\subsection{Differentiation of the terminal passage}
\label{sec:terminal-variation}

The auxiliary height separates two parts of a single orbit. Its
choice can therefore be held fixed when differentiating the full
terminal return. We first establish this local independence,
including ordinary neighborhoods of the initial data. We then
estimate the outgoing derivatives. Although the outgoing clock
interval grows like $\log(1/\alpha)$, its integrated Jacobian is
bounded. First derivatives consequently remain bounded, while
second derivatives grow at most linearly in the clock length.

Write $Y=(R,u,w)$ for the downward-exit parameters. Make the height
dependence in Section~\ref{sec:terminal-passage} explicit by setting
$z(h,Y)=\mathsf C_p y_h(Y)$ and
\begin{equation}
 \begin{aligned}
  \mathcal T(h,Y)&=S_h(Y)+T_{\rm in}(z(h,Y)),\\
  \mathcal E(h,Y)&=\mathcal P(z(h,Y)).
 \end{aligned}
 \label{eq:terminal-height-maps}
\end{equation}
All the time variables in this definition are original normalized
times.

\subsubsection{Local independence of the section height}

\begin{lemma}[The height can be held fixed]
\label{lem:terminal-height-freezing}
At every positive-wall input in
Proposition~\ref{prop:terminal-first-arrival}, put
$Y_0=(2+2\alpha,u,w)$ and $h_0=\eta\alpha^J$.
The functions $\mathcal T$ and $\mathcal E$ are jointly $C^2$
near $(h_0,Y_0)$. On a neighborhood of that point,
\begin{equation}
 \mathcal T(h,Y)=\mathcal T(h_0,Y),\qquad
 \mathcal E(h,Y)=\mathcal E(h_0,Y).
 \label{eq:terminal-local-height-independence}
\end{equation}
In particular, their height derivatives vanish.
For any function $a$ continuous at $Y_0$ with $a(Y_0)=h_0$,
the maps $\mathcal T(a(Y),Y)$ and $\mathcal E(a(Y),Y)$ agree
near $Y_0$ with their fixed-height versions and are $C^2$ there.
\end{lemma}

\begin{proof}
Consider first the outgoing event equation
$V(\phi_t x(Y))-h=0$, where $x(Y)$ is the normalized state
corresponding to $y_-(Y)$. Its time derivative is strictly negative
at $t=S_{h_0}(Y_0)$. The implicit function theorem, with $(h,Y)$
as parameters, gives a jointly $C^2$ event time. The chart and
strict negativity of $dV/dt$ persist on the whole compact segment,
including a small extension beyond its endpoints. For nearby
$(h,Y)$, they therefore persist up to the moving event time.
The event is still the first arrival at $V=h$. Hence this local
implicit time equals $S_h(Y)$, and its state is jointly $C^2$.

The cyclic denominator is positive at the endpoint. Moreover, all
incoming conditions hold there with strict margins. In particular,
\eqref{eq:terminal-entrance-smallness} and $k^q\geq1$ give
\[
 0<u_e,w_e\leq\varepsilon_{\rm in}/2<10^{-4},
 \qquad 0<v_e<1,\qquad k>7/4,\qquad m_e>1.
\]
They persist near $(h_0,Y_0)$. The rational cyclic map and the
incoming arrival are thus $C^2$ throughout a smaller neighborhood,
which proves joint regularity in~\eqref{eq:terminal-height-maps}.

We also need to identify these nearby compositions as physical
first arrivals. Along the original outgoing segment, the cyclic
chart is positive and its active wall is less than one.
These are open conditions on a compact segment. By continuity of
the flow and of $S_h(Y)$, they continue to hold up to the nearby
moving endpoint. The incoming segment then stays below its own
arrival section until its first crossing. The proof of
Proposition~\ref{prop:terminal-first-arrival} applies to every
point of this neighborhood. Thus $\mathcal T(h,Y)$ and
$\mathcal T(h_0,Y)$ are both the first positive arrival time of
the same cyclic orbit at $V=1$. They are equal, and their states
are equal as well. This proves~\eqref{eq:terminal-local-height-independence}.

