\subsection{The differential of a long reflection}
\label{sec:connection-cone}

The matching phase differs from the radial phase by a constant on
each single-wall boundary. At the actual entrance and exit, the
remaining wall produces a small norm defect. The same defects enter
the derivative, where their wall factors are essential: they absorb
the large logarithmic derivatives from the preceding subsection. We
first recover the radial derivative, then combine it with transverse
contraction to show that a long connection preserves the square-root
cone.

Retain the connection parameters $p=(\mu,u,w)$, with
$\mu\geq4$, $0<u\leq10^{-4}/\mu$ and $0<w\leq10^{-4}$.
Write $L=\log(1/w)$ and
\begin{equation}
 r_-=H(\mu+1,1),\qquad r_+=H(m_*-1,1),\qquad
 \Delta(p)=r_+-r_-+2.
 \label{eq:connection-phase-error-definition}
\end{equation}
The output in core coordinates is $\mathcal C(p)=(r_+,u_+,w_+)$.

\subsubsection{The endpoint factors}

The exponential bounds~\eqref{eq:connection-output-decay} give
three useful endpoint estimates:
\begin{equation}
 \frac{u_+}{w}\leq\frac{8u}{\mu},\qquad
 w_+^2\leq w^2,\qquad (1+L)w_+^2\leq\mu w^2.
 \label{eq:connection-endpoint-envelopes}
\end{equation}
Here is a direct verification that also explains the loss of one
radial power in the last estimate. The inequality
$(1+x)e^{-x}\leq1$, valid for $x\geq0$, implies
$xe^{-x}\leq1$. Since $L\geq1/2$ and $\mu\geq4$,
\[
 \frac{u_+}{w}\leq u e^{-\mu L/2+L}
       \leq u e^{-\mu L/4}\leq\frac{8u}{\mu}.
\]
Also $w_+^2\leq w^2e^{-4L/\mu}$. Multiplying by $1+L$ and
using $1+L\leq\mu(1+4L/\mu)$ proves the last assertion in
\eqref{eq:connection-endpoint-envelopes}.

The moving-clock term in the phase derivative is small for the same
reason. At the exit $W=1$, the phase-rate bound
\eqref{eq:connection-matching-rate-bound}, divided by $q\geq\mu/2$,
yields
\[
 |G_*|\leq10u_++\frac{8w_+^2}{\mu^3}.
\]
Consequently
\begin{equation}
 \frac{|G_*|}{w}\leq\frac{80u+8w}{\mu}
                   \leq\frac{100(u+w)}{\mu}.
 \label{eq:connection-phase-clock-correction}
\end{equation}

To convert to the radius, put $v_*=w_+$ and define
\begin{equation}
 \chi=\frac{\mu+1}{r_-},\qquad
 \sigma=\frac{(m_*-1)/r_+}{S(m_*,v_*,1)},\qquad
 \omega=\frac{v_*^2}{H(m_*+1,v_*)}.
 \label{eq:connection-endpoint-factors}
\end{equation}
The input radius has derivative $Dr_-v=\chi v_\mu$.
At the output, differentiating $J_*=J(m_*,v_*,1)$ gives
\[
 DJ_*v=S_*Dm_*v+\frac{v_*^2}{H(m_*+1,v_*)}
                                      D\log w_+v.
\]
Eliminating $Dm_*v$ from $Dr_+v=((m_*-1)/r_+)Dm_*v$ thus gives
the exact identity
\begin{equation}
 D\Delta(p)v=\sigma\bigl(\widehat Z_v-\omega\widehat y_v\bigr)
                                  -\chi v_\mu,
 \quad \widehat Z_v=DJ_*v,\quad
 \widehat y_v=D\log w_+v.
 \label{eq:connection-phase-error-differential}
\end{equation}
The hats distinguish actual arrival derivatives from the fixed-clock
variations of Section~\ref{sec:connection-variation}.

