\subsection{Differential bounds for the stable phase}
\label{sec:stable-phase-derivatives}

The finite outgoing segment has now been matched to the stable phase.
To follow that phase through the incoming segment, we need its response
to both shear and wall variations. These responses have different
orders in the wall amplitudes. They also become smaller as the Kasner
parameter increases. The following estimates retain both effects.
Here $D_s$ denotes differentiation in the two wall coordinates $(u,w)$,
with their supremum norm.

\begin{proposition}[Refined phase derivatives]
\label{prop:stable-refined-derivatives}
There is a universal constant $C>0$ such that, for
$m\geq2$, $|u|,|w|<1/200$, $\ell=m+1$, and $b=|u|+|w|$,
\begin{align}
 |P_m-1|&\leq C\frac{b^2}{\ell^2},&
 \|D_sP\|&\leq C\frac b\ell,
 \label{eq:stable-refined-first}\\
 |P_{mm}|&\leq C\frac{b^2}{\ell^3},&
 \|D_sP_m\|&\leq C\frac b{\ell^2},&
 \|D_s^2P\|&\leq\frac C\ell.
 \label{eq:stable-refined-second}
\end{align}
All derivatives are evaluated at $(m,u,w)$.
\end{proposition}

We prove the proposition by differentiating the residual integral for
$P-P_0$. The strong wall in that residual decays on the scale $K^{-1}$
of the rescaled time~\eqref{eq:stable-fixed-rescaling}. Its first two
variations retain this decay, which supplies an additional factor
$K^{-1}$ after integration. Throughout this subsection, $C$ denotes a
positive universal constant whose value may increase between estimates.

\subsubsection{Variations retaining the wall amplitudes}

Fix $c\geq2$, put $K=c+1$, and take $y=(m,u,w)\in\Omega_c$.
As in the preceding subsection, $K$ is fixed when the initial state varies.
Write $Y(s)=Y(s;y)$, $J_h(s)=D_yY(s;y)h$, and
$H_{h,z}(s)=D_y^2Y(s;y)[h,z]$. For $b=|u|+|w|$ define
\begin{align*}
 \omega_b(h)&=b\|h\|+\|h_s\|,\\
 \nu_b(h,z)&=b\|h\|\|z\|
       +\|h_s\|\|z\|+\|h\|\|z_s\|.
\end{align*}
The first expression records that a pure shear variation forces the
walls only through their existing amplitudes. The elementary relation
\begin{equation}
 b\nu_b(h,z)\leq\omega_b(h)\omega_b(z)
 \label{eq:stable-input-product}
\end{equation}
will preserve that distinction in the second derivative of the residual.

\begin{lemma}[Weighted boundary variations]
\label{lem:stable-weighted-variations}
For every $s\geq0$ and all directions $h,z$,
\begin{align}
 \|J_h(s)\|&\leq C\|h\|,&
 \|(J_h(s))_s\|&\leq C\omega_b(h)e^{-s/4},
 \label{eq:stable-weighted-first}\\
 |(J_h(s))_u|&\leq C\omega_b(h)e^{-3Ks/16},
 \label{eq:stable-fast-first}\\
 \|H_{h,z}(s)\|&\leq C\nu_b(h,z),&
 |(H_{h,z}(s))_u|&\leq\frac C K\nu_b(h,z)e^{-3Ks/32}.
 \label{eq:stable-weighted-second}
\end{align}
\end{lemma}