For fixed $Y_0$ the two functions are locally constant in $h$,
so their height derivatives vanish. Finally,
$Y\mapsto(a(Y),Y)$ is continuous at $Y_0$ and takes that point
to $(h_0,Y_0)$. Equation~\eqref{eq:terminal-local-height-independence}
therefore holds along this graph near $Y_0$. The fixed-height
functions are $C^2$, giving the last assertion without any
differentiability requirement on $a$.
\end{proof}

For the choice used in the terminal passage,
$a(R,u,w)=\eta((R-2)/2)^J$. The lemma identifies its derivatives
with those computed at fixed numerical height. The same local
identity also gives zero derivative with respect to the logarithmic
height $\ell$ when $h=e^{-\ell}$.

Uniform estimates over a range of heights follow by stopping one
longer outgoing segment earlier. This avoids choosing a different
initial neighborhood for each height.

\begin{lemma}[Uniformity above a lower height]
\label{lem:terminal-height-uniform}
Fix $0<\eta_0\leq1/100$ and $J\geq2$.
The constants $C_*$ and $\varepsilon$ in
Lemma~\ref{lem:terminal-cycle-entrance} can be chosen to work
simultaneously for every $\eta\in[\eta_0,1/100]$.
If $q+2\leq J$ and $0<\varepsilon_{\rm in}\leq10^{-4}$ are
also fixed, then one input threshold gives all the conclusions
of Proposition~\ref{prop:terminal-first-arrival} for every
\begin{equation}
 \eta_0\leq\eta\leq
 \min\left(\frac1{100},
              \frac{\varepsilon_{\rm in}}{4(11/5)^q}\right).
 \label{eq:terminal-height-interval}
\end{equation}
The local height independence of
Lemma~\ref{lem:terminal-height-freezing} holds at each such
input and height.
\end{lemma}

\begin{proof}
Construct the outgoing segment down to
$h_0=\eta_0\alpha^J$. Its strictly decreasing active wall crosses
each $h=\eta\alpha^J$ with $\eta\geq\eta_0$ and $\eta<1$
exactly once no later than that endpoint. The intermediate value theorem
gives the crossing, and strict decrease proves its first-arrival
property. Thus
\begin{equation*}
 S_{\eta\alpha^J}(Y)\leq S_{\eta_0\alpha^J}(Y).
\end{equation*}
All estimates along the longer segment apply up to the earlier
endpoint. Reduce $\varepsilon$ so that $C_*\varepsilon\leq1/10000$.
Then, with the fixed constant $C_*$ from that segment,
\[
 |\rho-R_0|\leq C_*b,\qquad |U|+|W|\leq C_*b,\qquad
 C_*b\leq\alpha^{2J}/10000.
\]
At the earlier endpoint, $V=\eta\alpha^J\leq\alpha/100$.
The algebra in Lemma~\ref{lem:terminal-cycle-entrance}
therefore gives exactly the same denominator and radius bounds,
and the same wall bounds with this value of $\eta$.

For the weighted incoming walls, reduce $\varepsilon$ so that
$C_*\varepsilon\leq\varepsilon_{\rm in}/[4(11/5)^q]$.
The proof of~\eqref{eq:terminal-entrance-smallness} is then uniform
over~\eqref{eq:terminal-height-interval}. The entire cyclic
outgoing segment still has active wall at most $1/5000$,
since it is a subsegment of the longer one. Composition with the
incoming passage gives the actual first arrival. The strict
margins used in Lemma~\ref{lem:terminal-height-freezing} hold
at each input, proving its local conclusion as well.
\end{proof}

\subsubsection{The Jacobian along the outgoing segment}

Let $Y=(\rho,U,W)$ and regard $V$ as a prescribed clock value.
Write the vector field in~\eqref{eq:terminal-outgoing-field} as
\[
 f_V(Y)=\bigl(f,a_U U,a_W W\bigr),\qquad
 f=\frac{-C\rho-g(U-W)^2/\rho}{q_-}.
\]
All derivatives in these estimates hold $V$ fixed. Norms on
$\R^3$ are maximum norms, with their induced operator norms.

