\subsection{Variations of the incoming passage}
\label{sec:entrance-variation}

The phase error is small because its rate contains decaying wall
products. Its derivative also depends on how the walls respond to a
change of the incoming phase. We estimate that response at equal
values of $s=\log(V/v)$. The strong and weak logarithmic wall
variations have different radial scales; weighting the latter by
$k^2$ makes their coupling uniformly integrable.

Fix an incoming point $y=(m,u,v,w)$ as in
Proposition~\ref{prop:incoming-passage}, and choose the local inverse
$\Psi$ of Lemma~\ref{lem:entrance-radial-chart}. In the input coordinates
$z=(r,u,v,w)$, define
\begin{equation}
 Q(z)=r\bigl(\mathcal P(\Psi(z))\bigr),\qquad
 \ell_U(z)=\log\mathcal P_U(\Psi(z)),\qquad
 \ell_W(z)=\log\mathcal P_W(\Psi(z)).
 \label{eq:entrance-return-observables}
\end{equation}
Thus $Q$ is the stable radial phase evaluated at the actual arrival.
The parameter $k=H(m-1,v)$ and the weight $k^2$ are fixed at the
reference input throughout differentiation.
All vector norms below are maximum norms, with the induced operator
norms on differentials.

\begin{proposition}[Three arrival columns]
\label{prop:entrance-three-columns}
There are universal constants $C>0$ and $0<\varepsilon\leq10^{-4}$
such that, for every input satisfying
\[
 k\geq7/4,\qquad m\geq1,\qquad
 0<u,w\leq\varepsilon,\qquad 0<v<1,
\]
the functions in~\eqref{eq:entrance-return-observables} are locally
$C^2$. At $z=(r(y),u,v,w)$, with $L=\log(1/v)$, they satisfy
\begin{align}
 |\partial_rQ-1|&\leq\frac C{k^2}\bigl[uv+(vw)^2\bigr],
 &|\partial_uQ|&\leq\frac{Cv}k,\notag\\
 |\partial_wQ|&\leq\frac Ck(uv+v^2w),
 \label{eq:entrance-three-phase-columns}
\end{align}
and
\begin{align}
 |\partial_r\ell_U|+k^2|\partial_r\ell_W|&\leq CL,\notag\\
 |\partial_u\ell_U-1/u|+k^2|\partial_u\ell_W|&\leq C,\notag\\
 |\partial_w\ell_U|+k^2|\partial_w\ell_W-1/w|&\leq C.
 \label{eq:entrance-three-wall-columns}
\end{align}
\end{proposition}

The wall derivatives retain the logarithmic singularity of their
initial amplitudes. After that term is removed, the two wall columns
are uniformly bounded, even as $v$ tends to zero. The radial column
grows at most with the logarithmic waiting length.

\subsubsection{Comparison at equal clock values}

Write $f_{\rm p}$ for the projective vector field
in~\eqref{eq:projective-flow}, and put
\[
 \mathcal V=\frac{f_{\rm p}}q,\qquad
 \Dt\gamma=A f_{\rm p}(\gamma),\qquad
 A=\frac6D,\qquad q=M-1-U(V+W-U).
\]
The vector field
$\mathcal V$ describes the same orbit in the clock $s$, and
$\mathcal V_V=V$.
Let $\gamma(t;y)$ be the actual flow starting at $y$, and let
$J_h(t)=D_y\gamma(t;y)h$ be its variation at fixed original time.
Set
\begin{equation}
 \zeta=\frac{h_v}{v},\qquad
 \xi_h(t)=J_h(t)
       -\left(\frac{(J_h)_V(t)}{V(t)}-\zeta\right)\mathcal V(\gamma(t)).
 \label{eq:entrance-clock-projection}
\end{equation}
Then
\begin{equation}
 (\xi_h)_V=\zeta V,\qquad \xi_h(0)=h,\qquad
 \frac{d\xi_h}{ds}=D\mathcal V(\gamma)\xi_h.
 \label{eq:entrance-clock-variation}
\end{equation}
For the differential identity, let
$\lambda=(J_h)_V/V-\zeta$ and $a=Aq$.
Since the original vector field is $a\mathcal V$ and its $V$
component is $aV$, differentiation gives
$\Dt\lambda=da[J_h]$. Consequently
\[
 \Dt\xi_h
 =D(a\mathcal V)J_h-\Dt\lambda\,\mathcal V
                    -\lambda D\mathcal V(a\mathcal V)
 =aD\mathcal V\,\xi_h.
\]
Division by the positive clock speed $a$ proves
\eqref{eq:entrance-clock-variation}.

