\subsection{Phase transport into the next core}
\label{sec:entrance-phase-transport}

The stable phase absorbs the shear motion caused by the two decaying
walls. During the incoming passage, its remaining motion is coupled
to the active wall. We now estimate this motion up to the actual
arrival at $V=1$. The useful quantities are the products $UV$ and
$VW$: although $V$ grows, both products decay, and their integrals
control the accumulated phase error.

Retain the incoming map $\mathcal P$ and the notation of
Proposition~\ref{prop:incoming-passage}. Thus $y=(m,u,v,w)$ satisfies
\eqref{eq:incoming-domain}, $k=H(m-1,v)$, and $L=\log(1/v)$.
Write
\[
 r(y)=\widehat\rho(y)=H(P(m,u,w)-1,v),\qquad
 \rho_+(y)=H(\mathcal P_M(y)-1,1).
\]
The first quantity is the stable radial phase at entry, and the second
is the radius at the next core entrance.

\begin{proposition}[Incoming phase error]
\label{prop:incoming-phase-value}
There is a universal constant $C>0$ such that
\begin{equation}
 |\rho_+(y)-r(y)|
 \leq\frac Ck\left[uv+(vw)^2(1+e^{-2L/k})\right]
 \leq\frac{2C}k\left[uv+v^2(u+w)^2\right].
 \label{eq:incoming-physical-phase-value}
\end{equation}
The estimate includes $u=0$ and $w=0$ and holds for the same locally
$C^2$ first-arrival map $\mathcal P$.
\end{proposition}

We first describe the radial coordinate in which this phase can be
varied. The coefficient estimates below also prepare the differential
analysis of the arrival map. The constant $C$ will be universal and
may increase from one estimate to the next.

\subsubsection{A radial coordinate adapted to the boundary}

For the local estimates, fix $k\geq7/4$ and let
$y=(M,U,V,W)$ belong to~\eqref{eq:incoming-region}, with $V>0$.
Set
\begin{equation}
 \chi=P(M,U,W)-1,\qquad r=H(\chi,V),\qquad
 b=U+W,\qquad q=M-1-U(V+W-U).
 \label{eq:entrance-phase-notation}
\end{equation}
Subscripts on $P$ always denote its ordinary derivatives in
$(M,U,W)$. The reference parameter $k$ labels the region and is held
fixed in the estimates; the new radial variable is $r$.

\begin{lemma}[Radial coordinates]
\label{lem:entrance-radial-chart}
At every such point the map
\[
 (M,U,V,W)\longmapsto(r,U,V,W)
\]
has a $C^2$ local inverse, which preserves the last three coordinates.
In the inverse coordinates, the first coordinate has derivatives
\begin{equation}
 M_r=\frac r{\chi P_M},\qquad
 M_U=-\frac{P_U}{P_M},\qquad
 M_V=-\frac V{\chi P_M},\qquad
 M_W=-\frac{P_W}{P_M}.
 \label{eq:entrance-inverse-columns}
\end{equation}
They satisfy
\begin{equation}
 \frac23\leq M_r\leq8,\qquad
 |M_U|+|M_W|\leq\frac{Cb}k,\qquad
 |M_V|\leq\frac{4V}k.
 \label{eq:entrance-inverse-bounds}
\end{equation}
\end{lemma}

\begin{proof}
The phase band~\eqref{eq:incoming-phase-band} gives $M>2$.
Since $b\leq1/5000$, the explicit estimates
\eqref{eq:stable-phase-tail} and~\eqref{eq:stable-phase-differential}
give
\[
 |P-M|\leq\frac{8b^2}{M+1}\leq\frac1{100},\qquad
 |P_M-1|\leq66000b^2e^{512b}\leq\frac12.
\]
For the last inequality one may use $e^{512b}\leq2$.
Combining these bounds with the phase band and $0<V\leq1$ yields
\begin{equation}
 \frac k2\leq\chi\leq\frac{3k}2,\qquad
 \frac k2\leq r\leq2k,\qquad
 \frac12\leq P_M\leq\frac32.
 \label{eq:entrance-stable-radius-band}
\end{equation}
Indeed, $\chi\leq r$ and $r^2=\chi^2+V^2$ give the radius bounds.
The ordinary $M$ derivative is
\[
 \partial_M r=\frac{\chi P_M}r\in[1/8,3/2].
\]
The coordinate change is $C^2$ by
Proposition~\ref{prop:stable-phase-regularity}. Its triangular
Jacobian is invertible, so the inverse function theorem gives the
claimed local inverse. Solving the differential of
$r^2=(P-1)^2+V^2$ proves~\eqref{eq:entrance-inverse-columns}.
The radial and active-wall bounds follow from
\eqref{eq:entrance-stable-radius-band}. For the two other columns,
use Proposition~\ref{prop:stable-refined-derivatives} and
$M+1\geq k/2$. Two local inverses agree on a neighborhood of
their common base point, so these derivatives do not depend on
the chosen inverse.
\end{proof}