\begin{lemma}[An integrable Jacobian]
\label{lem:terminal-jacobian}
On the signed region~\eqref{eq:terminal-clock-region},
with $\beta=|U|+|W|$,
\begin{equation}
 \|D_Y f_V\|\leq |a_U|+|a_W|+2048(V+\beta).
 \label{eq:terminal-jacobian-norm}
\end{equation}
The field and its first two state derivatives are uniformly
bounded on this region, including at $V=0$.
\end{lemma}

\begin{proof}
The small factors in the radial row and in the off-diagonal
wall entries are visible before taking norms. Put
$c_U=V+W-2U$. Quotient differentiation gives
\begin{align*}
 \partial_\rho f
   &=\frac{-C-(1/g-g/\rho^2)(U-W)^2-f\rho/g}{q_-},\\
 \partial_U f
   &=\frac{-c_U\rho-2g(U-W)/\rho-fc_U}{q_-},\\
 \partial_W f
   &=\frac{-U\rho+2g(U-W)/\rho-fU}{q_-}.
\end{align*}
The corresponding rate derivatives are
\begin{align*}
 \partial_\rho a_U&=\frac{\rho/g-\rho-a_U\rho/g}{q_-},&
 \partial_\rho a_W&=\frac{(1-a_W)\rho/g}{q_-},\\
 \partial_U a_U&=\frac{U-a_Uc_U}{q_-},&
 \partial_U a_W&=-\frac{(1+a_W)c_U}{q_-},\\
 \partial_W a_U&=\frac{V-W-a_UU}{q_-},&
 \partial_W a_W&=-\frac{(1+a_W)U}{q_-}.
\end{align*}
To bound them, use $q_-\geq1/2$, $g\geq1$,
$1\leq\rho\leq5$, $g/\rho\leq1$, $\rho/g\leq5$,
and
\[
 |c_U|\leq V+2\beta,\qquad
 |1/g-g/\rho^2|\leq2.
\]
Equation~\eqref{eq:terminal-perturbation-rates} and
$\beta\leq1/50$ give $|f|\leq1$ and $|a_U|,|a_W|\leq32$.
Keeping the sharper bound
$|f|\leq20(|U|V+\beta^2)$ in the radial derivative gives
\begin{align}
 |\partial_\rho f|&\leq256(|U|V+\beta^2),&
 |\partial_U f|&\leq32(V+\beta),&
 |\partial_W f|&\leq16\beta,\notag\\
 |\partial_\rho a_j|&\leq512,&
 |\partial_U a_j|,\ |\partial_W a_j|
       &\leq256(V+\beta),&& j=U,W.
 \label{eq:terminal-gradient-bounds}
\end{align}
For example, the three numerator terms in $\partial_\rho f$
are bounded by $H$, $2\beta^2$ and $100H$, respectively,
where $H=|U|V+\beta^2$. Division by $q_-\geq1/2$
gives the stated $256H$. The other bounds follow from the same
displayed formulas: the radial $U$ numerator is at most
$6V+14\beta$, and the radial $W$ numerator at most $8\beta$;
the rate numerators are bounded by $256$ in the radial direction
and $128(V+\beta)$ in either wall direction.

The Jacobian entries in a wall row retain the factor of that wall:
\[
 D(a_UU)[\xi]=U\,Da_U[\xi]+a_U\xi_U,\qquad
 D(a_WW)[\xi]=W\,Da_W[\xi]+a_W\xi_W.
\]
Since $H\leq V+\beta$ and $V+\beta\leq2$,
the sum of absolute values in the radial row is at most
$512(V+\beta)$, while each rate gradient has row sum at most
$1536$. Each wall row is consequently bounded by
$|a_j|+1536\beta$. Taking the maximum of the three row sums proves
\eqref{eq:terminal-jacobian-norm}.