This projection is the derivative of nearby actual trajectories.
At any point of the passage, the equation
\[
 \log V(\gamma(t;\widetilde y))-\log\widetilde v=s
\]
has a locally $C^2$ solution for $t$, because its time derivative
is $Aq>0$. Implicit differentiation and the chain rule give exactly
\eqref{eq:entrance-clock-projection}. The resulting local branches
agree wherever they overlap.

For a direction $d=(d_r,d_u,d_v,d_w)$ in the input radial chart,
take $h=D\Psi(z)d$. The inverse chart preserves the three walls, so
$\zeta=d_v/v$. The arrival clock $L=\log(1/v)$ has derivative
$-d_v/v$, and hence
\begin{equation}
 D(\mathcal P\circ\Psi)(z)d
    =\xi_h(L)-\frac{d_v}{v}\mathcal V(\mathcal P(y)).
 \label{eq:entrance-arrival-clock-correction}
\end{equation}
In this formula and below, the variation is evaluated using the
clock $s$. In particular, the correction vanishes for the three
directions considered in Proposition~\ref{prop:entrance-three-columns}.

\subsubsection{The weighted wall equation}

Assume $d_v=0$ and $u,w>0$. Positivity of the walls allows us to set
\begin{equation}
 X=\frac{(\xi_h)_U}{U},\qquad
 Y=\frac{(\xi_h)_W}{W},\qquad
 Z=dr(\gamma)[\xi_h],\qquad
 \mathbf x=(X,k^2Y).
 \label{eq:entrance-variation-variables}
\end{equation}
Their initial values are
\[
 X_0=d_u/u,\qquad Y_0=d_w/w,\qquad Z_0=d_r.
\]
The stable logarithmic rates in the clock $s$ are
\begin{equation}
 \alpha=\frac{1-M^2-(V-W)^2+U^2}{2q},\qquad
 \beta=-1-\frac{2(1+\mathcal C)}q,\qquad
 \mathcal C=U(V+W-U).
 \label{eq:entrance-log-rates}
\end{equation}
Thus $U'=\alpha U$ and $W'=\beta W$.
All partial derivatives of these rates in the next formulas are
in the local coordinates $(r,U,V,W)$. Differentiating the two
logarithmic equations gives
\begin{equation}
 \mathbf x'=B\mathbf x+\mathbf b Z,\qquad
 B=\begin{pmatrix}
 U\partial_U\alpha&(W/k^2)\partial_W\alpha\\
 k^2U\partial_U\beta&W\partial_W\beta
 \end{pmatrix},
 \qquad
 \mathbf b=\binom{\partial_r\alpha}{k^2\partial_r\beta}.
 \label{eq:entrance-weighted-wall-system}
\end{equation}
The derivatives defining these coefficients are independent of the
choice of local inverse, so the coefficients are continuous along
the full passage.

\begin{lemma}[Wall response]
\label{lem:entrance-wall-response}
There is a universal constant $T>0$ such that, for
$0\leq a\leq b\leq L$,
\begin{equation}
\begin{split}
 |X(b)-X(a)|+k^2|Y(b)-Y(a)|
 \leq T\left[u|X(a)|+w|Y(a)|
                         +\int_a^b|Z(s)|\dd s\right].
\end{split}
\label{eq:entrance-wall-response}
\end{equation}
\end{lemma}

