\subsection{Transverse variation of the remaining core}
\label{sec:partial-core-multipliers}

The radius estimate controls the direction in which an initial
curve reaches the outgoing section. We now compare its two
transverse derivatives with this radial speed. The strong wall
decays exponentially during the remaining passage. The weak
wall needs no comparable contraction: its relative derivative
grows more slowly than the radial speed. Together these estimates
place the outgoing curve in a narrow radial cone.

For an initial state $y_0$ satisfying
\eqref{eq:partial-core-initial-state}, keep the notation
$k,u_0,P_0,\alpha,\beta$ of the preceding subsection. Write
\[
 (r_*,u_*,w_*)=\mathcal R_\downarrow(y_0).
\]
The last coordinate is the outgoing wall $W$, since $V=1$
at the return. Let $L_U,L_P$ be the angular rates in
\eqref{eq:core-logarithmic-rates}. Along the actual orbit, put
$q=dz/dt=ARd$ and define
\begin{equation}
 \mu_j(y)=\exp\left(
   \int_0^{\tau_\downarrow(y)}
       q(t;y)L_j(z(t;y),\mathcal Y(t;y))\dd t\right),
 \qquad j\in\{U,P\}.
 \label{eq:partial-core-multipliers}
\end{equation}
Here $\mathcal Y=(R,U,P)$ and the downward return time is the
one defined in \eqref{eq:partial-core-oriented-return}.
The scalar wall equations give
\begin{equation}
 u_*(y)=U(y)\mu_U(y),\qquad
 w_*(y)=V(y)W(y)\mu_P(y).
 \label{eq:partial-core-multiplier-formula}
\end{equation}

These identities hold on an ordinary neighborhood of $y_0$.
Indeed, the compact base segment has positive $V$, projective
denominator and angular denominator $d$. Continuous dependence
and the smooth return time preserve these inequalities along
every nearby segment. The original flow and the integrands in
\eqref{eq:partial-core-multipliers} are jointly $C^1$ there.
Thus $\mu_U,\mu_P$ and their logarithms are $C^1$ on that
neighborhood, and both multipliers are strictly positive.
This construction includes the incoming boundary $V_0=1$
and zero transverse walls.

\begin{proposition}
\label{prop:partial-core-log-multipliers}
For $w\in\R^4$, let $v=D\mathcal C(y_0)w$,
$\delta z=Dz(y_0)w$, and
\[
 Q_0=|v_R|+k|v_U|+|v_P|+(k^2u_0+P_0/k)|\delta z|.
\]
Then
\begin{align}
 |D\log\mu_U(y_0)w|
     &\leq40K_v\log(4k)Q_0+2k|\delta z|,\notag\\
 |D\log\mu_P(y_0)w|
     &\leq236K_v\frac{\log(4k)}{k^2}Q_0
                              +\frac4k|\delta z|.
 \label{eq:partial-core-log-multipliers}
\end{align}
\end{proposition}

\begin{proof}
Let $\xi$ be the variation at fixed angle. By
\eqref{eq:partial-core-variation-growth},
$\|S_k\xi\|_\infty\leq Q=4K_vQ_0$ throughout the passage.
With $x=k\xi_U$, the rate bounds
\eqref{eq:core-log-rate-variation} give
\[
 |D_{\mathcal Y}L_U\xi|\leq3Q,\qquad
 |D_{\mathcal Y}L_P\xi|
             \leq\frac{9Q}{k^2}+\frac8k|x|.
\]
The angle interval has length at most $2\log(4k)$.
Moreover, \eqref{eq:partial-core-strong-integral} implies
\[
 \frac8k\int_\alpha^\beta|x|\dd z
 \leq\frac{32|x(\alpha)|+384u_0Q}{k^2}
 \leq\frac{33Q}{k^2},
\]
because $|x(\alpha)|\leq Q$ and $u_0\leq1/60000$.
Consequently, for
$I_j=\int_\alpha^\beta D_{\mathcal Y}L_j\xi\dd z$,
\begin{equation}
 |I_U|\leq6\log(4k)Q,\qquad
 |I_P|\leq51\frac{\log(4k)}{k^2}Q.
 \label{eq:partial-core-log-integrals}
\end{equation}