The factors in~\eqref{eq:connection-endpoint-factors} satisfy
\begin{equation}
 0\leq\sigma\leq\frac43,\qquad
 |\sigma-1|\leq\frac{4v_*^2}{\mu^2},\qquad
 0\leq\omega\leq\frac{v_*^2}{\mu},\qquad
 |S_0-\chi|\leq\frac{2w^2}{\mu^2},
 \label{eq:connection-endpoint-factor-bounds}
\end{equation}
where $S_0=S(\mu,1,w)$. Indeed, the ratio-defect identity used in
the phase cancellation gives
\begin{equation}
 \sigma-1=
  \frac{v_*^2 K_+(m_*,v_*,1)}{S(m_*,v_*,1)},\qquad
 S_0-\chi=-w^2K_-(\mu,1,w).
 \label{eq:connection-factor-defects}
\end{equation}
Now use $S\geq3/4$, $K_\pm\leq2/\mu^2$ and
$H(m_*+1,v_*)\geq\mu$. These formulas retain the square of
the small endpoint wall even when the waiting time is arbitrarily
large.

\begin{proposition}[Derivative of the radial error]
\label{prop:connection-phase-error-derivatives}
For $\mu\geq4$, $0\leq u\leq10^{-4}/\mu$ and
$0<w\leq10^{-4}$, the actual first-arrival error in
\eqref{eq:connection-phase-error-definition} satisfies
\begin{equation}
 \begin{aligned}
 |\partial_\mu\Delta|&\leq\frac{50000(u+w^2)}{\mu^2},&
 |\partial_u\Delta|&\leq\frac{50000}{\mu},\\
 |\partial_w\Delta|&\leq\frac{50000(u+w)}{\mu}.&&
 \end{aligned}
 \label{eq:connection-phase-error-derivatives}
\end{equation}
\end{proposition}

\begin{proof}
First assume $u>0$. For $i=\mu,u,w$, write
$\widehat Z_i=\partial_iJ_*$ and
$\widehat y_i=\partial_i\log w_+$. The phase and strong-wall
responses in Proposition~\ref{prop:connection-phase-strong-responses}
have no clock correction. They give
\begin{equation}
 |\widehat Z_\mu|\leq\frac{16}{15},\qquad
 |\widehat Z_\mu-S_0|\leq\frac{35000(u+w^2)}{\mu^2},\qquad
 |\widehat Z_u|\leq\frac{70}{\mu}.
 \label{eq:connection-arrival-phase-responses}
\end{equation}
For the middle estimate, multiply the expression for $\Lambda$ in
\eqref{eq:connection-feedback-integral} by $16/15$ and use
$\mu^{-3}\leq\mu^{-2}/4$. In the growing-wall direction,
Proposition~\ref{prop:connection-growing-response} and the
actual clock correction~\eqref{eq:connection-phase-clock-correction}
give
\begin{equation*}
 |\widehat Z_w|\leq\frac{8100(u+w)}{\mu}.
\end{equation*}

The weak logarithmic derivatives enter multiplied by $v_*^2/\mu$.
Equations~\eqref{eq:connection-log-amplitude-derivatives} and
\eqref{eq:connection-endpoint-envelopes} imply
\begin{equation}
 \frac{v_*^2}{\mu}|\widehat y_\mu|
       \leq\frac{137w^2}{\mu^2},\qquad
 \frac{v_*^2}{\mu}|\widehat y_u|
       \leq\frac{9000w^2}{\mu^2},\qquad
 \frac{v_*^2}{\mu}|\widehat y_w|
       \leq\frac{9w}{\mu}.
 \label{eq:connection-weighted-weak-responses}
\end{equation}
For the first two bounds, use
\[
 |Y|\leq\frac{C(1+L)}{\mu^2}
 \quad\Longrightarrow\quad
 \frac{v_*^2}{\mu}|Y|\leq\frac{Cw^2}{\mu^2}.
\]
For the third, $1+32/\mu\leq9$ and $v_*^2\leq w^2$ suffice.