In what follows, $\partial_r,\partial_U,\partial_V,\partial_W$
denote derivatives in this radial chart. We use $d$ for a differential
in the ordinary coordinates and $D_z$ for a differential in
$z=(r,U,V,W)$. This distinction matters for the cancellations below.
The boundary estimates, transferred using $M+1\geq k/2$, read
\begin{align}
 |P-M|&\leq Cb^2/k,& |P_M-1|&\leq Cb^2/k^2,&
 |P_U|+|P_W|&\leq Cb/k,\notag\\
 |P_{MM}|&\leq Cb^2/k^3,&
 |P_{MU}|+|P_{MW}|&\leq Cb/k^2.
 \label{eq:entrance-projection-scales}
\end{align}
The full norms of $dP$ and $d^2P$ are uniformly bounded.

\subsubsection{The exact radial drift}

The cancellation of the boundary motion is expressed by two scalar
coefficients. Define
\begin{align}
 F&=\chi-(M-1)P_M,\notag\\
 N&=M-1-U(W-U)-\chi P_M,\notag\\
 E&=\chi\left[F+W(P_U-P_W)-\frac V2P_U\right],
 \label{eq:entrance-radial-numerators}\\
 \mathcal A&=-\frac rq+\frac E{rq},&
 \mathcal B&=\frac N{rq}.
 \label{eq:entrance-radial-coefficients}
\end{align}
The denominators are positive by
\eqref{eq:incoming-clock-speed} and
\eqref{eq:entrance-stable-radius-band}.
Along an actual incoming orbit, with $s=\log(V/v)$,
\begin{equation}
 \frac{dr}{ds}=G,\qquad
 G=UV\mathcal A+V^2\mathcal B.
 \label{eq:entrance-stable-radial-equation}
\end{equation}
To verify this identity, insert~\eqref{eq:stable-active-phase-rate}
in the following expression, where a dot denotes projective time:
\[
 \frac{dr}{ds}=\frac{\chi\dot P+V^2q}{rq}.
\]
The coefficient of $V^2$ is $N/(rq)$ after removing the term
$-UV^3$. The numerator of the remaining $UV$ coefficient is
\[
 \chi\bigl((1-M)P_M+W(P_U-P_W)\bigr)-V^2-\frac{\chi V}2P_U.
\]
Using $r^2=\chi^2+V^2$, this equals $-r^2+E$, which gives
\eqref{eq:entrance-radial-coefficients}.

\begin{proposition}[Radial coefficient bounds]
\label{prop:entrance-radial-coefficients}
The coefficients are $C^1$ in every local radial chart, and
\begin{align}
 |\mathcal A|+\|D_z\mathcal A\|&\leq C,&
 |\partial_r\mathcal A|&\leq C/k^2,
 \label{eq:entrance-strong-coefficient-bounds}\\
 |\mathcal B|&\leq Cb^2/k^2,&
 |\partial_r\mathcal B|&\leq Cb^2/k^3,&
 |\partial_U\mathcal B|+|\partial_W\mathcal B|&\leq Cb/k^2.
 \label{eq:entrance-weak-coefficient-bounds}
\end{align}
These bounds hold on the full region under consideration, including
zero stable walls.
\end{proposition}

\begin{proof}
All expressions involve at most first derivatives of the $C^2$
function $P$ and positive denominators, so they are $C^1$ on an
ordinary open neighborhood of each point.

