\subsection{Statistics along angular families}
\label{sec:angular-statistical-families}

The phase at the next entrance is strictly increasing along the
angular radial coordinate. We can therefore transfer the
statistical estimate to this coordinate, including at zero
shear. To do so, we first extend each returned curve to an
admissible graph and keep track of its amplitude cost.

For fixed $h,L,q,P,a$ as above, put
\begin{equation}
 I=[a,2a],\qquad
 \Lambda_a=[-L,L]\times(0,a^P],\qquad
 B_{a,\zeta}=6400M_L\frac\zeta{a^3}.
 \label{eq:angular-statistical-labels}
\end{equation}
For $\ell=(\theta,\zeta)\in\Lambda_a$, abbreviate the
returned state and phase by $G_\ell$ and $\xi_\ell$.

\begin{proposition}
\label{prop:angular-statistical-graphs}
There are positive $C^1$ functions
$\widetilde u_\ell,\widetilde w_\ell:\R\to\R$ such that
\begin{equation}
 \widetilde u_\ell+\widetilde w_\ell\leq B_{a,\zeta},\qquad
 |\widetilde u_\ell'|\leq\sqrt{\widetilde u_\ell},\qquad
 |\widetilde w_\ell'|\leq\sqrt{\widetilde w_\ell}.
 \label{eq:angular-extended-graph-bounds}
\end{equation}
The graph $\Gamma_\ell(r)=(r,\widetilde u_\ell(r),
\widetilde w_\ell(r))$ agrees with the actual output:
\begin{equation}
 \Gamma_\ell(g_\ell(\rho))=G_\ell(\rho),\qquad
 \widetilde u_\ell'(g_\ell(\rho))
       =\frac{\widehat u_\ell'(\rho)}{g_\ell'(\rho)},\qquad
 \widetilde w_\ell'(g_\ell(\rho))
       =\frac{\widehat w_\ell'(\rho)}{g_\ell'(\rho)}
 \label{eq:angular-extended-graph-match}
\end{equation}
for all $\rho\in I$. The two extended amplitudes and their
derivatives are jointly Borel in $(\ell,r)$.
For any prescribed $B_*>0$, the scale can be decreased so
that, on all labels and radial parameters,
\begin{equation}
 0<B_{a,\zeta}\leq B_*,\qquad
 g_\ell\geq4,\qquad 0<\xi_\ell<1,
 \qquad \Gamma_\ell(2/\xi_\ell)=G_\ell.
 \label{eq:angular-statistical-preparation}
\end{equation}
\end{proposition}

\begin{proof}
The radius $g_\ell$ is $C^2$ and strictly decreasing on
$I$, with nonzero ordinary derivatives at its endpoints.
It has a continuous inverse $j_\ell$ on
$J_\ell=[g_\ell(2a),g_\ell(a)]$, whose derivative is
$1/g_\ell'\circ j_\ell$, including one-sided derivatives
at the two endpoints. Define on this interval
\[
 u_\ell=\widehat u_\ell\circ j_\ell,\qquad
 w_\ell=\widehat w_\ell\circ j_\ell.
\]
The cone inequalities~\eqref{eq:angular-return-cone} give
$|u_\ell'|\leq\sqrt{u_\ell}$ and
$|w_\ell'|\leq\sqrt{w_\ell}$.
The sum of their amplitudes is at most
$4096M_L\zeta/a^3$. Apply the explicit extension of
Lemma~\ref{lem:positive-amplitude-extension} with cone
constant one. It preserves values and derivatives on the
closed interval, and multiplies the sum bound by $25/16$.
Since $(25/16)4096=6400$, this proves
\eqref{eq:angular-extended-graph-bounds}
and~\eqref{eq:angular-extended-graph-match}.

For measurability, the actual return is jointly $C^2$
on ordinary neighborhoods by
Theorem~\ref{thm:angular-actual-return}. Thus the endpoints
of $J_\ell$ are Borel functions of $\ell$.
The map $(\ell,\rho)\mapsto(\ell,g_\ell(\rho))$
is an injective Borel map between standard Borel spaces,
and its inverse on its image is Borel. The transported
radius graphs are consequently jointly Borel.
The interior-sampling formulas
\eqref{eq:shear-interior-sampling-step}--\eqref{eq:shear-interior-sampling-limits}
apply to these varying intervals: both endpoints are Borel,
the interval lengths are positive, and the one-sided
derivatives are continuous. They give Borel endpoint
values and slopes, as well as the clamped interior values
and slopes. The explicit square-root extension uses only
these data and Borel case distinctions. It follows that
the extensions and their derivatives are jointly Borel,
as claimed. This argument also permits extension by zero
off the Borel label set $\Lambda_a$.