\begin{proof}
Set $E=e^{512b}\leq e^6$. The two bounds in
\eqref{eq:stable-weighted-first} follow directly from
\eqref{eq:stable-first-growth} and~\eqref{eq:stable-first-decay}.
To retain the faster decay of the strong component, use its scalar
variational equation. On~\eqref{eq:stable-band}, the Jacobian
formula~\eqref{eq:stable-jacobian-factor} gives
\begin{equation}
 \mathcal A_{uu}\leq-\frac{3K}{16},\qquad
 |\mathcal A_{uw}|\leq18|u|(|u|+|w|),\qquad
 |\mathcal A_{um}v_m+\mathcal A_{uw}v_w|
       \leq96|u|\|v\|.
 \label{eq:stable-strong-row}
\end{equation}
The second inequality uses the factor $u$ in both
$(Df)_{uw}=-uw$ and $f_u$; the last also uses
\eqref{eq:stable-center-coupling}.
Thus the forcing of $(J_h)_u$ has magnitude at most
$96|u|E\|h\|e^{-3Ks/8}$. Scalar comparison yields
\begin{equation}
 |(J_h(s))_u|
 \leq\left(|h_u|+\frac{512}{K}|u|E\|h\|\right)e^{-3Ks/16}.
 \label{eq:stable-strong-first-explicit}
\end{equation}
Since $|u|\leq b$ and $K\geq3$, this proves
\eqref{eq:stable-fast-first}.

For the full second variation, first suppose that $h_s=z_s=0$.
The stable first variations in~\eqref{eq:stable-first-decay} then
carry the factor $b$. Substitution in
\eqref{eq:stable-hessian-bound}, using
$|Y_u|+|Y_w|\leq b e^{-s/2}$, bounds the second-order forcing by
\[
 \|K D^2g(Y)[J_h,J_z]\|
       \leq CbE^2\|h\|\|z\|e^{-s/4}.
\]
The homogeneous growth on every subinterval is at most $E$, and
$H_{h,z}(0)=0$. Integration therefore gives
$\|H_{h,z}\|\leq CbE^3\|h\|\|z\|$ for these two center inputs.
For general inputs, decompose $h=h_c+h_s^0$ and $z=z_c+z_s^0$,
where the superscript denotes inclusion of the stable pair in
$\R^3$. Bilinearity gives
\[
 H_{h,z}=H_{h_c,z_c}+H_{h_s^0,z}+H_{h_c,z_s^0}.
\]
Apply the improved bound to the first term and
\eqref{eq:stable-second-growth} to the other two. This proves
the first bound in~\eqref{eq:stable-weighted-second}.

The strong component of the Hessian has more structure than its
full norm. Put $\beta=|u|+|w|$. On the same band,
\begin{equation}
\begin{split}
 |(K D^2g(y)[h,z])_u|\leq C\bigl(&|u|\|h\|\|z\|
       +|h_m||z_u|+|h_u||z_m|\\
       &+\beta(|h_u|\|z\|+\|h\||z_u|)\bigr).
\end{split}
\label{eq:stable-strong-hessian-structure}
\end{equation}
For completeness, its polynomial part is
\[
\begin{split}
 (D^2f[h,z])_u={}&-u h_mz_m-m(h_mz_u+h_uz_m)+3u h_uz_u\\
                 &-w(h_uz_w+h_wz_u)-u h_wz_w.
\end{split}
\]
In the quotient terms of~\eqref{eq:stable-hessian-factor}, use
\[
 |dD[h]|\leq2K|h_m|+4\beta\|h\|,\qquad
 |(\mathcal A z)_u|\leq96|u|\|z\|+18K|z_u|,
 \qquad |g_u|\leq6|u|.
\]
Together with $K/D\leq1$, $K^2/D\leq3$, and the bound for $D^2D$,
these inequalities give~\eqref{eq:stable-strong-hessian-structure}.
In particular, each term contains either the strong wall or a
strong component of a direction.

Insert $Y,J_h,J_z$ in this estimate. Equations
\eqref{eq:stable-rescaled-decay} and
\eqref{eq:stable-strong-first-explicit} give
\[
 |(K D^2g(Y)[J_h,J_z])_u|
       \leq C\nu_b(h,z)e^{-3Ks/16}.
\]
For example, $\|h\|\omega_b(z)$ and
$\omega_b(h)\|z\|$ are each bounded by $\nu_b(h,z)$.
The off-diagonal linear forcing of $(H_{h,z})_u$ is bounded by
$96|Y_u|\|H_{h,z}\|$, so it obeys the same estimate.
Use the diagonal bound in~\eqref{eq:stable-strong-row} with the
weaker decay rate $3K/32$. The total forcing decays at twice that
rate and the initial value is zero. Its convolution with the
scalar decay is at most
\[
 C\nu_b(h,z)e^{-3Ks/32}
       \int_0^s e^{-3Kr/32}\dd r
 \leq\frac C K\nu_b(h,z)e^{-3Ks/32}.
\]
This proves the remaining bound.
\end{proof}

