\subsection{Accumulated error and the terminal core}
\label{sec:era-accumulation}

A core followed by a reflection lowers the radius by approximately two.
Its radius error has the scale of the lost transverse amplitude, and
its logarithmic derivative error has the scale of the lost square root
of that amplitude. Applying Lemma~\ref{lem:amplitude-loss} to these two
powers controls the entire finite sequence and locates its terminal core.

Write $F$ for the core return of
Proposition~\ref{prop:uniform-core-estimates}, and retain
$\mathcal F$ for the reflection of
Theorem~\ref{thm:uniform-reflection}. Their composition
\begin{equation}
 G=\mathcal F\circ F
 \label{eq:core-reflection-composition}
\end{equation}
takes an upward core entrance to the next upward entrance,
in the reflected frame. Its elapsed original time is
$T_G(Y)=T_c(Y)+T_r(F(Y))$, where $T_c$ and $T_r$ are the
core and reflection times. We use the embeddings
\[
 \iota_\pm(r,u,w)=(1\pm\sqrt{r^2-1},u,1,w)
\]
for upward and downward core states. Denote by $\Phi_t$ the
original normalized flow expressed in projective coordinates,
on its regular coordinate domain.

\begin{lemma}
\label{lem:core-reflection-step}
There are constants $0<\varepsilon\leq10^{-4}$, $C>0$,
and $0<c\leq1/2$ with the following properties.
Let $Y=(r,u,w)$ satisfy
\begin{equation}
 r\geq2,\qquad 0<u,w\leq\varepsilon/r,
 \qquad b=u+w,
 \label{eq:core-reflection-input}
\end{equation}
and put $Z=F(Y)=(R,U,W)$. The core and its time are
locally $C^1$, and
\begin{equation}
 |R-r|\leq\min(1/8,Cb/r),\qquad
 0<U\leq e^{-c/r}u,\qquad0<W\leq e^{-c/r}w.
 \label{eq:core-reflection-core-bounds}
\end{equation}
If $R\geq31/8$, then $G(Y)=(r',u',w')$ and $T_G$ are
locally $C^1$, $T_G>0$, and
\begin{equation}
 \iota_+(G(Y))=\mathsf S\Phi_{T_G(Y)}(\iota_+(Y)).
 \label{eq:core-reflection-actual-orbit}
\end{equation}
Moreover,
\begin{align}
 |R-r|+|r'-(R-2)|&\leq Cb/r,&
 |r'-(r-2)|&\leq Cb/r,\notag\\
 3/2\leq r-r'&\leq5/2,&
 0<u'\leq U,&\qquad0<w'\leq W.
 \label{eq:core-reflection-step-values}
\end{align}
If $r'\geq2$, then $u',w'\leq\varepsilon/r'$.
For either $H=F$ or, when defined, $H=G$, every direction
$v=(1,v_u,v_w)$ with $|v_u|\leq\sqrt u$ and
$|v_w|\leq\sqrt w$ has image $d=DH(Y)[v]$ satisfying
\begin{equation}
 d_r>0,\qquad |\log d_r|\leq C\sqrt b/r,\qquad
 |d_u|\leq\sqrt{H_u(Y)}\,d_r,\quad
 |d_w|\leq\sqrt{H_w(Y)}\,d_r.
 \label{eq:core-reflection-step-cone}
\end{equation}
\end{lemma}

\begin{proof}
Let $K,c,\varepsilon_c$ be the constants for the uniform
core return, and let $L,\varepsilon_r$ be those for the
uniform reflection. Set $C=K+2L$. Choose
\[
 \varepsilon\leq
 \min\{\varepsilon_c,\varepsilon_r/2,10^{-4},1/(4K),1/C\}.
\]
Then $b\leq2\varepsilon/r\leq\varepsilon$. The core estimate
gives $|R-r|\leq K\varepsilon/r\leq1/8$, hence
\begin{equation*}
 r/2\leq R\leq2r.
\end{equation*}
Its walls do not increase, so
\[
 U,W\leq\varepsilon/r\leq\varepsilon_r/R.
\]
For $R\geq31/8$, the reflection is therefore defined.
Its radius error is at most $L(U+W)/R\leq2Lb/r$.
Adding the core error proves the first two estimates
in~\eqref{eq:core-reflection-step-values}. Their right-hand
side is at most $C\varepsilon/2\leq1/2$, which gives the
stated decrease in radius. The walls remain positive and
decrease through the reflection. When $r'\geq2$, the
inequality $r'<r$ gives
$u'\leq u\leq\varepsilon/r\leq\varepsilon/r'$, and
similarly for $w'$.