In the radial direction, expand the exact formula as
\[
 \partial_\mu\Delta=(\widehat Z_\mu-S_0)
       +(\sigma-1)\widehat Z_\mu
       -\sigma\omega\widehat y_\mu+(S_0-\chi).
\]
Its absolute value is at most
\begin{equation*}
 |\widehat Z_\mu-S_0|
  +\frac{4v_*^2}{\mu^2}|\widehat Z_\mu|
  +\frac43\frac{v_*^2}{\mu}|\widehat y_\mu|
  +\frac{2w^2}{\mu^2}.
\end{equation*}
The second term is at most $5w^2/\mu^2$. Substitution of
\eqref{eq:connection-arrival-phase-responses} and
\eqref{eq:connection-weighted-weak-responses} proves the first
bound in~\eqref{eq:connection-phase-error-derivatives}.
For the other two directions the last term of
\eqref{eq:connection-phase-error-differential} is zero, so
\[
 |\partial_i\Delta|\leq\frac43
       \left(|\widehat Z_i|+\frac{v_*^2}{\mu}|\widehat y_i|\right),
 \qquad i=u,w.
\]
The remaining estimates follow from the same bounds, using
$w^2/\mu^2\leq1/\mu$ in the strong direction.

At $u=0$, the actual return is locally $C^2$ by
Proposition~\ref{prop:long-connection-arrival}; the two norms
defining $\Delta$ are smooth there. Apply the positive-wall
estimates to
\[
 p_n=\left(\mu,\frac{10^{-4}}{\mu(n+1)},w\right),
 \qquad n\geq0.
\]
They tend to $(\mu,0,w)$ within the admissible region. Continuity
of $D\Delta$ and of the right-hand sides proves all three bounds
on this boundary.
\end{proof}

The derivative estimates also retain the actual amplitude on a
square-root direction. If
\begin{equation}
 \chi v_\mu=1,\qquad |v_u|\leq\sqrt u,
                 \qquad |v_w|\leq\sqrt w,
 \label{eq:connection-unit-phase-direction}
\end{equation}
then $|v_\mu|\leq2$, since $\chi\geq7/8$.
Linearity and~\eqref{eq:connection-phase-error-derivatives} give
\begin{equation}
 |D\Delta(p)v|\leq\frac{50000}{\mu}
   \left(\frac{2(u+w^2)}{\mu}+\sqrt u+(u+w)\sqrt w\right)
 \leq\frac{150000\sqrt{u+w}}{\mu}.
 \label{eq:connection-phase-cone-amplitude}
\end{equation}
For the last step put $b=u+w\leq1$. Each of the three terms
in parentheses is at most $\sqrt b$, using $b\leq\sqrt b$,
$w^2\leq w$ and $\mu\geq4$.
If instead the size is expressed through
$u\leq\varepsilon/\mu$, $w\leq\varepsilon\leq1$, the same
parenthesis is at most $4\sqrt\varepsilon$. In this form,
\begin{equation}
 |D\Delta(p)v|\leq\frac{200000\sqrt\varepsilon}{\mu}.
 \label{eq:connection-phase-cone-threshold}
\end{equation}
Indeed, its first term is at most $\sqrt\varepsilon$, its
second at most $\sqrt\varepsilon$, and its last at most
$2\sqrt\varepsilon$.

\subsubsection{An invariant cone through the connection}

Choose $\varepsilon_\ell>0$ so that
\begin{equation}
 \varepsilon_\ell\leq10^{-4},\qquad
 \sqrt{\varepsilon_\ell}\leq10^{-6},\qquad
 \varepsilon_\ell\leq e^{-256}.
 \label{eq:long-connection-cone-threshold}
\end{equation}
For example, the minimum of $10^{-12}$ and $e^{-256}$ has
these properties. The small threshold supplies a lower bound on
the waiting length, while the estimates remain uniform as both
$\mu$ and $L$ become unbounded.