\subsubsection{Differentiating the residual integral}

Let $r_4,r_3,r_2,r_1$ be the four coefficients in
\eqref{eq:stable-quartic-identity}. In the fixed time scale used here,
the residual is
\begin{equation}
 \begin{aligned}
 S_K(y)&=\frac{6K}{D(y)}DP_0(y)f(y)=uG_K(y),\\
 G_K&=\frac{6K}{D}(r_4u^3+r_3u^2w+r_2uw^2+r_1w^3).
 \end{aligned}
 \label{eq:stable-scaled-residual}
\end{equation}
The coefficient bounds must hold after differentiation as well.
A reciprocal coordinate makes their uniformity in $K$ transparent.

\begin{lemma}[Residual coefficients]
\label{lem:stable-residual-coefficients}
On~\eqref{eq:stable-band}, every coefficient $(6K/D)r_i$ is $C^2$,
and its value, first derivative norm, and second derivative norm
are bounded by $C/K^2$.
\end{lemma}

\begin{proof}
Use the reciprocal coordinate $x=(m+1)^{-1}$ and put
$\varrho=H(1,xw)$. The coefficients have the form
$r_i(m,w)=x\phi_i(x,w)$, where
\begin{align*}
 \phi_4&=-\frac{2(1-x)x^2}{(1-2x)^2},&
 \phi_3&=\frac{2(3-4x)x^2}{(1-2x)^2},\\
 \phi_2&=-\frac{x((1-x)^2+x^2)}{(1-2x)^2}
          -\frac{1-x}{1-2x}
          +\frac{\varrho+x}{\varrho(\varrho+1)},&
 \phi_1&=2x+1-\frac1{\varrho+1}.
\end{align*}
These identities follow by substituting
$m=x^{-1}-1$ and $H(m+1,w)=(m+1)H(1,xw)$ in the coefficients.
All four profiles are smooth on a neighborhood of
$[0,2/5]\times[-1,1]$: their denominators stay positive there.
Their first two derivatives are therefore uniformly bounded.
Since $m\geq3/2$ on the band and
\[
 \partial_m x=-x^2,\qquad \partial_m^2x=2x^3,
\]
the product and chain rules give
\[
 |r_i|+\|Dr_i\|+\|D^2r_i\|
       \leq\frac C{m+1}\leq\frac{2C}K.
\]
The derivatives here may also be taken in $(m,u,w)$, as $r_i$
does not depend on $u$.

For $q_K=6K/D$, direct differentiation gives
\[
 Dq_K[h]=-q_K\frac{dD[h]}D,\qquad
 D^2q_K[h,z]=q_K\left(
       2\frac{dD[h]dD[z]}{D^2}-\frac{D^2D[h,z]}D\right).
\]
The band estimates give $q_K\leq18/K$,
$|dD[h]|/D\leq6\|h\|$, and
$|D^2D[h,z]|/D\leq10\|h\|\|z\|$.
Hence $q_K$ and its first two derivatives are bounded by $C/K$.
The product rule proves the assertion.
\end{proof}

Define the residual along a boundary orbit by
\[
 q(s,y)=S_K(Y(s;y))=\frac d{ds}P_0(Y(s;y)).
\]
Here $q$ is a scalar function; the time scale $K$ remains fixed.
Its decay and the decay of its derivatives can now be estimated
in the same integrable expression.