Finally, $P>4$ and $a\leq1$ imply
\[
 B_{a,\zeta}\leq6400M_La^{P-3}\leq6400M_La.
\]
Choose the scale so this is at most $B_*$, and so that
$a\leq1/(8B_L)$. The lower radius bound in
\eqref{eq:angular-return-amplitudes} gives $g_\ell\geq4$.
Taking its reciprocal and using
\eqref{eq:angular-extended-graph-match} proves
\eqref{eq:angular-statistical-preparation}.
\end{proof}

Fix the statistical parameters
\[
 0<\eta\leq\delta,\qquad J\geq20,\qquad
 0<\sigma<1,\qquad 2J\sigma<1,
\]
where $\delta\leq1/40000$ is as in
Theorem~\ref{thm:era-graph-statistics}. Choose its
$B_*,B_{\mathrm{cap}}$ and use that $B_*$ in
Proposition~\ref{prop:angular-statistical-graphs}.
Define
\begin{equation}
 \mathcal F_\ell
   =\{\rho\in I:(B_{a,\zeta},G_\ell(\rho))
                         \notin\mathcal S_{\mathrm{stat}}\}.
 \label{eq:angular-radial-failure}
\end{equation}

\begin{theorem}
\label{thm:angular-radial-statistics}
At all sufficiently small scales, the relation
$\rho\in\mathcal F_\ell$ is jointly Borel on
$\Lambda_a\times I$. For every Borel $A\subset I$,
nonnegative Borel weight $\omega$ and constant $M_\omega\geq0$
with $\omega\leq M_\omega$ on $A$,
\begin{equation}
 \int_{A\cap\mathcal F_\ell}\omega\dd\rho
 \leq\min\left\{\int_A\omega\dd\rho,
                  640\sqrt3 M_\omega B_{a,\zeta}^{\sigma}\right\}.
 \label{eq:angular-radial-statistical-loss}
\end{equation}
For $\rho\in I\setminus\mathcal F_\ell$, each of the two
section points
\[
 y=(\theta\rho,-\log\rho,-\log\zeta),\qquad \mathsf s y
\]
has a statistical entrance at the positive actual time
$t_{\mathrm a}$. Their complete eras obey all six limits
of Theorem~\ref{thm:main}.
\end{theorem}

\begin{proof}
The pair $(B_{a,\zeta},G_\ell(\rho))$ is jointly Borel,
so~\eqref{eq:angular-radial-failure} is the inverse image
of the complement of the common Borel statistical set.
The graph $\Gamma_\ell$ is admissible at amplitude
$B_{a,\zeta}\leq B_*$, and
\eqref{eq:angular-statistical-preparation} identifies it
with the actual entrance on $I$.
Theorem~\ref{thm:angular-actual-return} gives
$\xi_\ell'\geq1/(8\sqrt3)$.
Apply Theorem~\ref{thm:radial-family-statistics}, or its
single-fiber estimate~\eqref{eq:radial-weighted-loss},
with this slope and the excluded phase length
$80B_{a,\zeta}^\sigma$. The coefficient is
$80/(1/(8\sqrt3))=640\sqrt3$.

Outside $\mathcal F_\ell$, the returned state belongs to
$\mathcal S_{\mathrm{stat}}$ by its definition.
Equations~\eqref{eq:angular-return-orbit}
and~\eqref{eq:angular-reflected-return-orbit} give the
two actual statistical entrances. The complete-era and
time-divergence conclusions follow from
Corollary~\ref{cor:complete-eras-from-survivor}.
\end{proof}

In particular,
\begin{equation}
 \lambda(\mathcal F_{\theta,\zeta})
       \leq\min\{a,640\sqrt3 B_{a,\zeta}^{\sigma}\}.
 \label{eq:angular-failed-length}
\end{equation}
The minimum retains the total radial length, which will
control the integral over the angular and weak-wall labels.