\begin{proof}
We first record the coefficient bounds that make the estimate
independent of $L$. In ordinary coordinates,
\begin{align*}
 d\alpha&=\frac{-M\,dM-(V-W)(dV-dW)+U\,dU-\alpha\,dq}{q},
 \\
 d\beta&=\frac{2(1+\mathcal C)\,dM-2M\,d\mathcal C}{q^2},
 \\
 d\mathcal C&=(V+W-2U)\,dU+U(dV+dW),\qquad
 dq=dM-d\mathcal C.
\end{align*}
On the incoming region, $M\leq3k$, $q\geq k/2$,
$|1+\mathcal C|\leq2$, and $|\alpha|\leq12k$.
The first formula therefore gives $\|d\alpha\|\leq180$.
Composing with~\eqref{eq:entrance-inverse-columns} bounds all radial
chart derivatives of $\alpha$ by a universal constant.
For $\beta$, the same inverse columns give
\begin{align*}
 |\partial_r\beta|&\leq128/k^2,\\
 |\partial_U\beta|&\leq C(U+W)/k^3+48(V+U+W)/k,\\
 |\partial_W\beta|&\leq C(U+W)/k^3+24U/k.
\end{align*}
Indeed, the three numerators are respectively
$2(1+\mathcal C)M_r$,
$2(1+\mathcal C)M_U-2M(V+W-2U)$, and
$2(1+\mathcal C)M_W-2MU$.
The estimates for $M_r,M_U,M_W$ and $q^2\geq k^2/4$
give the displayed constants and scales.

It follows that $\|\mathbf b\|\leq C$ and
\begin{align}
 |B_{11}|&\leq CU,&
 |B_{12}|&\leq CW/k^2,\notag\\
 |B_{21}|&\leq CkU(V+U+W),&
 |B_{22}|&\leq C\left(\frac{UW}k+\frac{W(U+W)}{k^3}\right).
 \label{eq:entrance-wall-matrix-bounds}
\end{align}
Let $c_1,c_2$ be the sums of the absolute values in the two
columns of $B$. Their integrals satisfy
\begin{equation}
 \int_a^b c_1\dd s\leq C_0u,\qquad
 \int_a^b c_2\dd s\leq C_0w/k^2
 \quad(0\leq a\leq b\leq L).
 \label{eq:entrance-wall-column-integrals}
\end{equation}
To see this, the decay in~\eqref{eq:incoming-decay} gives
$\int_0^L U\dd s\leq3u/k$ and
$\int_0^L W\dd s\leq w$, while $W\leq w$.
Using $U,V,W\leq1$, the two envelopes in
\eqref{eq:entrance-wall-matrix-bounds} obey
\begin{align*}
 \int_0^L[U+kU(V+U+W)]\dd s
      &\leq(1+3k)\frac{3u}k\leq12u,\\
 \int_0^L\left[\frac W{k^2}+\frac{UW}k
                      +\frac{W(U+W)}{k^3}\right]\dd s
      &\leq\frac wk\frac{3u}k
                  +\left(\frac1{k^2}+\frac2{k^3}\right)w
       \leq\frac{6w}{k^2}.
\end{align*}
Nonnegativity gives the same bounds on every subinterval.

For completeness, apply a scalar growth estimate to the change
$\mathbf h(s)=\mathbf x(s)-\mathbf x(a)$. It satisfies
\[
 \mathbf h'=B\mathbf h+B\mathbf x(a)+\mathbf b Z,\qquad
 \mathbf h(a)=0.
\]
In the maximum norm, every row sum of absolute values of $B$
is at most $c_1+c_2$, and
\[
 \|B\mathbf x(a)\|
     \leq c_1|X(a)|+c_2 k^2|Y(a)|.
\]
Multiplication by the scalar integrating factor therefore gives
\[
 \|\mathbf h(b)\|
 \leq e^{\int_a^b(c_1+c_2)\dd s}
       \int_a^b\bigl[c_1|X(a)|+c_2k^2|Y(a)|
                                      +C|Z|\bigr]\dd s.
\]
This comparison also follows at zeros of a component by taking
upper one-sided derivatives of the absolute values.
The exponent is at most $2C_0$, since $u,w\leq1$ and $k\geq1$.
Substitute~\eqref{eq:entrance-wall-column-integrals} and use
$|h_1|+|h_2|\leq2\|\mathbf h\|$ to obtain
\eqref{eq:entrance-wall-response}.
\end{proof}

