\subsection{Cone control and uniform reflections}
\label{sec:reflection-uniform}

The phase estimate controls how a connection changes the radial
parameter. To carry a family of orbits through the connection,
we also need its transverse slope. The powers of the small growing
wall in~\eqref{eq:compact-reflection-wall-powers} provide enough
decay to absorb both the logarithmic waiting length and the
singular derivatives of the initial amplitudes. We first prove this
at bounded radii, then combine the result with the long reflection.

For now let $31/8\leq R\leq6$ and fix an intermediate height
$\eta$ small enough for the incoming estimates. Write
\[
 \mathcal F(R,u,w)=(R_+,u_+,w_+),\qquad b=u+w.
\]
All wall amplitudes are positive. The directions considered below
are the normalized vectors in the radial cone
\eqref{eq:core-radial-cone}:
\begin{equation}
 h=(1,\xi,\zeta),\qquad
 |\xi|\leq\sqrt u,\qquad |\zeta|\leq\sqrt w.
 \label{eq:compact-reflection-directions}
\end{equation}
We decrease the wall threshold as needed, always uniformly in $R$.
Constants may depend on the fixed height $\eta$.

\begin{lemma}[Logarithmic wall response]
\label{lem:compact-reflection-logs}
For sufficiently small positive $u,w$, the actual output walls
satisfy, in every direction~\eqref{eq:compact-reflection-directions},
\begin{align}
 |D\log u_+[h]|
  &\leq C_\eta\left(1+|\log w|+\frac1{\sqrt u}
                                      +\frac1{\sqrt w}\right),
 \notag\\
 |D\log w_+[h]|
  &\leq C_\eta\left(1+|\log w|+\frac1{\sqrt w}\right).
 \label{eq:compact-reflection-log-bounds}
\end{align}
\end{lemma}

\begin{proof}
Recall the reflected entrance coordinates
\[
 z_\eta=(F_\eta,\widetilde u,\widetilde v,\eta),
 \qquad \widetilde u=uB_U,\quad \widetilde v=wB_W,
\]
from~\eqref{eq:compact-radial-entrance}. The finite factors,
their inverses, and the differentials of their logarithms are
uniformly bounded on a fixed neighborhood of the reference
segment. The same is true of $DF_\eta$ and $DB_U$.
Since $u,w\leq1$, every direction in
\eqref{eq:compact-reflection-directions} has maximum norm one.
It follows that, for $d=Dz_\eta[h]$, there is $A_\eta\geq1$ such that
\begin{align}
 |d_r|,\ |d_u|&\leq A_\eta,\qquad d_w=0,\notag\\
 |d_u|&\leq\widetilde u
                    \left(A_\eta+\frac1{\sqrt u}\right),&
 |d_v|&\leq\widetilde v
                    \left(A_\eta+\frac1{\sqrt w}\right).
 \label{eq:compact-entrance-tangent}
\end{align}
For the relative bounds, differentiate the factored coordinates:
\[
 d_u=B_U\bigl(\xi+u\,D\log B_U[h]\bigr),\qquad
 d_v=B_W\bigl(\zeta+w\,D\log B_W[h]\bigr).
\]
Then use $\sqrt u=u/\sqrt u$ and
$\sqrt w=w/\sqrt w$. The separate absolute bound on $d_u$
follows directly from $d_u=B_U\xi+u\,DB_U[h]$.

The incoming waiting length satisfies
\begin{equation}
 0\leq L_\eta=\log(1/\widetilde v)
       \leq A_\eta+|\log w|.
 \label{eq:compact-entrance-log-length}
\end{equation}
Indeed, $-\log B_W\leq B_W^{-1}$, and the inverse factor is
bounded. The entrance has $7/4\leq k_\eta<4$.
Apply Propositions~\ref{prop:entrance-three-columns}
and~\ref{prop:entrance-active-column} in its radial chart.
For a universal $B>0$, the columns used in this composition obey
\begin{align*}
 |(\ell_U)_r|&\leq BL_\eta,&
 |(\ell_U)_u-1/\widetilde u|&\leq B,&
 |(\ell_U)_v|&\leq4B/\widetilde v,\\
 |(\ell_W)_r|&\leq BL_\eta,&
 |(\ell_W)_u|&\leq B,&
 |(\ell_W)_v|&\leq B/\widetilde v.
\end{align*}
The coordinate $w$ in this chart equals the fixed height $\eta$,
so its column makes no contribution.