To identify the actual core endpoint, consider a downward
$V=1$ crossing with radius $R\geq2$ and $0\leq U,W\leq1/4$.
The crossing condition gives $M-1<U(1+W-U)\leq5/16$.
If $M\geq1$, this would give $0\leq M-1<5/16$, contrary
to $(M-1)^2+1=R^2\geq4$. Hence $M<1$, so the state is
$\iota_-(R,U,W)$. Our exit satisfies these bounds. Thus
$\iota_-(Z)=\Phi_{T_c(Y)}(\iota_+(Y))$,
and the reflected arrival is
$\iota_+(G(Y))=\mathsf S\Phi_{T_r(Z)}(\iota_-(Z))$.
The flow restart identity gives
\eqref{eq:core-reflection-actual-orbit}, with positive
elapsed time. Composition gives the local $C^1$ assertions.

It remains to combine the derivatives. Put $z=DF(Y)[v]$.
The core estimates give $z_r>0$ and place
$\widehat z=z/z_r$ in the normalized radial cone at $Z$.
Thus, with $q=D\mathcal F(Z)[\widehat z]$,
\[
 DG(Y)[v]=z_rq,\qquad q_r>0,
 \qquad |\log q_r|\leq L\sqrt{U+W}/R.
\]
The transverse inequalities are homogeneous. For the radial
component, positivity allows the logarithms to add:
\begin{align*}
 |\log(z_rq_r)|
 &\leq|\log z_r|+|\log q_r|\\
 &\leq K\sqrt b/r+2L\sqrt b/r=C\sqrt b/r.
\end{align*}
This proves~\eqref{eq:core-reflection-step-cone} for $G$;
the assertion for $F$ follows directly from the core estimate.
\end{proof}

Consider a finite sequence $Y_{j+1}=G(Y_j)$ and write
\begin{equation}
 Y_j=(r_j,u_j,w_j),\qquad F(Y_j)=(R_j,U_j,W_j),
 \qquad b_j=u_j+w_j,\quad B_j=U_j+W_j.
 \label{eq:era-prefix-notation}
\end{equation}
We shall use the constants
\begin{equation}
 A_p=\frac{1+pc/2}{pc}\quad(p>0),\qquad
 A=CA_1,\qquad D_0=CA_{1/2},
 \label{eq:era-loss-constants}
\end{equation}
where $C,c$ are from Lemma~\ref{lem:core-reflection-step}.