For the second derivatives, consider the compact box
\[
 7/4\leq\rho\leq17/4,\qquad
 |U|,|W|\leq1/100,\qquad0\leq V\leq1.
\]
On it, $\rho^2-V^2$ is bounded away from zero, $\rho>0$,
and $q_-\geq1/2$. The formulas defining $f_V$ are therefore
smooth on a neighborhood of the box. Their derivatives through
order two are continuous there and have a finite common bound.
\end{proof}

Fix $\eta_0\in(0,1)$ and $J\geq1$, and set
$L_0=\log(1/\eta_0)+J\log(1/\alpha)$. For the outgoing orbit
constructed at this lowest height, denote its clock state by
$Y(\sigma)$ and its state Jacobian by
$L_Y(\sigma)=D_Y f_{e^{-\sigma}}(Y(\sigma))$.
The pointwise bounds~\eqref{eq:terminal-boundary-rate-bound}
and~\eqref{eq:terminal-perturbation-rates} give, for $T\leq L_0$,
\begin{align}
 \int_0^T |a_j|\dd\sigma
   &\leq\mathcal R(T),\notag\\
 \mathcal R(T)
   &=10\alpha T+800BbT+100(Bb)^2T+100Bb+5.
 \label{eq:terminal-absolute-rate-integral}
\end{align}
Here $B$ is chosen using $\eta_0$ as in
Lemma~\ref{lem:terminal-outgoing}. Integrating
\eqref{eq:terminal-jacobian-norm} now yields
\begin{equation}
 \int_0^T\|L_Y(\sigma)\|\dd\sigma
       \leq2\mathcal R(T)+2048(1+BbT).
 \label{eq:terminal-integrated-jacobian}
\end{equation}
With the same smallness bounds as before,
$\mathcal R(T)\leq10K+6$, where
$K=\log(1/\eta_0)+J/e$. Reducing the input threshold if necessary
also gives $bT\leq1$. Thus the right-hand side is at most
\begin{equation}
 B_*=2(10K+6)+2048(1+B),
 \label{eq:terminal-jacobian-budget}
\end{equation}
uniformly in $\alpha$, the signed wall inputs, and the duration
$T\leq L_0$.

For fixed $h$, express the outgoing state in its radius and two
small walls:
\begin{equation}
 E_h(Y)=\bigl(H(M_h-1,h),U_h,W_h\bigr).
 \label{eq:terminal-outgoing-radius-map}
\end{equation}
The active wall has the fixed value $h$ at this event.

\begin{proposition}[First and second outgoing derivatives]
\label{prop:terminal-outgoing-derivatives}
Fix $0<\eta_0<1$ and $J\geq1$. There are $C>0$ and
$\varepsilon>0$ such that, for
\begin{equation}
 0<\alpha\leq1,\qquad
 0<b\leq\varepsilon\alpha^J,\qquad |u|+|w|\leq b,
 \label{eq:terminal-variation-domain}
\end{equation}
and every $\eta\in[\eta_0,1)$, the map $E_h$, with
$h=\eta\alpha^J$ held fixed, is $C^2$ near $Y_0=(2+2\alpha,u,w)$.
At this input,
\begin{equation}
 \|DE_h(Y_0)\|\leq C,\qquad
 \|D^2E_h(Y_0)\|\leq C\log(1/h).
 \label{eq:terminal-outgoing-two-bounds}
\end{equation}
More generally, the outgoing state at any fixed clock value
$0\leq\sigma\leq L_0$ has these bounds with $\log(1/h)$ replaced
by $\sigma$.
\end{proposition}

\begin{proof}
We first construct the clock family from the actual flow.
Along the segment down to $\eta_0\alpha^J$, the functions
$A,V,q_-$ are positive and $M<1$. Extend the compact segment
slightly in original time. Smooth dependence gives a common open
interval and a neighborhood of $Y_0$ on which these strict
conditions persist. The map
\[
 (Y,t)\longmapsto (Y,-\log V(\phi_t x(Y)))
\]
has positive time derivative $Aq_-$. The inverse function theorem
gives a joint $C^2$ inverse time $t=t(Y,\sigma)$.
For each nearby $Y$, the clock is strictly increasing on the
common interval, so the local inverses agree wherever they overlap.
At $Y_0$, the extended interval has clock image containing
$[0,L_0]$ in its interior. This remains true after restricting
the initial neighborhood, so the inverses cover that entire
clock interval for every nearby input.