The actual composition holds on a neighborhood of the input.
Its chain rule identifies
$D\log u_+[h]=D\ell_U[d]$ and
$D\log w_+[h]=D\ell_W[d]$. Thus
\begin{align*}
 |D\log u_+[h]|
 &\leq BA_\eta L_\eta
       +\frac{|d_u|}{\widetilde u}+B|d_u|
       +4B\frac{|d_v|}{\widetilde v},\\
 |D\log w_+[h]|
 &\leq BA_\eta L_\eta+B|d_u|
       +B\frac{|d_v|}{\widetilde v}.
\end{align*}
Substitution of~\eqref{eq:compact-entrance-tangent}
and~\eqref{eq:compact-entrance-log-length} proves the result.
In particular, the weak output uses the absolute bound on $d_u$;
its estimate has no inverse power of $u$.
\end{proof}

\begin{lemma}[Transverse response]
\label{lem:compact-reflection-weighted-walls}
After reducing the input threshold if necessary, there is
$C_\eta>0$ such that
\begin{equation}
 |Du_+[h]|\leq C_\eta b^{19/24}\sqrt{u_+},\qquad
 |Dw_+[h]|\leq C_\eta b^{2/9}\sqrt{w_+}
 \label{eq:compact-weighted-wall-bounds}
\end{equation}
for every direction~\eqref{eq:compact-reflection-directions}.
\end{lemma}

\begin{proof}
For $0<w\leq1$,
\begin{equation*}
 \sqrt w(1+|\log w|)\leq2.
\end{equation*}
To prove this, set $s=\sqrt w$. The inequality
$\log s\geq1-1/s$ gives
$s(1-2\log s)\leq2-s\leq2$.
The wall decay of Proposition~\ref{prop:compact-reflection-passage}
gives, with a fixed $A_\eta>0$,
\begin{equation}
 \sqrt{u_+}\leq\sqrt{A_\eta}\sqrt u\,w^{19/24},\qquad
 \sqrt{w_+}\leq\sqrt{A_\eta}\,w^{13/18}.
 \label{eq:compact-output-square-roots}
\end{equation}

For the strong wall, the factor in
\eqref{eq:compact-reflection-log-bounds} has the decomposition
\begin{align*}
 &\sqrt u\,w^{19/24}
      \left(1+|\log w|+\frac1{\sqrt u}+\frac1{\sqrt w}\right)
 \\
 &\qquad=\sqrt u\,w^{7/24}
             \bigl[\sqrt w(1+|\log w|)+1\bigr]+w^{19/24}
 \\
 &\qquad\leq3\sqrt u\,w^{7/24}+w^{19/24}
       \leq4b^{19/24}.
\end{align*}
The last step uses
$\sqrt u\,w^{7/24}\leq b^{1/2}b^{7/24}$ and $w\leq b$.
For the weak wall the corresponding calculation is
\begin{align*}
 w^{13/18}\left(1+|\log w|+\frac1{\sqrt w}\right)
 &=w^{2/9}\bigl[\sqrt w(1+|\log w|)+1\bigr]\\
 &\leq3w^{2/9}\leq3b^{2/9}.
\end{align*}
Multiply the logarithmic response bounds by the appropriate
square root in~\eqref{eq:compact-output-square-roots}.
Since $Du_+=u_+D\log u_+$ and $Dw_+=w_+D\log w_+$,
one further multiplication by the output square root proves
\eqref{eq:compact-weighted-wall-bounds}.
\end{proof}

These estimates are uniform over the ratio of the two incoming
walls. In particular, the exponent $2/9$ in the weak output
comes from $13/18-1/2$: the available wall decay exceeds the
square-root singularity of its input derivative.