Differentiating the original-time integral in
\eqref{eq:partial-core-multipliers} yields
\begin{equation*}
 D\log\mu_j(y_0)w
   =I_j+L_j(\beta,\mathcal Y_*)D\beta(y_0)w
             -L_j(\alpha,\mathcal Y_0)\delta z.
\end{equation*}
This is the moving-endpoint identity
\eqref{eq:core-moving-multiplier-derivative} with the actual
initial angle. More explicitly, if $\zeta$ is the angle
variation at fixed original time, the identity
\[
 \delta(qL_j)=qD_{\mathcal Y}L_j\xi
                       +\frac{d}{dt}(L_j\zeta)
\]
leaves the lower term $-L_j(\alpha,\mathcal Y_0)\delta z$.
At the upper endpoint,
$\zeta(t_*)+q(t_*)D\tau_\downarrow(y_0)w
=D\beta(y_0)w$.

The exit tangency calculation used in
\eqref{eq:core-moving-angle-bounds} applies to this same
downward branch. It gives
\[
 |D\beta(y_0)w|\leq\frac{2|\xi_R(\beta)|}{k}
                                     \leq\frac{2Q}{k}.
\]
Together with $|L_U|\leq2k$ and $|L_P|\leq4/k$, this bounds
the upper endpoint terms by $4Q$ and $8Q/k^2$, respectively.
The lower terms are bounded by $2k|\delta z|$ and
$4|\delta z|/k$. Combining these estimates with
\eqref{eq:partial-core-log-integrals} and
$\log(4k)\geq1$ gives
\[
 |D\log\mu_Uw|\leq10\log(4k)Q+2k|\delta z|,
 \qquad
 |D\log\mu_Pw|\leq59\frac{\log(4k)}{k^2}Q
                                      +\frac4k|\delta z|.
\]
Substitute $Q=4K_vQ_0$.
\end{proof}

Let $Y(\rho)$ be an ordinarily differentiable family of
projective states. As in
Proposition~\ref{prop:partial-core-radial-family}, write
\[
 k=r(Y(\rho)),\quad U=Y_1(\rho),\quad
 \Pin=Y_2(\rho)Y_3(\rho),\quad z=z(Y(\rho)),\quad
 (f,u_+,w_+)=\mathcal R_\downarrow(Y(\rho)).
\]
Primes in the rest of this subsection denote ordinary derivatives
with respect to $\rho$ at the point under consideration.
The scale $a$ is held fixed.

\begin{proposition}
\label{prop:partial-core-radial-multipliers}
Suppose that $Y(\rho)$ satisfies
\eqref{eq:partial-core-initial-state}, $0<a\leq1$, $K\geq1$,
and
\begin{equation}
 \begin{gathered}
 \frac1k\leq Ka,\qquad k\leq\frac Ka,\qquad
 |k'|\leq\frac K{a^2},\\
 |U'|\leq\frac{KU}{a},\qquad
 |\Pin'|\leq\frac{K\Pin}{a},\qquad |z'|\leq\frac Ka.
 \end{gathered}
 \label{eq:partial-core-multiplier-input}
\end{equation}
Set
\begin{equation}
 A_m=(200K_v+2)K^2,\qquad
 B_m=1180K_vK^4+4K^2,\qquad L=\log(4k).
 \label{eq:partial-core-radial-multiplier-constants}
\end{equation}
The two outgoing walls are nonnegative and ordinarily
differentiable, and their multipliers satisfy
\begin{equation}
 |(\log\mu_U\circ Y)'|\leq\frac{A_mL}{a^2},
 \qquad
 |(\log\mu_P\circ Y)'|\leq B_mL.
 \label{eq:partial-core-radial-log-bounds}
\end{equation}
Moreover,
\begin{align}
 |u_+'|&\leq u_+\left(\frac Ka+\frac{A_mL}{a^2}\right),
 &|w_+'|&\leq w_+\left(\frac Ka+B_mL\right).
 \label{eq:partial-core-radial-wall-bounds}
\end{align}
If $U=0$, then $u_+=u_+'=0$. If $\Pin=0$, then
$w_+=w_+'=0$.
\end{proposition}