\begin{proposition}[Uniform control of a finite sequence]
\label{prop:core-reflection-prefix}
Suppose $Y_0$ satisfies~\eqref{eq:core-reflection-input} and
the first $n$ steps satisfy
\[
 r_j\geq2,\qquad R_j\geq31/8\quad(0\leq j<n).
\]
All the walls at $Y_0,\ldots,Y_n$ are positive, and
\begin{equation}
 \sum_{j=0}^{n-1}\frac{b_j^p}{r_j}
       \leq A_p(b_0^p-b_n^p)\qquad(p>0).
 \label{eq:era-prefix-power-sum}
\end{equation}
The accumulated radius error obeys
\begin{align}
 \sum_{j=0}^{n-1}
   \bigl(|R_j-r_j|+|r_{j+1}-(R_j-2)|\bigr)
       &\leq A(b_0-b_n),\notag\\
 |r_n-(r_0-2n)|&\leq A(b_0-b_n).
 \label{eq:era-prefix-radius-error}
\end{align}
The iterate $G^n$ is locally $C^1$ at $Y_0$. For every
normalized radial cone direction $v$ there, its differential
$d=D(G^n)(Y_0)[v]$ satisfies
\begin{equation}
 d_r>0,\qquad
 |\log d_r|\leq D_0(\sqrt{b_0}-\sqrt{b_n}),\qquad
 |d_u|\leq\sqrt{u_n}\,d_r,\quad
 |d_w|\leq\sqrt{w_n}\,d_r.
 \label{eq:era-prefix-differential}
\end{equation}
If also $r_n\geq2$, the last core is defined and
$\mathcal H_n=F\circ G^n$ and $T_c\circ G^n$ are locally
$C^1$ at $Y_0$. Its output walls are positive, $B_n\leq b_0$,
and
\begin{equation}
 |R_n-(r_0-2n)|\leq A(b_0-B_n).
 \label{eq:era-prefix-final-core-value}
\end{equation}
For $e=D\mathcal H_n(Y_0)[v]$,
\begin{equation}
 e_r>0,\qquad
 |\log e_r|\leq D_0(\sqrt{b_0}-\sqrt{B_n}),\qquad
 |e_u|\leq\sqrt{U_n}\,e_r,\quad
 |e_w|\leq\sqrt{W_n}\,e_r.
 \label{eq:era-prefix-final-core-differential}
\end{equation}
\end{proposition}

\begin{proof}
The amplitude bound in Lemma~\ref{lem:core-reflection-step}
propagates the input bound $u_j,w_j\leq\varepsilon/r_j$
whenever the next radius is at least two. Induction therefore
applies the lemma at every indicated step. In particular,
\begin{equation*}
 0<b_{j+1}\leq B_j\leq e^{-c/r_j}b_j.
\end{equation*}
Lemma~\ref{lem:amplitude-loss} now gives
\eqref{eq:era-prefix-power-sum}. Sum the first estimate
in~\eqref{eq:core-reflection-step-values} and use $p=1$
to obtain the first line of~\eqref{eq:era-prefix-radius-error}.
For the second line, sum the identity
\[
 r_{j+1}-(r_j-2)
       =(R_j-r_j)+(r_{j+1}-(R_j-2)).
\]
The left side telescopes to $r_n-(r_0-2n)$.

Each map $G$ is $C^1$ on a neighborhood of the point where
it is used. A finite sequence of compositions is consequently
$C^1$ near $Y_0$. To estimate its derivative, put
$d_j=D(G^j)(Y_0)[v]$, so $d_0=v$. Inductively $d_{j,r}>0$,
and $d_j/d_{j,r}$ lies in the normalized cone at $Y_j$.
If
\[
 q_j=DG(Y_j)[d_j/d_{j,r}],
\]
then $d_{j+1}=d_{j,r}q_j$. The cone inequalities propagate,
and
\[
 |\log d_{n,r}|
 \leq\sum_{j=0}^{n-1}|\log q_{j,r}|
 \leq C\sum_{j=0}^{n-1}\frac{\sqrt{b_j}}{r_j}
 \leq D_0(\sqrt{b_0}-\sqrt{b_n}).
\]
This proves~\eqref{eq:era-prefix-differential}, including
$n=0$, where the sum is empty and the derivative is the identity.

When $r_n\geq2$, the propagated amplitude bound also permits
the last core. Its loss satisfies
$B_n\leq e^{-c/r_n}b_n$. Applying
\eqref{eq:one-core-loss} to this last passage gives
\[
 \frac{b_n}{r_n}\leq A_1(b_n-B_n),\qquad
 \frac{\sqrt{b_n}}{r_n}
       \leq A_{1/2}(\sqrt{b_n}-\sqrt{B_n}).
\]
The first inequality bounds $|R_n-r_n|$ and extends the
radius telescope to $B_n$. The second extends the logarithmic
derivative bound: normalize the derivative of $G^n$ by its
positive radial component, apply the core estimate, and add
the two logarithms. This gives
\eqref{eq:era-prefix-final-core-value}
and~\eqref{eq:era-prefix-final-core-differential}.
Local regularity of the last core and its time gives the
remaining composition statements.
\end{proof}