The phase derivative has a positive lower bound on the same
directions. This supplies the normalization needed to compare
input and output slopes.

\begin{proposition}[Strict cone preservation at bounded radii]
\label{prop:compact-reflection-cone}
There are $\varepsilon>0$ and $C>0$ such that, for
$31/8\leq R\leq6$ and $0<u,w\leq\varepsilon$, the actual
reflection is locally $C^2$, and for every
$h$ in~\eqref{eq:compact-reflection-directions}, with
$d=D\mathcal F[h]$,
\begin{align}
 d_R&\geq\tfrac12,&
 |d_R-1|&\leq C(\sqrt u+w\sqrt w),\notag\\
 |d_u|&\leq Cb^{2/9}\sqrt{u_+}\,d_R,&
 |d_w|&\leq Cb^{2/9}\sqrt{w_+}\,d_R,
 \label{eq:compact-reflection-cone}\\
 Cb^{2/9}&<\tfrac12.&&\notag
\end{align}
Consequently the radial cone is strictly preserved.
\end{proposition}

\begin{proof}
Let $P$ be a common constant for the phase bounds of
Proposition~\ref{prop:compact-connection-phase}.
For $u,w\leq1$, those bounds give
\[
 |(R_+)_R-1|\leq P(u+w^2),\qquad
 |(R_+)_u|\leq1+2P,\qquad |(R_+)_w|\leq P(u+w),
\]
where $2/R\leq1$ was used in the strong column.
The differential on $h=(1,\xi,\zeta)$ therefore satisfies
\begin{align}
 |d_R-1|
 &\leq P(u+w^2)+(1+2P)\sqrt u+P(u+w)\sqrt w\notag\\
 &\leq(4P+1)(\sqrt u+w\sqrt w).
 \label{eq:compact-phase-cone-error}
\end{align}
Here $u\leq\sqrt u$, $w^2\leq w\sqrt w$, and
$u\sqrt w\leq\sqrt u$.
Put $K=4P+1$ and choose the threshold so that
\[
 \varepsilon\leq\tfrac12,\qquad
 \sqrt\varepsilon\leq\frac1{4K}.
\]
Then $\sqrt u+w\sqrt w\leq2\sqrt\varepsilon$, and
\eqref{eq:compact-phase-cone-error} implies $d_R\geq1/2$.

Let $W$ be the constant in
Lemma~\ref{lem:compact-reflection-weighted-walls}.
Since $b\leq1$ and $19/24>2/9$, both transverse estimates
are bounded by $Wb^{2/9}$ times the respective output square
root. Choose $C\geq\max(K,2W)$.
The lower bound on $d_R$ gives $W\leq Cd_R$, which proves
the two slope estimates in~\eqref{eq:compact-reflection-cone}.
Finally reduce $\varepsilon$ so that
$C(2\varepsilon)^{2/9}<1/2$.
For a nonzero vector of the full radial cone, division by
its positive radial component reduces to
\eqref{eq:compact-reflection-directions}; homogeneity gives
the asserted strict preservation.
\end{proof}

The connection is now controlled across the whole reflection
range. We state the result in the same coordinates as the
uniform core return. The arrival time is measured in the original
normalized time $t$.