\subsubsection{Closing the phase estimate}

The radial equation $r'=G$ gives a scalar companion to
\eqref{eq:entrance-weighted-wall-system}:
\begin{equation}
 Z'=a_0Z+a_1X+a_2Y,\qquad
 a_0=\partial_rG,\quad a_1=U\partial_UG,\quad a_2=W\partial_WG.
 \label{eq:entrance-phase-variation}
\end{equation}
This identity follows by differentiating $dr[\mathcal V]=G$.
The $C^2$ regularity of $r$ justifies the two derivatives, and
$(\xi_h)_V=0$ removes the active column.
Corollary~\ref{cor:entrance-drift-bounds} gives
\begin{equation}
\begin{split}
 |a_0|&\leq C\left(\frac{UV}{k^2}+\frac{(VW)^2}{k^3}\right),
 \qquad |a_1|\leq CUV,\\
 |a_2|&\leq C\left(UVW+\frac{(VW)^2}{k^2}\right).
\end{split}
\label{eq:entrance-phase-variation-coefficients}
\end{equation}

\begin{lemma}[Phase response at fixed clock]
\label{lem:entrance-phase-response}
There are universal $K,F>0$ such that, after reducing the allowed
stable amplitudes to a universal $\varepsilon>0$, the following holds.
Put
\begin{align}
 A_0&=u|X_0|+w|Y_0|,\notag\\
 I_0&=uv|X_0|+\bigl[uvw+(vw)^2\bigr]|Y_0|,\notag\\
 B_0&=KI_0/k,\qquad
 \Lambda=F\bigl[uv+(vw)^2\bigr]/k^2.
 \label{eq:entrance-response-inputs}
\end{align}
Then $0\leq\Lambda\leq1/2$, and for $0\leq s\leq L$,
\begin{align}
 |Z(s)-Z_0|&\leq2B_0+2\Lambda|Z_0|,\notag\\
 |Z(s)|&\leq(1+2\Lambda)|Z_0|+2B_0,\notag\\
 |X(s)-X_0|+k^2|Y(s)-Y_0|
     &\leq T\bigl[A_0+2s(|Z_0|+B_0)\bigr].
 \label{eq:entrance-fixed-clock-response}
\end{align}
Here $T$ is the constant of Lemma~\ref{lem:entrance-wall-response}.
\end{lemma}

\begin{proof}
The moments~\eqref{eq:entrance-product-moments} and $W\leq w$
give the three integral bounds
\begin{align}
 \int_0^L|a_1|\dd s&\leq Cuv/k,\notag\\
 \int_0^L|a_2|\dd s&\leq C\bigl[uvw+(vw)^2\bigr]/k,\notag\\
 \int_0^L\left[|a_0|
       +s\left(|a_1|+\frac{|a_2|}{k^2}\right)\right]\dd s
       &\leq C\bigl[uv+(vw)^2\bigr]/k^2.
 \label{eq:entrance-phase-integral-bounds}
\end{align}
The last integral is at most a universal constant times
\[
 \frac{3uv}{k^3}+\frac{(vw)^2}{2k^2}
       +\left(1+\frac w{k^2}\right)\frac{9uv}{k^2}
       +\frac{(vw)^2}{4k^2}.
\]
Thus the linear growth in $s$ allowed by the wall response is
still integrable in the phase equation.

The inhomogeneous terms from the initial wall data also preserve
their amplitudes. More precisely,
\begin{equation}
 A_0\left[uv+\frac{uvw+(vw)^2}{k^2}\right]\leq3I_0.
 \label{eq:entrance-initial-mixing}
\end{equation}
To verify this, write $b_0=uv$ and $j=uvw+(vw)^2$.
Then $b_0\leq v$, $j\leq2v$,
$u(b_0+j/k^2)\leq3b_0$, and
$w(b_0+j/k^2)\leq wb_0+wj\leq2j$.
Multiplication by $|X_0|$ and $|Y_0|$ proves
\eqref{eq:entrance-initial-mixing}.
It follows from~\eqref{eq:entrance-phase-integral-bounds} that
\begin{align}
 \int_0^L\biggl[
 |a_1|(|X_0|+TA_0)
 +|a_2|\left(|Y_0|+\frac{TA_0}{k^2}\right)
 \biggr]\dd s&\leq B_0,\notag\\
 \int_0^L\left[|a_0|
        +Ts\left(|a_1|+\frac{|a_2|}{k^2}\right)\right]\dd s
        &\leq\Lambda
 \label{eq:entrance-response-integrals}
\end{align}
for universal choices of $K$ and $F$.
Choose
$\varepsilon\leq\min(10^{-4},1/(4F))$.
Since $uv\leq\varepsilon$ and $(vw)^2\leq\varepsilon$,
this makes $\Lambda\leq1/2$.