\begin{proposition}[The connection cone]
\label{prop:long-connection-cone}
Suppose that
\[
 \mu\geq4,\qquad 0<u\leq\varepsilon_\ell/\mu,
                    \qquad0<w\leq\varepsilon_\ell.
\]
For a direction~\eqref{eq:connection-unit-phase-direction}, put
$d=D\mathcal C(p)v=(d_r,d_u,d_w)$ and
$\gamma=1-e^{-L/(2\mu)}$. Then
\begin{align}
 d_r&\geq\frac34,& |d_r-1|&\leq\frac1{4\mu},
 \label{eq:long-connection-radial-margin}\\
 |d_u|&\leq\frac13\sqrt{u_+}\,d_r,&
 |d_w|&\leq(1-\gamma/2)\sqrt{w_+}\,d_r.
 \label{eq:long-connection-cone-margins}
\end{align}
In particular the cone
$|v_u|\leq\sqrt u\,|\chi v_\mu|$,
$|v_w|\leq\sqrt w\,|\chi v_\mu|$ is mapped into the
corresponding square-root cone at the output.
\end{proposition}

\begin{proof}
The physical differential has components
\begin{equation}
 d_r=1+D\Delta(p)v,\qquad
 d_u=u_+D\log u_+(p)v,\qquad
 d_w=w_+D\log w_+(p)v.
 \label{eq:connection-physical-differential}
\end{equation}
The first follows by differentiating $r_+=r_--2+\Delta$;
the others use positivity of the returned walls.
Equation~\eqref{eq:connection-phase-cone-threshold} gives
$|d_r-1|\leq1/(4\mu)$, hence $d_r\geq3/4$.

Let $\ell_u=D\log u_+(p)v$ and $\ell_w=D\log w_+(p)v$.
The six amplitude derivatives and $|v_\mu|\leq2$ imply
\begin{align}
 |\ell_u|&\leq274L+9000\left(\frac1u+\frac L\mu\right)\sqrt u
                                      +\frac{32\mu}{w}\sqrt w,
 \notag\\
 |\ell_w|&\leq\frac{274L}{\mu^2}
        +\frac{9000(1+L)}{\mu^2}\sqrt u
        +\frac{1+32/\mu}{w}\sqrt w.
 \label{eq:connection-cone-log-rows}
\end{align}
The wall decay supplies
\begin{equation}
 \sqrt{u_+}\leq\sqrt u\,e^{-\mu L/4},\qquad
 \sqrt{w_+}\leq\sqrt w\,e^{-L/\mu}.
 \label{eq:connection-square-root-decay}
\end{equation}
We estimate these products before dividing by the radial derivative.

For the strong wall, set $E=e^{-\mu L/4}$.
Multiplying its row in~\eqref{eq:connection-cone-log-rows} by
$\sqrt u E$ gives
\begin{equation*}
 \sqrt{u_+}|\ell_u|\leq
 274\sqrt u\,LE+9000E+9000\frac u\mu LE
       +32\sqrt u\,\mu e^{(-\mu/4+1/2)L}.
\end{equation*}
Here $w=e^{-L}$ was used in the last term. Since $L\geq256$,
we have $E\leq w$, $LE\leq1$, and
\[
 \mu e^{(-\mu/4+1/2)L}\leq\mu e^{-\mu/8}\leq8.
\]
The first inequality uses $\mu\geq4$ and $L\geq1$; the
second again follows from $xe^{-x}\leq1$.
As $\sqrt u\leq\sqrt{\varepsilon_\ell}$,
$E\leq\varepsilon_\ell\leq\sqrt{\varepsilon_\ell}$ and
$u/\mu\leq\sqrt{\varepsilon_\ell}$, the four terms sum to at
most $20000\sqrt{\varepsilon_\ell}$. Thus
\begin{equation}
 \sqrt{u_+}|\ell_u|\leq\frac14\leq\frac13d_r.
 \label{eq:connection-strong-cone-margin}
\end{equation}

For the weak wall, $u\leq1$ bounds the first two terms of its
logarithmic row by $10000(1+L)/\mu^2$. Consequently
\[
 \sqrt{w_+}|\ell_w|
 \leq e^{-L/\mu}
       \left(1+\frac{32}{\mu}
                   +\frac{10000(1+L)\sqrt w}{\mu^2}\right).
\]
The threshold gives $40000\sqrt w\leq1$. Together with
$\mu\geq4$ and $L\geq256$, this implies
\[
 \frac{32}{\mu}+\frac{10000(1+L)\sqrt w}{\mu^2}
 \leq\frac{32}{\mu}+\frac{1+L}{4\mu^2}
 \leq\frac L{4\mu}.
\]
The last inequality is $128\mu+1+L\leq\mu L$.
Using $1+x\leq e^x$ now gives
\begin{equation}
 \sqrt{w_+}|\ell_w|
       \leq e^{-L/\mu}(1+L/(4\mu))
       \leq e^{-3L/(4\mu)}\leq e^{-L/(2\mu)}=1-\gamma.
 \label{eq:connection-weak-cone-product}
\end{equation}