\begin{samepage}
\begin{lemma}[Differentiable residual tails]
\label{lem:stable-residual-tails}
For every $a\geq0$, the function
\[
 T_a(y)=P(y)-P_0(Y(a;y))
\]
is $C^2$ on $\Omega_c$. Its derivatives are obtained by
differentiating
\begin{equation}
 T_a(y)=\int_a^\infty q(s,y)\dd s
 \label{eq:stable-tail-integral}
\end{equation}
under the integral sign. Moreover,
\begin{align}
 |T_a(y)|&\leq\frac C{K^3}|u|b^3e^{-Ka/16},\notag\\
 |DT_a(y)h|&\leq\frac C{K^3}b^3\omega_b(h)e^{-Ka/16},
 \label{eq:stable-differentiated-tail}\\
 |D^2T_a(y)[h,z]|&\leq\frac C{K^3}
       b^2\omega_b(h)\omega_b(z)e^{-Ka/16}.\notag
\end{align}
\end{lemma}
\end{samepage}

\begin{proof}
Write $\beta=|Y_u|+|Y_w|\leq b$. Product differentiation of
the cubic expression $G_K$ in~\eqref{eq:stable-scaled-residual}
gives, for arbitrary directions $v,z$,
\begin{align*}
 |G_K(Y)|&\leq\frac C{K^2}\beta^3,\\
 |DG_K(Y)v|&\leq\frac C{K^2}
       (\beta^3\|v\|+3\beta^2\|v_s\|),\\
 |D^2G_K(Y)[v,z]|&\leq\frac C{K^2}\bigl(
       \beta^3\|v\|\|z\|
       +3\beta^2(\|v\|\|z_s\|+\|v_s\|\|z\|)
       +6\beta\|v_s\|\|z_s\|\bigr).
\end{align*}
The three contributions correspond to differentiating a coefficient
twice, once, or not at all. Lemma~\ref{lem:stable-residual-coefficients}
bounds each coefficient and its derivatives by the same $C/K^2$.
With the weighted variation bounds, these imply
\begin{align*}
 |DG_K(Y)J_h|&\leq\frac C{K^2}b^2\omega_b(h),\\
 |D^2G_K(Y)[J_h,J_z]|&\leq\frac C{K^2}
       b\omega_b(h)\omega_b(z),\\
 |DG_K(Y)H_{h,z}|&\leq\frac C{K^2}b^2\nu_b(h,z).
\end{align*}
In the last line we used $b\leq1$ and
$\|(H_{h,z})_s\|\leq\|H_{h,z}\|$.

Put $Q(s)=e^{-Ks/16}$. The stronger decay rates already proved
give $|Y_u|\leq|u|Q$, $|(J_h)_u|\leq C\omega_b(h)Q$, and
$|(H_{h,z})_u|\leq C\nu_b(h,z)Q$.
The exact product and chain rules for $q=Y_uG_K(Y)$ are
\begin{align*}
 D_yq[h]={}&(J_h)_uG_K(Y)+Y_uDG_K(Y)J_h,\\
 D_y^2q[h,z]={}&(H_{h,z})_uG_K(Y)
        +(J_h)_uDG_K(Y)J_z+(J_z)_uDG_K(Y)J_h\\
        &+Y_u\bigl(D^2G_K(Y)[J_h,J_z]+DG_K(Y)H_{h,z}\bigr).
\end{align*}
Thus
\begin{align}
 |q(s,y)|&\leq\frac C{K^2}|u|b^3Q(s),\notag\\
 |D_yq(s,y)h|&\leq\frac C{K^2}b^3\omega_b(h)Q(s),
 \label{eq:stable-residual-jet-decay}\\
 |D_y^2q(s,y)[h,z]|&\leq\frac C{K^2}
       b^2\omega_b(h)\omega_b(z)Q(s).\notag
\end{align}
For the terms containing $H_{h,z}$, the required conversion is
$b^3\nu_b(h,z)\leq b^2\omega_b(h)\omega_b(z)$,
which is~\eqref{eq:stable-input-product}. This step also covers
$b=0$ without division by an amplitude.