Fix $S\leq L$ and let
$M_S=\max_{0\leq s\leq S}|Z(s)|$.
Lemma~\ref{lem:entrance-wall-response} gives, on $[0,S]$,
\[
 |X(s)|\leq|X_0|+T(A_0+sM_S),\qquad
 |Y(s)|\leq|Y_0|+\frac T{k^2}(A_0+sM_S).
\]
Insert these two inequalities in~\eqref{eq:entrance-phase-variation}
and integrate. By~\eqref{eq:entrance-response-integrals},
\[
 |Z(s)-Z_0|\leq B_0+\Lambda M_S,\qquad
 M_S\leq|Z_0|+B_0+\Lambda M_S.
\]
Therefore
$M_S\leq(|Z_0|+B_0)/(1-\Lambda)\leq2(|Z_0|+B_0)$.
Substituting this bound into the first inequality, and using
$2\Lambda B_0\leq B_0$, proves the first two lines of
\eqref{eq:entrance-fixed-clock-response}.
The wall response with $a=0$ gives the last line.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:entrance-three-columns}]
The arrival map is locally $C^2$ by
Proposition~\ref{prop:incoming-passage}. The radial coordinate is
$C^2$, and the positive output walls remain positive on a
neighborhood of the given input. This proves the asserted regularity.

For any direction $d$ with $d_v=0$, set
\[
 J_0=v|d_u|+(uv+v^2w)|d_w|,\qquad H_0=uv+(vw)^2.
\]
The initial values in~\eqref{eq:entrance-variation-variables} give
$A_0=|d_u|+|d_w|$ and $I_0=J_0$.
At $s=L$, equation~\eqref{eq:entrance-arrival-clock-correction}
identifies $Z(L)$ with $DQ[d]$, and $X(L),Y(L)$ with
$D\ell_U[d],D\ell_W[d]$. Hence
\begin{align}
 |DQ[d]-d_r|
   &\leq\frac{2K}kJ_0+\frac{2F}{k^2}H_0|d_r|,
 \label{eq:entrance-ordinary-phase-response}\\
 |D\ell_U[d]-d_u/u|+k^2|D\ell_W[d]-d_w/w|
   &\leq T\left[|d_u|+|d_w|
                  +2L\left(|d_r|+\frac KkJ_0\right)\right].
 \label{eq:entrance-ordinary-wall-response}
\end{align}
Apply these inequalities to the three coordinate directions.
The phase bounds follow immediately. For the radial wall column,
$J_0=0$ and $d_r=1$, giving $2TL$.
For the strong wall column, $J_0=v$ and
$Lv\leq1$, so its remainder is at most $T(1+2K)$.
For the weak wall column, $J_0=uv+v^2w$ and
$LJ_0\leq2$, giving a remainder at most $T(1+4K)$.
Here $Lv\leq1$ follows by writing $v=e^{-L}$ and using
$L\leq e^L$; the second bound follows from $u,w,v\leq1$.
A common larger constant proves
\eqref{eq:entrance-three-phase-columns} and
\eqref{eq:entrance-three-wall-columns}.
\end{proof}

Flow transport determines the fourth column from the three already
estimated. Moving the input along its orbit preserves the arrival
state. In differential form,
\begin{equation}
 D\mathcal P(y)\mathcal V(y)=0.
 \label{eq:entrance-return-flow-kernel}
\end{equation}
Indeed, the differential of a flow transports its vector field to
the vector field at the endpoint. In the derivative of a transverse
event map, the change of arrival time subtracts precisely this
endpoint vector. The positive change of clock preserves the identity.