It remains to compare this last bound with $d_r$.
For $0\leq x\leq1$,
$e^{-x}\leq(1+x)^{-1}\leq1-x/2$.
Apply this with $x=2/\mu$ and use $L\geq4$ to obtain
\begin{equation}
 \frac1\mu\leq\gamma\leq1,
 \qquad d_r\geq1-\frac\gamma4.
 \label{eq:connection-weak-gap}
\end{equation}
It follows that
\[
 (1-\gamma/2)d_r
    \geq(1-\gamma/2)(1-\gamma/4)\geq1-\gamma.
\]
Equations~\eqref{eq:connection-physical-differential},
\eqref{eq:connection-strong-cone-margin} and
\eqref{eq:connection-weak-cone-product} prove both margins.

For a general cone direction with $\chi v_\mu\ne0$, divide
by $\chi v_\mu$ to reduce to the unit direction just treated,
then use linearity and absolute values. If $\chi v_\mu=0$,
the cone inequalities force $v=0$. This proves the full cone
assertion, including both orientations.
\end{proof}

The strong component has a fixed margin. The weak margin retains
its dependence on $L/\mu$ and can become small at large radius.
Its comparison with the radial error in
\eqref{eq:connection-weak-gap} is what allows the same cone to
pass through the whole unbounded range.

For later control of families, it is useful to know how much slope
a radial direction acquires. This follows from the same amplitude
derivatives without repeating the variational argument.

\begin{corollary}[Slope of a narrower direction]
\label{cor:long-connection-narrow-cone}
Under the hypotheses of Proposition~\ref{prop:long-connection-cone},
let $0\leq\theta\leq1$, $\chi v_\mu=1$ and
\[
 |v_u|\leq\theta\sqrt u,\qquad
 |v_w|\leq\theta\sqrt w.
\]
Put $d=D\mathcal C(p)v$ and
\[
 S_\ell=\frac{\sqrt u+(1+L)\sqrt w}{\mu^2}.
\]
Then $d_r\geq3/4$ and
\begin{equation}
 \max\left\{\frac{|d_u|}{\sqrt{u_+}\,d_r},
             \frac{|d_w|}{\sqrt{w_+}\,d_r}\right\}
       \leq30000(\theta+S_\ell).
 \label{eq:long-connection-narrow-cone}
\end{equation}
\end{corollary}

\begin{proof}
First consider the radial direction $v^0=(\chi^{-1},0,0)$.
Its logarithmic responses are bounded by $274L$ and
$274L/\mu^2$. For $L\geq1$,
\begin{equation}
 Le^{-\mu L/4}\leq\frac{64}{\mu^2}.
 \label{eq:connection-radial-exponential}
\end{equation}
To see this, let $E_0=e^{-\mu L/8}$. The elementary exponential
inequality gives $\mu LE_0\leq8$; since $L\geq1$, also
$\mu E_0\leq8$. Multiplication proves
\eqref{eq:connection-radial-exponential}.
Combining the logarithmic derivatives with
\eqref{eq:connection-square-root-decay} therefore gives
\begin{equation}
 \begin{aligned}
 |D\mathcal C_u(p)v^0|&\leq
           \frac{17536\sqrt u}{\mu^2}\sqrt{u_+},\\
 |D\mathcal C_w(p)v^0|&\leq
           \frac{274L\sqrt w}{\mu^2}\sqrt{w_+}.
 \end{aligned}
 \label{eq:connection-radial-wall-responses}
\end{equation}
Both coefficients are at most $20000S_\ell$.