\begin{theorem}[Uniform reflection estimates]
\label{thm:uniform-reflection}
There are constants $C>0$ and $0<\varepsilon\leq10^{-4}$
such that, for
\begin{equation}
 R\geq31/8,\qquad
 0<u,w\leq\varepsilon/R,
 \label{eq:uniform-reflection-domain}
\end{equation}
the exact orbit from
$(1-\sqrt{R^2-1},u,1,w)$ has a first positive arrival
at the old wall $W=1$. Its time $T$ and the reflected return
$\mathcal F(R,u,w)=(R_+,u_+,w_+)$ are locally $C^2$.
The reflected arrival state is
$(1+\sqrt{R_+^2-1},u_+,1,w_+)$, and
\begin{equation}
 |R_+-(R-2)|\leq\frac{C(u+w)}R,\qquad
 0<u_+\leq u,\qquad0<w_+\leq w.
 \label{eq:uniform-reflection-values}
\end{equation}
For every $h$ in~\eqref{eq:compact-reflection-directions},
write $d=D\mathcal F[h]$. Then
\begin{align}
 d_R&\geq\tfrac12,&
 |d_R-1|&\leq\frac{C\sqrt{u+w}}R,\notag\\
 |\log d_R|&\leq\frac{C\sqrt{u+w}}R,&
 |d_u|&\leq\sqrt{u_+}\,d_R,\qquad
 |d_w|\leq\sqrt{w_+}\,d_R.
 \label{eq:uniform-reflection-derivatives}
\end{align}
\end{theorem}

\begin{proof}
First put the estimates for bounded radii in the stated form.
Fix a height $\eta$ allowed by the preceding results, and take
a common threshold for their existence, phase, wall and cone
estimates. Restrict it further so that $b=u+w\leq1$.
With $P$ denoting the constant in
Proposition~\ref{prop:compact-connection-phase}, its expansion gives
\[
 |R_+-(R-2)|\leq P b^2+2u/R\leq(P+1)b.
\]
As $R\leq6$, this is at most $6(P+1)b/R$.
Likewise
\[
 \sqrt u+w\sqrt w\leq2\sqrt b
 \quad\Longrightarrow\quad
 |d_R-1|\leq 12K\sqrt b/R
\]
for a suitable constant $K$ from
Proposition~\ref{prop:compact-reflection-cone}.
Its transverse factor is less than $1/2$, hence at most one.
The wall nonincrease is
\eqref{eq:compact-reflection-wall-powers}.
The value, additive derivative and cone bounds therefore hold
on $[31/8,6]$, with some constants $\varepsilon_c,C_c>0$.

On $R\geq11/2$, Corollary~\ref{cor:long-reflection-estimates}
gives the same conclusions with threshold
$\varepsilon_\ell/R$ and error constant at most $300000$.
Both constructions describe the first arrival at the old
$W=1$ section. The arrival remains valid on neighborhoods,
so uniqueness identifies their times and maps there, including
on the overlapping interval. Their differentials thus belong
to this single reflection map.

Take
\[
 \varepsilon=\min(\varepsilon_c,\varepsilon_\ell,10^{-4}),
 \qquad C_0=\max(C_c,300000).
\]
For $R\geq11/2$ use the long estimate.
For $31/8\leq R<11/2$, the hypothesis
$u,w\leq\varepsilon/R$ implies $u,w\leq\varepsilon_c$,
and the estimate for bounded radii applies.
This proves the value and additive derivative estimates with
constant $C_0$, together with the regularity and cone assertions.
At the actual arrival the reflected shear satisfies $M_+>1$
and the active wall equals one. Solving
$R_+^2=(M_+-1)^2+1$ yields the stated reconstruction.

Finally, for every $a\geq1/2$,
\begin{equation}
 |\log a|\leq2|a-1|.
 \label{eq:positive-log-error}
\end{equation}
Indeed, $1/a-1\leq2|a-1|$, while
$1-1/a\leq\log a\leq a-1$.
Apply~\eqref{eq:positive-log-error} to $a=d_R$
and replace $C_0$ by $C=2C_0$.
\end{proof}

Core passages and reflections now preserve the same radial
cone. Their phase errors are controlled by the amplitudes, and
their derivative errors by the square roots of those amplitudes.
The logarithmic estimate is suited to successive passages,
because logarithms turn the product of radial derivatives into
a sum. To close an era, it remains to estimate the final connection,
where the change of frame differs from a reflection.