\subsection{The remaining core passage for angular families}
\label{sec:angular-remaining-core}

The active wall places the angular families inside a large core
with a bounded initial angle. A definite part of the core passage
therefore remains after the fixed time $T$. Its contraction makes
the outgoing curves almost radial. We establish the initial
directional estimates and then apply the general core results of
Sections~\ref{sec:small-section-partial-core}--\ref{sec:partial-core-connection}.

Throughout this subsection, retain
\[
 Z_{\theta,\zeta}(\rho)=(\theta\rho,\rho,\zeta),\qquad
 T=\frac q2\log\frac1a,\qquad M_L=8(L+1),\qquad B_L=M_L+1.
\]
Write $Y=(M,U,V,W)$ for the actual projective state at $T$, and put
\begin{equation}
 k=r(Y),\qquad p=VW,\qquad
 z=\operatorname{arsinh}\frac{1-M}{V},\qquad
 Z_L=\operatorname{arsinh}(4M_L).
 \label{eq:angular-core-radial-data}
\end{equation}
Primes always denote ordinary derivatives in $\rho$, with
$a,T,\theta,\zeta$ fixed. We use $p$ for the wall product so that
$P$ can denote the exponent in the restriction $\zeta\leq a^P$.

\begin{proposition}
\label{prop:angular-core-radial-data}
There is $h_*>0$ such that, after fixing $0<h\leq h_*$ and
$L\geq0$, constants $K\geq1$ and $D=4B_L^2$ can be chosen
before $q>3$ and $P>4$. For every $0<\epsilon\leq10^{-4}$,
all sufficiently small $a>0$ have the following properties,
uniformly for $|\theta|\leq L$, $0\leq\zeta\leq a^P$ and
$\rho\in[a,2a]$:
\begin{align}
 &k\geq\max\{6,2\cosh Z_L\},\qquad
   V\geq2,\qquad -\operatorname{arcosh}k\leq z\leq Z_L,
       \notag\\
 &0<U,\quad 0\leq p,\quad U+p\leq\epsilon/k,
       \label{eq:angular-core-smallness}\\
 &\frac2{B_La}\leq k\leq\frac{32}a,\qquad
   \frac1{Da^2}\leq-k'\leq\frac{512(2L+4)}{a^2},
       \label{eq:angular-core-radius-direction}\\
 &\frac2{Ka}\leq k\leq\frac Ka,\qquad
   |k'|\leq\frac K{a^2},\qquad
   |U'|\leq\frac{KU}a,\quad |p'|\leq\frac{Kp}a,
   \quad |z'|\leq\frac Ka.
       \label{eq:angular-core-family-direction}
\end{align}
The state lies in the core region of
Theorem~\ref{thm:partial-core-first-exit}. If $\zeta>0$ then
$p>0$. If $\zeta=0$ then $W=p=W'=p'=0$.
\end{proposition}

\begin{proof}
First take the common height threshold from
Sections~\ref{sec:angular-core-coordinates}
and~\ref{sec:angular-projected-phase}. The affine reference
phase obeys
\[
 a/8\leq F_{\theta,\zeta}\leq M_La.
\]
For the lower bound, its second component has magnitude
$C(\rho-\zeta)\geq a/2$. For the upper bound, use
$\varrho(\Pi_h Z)\leq4\|Z\|$ and
$\|Z\|\leq2(L+1)a$. Here $a$ is small enough that
$\zeta\leq a^2\leq a/9$.

Let $x=2/k$. Theorem~\ref{thm:angular-projected-phase},
after decreasing the scale, gives
\[
 a/16\leq x\leq B_La,\qquad
 1/8\leq x'\leq2L+4.
\]
The identities $k=2/x$ and $-k'=2x'/x^2$ prove
\eqref{eq:angular-core-radius-direction}. In particular $k$
eventually exceeds both $6$ and $2\cosh Z_L$.

We next control the other directions without dividing by a
small wall. Put $D_0=2(L+1)$ and let $C\geq32$ be the
constant in Proposition~\ref{prop:angular-cutoff-state}.
For the physical height $A$, the bounds
in~\eqref{eq:angular-height-bounds} imply
\[
 |A'|\leq32M_LD_0a
       \leq F_0 A/a,\qquad F_0=512M_LD_0.
\]
Each physical wall $v=n,w_2,w_3$ satisfies
$|v'|\leq E_0v/a$ with $E_0=8C(L+1)$.
For the active wall this follows from $|w_2'|\leq D_0$
and $w_2\geq a/2$; the other two bounds were obtained
with their wall factors in Proposition~\ref{prop:angular-cutoff-state}.
Differentiating $\sqrt3v/A$ therefore gives, with
$H_0=E_0+F_0$,
\[
 |U'|\leq H_0U/a,\qquad |V'|\leq H_0V/a,
 \qquad |W'|\leq H_0W/a,
 \qquad |p'|\leq2H_0p/a.
\]

For the angle use its physical expression
$z=\operatorname{arsinh}(b/w_2)$, where
$b=A/\sqrt3-s$. The same estimates, with $M_La\leq1/16$,
give $|b|\leq2M_La$, $|b'|\leq3D_0$,
$a/2\leq w_2\leq M_La$ and $|w_2'|\leq D_0$.
Thus
\[
 |z'|\leq\left|\left(\frac b{w_2}\right)'\right|
      \leq\frac{20M_LD_0}a.
\]
Choose $K$ at least
$1+B_L+32+512(2L+4)+3H_0+20M_LD_0$.
This proves~\eqref{eq:angular-core-family-direction}, with
constants independent of $q,P$.