For any unit-phase direction in the full cone,
Proposition~\ref{prop:long-connection-cone} gives
$d_r\leq2$ and hence
$|d_u|\leq2\sqrt{u_+}$, $|d_w|\leq2\sqrt{w_+}$.
If $\theta>0$, write
\[
 v=\theta z+(1-\theta)v^0,\qquad
 z=v^0+\theta^{-1}(v-v^0).
\]
The vector $z$ is in that full cone and has unit input phase.
Linearity and~\eqref{eq:connection-radial-wall-responses} imply,
for either output wall $j=u,w$,
\[
 |d_j|\leq(2\theta+20000S_\ell)\sqrt{j_+}.
\]
For $\theta=0$ the same assertion follows directly from $v=v^0$.
Finally $d_r\geq3/4$ and
$2\theta+20000S_\ell\leq30000(\theta+S_\ell)d_r$.
The output walls are positive, so division proves
\eqref{eq:long-connection-narrow-cone}.
\end{proof}

We can now express the estimates in the radius used at successive
cores. The reflection map $\mathcal F$ was defined in
\eqref{eq:long-reflection-map}; its input is the downward core
exit $(R,u,w)$, and its output is the next upward core entrance.

\begin{corollary}[Long reflection in core coordinates]
\label{cor:long-reflection-estimates}
There exists $\varepsilon_\ell>0$ such that, for
\[
 R\geq\frac{11}{2},\qquad
 0<u\leq\frac{\varepsilon_\ell}{R},\qquad
 0<w\leq\frac{\varepsilon_\ell}{R},
\]
the actual reflection time and map $\mathcal F=(R_+,u_+,w_+)$
are locally $C^2$. The value estimates are
\begin{equation}
 |R_+-(R-2)|\leq\frac{48(u+w)}R,
 \qquad0<u_+\leq u,\qquad0<w_+\leq w.
 \label{eq:long-reflection-values-complete}
\end{equation}
For every $v=(1,a,b)$ with $|a|\leq\sqrt u$ and
$|b|\leq\sqrt w$, the differential $d=D\mathcal F(R,u,w)v$
satisfies
\begin{equation}
 \begin{aligned}
 d_R&\geq\frac12,&
 |d_R-1|&\leq\frac{300000\sqrt{u+w}}R,\\
 |d_u|&\leq\sqrt{u_+}\,d_R,&
 |d_w|&\leq\sqrt{w_+}\,d_R.
 \end{aligned}
 \label{eq:long-reflection-differential}
\end{equation}
\end{corollary}

\begin{proof}
Choose the threshold in~\eqref{eq:long-connection-cone-threshold}.
The coordinate change
\[
 (R,u,w)\longmapsto p=(\sqrt{R^2-1}-1,u,w)
\]
has $\mu\geq4$ and $R/2\leq\mu<R$ on this range.
Thus $u\leq\varepsilon_\ell/\mu$ and
$w\leq\varepsilon_\ell$, as required for the connection.
The first arrival and its local $C^2$ dependence were proved in
Proposition~\ref{prop:long-connection-arrival}. By reflection
equivariance, its output is the next positive entrance, as in
\eqref{eq:long-reflection-map}. The value estimates follow from
\eqref{eq:long-reflection-values} and positivity of the walls.

The coordinate differential leaves the two wall components
unchanged and sends a unit radial input to a vector with
$\chi v_\mu=1$, because $H(\mu+1,1)=R$.
The chain rule therefore identifies $d$ with the differential
in Proposition~\ref{prop:long-connection-cone}. Its radial lower
bound is at least $3/4$, and both transverse margins are at most
one. The error bound follows from
\eqref{eq:connection-phase-cone-amplitude} and
\[
 \frac{150000\sqrt{u+w}}{\mu}
       \leq\frac{300000\sqrt{u+w}}R.
\]
\end{proof}

The connection now has the same form of phase control as the core:
a value error proportional to the incoming amplitude, a derivative
error proportional to its square root, and preservation of the same
cone. To use this control throughout an era, the reflection estimates
must also cover the remaining bounded range of exit radii. There
the waiting time can still be long, so the small growing wall must
remain part of the estimates.