The finite-sequence estimate also constructs the sequence.
A fixed margin from the endpoints of a radius interval
prevents a perturbed orbit from changing its number of
reflections.

\begin{proposition}[The terminal index]
\label{prop:core-itinerary}
There are $0<\varepsilon\leq10^{-4}$ and $K>0$ such that
the following holds for every integer $m\geq0$ and
$0<\beta<1/2$. Let $X=(r,u,w)$ satisfy
\begin{equation}
 0<u,w\leq\varepsilon/r,\qquad
 m+1+\beta\leq r/2\leq m+2-\beta,\qquad
 K(u+w)<\beta/2.
 \label{eq:core-itinerary-input}
\end{equation}
Then all $m+1$ cores at $Y_j=G^j(X)$, $0\leq j\leq m$,
exist. In the notation~\eqref{eq:era-prefix-notation},
\begin{align}
 r_j&>2+3\beta/2 &&(0\leq j\leq m),\notag\\
 R_j&>4+3\beta/2 &&(0\leq j<m),\notag\\
 2+3\beta/2&<R_m<4-3\beta/2.
 \label{eq:core-itinerary-ranges}
\end{align}
The $m$ intervening reflections are the actual positive-time
passages in~\eqref{eq:core-reflection-actual-orbit}.
For every $j\leq m$, $G^j$, $\mathcal H_j=F\circ G^j$,
and $T_c\circ G^j$ are locally $C^1$ at $X$, their core
output walls are positive, and
\begin{equation}
 B_j\leq b_0=u+w,\qquad
 |r_j-(r-2j)|+|R_j-(r-2j)|\leq Kb_0.
 \label{eq:core-itinerary-errors}
\end{equation}
For every normalized cone direction $v$ at $X$, put
$d=D\mathcal H_j(X)[v]$. Then
\begin{equation}
 d_r>0,\qquad |\log d_r|\leq K\sqrt{b_0},\qquad
 |d_u|\leq\sqrt{U_j}\,d_r,\quad
 |d_w|\leq\sqrt{W_j}\,d_r.
 \label{eq:core-itinerary-differential}
\end{equation}
Finally,
\begin{equation}
 \lfloor r/2\rfloor=m+1,\qquad
 \frac{3\beta}{4}<\alpha:=\frac{R_m-2}{2}
                         <1-\frac{3\beta}{4}.
 \label{eq:core-itinerary-terminal-phase}
\end{equation}
\end{proposition}

\begin{proof}
Use the amplitude threshold of
Proposition~\ref{prop:core-reflection-prefix} and put
$K=2A+D_0$, with~\eqref{eq:era-loss-constants}.
Suppose the first $j$ reflections have been constructed,
where $j\leq m$. The prefix estimate gives
\[
 |r_j-(r-2j)|\leq Ab_0<\beta/2.
\]
Since $r-2j\geq2+2\beta$, this shows
$r_j>2+3\beta/2$ and permits the next core.
The last-core estimate then gives
\[
 |R_j-(r-2j)|\leq Ab_0<\beta/2.
\]
For $j<m$, the reference radius is at least $4+2\beta$.
Thus $R_j>4+3\beta/2>31/8$, so the next reflection exists.
This closes the induction, starting from $j=0$ with an
empty reflection sequence.

At $j=m$, the reference radius lies in
$[2+2\beta,4-2\beta]$. The same error bound places $R_m$
in the interval in~\eqref{eq:core-itinerary-ranges}.
All the regularity, positivity, and cone statements follow
from Proposition~\ref{prop:core-reflection-prefix}.
The two radius errors sum to at most $2Ab_0\leq Kb_0$,
and the logarithmic error is at most $D_0\sqrt{b_0}\leq K\sqrt{b_0}$.
The input interval gives $m+1<r/2<m+2$, proving the floor
identity; dividing the final radius interval by two after
subtracting two gives the bounds on $\alpha$.
\end{proof}

The constants are independent of $m$. The entrance condition
scales the walls with $r$, while the accumulated error is
controlled by the amplitude lost along the actual sequence.