\begin{proof}
Apply Proposition~\ref{prop:partial-core-log-multipliers}
with $w=Y'(\rho)$. The input derivatives are
$k',U',\Pin',z'$. Since $kU\leq1$ and $\Pin\leq1$, the five terms in
\[
 Q_0=|k'|+k|U'|+|\Pin'|+k^2U|z'|+\frac \Pin k|z'|
\]
are each at most $K^2/a^2$. Hence $Q_0\leq5K^2/a^2$.
For the strong multiplier,
\[
 40K_vLQ_0+2k|z'|
       \leq(200K_v+2)\frac{K^2L}{a^2}.
\]
For the weak multiplier, use $1/k^2\leq K^2a^2$:
\[
 \frac{236K_vLQ_0}{k^2}+\frac4k|z'|
        \leq1180K_vK^4L+4K^2
        \leq B_mL.
\]
These prove \eqref{eq:partial-core-radial-log-bounds}.

The local identities \eqref{eq:partial-core-multiplier-formula}
give
\[
 u_+'=\mu_U\bigl(U'+U(\log\mu_U\circ Y)'\bigr),\qquad
 w_+'=\mu_P\bigl(\Pin'+\Pin(\log\mu_P\circ Y)'\bigr).
\]
Their factors are positive, so the relative input bounds
in \eqref{eq:partial-core-multiplier-input} prove
\eqref{eq:partial-core-radial-wall-bounds}.
If an input wall is zero, its assumed derivative bound
forces its derivative to vanish as well. The displayed
product identities then give both zero-wall conclusions.
\end{proof}

To obtain a small transverse slope, the initial point must
leave a definite angular interval before the exit. The
positive-shear families from the small section have
$z\leq0$, so they satisfy the following condition with $Z=0$.

\begin{lemma}
\label{lem:partial-core-wall-contraction}
Suppose that $y_0$ satisfies
\eqref{eq:partial-core-initial-state}. If
$Z\geq0$, $2\cosh Z\leq k$ and $\alpha\leq Z$, then
\begin{equation}
 0\leq u_*\leq u_0e^{-k/12},\qquad
 0\leq w_*\leq P_0.
 \label{eq:partial-core-wall-contraction}
\end{equation}
\end{lemma}

\begin{proof}
At the downward exit, $\beta>0$ and
$\cosh\beta=r_*\geq99k/100$. Thus $Z\leq\beta$ and
\[
 \frac{99}{50}\leq\frac{\cosh\beta}{\cosh Z}
                      \leq e^{\beta-Z}.
\]
The second inequality follows from the addition formula
for $\cosh$ and $0\leq\tanh Z\leq1$.
Since $\log x\geq1-1/x$ for $x>0$,
\[
 \beta-\alpha\geq\log(99/50)\geq49/99>1/3.
\]
Integrating the coefficient bounds
$L_U\leq-k/4$ and $L_P\leq0$ proves
\eqref{eq:partial-core-wall-contraction}.
\end{proof}

\begin{theorem}
\label{thm:partial-core-radial-cone}
Assume the hypotheses of
Proposition~\ref{prop:partial-core-radial-family}, and let
$A_m,B_m$ be as in
\eqref{eq:partial-core-radial-multiplier-constants}.
Suppose in addition that, for some $D>0$ and $Z\geq0$,
\begin{equation}
 -k'\geq\frac1{Da^2},\qquad
 2\cosh Z\leq k,\qquad z\leq Z,\qquad
 4C_pK^2(U+a\Pin)\leq\frac1{2D}.
 \label{eq:partial-core-cone-input}
\end{equation}
Then
\begin{equation}
 -f'\geq\frac1{2Da^2},\qquad
 |f'|\leq\frac{K+1/(2D)}{a^2},
 \label{eq:partial-core-cone-radial-speed}
\end{equation}
and $0\leq u_+\leq Ue^{-k/12}$, $0\leq w_+\leq \Pin$.
The transverse derivatives satisfy
\begin{align}
 |u_+'|
   &\leq 2D(K+24A_m)\sqrt U\,\sqrt{u_+}\,|f'|,
       \notag\\
 |w_+'|
   &\leq 2DK(1+B_m)a\sqrt \Pin\,\sqrt{w_+}\,|f'|.
 \label{eq:partial-core-radial-cone}
\end{align}
The zero-wall conclusions of
Proposition~\ref{prop:partial-core-radial-multipliers}
hold as well.
\end{theorem}