For any differentiable scalar observable $p$ at the arrival, let
$F_p=p\circ\mathcal P\circ\Psi$.
The radial coordinates of $\mathcal V(y)$ are
$(G_0,u\alpha_0,v,w\beta_0)$, where
$G_0=G(y)$, $\alpha_0=\alpha(y)$ and $\beta_0=\beta(y)$.
Equation~\eqref{eq:entrance-return-flow-kernel} gives
\begin{equation}
 v\,\partial_vF_p
   =-DF_p[d_*],\qquad d_*=(G_0,u\alpha_0,0,w\beta_0).
 \label{eq:entrance-active-transport}
\end{equation}
The direction on the right has zero active component, so the
mixed response estimates~\eqref{eq:entrance-ordinary-phase-response}
and~\eqref{eq:entrance-ordinary-wall-response} apply to it.

\begin{proposition}[The active arrival column]
\label{prop:entrance-active-column}
On the domain of Proposition~\ref{prop:entrance-three-columns},
after enlarging its universal constant if necessary,
\begin{equation}
 |\partial_vQ|\leq C(u+vw^2).
 \label{eq:entrance-active-phase-column}
\end{equation}
With $H_0=uv+(vw)^2$, the logarithmic wall derivatives satisfy
\begin{equation}
 |v\partial_v\ell_U+\alpha_0|
       +k^2|v\partial_v\ell_W+\beta_0|
       \leq C(ku+w+LH_0).
 \label{eq:entrance-active-wall-response}
\end{equation}
In particular,
\begin{equation}
 |\partial_v\ell_U|\leq Ck/v,\qquad
 |\partial_v\ell_W|\leq C/v.
 \label{eq:entrance-active-wall-columns}
\end{equation}
\end{proposition}

\begin{proof}
The initial drift bound in~\eqref{eq:entrance-drift-radial-bounds}
gives $|G_0|\leq C_1H_0$. Also
$|\alpha_0|\leq12k$ and $|\beta_0|\leq9$ by
\eqref{eq:entrance-log-rates} and the incoming bounds.
For the direction $d_*$, the quantities entering the mixed
response obey
\begin{align}
 |(d_*)_u|+|(d_*)_w|&\leq12ku+9w,\notag\\
 J_*:=v|(d_*)_u|+(uv+v^2w)|(d_*)_w|
     &\leq12kuv+9uvw+9v^2w^2\leq21kH_0.
 \label{eq:entrance-transport-weights}
\end{align}
Since $H_0\leq2$ and $k\geq1$,
\eqref{eq:entrance-ordinary-phase-response} yields
\[
 |DQ[d_*]-G_0|
       \leq42KH_0+4F|G_0|,
 \qquad |DQ[d_*]|\leq C_2H_0.
\]
Apply~\eqref{eq:entrance-active-transport} with $p=r$ and divide
by $v>0$. The identity $H_0/v=u+vw^2$ proves
\eqref{eq:entrance-active-phase-column}.

For the walls, $(d_*)_u/u=\alpha_0$ and
$(d_*)_w/w=\beta_0$. Equations
\eqref{eq:entrance-ordinary-wall-response} and
\eqref{eq:entrance-transport-weights} give
\[
 |D\ell_U[d_*]-\alpha_0|
       +k^2|D\ell_W[d_*]-\beta_0|
 \leq T\bigl[12ku+9w+2L(C_1+21K)H_0\bigr].
\]
The transport identity for each logarithmic observable proves
\eqref{eq:entrance-active-wall-response}, including its signs.
Finally $Lv\leq1$ implies
$LH_0\leq u+w\leq2$, and hence
$ku+w+LH_0\leq4k$.
The first term of~\eqref{eq:entrance-active-wall-response} is
therefore at most $4Ck$; the second, after division by $k^2$,
is at most $4C$.
Adding $|\alpha_0|\leq12k$ or $|\beta_0|\leq9$ and dividing
by $v$ proves~\eqref{eq:entrance-active-wall-columns}.
\end{proof}

The two propositions estimate every input direction of the stable
radial phase and both wall amplitudes at the same arrival.
The physical core radius is obtained from this phase by subtracting
the endpoint radius correction of
Lemma~\ref{lem:entrance-radius-correction}.