eqref{eq:era-radius-amplitudes} on $L_\beta$
to all of $\R$. These extensions are given by the
construction in Lemma~\ref{lem:positive-amplitude-extension}.
With $\theta=CB^{7/24}<1/2$, they satisfy
\begin{equation}
 U^++W^+\leq B^{4/3},\qquad
 |(U^+)'|\leq\theta\sqrt{U^+},\qquad
 |(W^+)'|\leq\theta\sqrt{W^+}.
 \label{eq:era-successor-bounds}
\end{equation}
For every $y\in K_\beta$,
\begin{gather}
 k(\zeta_m(y))=2/y,\qquad
 U^+(2/y)=u_0(\zeta_m(y)),\quad
 W^+(2/y)=w_0(\zeta_m(y)),\notag\\
 (U^+)'(2/y)=\frac{u_0'(\zeta_m(y))}{k'(\zeta_m(y))},
 \qquad
 (W^+)'(2/y)=\frac{w_0'(\zeta_m(y))}{k'(\zeta_m(y))}.
 \label{eq:era-successor-match}
\end{gather}
\end{proposition}

\begin{proof}
Take common constants for
Theorems~\ref{thm:era-output} and~\ref{thm:era-interval-inverse},
enlarging $C$ if necessary. Shrink $\varepsilon$ so that
\begin{equation*}
 C\varepsilon^{7/24}<\tfrac12,\qquad
 \frac{25}{16}C\varepsilon^{17/30}\leq1.
\end{equation*}
The hypotheses imply $B\leq\varepsilon$. At every
$r\in I_m$, the graph tangent is a normalized radial
cone vector. The chain rule and
Theorem~\ref{thm:era-output} therefore give
\begin{gather*}
 k'(r)<0,\qquad u_0(r),w_0(r)>0,\qquad
 u_0(r)+w_0(r)\leq CB^{19/10},\\
 |u_0'(r)|\leq\theta\sqrt{u_0(r)}\,|k'(r)|,\qquad
 |w_0'(r)|\leq\theta\sqrt{w_0(r)}\,|k'(r)|.
\end{gather*}
All three functions are locally $C^1$ on $I_m$.
The function $k$ is positive there, and is strictly
decreasing by the mean value theorem. Consequently it
has an inverse $\xi_m$ on $[k(a_+),k(a_-)]$.

The strict phase endpoint bounds
\eqref{eq:era-interval-endpoints}, together with
$\varphi_m=2/k$, give
\begin{equation*}
 k(a_+)<l_\beta<r_\beta<k(a_-).
\end{equation*}
The middle inequality follows from $\beta<1/4$.
In particular, $L_\beta$ lies strictly inside the image
of $k$. At every point of this window, its interval
inverse agrees near that point with the ordinary local
inverse, by injectivity of $k$. Thus $\xi_m$ is $C^1$
near $L_\beta$ and
\[
 \xi_m'(t)=\frac1{k'(\xi_m(t))}.
\]
The functions in~\eqref{eq:era-radius-amplitudes} are
positive and $C^1$ there, with
\begin{gather}
 u'(t)=\frac{u_0'(\xi_m(t))}{k'(\xi_m(t))},\qquad
 w'(t)=\frac{w_0'(\xi_m(t))}{k'(\xi_m(t))},\notag\\
 u(t)+w(t)\leq CB^{19/10},\qquad
 |u'(t)|\leq\theta\sqrt{u(t)},\quad
 |w'(t)|\leq\theta\sqrt{w(t)}.
 \label{eq:era-reparametrized-bounds}
\end{gather}
Their derivatives are continuous through the endpoints
of the window.

Apply Lemma~\ref{lem:positive-amplitude-extension} on
$L_\beta$, with $A=CB^{19/10}$, to define $U^+,W^+$.
It preserves the cone constant $\theta$ and gives
\[
 U^++W^+\leq\frac{25}{16}CB^{19/10}
   =\left(\frac{25}{16}CB^{17/30}\right)B^{4/3}
   \leq B^{4/3}.
\]
This proves~\eqref{eq:era-successor-bounds}.

Finally, $\varphi_m(\zeta_m(y))=y>0$ gives
$k(\zeta_m(y))=2/y$, and $y\in K_\beta$ implies
$2/y\in L_\beta$. Uniqueness of the radius inverse
then gives $\xi_m(2/y)=\zeta_m(y)$.
The extension preserves both values and derivatives
on the closed window, so
\eqref{eq:era-reparametrized-bounds} yields all the
identities in~\eqref{eq:era-successor-match}.
\end{proof}

We can now combine the retained phase intervals with
the successor graphs. The parameters $\eta,J,\sigma$
remain fixed as in~\eqref{eq:era-margin-choice}.

\begin{theorem}
\label{thm:era-curve-step}
There are $C\geq1$ and $B_*\in(0,1)$ such that every
graph $\Gamma$ admissible at amplitude $0<B\leq B_*$
has the following properties. Set $\beta=B^\sigma$ and
$N=\lfloor1/\beta\rfloor$. The retained inverse branches
$h_m$, $0\leq m<N$, satisfy
\eqref{eq:era-gauss-branch-estimates}; their open images
are disjoint and omit length at most $10B^\sigma$
from $(0,1)$.

For each such $m$ there is a successor graph
\[
 \Gamma_m^+(t)=(t,\widehat U_m(t),\widehat W_m(t))
\]
admissible at amplitude $B^{4/3}$, defined by the
extension in Proposition~\ref{prop:era-amplitude-extension}.
For $y\in K_\beta$, the actual era segment starting
at $\Gamma(2/h_m(y))$ has next core entrance
$\Gamma_m^+(2/y)$ in core coordinates.
It consists of $m+1$ successive cores, $m$ reflections,
and the following terminal connection. The two output
wall derivatives also satisfy
\eqref{eq:era-successor-match}.
\end{theorem}

\begin{proof}
Take the constants of Theorem~\ref{thm:era-retained-phases}
and Proposition~\ref{prop:era-amplitude-extension},
and enlarge $C$ to dominate both. Apply
Lemma~\ref{lem:era-parameter-choice} to the latter
proposition's input constants. Intersect the resulting
amplitude threshold with that of the retained-phase
theorem. All choices are independent of the graph and
of $m<N$.

The retained-phase theorem gives the branches,
disjointness, and omitted-length estimate. For each
branch, the parameter lemma verifies the interval
conditions for the extension proposition. Its positive
global extensions satisfy
$\widehat U_m+\widehat W_m\leq B^{4/3}$ and the square-root slope
bounds with a coefficient smaller than $1/2$.
They therefore define an admissible successor graph.
Since $2/h_m(y)=\zeta_m(y)$, the value and derivative
identities~\eqref{eq:era-successor-match} identify this
graph with the actual output for every retained $y$.
The count and order of the intervening events are those
of Theorem~\ref{thm:era-bounded-phase}.
\end{proof}

The successor graph agrees with the actual era image
over $L_\beta$. Its extension supplies a common radius
domain for choosing the following branches. The
intersections in~\eqref{eq:actual-compositions} retain
only the parts that are images of the preceding actual
segments.