Start with the weak numerator. The identity
\[
 N=-(P-M)-U(W-U)-\chi(P_M-1)
\]
and~\eqref{eq:entrance-projection-scales} imply $|N|\leq Cb^2$.
Its ordinary partial derivatives are
\begin{align*}
 N_M&=1-P_M^2-\chi P_{MM},\\
 N_U&=2U-W-P_MP_U-\chi P_{MU},\\
 N_W&=-U-P_MP_W-\chi P_{MW},\qquad N_V=0.
\end{align*}
Since $1-P_M^2=-(P_M-1)(P_M+1)$, they satisfy
\begin{equation*}
 |N_M|\leq Cb^2/k^2,\qquad |N_U|+|N_W|\leq Cb.
\end{equation*}
The inverse columns~\eqref{eq:entrance-inverse-columns} therefore give
\[
 |\partial_r N|\leq Cb^2/k^2,\qquad
 |\partial_U N|+|\partial_W N|\leq Cb.
\]
Here and below an ordinary numerator is regarded as a function of
the radial chart when a $\partial$ is used.
Likewise, $\partial_r q=M_r$, and the other clock columns are
\begin{equation*}
 \partial_Uq=M_U-(V+W-2U),\qquad
 \partial_Vq=M_V-U,\qquad
 \partial_Wq=M_W-U.
\end{equation*}
In particular, $\|D_zq\|\leq C$.
For a direction $h$ in the radial chart, quotient differentiation gives
\begin{equation*}
 D_z\mathcal B[h]=\frac{D_zN[h]}{rq}
       -\mathcal B\left(\frac{h_r}r+\frac{D_zq[h]}q\right).
\end{equation*}
Use $r,q\geq k/2$ in this formula. The value bound is $Cb^2/k^2$.
For the radial column, the first term is at most $Cb^2/k^4$ and
the second at most $Cb^2/k^3$. For a stable-wall column they are
at most $Cb/k^2$ and $Cb^2/k^3$. Since $k\geq1$ and $b\leq1$,
these prove~\eqref{eq:entrance-weak-coefficient-bounds}.

For the strong coefficient, the useful ordinary-coordinate estimates are
\begin{equation}
 |F|\leq Cb^2/k,\qquad
 |E|\leq C(Vb+b^2),\qquad
 |E_M|\leq C(Vb+b^2)/k.
 \label{eq:entrance-strong-remainder-bounds}
\end{equation}
The first follows from
$F=(P-M)-(M-1)(P_M-1)$. For the third, the derivative
$F_M=-(M-1)P_{MM}$ retains the cancellation of the two first-order
terms. In full,
\[
\begin{split}
 E_M={}&P_M\left[F+W(P_U-P_W)-\frac V2P_U\right]\\
 &+\chi\left[-(M-1)P_{MM}
       +W(P_{MU}-P_{MW})-\frac V2P_{MU}\right].
\end{split}
\]
The bracket in the definition of $E$ is bounded by
$C(Vb+b^2)/k$, and the second bracket here by
$C(Vb+b^2)/k^2$. With $\chi\leq2k$ and $P_M\leq3/2$, this
proves the other two estimates in
\eqref{eq:entrance-strong-remainder-bounds}.
Consequently $|E|\leq C$ and
$|\partial_rE|=|E_M M_r|\leq C/k$.

The leading quotient $-r/q$ has the exact radial identity
\begin{equation*}
 r\partial_rq-q
   =\frac F{P_M}+\frac{V^2}{\chi P_M}+U(V+W-U).
\end{equation*}
Indeed, substitute $\partial_rq=r/(\chi P_M)$ and
$r^2=\chi^2+V^2$. The right side is bounded by
$C(V^2/k+UV+b^2)\leq C$.
Differentiating~\eqref{eq:entrance-radial-coefficients} gives
\begin{equation*}
\begin{split}
 \partial_r\mathcal A={}&\frac{r\partial_rq-q}{q^2}
       +\frac{\partial_rE}{rq}
       -\frac E{rq}\left(\frac1r+\frac{\partial_rq}q\right).
\end{split}
\end{equation*}
The three terms are bounded by $C/k^2$, $C/k^3$, and $C/k^3$,
respectively. The value bound $|\mathcal A|\leq C$ follows from
$r/q\leq4$ and $|E|/(rq)\leq C/k^2$.