Integrating the finite-time identity for $P_0(Y)$ and using
$P_0(Y(s;y))\to P(y)$ proves~\eqref{eq:stable-tail-integral}.
For differentiation, note that $q$ and its first two initial-state
derivatives are continuous in $(s,y)$ on every finite time interval.
Since $b\leq1$ and $\omega_b(h)\leq2\|h\|$, all three operator
bounds in~\eqref{eq:stable-residual-jet-decay} have the common
integrable majorant $C K^{-2}e^{-Ks/16}$, uniformly for
$y\in\Omega_c$. Dominated differentiation, applied twice on this
open box, therefore identifies the first two derivatives of the
improper integral.
Finally,
\[
 \int_a^\infty e^{-Ks/16}\dd s=\frac{16}K e^{-Ka/16}.
\]
Integrating the three bounds gives
\eqref{eq:stable-differentiated-tail}.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:stable-refined-derivatives}]
Let $R=P-P_0$. Take $a=0$ in
Lemma~\ref{lem:stable-residual-tails}. At a given point with $m\geq2$,
choose $c=m$, so $K=\ell=m+1$. The differentiation has already
been performed with $K$ fixed on $\Omega_c$; we now evaluate those
estimates at the chosen initial point. We obtain
\begin{align}
 |DR(y)h|&\leq\frac C{\ell^3}b^3
       (b\|h\|+\|h_s\|),\notag\\
 |D^2R(y)[h,z]|&\leq\frac C{\ell^3}b^2
       (b\|h\|+\|h_s\|)(b\|z\|+\|z_s\|).
 \label{eq:stable-remainder-derivatives}
\end{align}

It remains to differentiate the explicit approximation.
Put $\sigma=H(\ell,w)$,
$\alpha=m/(m^2-1)$, and $\gamma=-2/\ell$. Then
\[
 DP_0=\left(
 \frac\ell\sigma+\alpha'u^2+\gamma'uw,\quad
 2\alpha u+\gamma w,\quad \frac w\sigma+\gamma u\right),
\]
and the distinct Hessian entries are
\begin{align*}
 (P_0)_{mm}&=\frac{w^2}{\sigma^3}+\alpha''u^2+\gamma''uw,&
 (P_0)_{mu}&=2\alpha'u+\gamma'w,\\
 (P_0)_{mw}&=-\frac{\ell w}{\sigma^3}+\gamma'u,&
 (P_0)_{uu}&=2\alpha,\\
 (P_0)_{uw}&=\gamma,&
 (P_0)_{ww}&=\frac{\ell^2}{\sigma^3}.
\end{align*}
The partial-fraction identity
\[
 \alpha=\frac12\left(\frac1{m-1}+\frac1{m+1}\right)
\]
and $m+1\leq3(m-1)$ give
\[
 |\alpha|\leq\frac2\ell,\quad
 |\alpha'|\leq\frac5{\ell^2},\quad
 |\alpha''|\leq\frac{28}{\ell^3},\qquad
 |\gamma^{(j)}|=\frac{2j!}{\ell^{j+1}}\quad(0\leq j\leq2).
\]
Using $\sigma\geq\ell$ and
$1-\ell/\sigma=w^2/[\sigma(\sigma+\ell)]$ now yields
\begin{align*}
 |(P_0)_m-1|&\leq8b^2/\ell^2,&
 \|D_sP_0\|&\leq9b/\ell,\\
 |(P_0)_{mm}|&\leq33b^2/\ell^3,&
 \|D_s(P_0)_m\|&\leq15b/\ell^2,&
 \|D_s^2P_0\|&\leq9/\ell.
\end{align*}
For a center direction, $\omega_b(h)=b\|h\|$; for a stable
direction, $\omega_b(h)\leq2\|h\|$.
Thus~\eqref{eq:stable-remainder-derivatives} contributes at most
$Cb^4/\ell^3$ and $Cb^3/\ell^3$ to the two first-derivative bounds,
and $Cb^4/\ell^3$, $Cb^3/\ell^3$, and $Cb^2/\ell^3$ to the
three second-derivative bounds. Since $b\leq1$ and $\ell\geq1$,
these are absorbed by the stated bounds for $P_0$.
\end{proof}

The phase is therefore close to the shear coordinate in a way that
also controls its differential. In the incoming passage, these
amplitude factors accompany the $UV$ and $V^2$ terms in
\eqref{eq:stable-active-phase-rate}. They will allow the phase
variation to be estimated through the long interval before the
active wall reaches the next core.