In radius and wall coordinates the resulting actual family,
denoted by $\Phi(Y,\sigma)$, satisfies
\begin{equation}
 \Phi(Y,0)=Y,\qquad
 \partial_\sigma\Phi(Y,\sigma)
       =f_{e^{-\sigma}}(\Phi(Y,\sigma)).
 \label{eq:terminal-actual-clock-family}
\end{equation}
For every fixed $h\in[\eta_0\alpha^J,1)$,
strict monotonicity and persistence of the first arrival identify
\begin{equation}
 E_h(Y)=\Phi(Y,\log(1/h))
 \label{eq:terminal-outgoing-clock-identification}
\end{equation}
on a neighborhood of $Y_0$. This is an equality of local
functions, so it identifies their first and second derivatives.

For an input direction $\xi$, put
$X(\sigma)=D_Y\Phi(Y_0,\sigma)[\xi]$. Differentiating
\eqref{eq:terminal-actual-clock-family} gives
\begin{equation*}
 X'=L_Y X,\qquad X(0)=\xi.
\end{equation*}
The estimate needed also applies to a forced equation
$X'=L_YX+P$. A signed row sum of a three-dimensional matrix
is bounded by $3\|L_Y\|$. As in
Lemma~\ref{lem:core-variation-growth}, the upper right derivative
of $\|X\|$ is therefore at most $3\|L_Y\|\|X\|+\|P\|$.
Multiply by the integrating factor
\[
 \exp\left(-3\int_a^t\|L_Y(s)\|\dd s\right).
\]
Integration, together with $\int_a^t\|L_Y\|\leq B_*$, gives
\begin{equation}
 \|X(t)\|\leq e^{3B_*}
       \left(\|X(a)\|+\int_a^t\|P(s)\|\dd s\right),
       \qquad 0\leq a\leq t\leq L_0.
 \label{eq:terminal-forced-variation}
\end{equation}
In particular,
\[
 \|D_Y\Phi(Y_0,\sigma)[\xi]\|
                    \leq e^{3B_*}\|\xi\|.
\]

For two input directions $\xi,\zeta$, let
$X=D_Y\Phi[\xi]$, $Z=D_Y\Phi[\zeta]$, and
$Q=D_Y^2\Phi[\xi,\zeta]$, all evaluated along the reference
clock orbit. The second variation satisfies
\begin{equation}
 Q'=L_YQ+D_Y^2 f_{e^{-\sigma}}(Y(\sigma))[X,Z],
 \qquad Q(0)=0.
 \label{eq:terminal-second-variation}
\end{equation}
This identity follows by differentiating the integral equation
for the first variation with respect to its input. On each
compact clock interval the first-variation integrand is $C^1$
in that input and has continuous derivatives. Differentiation
under the finite integral is therefore valid and gives
\eqref{eq:terminal-second-variation}.

Let $C_2$ bound the state Hessian in
Lemma~\ref{lem:terminal-jacobian}. The forcing in
\eqref{eq:terminal-second-variation} is bounded by
$C_2e^{6B_*}\|\xi\|\|\zeta\|$. Applying
\eqref{eq:terminal-forced-variation} once more gives
\[
 \|Q(\sigma)\|\leq
       C_2e^{9B_*}\sigma\|\xi\|\|\zeta\|.
\]
Taking operator norms and using
\eqref{eq:terminal-outgoing-clock-identification} proves the
result, with $C\geq\max(e^{3B_*},C_2e^{9B_*})$.
The argument takes place in ordinary initial coordinates and
allows signed or zero walls.
\end{proof}