It remains to check the core amplitude and geometry.
Corollary~\ref{cor:angular-projective-bounds} gives
\begin{equation}
 U\leq32h a^{q-2},\qquad
 p\leq2048M_L\frac\zeta{a^3},\qquad |z|\leq Z_L.
 \label{eq:angular-core-amplitude-budget}
\end{equation}
Consequently
\[
 k(U+p)\leq1024h a^{q-3}+65536M_La^{P-4}\longrightarrow0.
\]
Both exponents are positive, proving the required smallness.
The relation $k=V\cosh z$ and $V\geq2$ give
$|z|\leq\operatorname{arcosh}k$.
The initial core energy equals $k^2$ by definition, and
$U,p\leq10^{-4}/k$; these are precisely the initial-region
conditions in~\eqref{eq:partial-core-initial-state}.
Finally positivity and vanishing of the physical wall multipliers
give the assertions about $\zeta$. The relative derivative bounds
then give $W'=p'=0$ when that wall vanishes.
\end{proof}

Let $\tau_\downarrow(Y)$ be the first positive $V=1$ time
from $Y$, measured in the original normalized time, and write
\[
 (f,u,w)=\mathcal R_\downarrow(Y).
\]
The initial inequality $V\geq2$ separates this event from
the fixed time $T$.

\begin{proposition}
\label{prop:angular-core-exit}
Under the preceding thresholds, $\tau_\downarrow>0$ exists,
is a transverse downward crossing, and is locally $C^2$ in
the initial state. Along each angular family,
\begin{align}
 k/2&\leq f\leq2k,\qquad
 |f-k|\leq16K(U+ap^2),\notag\\
 |f'-k'|&\leq4C_pK^2\frac{U+ap}{a^2},\qquad
 -f'\geq\frac1{2Da^2},
 \label{eq:angular-core-exit-radius}\\
 0&\leq u\leq Ue^{-k/12},\qquad 0\leq w\leq p.
 \label{eq:angular-core-exit-amplitudes}
\end{align}
There is $C_{h,L}>0$, chosen before $q,P$, such that, with
\begin{equation}
 E_a=a^{(q-2)/2}+\sqrt{\zeta/a},
 \label{eq:angular-core-cone-rate}
\end{equation}
\begin{equation}
 |u'|\leq C_{h,L}E_a\sqrt u\,|f'|,\qquad
 |w'|\leq C_{h,L}E_a\sqrt w\,|f'|.
 \label{eq:angular-core-exit-cone}
\end{equation}
The factor $C_{h,L}E_a$ is uniformly smaller than any prescribed
positive number when the scale is sufficiently small.
On the original orbit, $T+\tau_\downarrow$ is the first
$V=1$ event strictly after $T$.
\end{proposition}

\begin{proof}
Apply Theorem~\ref{thm:partial-core-first-exit} and
Proposition~\ref{prop:partial-core-smooth-exit} to
\eqref{eq:angular-core-smallness}. The ordinary radial inputs
have just been checked, so
Proposition~\ref{prop:partial-core-radial-family} gives
the first two radius estimates and the derivative error.
Here $C_p$ is the constant of
\eqref{eq:partial-core-family-radius-error}.
Since $U+p\leq1/k\leq Ka/2$, taking
$a\leq1/(8C_pK^3D)$ ensures
$4C_pK^2(U+ap)\leq1/(2D)$.
Theorem~\ref{thm:partial-core-radial-cone} now applies with
$Z=Z_L$. It gives the negative radial speed and the amplitude
bounds, as well as
\[
 |u'|\leq C_U\sqrt U\sqrt u\,|f'|,
 \qquad |w'|\leq C_Pa\sqrt p\sqrt w\,|f'|,
\]
where $C_U=2D(K+24A_m)$ and $C_P=2DK(1+B_m)$.
Using~\eqref{eq:angular-core-amplitude-budget},
\[
 C_U\sqrt U+C_Pa\sqrt p
 \leq C_U\sqrt{32h}\,a^{(q-2)/2}
       +C_P\sqrt{2048M_L}\sqrt{\zeta/a}.
\]
This proves~\eqref{eq:angular-core-exit-cone}. Its uniform
smallness follows from
$\sqrt{\zeta/a}\leq a^{(P-1)/2}$.

The original flow shifted by $T$ has projective initial
state $Y$. Uniqueness identifies it with the core trajectory
throughout the passage. Since $V>1$ before the downward
exit, translating its first-event property gives the last
assertion. The event identity holds on ordinary neighborhoods;
the local time and output maps used for all derivatives are
therefore the actual first-event maps.
\end{proof}