\begin{proof}
The derivative error
\eqref{eq:partial-core-family-radius-error} and
\eqref{eq:partial-core-cone-input} give
\[
 |f'-k'|\leq\frac1{2Da^2}.
\]
Combining this with the lower bound for $-k'$ and the
upper bound for $|k'|$ proves
\eqref{eq:partial-core-cone-radial-speed}.
Lemma~\ref{lem:partial-core-wall-contraction} supplies the
two value bounds. The stronger lower radius bound
$2/(Ka)\leq k$ implies $1/k\leq Ka$, so
Proposition~\ref{prop:partial-core-radial-multipliers}
applies.

For the strong wall, take a square root in its value bound:
$\sqrt{u_+}\leq\sqrt U\,e^{-k/24}$.
For $k\geq6$, we have $\log(4k)\leq k$ and
$e^{-k/24}\log(4k)\leq24$. The latter follows from
$xe^{-x}\leq1$ at $x=k/24$.
Since $a\leq1$,
\[
 e^{-k/24}\left(\frac Ka+\frac{A_mL}{a^2}\right)
                   \leq\frac{K+24A_m}{a^2}.
\]
Factor one square root from $u_+$ in
\eqref{eq:partial-core-radial-wall-bounds} to obtain
\[
 |u_+'|\leq\sqrt{u_+}\sqrt U\,\frac{K+24A_m}{a^2}.
\]
For the weak wall, $\sqrt{w_+}\leq\sqrt \Pin$ and
$L\leq k\leq K/a$ give
\[
 |w_+'|\leq\sqrt{w_+}\sqrt \Pin\,\frac{K(1+B_m)}a.
\]
Finally, $1\leq2Da^2|f'|$ converts these two inequalities
into \eqref{eq:partial-core-radial-cone}.
All factors are nonnegative, and no division by a wall
amplitude has been used.
\end{proof}

\begin{corollary}
\label{cor:small-section-partial-core-cone}
The constants $h_*,K$ in
Corollary~\ref{cor:small-section-partial-core-response}
can also be used for the following conclusion.
For $0<h\leq h_*$, $q>3$, $\ell>4$ and sufficiently
small $a>0$, every positive-shear input in that
corollary with $V\geq1$ satisfies
\eqref{eq:partial-core-cone-radial-speed} and
\eqref{eq:partial-core-radial-cone} with $D=4096$.
Its outgoing walls satisfy
$0\leq u_+\leq Ue^{-k/12}$ and $0\leq w_+\leq \Pin$.
These are the walls on the original orbit at its first
$V=1$ event strictly after $T=(q/2)\log(1/a)$.
\end{corollary}

\begin{proof}
Proposition~\ref{prop:small-section-core-data} gives
$-k'\geq1/(4096a^2)$, $z\leq0$ and $U+\Pin\leq1/k$.
Use $Z=0$, for which $2\cosh Z\leq k$ follows from $k\geq6$.
The radius bound gives $1/k\leq Ka$, so
$U+a\Pin\leq U+\Pin\leq Ka$. Restrict the scale further to
\[
 a\leq\frac1{8C_pK^3\cdot4096}.
\]
Then $4C_pK^2(U+a\Pin)\leq1/(2\cdot4096)$.
All hypotheses of Theorem~\ref{thm:partial-core-radial-cone}
hold. The orbit and first-event identities follow from
the same time shift as in
Corollary~\ref{cor:small-section-partial-core-response}.
\end{proof}

The two cone coefficients depend on $\sqrt U$ and
$a\sqrt \Pin$, respectively. The exponential decay absorbs
the logarithmic derivative of the strong multiplier.
For the weak wall, the extra factor $a$ comes from
comparing a derivative of order $a^{-1}$ with a radial
speed of order $a^{-2}$. This is the quantitative
transverse control needed to continue the outgoing
curve through the following connection.