\subsubsection{The incoming phase at terminal scales}

The incoming portion of a terminal connection has a large radius
$k\gtrsim\alpha^{-1}$. Its weak wall is smaller than its strong
wall once the old input is sufficiently small. These two facts
simplify the estimates of
Proposition~\ref{prop:entrance-physical-bounds}.

\begin{lemma}[Terminal incoming phase]
\label{lem:terminal-incoming-phase}
There are constants $A_1\geq1$ and $0<\delta\leq10^{-4}$ with
the following property. For an incoming point $y=(m,u,v,w)$
satisfying
\[
 \begin{gathered}
  0<\alpha\leq1,\qquad k=H(m-1,v)\geq\max(7/4,1/\alpha),\\
  m\geq1,\qquad 0<w\leq u\leq\delta,\qquad0<v<1,
 \end{gathered}
\]
write its stable radial coordinates as $z=(r,u,v,w)$ and use the
physical arrival function $F$ of~\eqref{eq:entrance-physical-return}.
At $z$,
\begin{align}
 |F-r|&\leq A_1\alpha uv,&
 |\partial_rF-1|&\leq A_1uv,\notag\\
 |\partial_uF|&\leq A_1\alpha v,&
 |\partial_vF|&\leq A_1u,\notag\\
 |\partial_wF|&\leq A_1\alpha uv.
 \label{eq:terminal-incoming-phase-bounds}
\end{align}
The function $F$ is locally $C^2$.
\end{lemma}

\begin{proof}
Let $s=u+w$. By
\eqref{eq:entrance-terminal-clock}, both $\omega$ and
$(1+L/k)\omega$ are at most one. Enlarging the constant in
Proposition~\ref{prop:entrance-physical-bounds} gives its
following common bounds:
\begin{align*}
 |F-r|&\leq C(uv+v^2s^2)/k,&
 |\partial_rF-1|&\leq C(uv+v^2s^2)/k^2,\\
 |\partial_uF|&\leq C(v+v^2s)/k,&
 |\partial_vF|&\leq C(u+vs^2),\\
 |\partial_wF|&\leq C(uv+v^2s)/k.
\end{align*}
Here $w\leq u\leq1$ implies $s\leq2u$ and $s^2\leq4u$,
while $v^2\leq v$. Consequently the four numerators in these
bounds are at most $5uv$, $3v$, $5u$ and $3uv$, respectively.
Use $1/k\leq\alpha$ in the value and two decreasing-wall
columns, and $1/k^2\leq1$ in the radial column. Taking
$A_1=5C+1$ proves~\eqref{eq:terminal-incoming-phase-bounds}.
The $C^2$ conclusion is the one for the same physical arrival
function in Proposition~\ref{prop:entrance-physical-bounds}.
\end{proof}

The ordering $w_e\leq u_e$ required by this lemma holds uniformly
on a terminal height interval. Indeed, fix
$0<\eta_0\leq\delta_1:=\min(\delta/4,1/1000)$ and $J\geq2$.
Lemma~\ref{lem:terminal-height-uniform}, with
$q=0$ and $\varepsilon_{\rm in}=\delta$, puts every cyclic
endpoint with $\eta\in[\eta_0,\delta_1]$ in the same incoming
domain. Decrease the old input threshold further so that
$C_*\varepsilon\leq\eta_0/4$. Then
\begin{equation}
 w_e\leq\frac{\eta_0}{2}\alpha^{2J-2}
      \leq\frac{\eta}{2}\alpha^{J-2}\leq u_e.
 \label{eq:terminal-incoming-wall-order}
\end{equation}
Also~\eqref{eq:terminal-cycle-end-radius} gives $k>1/\alpha$.
Thus every one of these actual cyclic endpoints satisfies
\eqref{eq:terminal-incoming-phase-bounds}, with one constant
$A_1$. These estimates retain the growing-wall factor $v_e$.
To obtain derivatives in the original exit parameters, the
remaining step is to control the cyclic entrance coordinates
and their stable radial phase.