To control all remaining columns, a coarse bound suffices.
The uniform bounds on $dP,d^2P$ imply
$\|dF\|\leq Ck$. The bracket defining $E$ is bounded by $C$,
and its ordinary differential by $Ck$. Hence
$\|dE\|\leq Ck^2$. The inverse chart has bounded differential
by~\eqref{eq:entrance-inverse-bounds}, so
$\|D_zE\|\leq Ck^2$ as well. For an arbitrary radial-chart
direction $h$,
\[
 D_z\mathcal A[h]
 =\frac{rD_zq[h]-qh_r}{q^2}
       +\frac{D_zE[h]}{rq}
       -\frac E{rq}\left(\frac{h_r}r+\frac{D_zq[h]}q\right).
\]
The bounds $r,q\asymp k$, $\|D_zq\|\leq C$, and $|E|\leq C$
give $|D_z\mathcal A[h]|\leq C\|h\|$. This completes the proof.
\end{proof}

The distinction between the two coefficients now gives useful bounds
for the drift and its derivatives.

\begin{corollary}[Drift estimates]
\label{cor:entrance-drift-bounds}
In the radial chart,
\begin{align}
 |G|&\leq C\left(UV+\frac{(VW)^2}{k^2}\right),&
 |\partial_rG|&\leq C\left(\frac{UV}{k^2}
                         +\frac{(VW)^2}{k^3}\right),
 \label{eq:entrance-drift-radial-bounds}\\
 |U\partial_UG|&\leq CUV,&
 |W\partial_WG|&\leq C\left(UVW+\frac{(VW)^2}{k^2}\right).
 \label{eq:entrance-drift-wall-bounds}
\end{align}
For a positive wall the weighted derivative is its logarithmic
derivative; the displayed expressions also make sense when a wall
vanishes.
\end{corollary}

\begin{proof}
First use $G=UV\mathcal A+V^2\mathcal B$ and
Proposition~\ref{prop:entrance-radial-coefficients}, which give
\[
 |G|\leq C(UV+V^2b^2/k^2),\qquad
 |\partial_rG|\leq C(UV/k^2+V^2b^2/k^3).
\]
Since $U,W,V\leq1$,
\[
 b^2\leq3U+W^2,\qquad V^2U\leq UV.
\]
Substitution proves~\eqref{eq:entrance-drift-radial-bounds}.
For the wall derivatives, use the exact formulas
\begin{align*}
 \partial_UG&=V\mathcal A+UV\partial_U\mathcal A
                                  +V^2\partial_U\mathcal B,\\
 \partial_WG&=UV\partial_W\mathcal A+V^2\partial_W\mathcal B.
\end{align*}
Multiply by $U$ or $W$, respectively. The first line is bounded
by $C(UV+U^2V+UV^2b/k^2)\leq CUV$.
In the second, split $Wb=UW+W^2$ and use
$UV^2W/k^2\leq UVW$. This proves
\eqref{eq:entrance-drift-wall-bounds} without dividing by either wall.
\end{proof}

We return to the actual orbit from $y=(m,u,v,w)$.
The products in~\eqref{eq:entrance-drift-radial-bounds}
have simple dissipation inequalities in $s=\log(V/v)$:
\begin{equation}
 (UV)'\leq-\frac k3UV,\qquad
 ((VW)^2)'\leq-\frac2k(VW)^2.
 \label{eq:entrance-product-dissipation}
\end{equation}
These follow by adding the active logarithmic rate $1$ to each
of the two rates in~\eqref{eq:incoming-clock-rates}.
Their zeroth and first moments obey, for every $0\leq S\leq L$,
\begin{align}
 \int_0^S UV\dd s&\leq\frac{3uv}k,&
 \int_0^S sUV\dd s&\leq\frac{9uv}{k^2},\notag\\
 \int_0^S(VW)^2\dd s&\leq\frac k2(vw)^2,&
 \int_0^S s(VW)^2\dd s&\leq\frac{k^2}4(vw)^2.
 \label{eq:entrance-product-moments}
\end{align}
For example, the first product is bounded by $uv e^{-ks/3}$;
integrating that exponential and $s$ times it proves the first row.
The second row follows with decay rate $2/k$.
These are also the corresponding integrals along the normalized
flow with $\dd s=Aq\dd t$, since that clock is strictly increasing
throughout the passage.

Integrating~\eqref{eq:entrance-stable-radial-equation} with the
first bound in~\eqref{eq:entrance-drift-radial-bounds} gives
\begin{equation}
 |r(Y(S))-r(y)|\leq\frac Ck\left[uv+(vw)^2\right],
 \qquad 0\leq S\leq L.
 \label{eq:entrance-phase-integrated-drift}
\end{equation}
Here $Y(S)$ is the state at logarithmic clock value $S$.
To convert the endpoint to the core radius, we need the
difference between the two radial coordinates.

\begin{lemma}[Radius correction]
\label{lem:entrance-radius-correction}
On~\eqref{eq:incoming-region}, with $V>0$, put
$J=r-\rho$, where $\rho=H(M-1,V)$. Then $J$ is $C^2$ locally and
\begin{equation}
 |J|\leq Cb^2/k,\qquad
 |\partial_rJ|\leq Cb^2/k^2,\qquad
 |\partial_UJ|+|\partial_WJ|\leq Cb/k.
 \label{eq:entrance-radius-correction-bounds}
\end{equation}
\end{lemma}

\begin{proof}
Set $a=M-1>0$. Rationalization gives
\[
 r-\rho=\frac{(\chi-a)(\chi+a)}{r+\rho},\qquad
 |r-\rho|\leq|\chi-a|=|P-M|.
\]
This proves the value bound, while positivity of $r,\rho$ gives
local $C^2$ regularity. By~\eqref{eq:entrance-inverse-columns},
\begin{equation*}
 \partial_rJ=1-\frac{ar}{\rho\chi P_M},\qquad
 \partial_UJ=\frac{aP_U}{\rho P_M},\qquad
 \partial_WJ=\frac{aP_W}{\rho P_M}.
\end{equation*}
Let $\delta=Cb^2/k$ bound both $|r-\rho|$ and $|a-\chi|$.
Since $a\leq\rho$ and $\chi\geq k/2$,
\[
 |ar-\rho\chi|
 =|a(r-\rho)+\rho(a-\chi)|\leq2\rho\delta,
 \qquad
 \left|\frac{ar}{\rho\chi}-1\right|\leq\frac{4\delta}k.
\]
Finally,
\[
 1-\frac{ar}{\rho\chi P_M}
 =\frac{(P_M-1)-(ar/(\rho\chi)-1)}{P_M}.
\]
Use $P_M\geq1/2$ and~\eqref{eq:entrance-projection-scales} to
obtain the radial derivative bound. The wall bounds follow from
$a/\rho\leq1$, $1/P_M\leq2$, and the same projection estimates.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:incoming-phase-value}]
At the first arrival $S=L$, write $U_+,W_+$ for the two stable walls
and put $\omega=e^{-2L/k}$. The decay estimates
\eqref{eq:incoming-output} give
\[
 U_+^2\leq u^2v^{2+2k/3}\leq uv,\qquad
 W_+^2\leq w^2v^{2+2/k}=(vw)^2\omega.
\]
Thus
\begin{equation*}
 (U_++W_+)^2\leq2\left[uv+(vw)^2\omega\right].
\end{equation*}
The endpoint remains in the same region. Apply
Lemma~\ref{lem:entrance-radius-correction} there and combine it
with~\eqref{eq:entrance-phase-integrated-drift} at $S=L$.
The identity
\[
 \rho_+(y)-r(y)
   =r(\mathcal P(y))-r(y)-J(\mathcal P(y))
\]
proves the first inequality in
\eqref{eq:incoming-physical-phase-value}. Since $0<\omega\leq1$
and $w^2\leq(u+w)^2$, the second follows.
All estimates were obtained on the actual passage of
Proposition~\ref{prop:incoming-passage}; its first-arrival property
and local $C^2$ regularity therefore apply to this same map.
\end{proof}

At fixed incoming shear and stable walls, the phase error tends to zero
with the height of the entering active wall, even though the passage
time diverges. The first moments in~\eqref{eq:entrance-product-moments}
will also control the additional time factors produced by variations
of this passage.