\subsection{Statistical entrances on the whole small section}
\label{sec:whole-small-section}

The angular families cover the region where shear is at most
a fixed multiple of the larger weak wall. Points not selected
by the weak-wall power condition occupy a finite logarithmic
strip. Adding that strip to the excluded set gives
coverage of the entire angular region. Together with the
dominant-shear construction, this covers the whole small section.

As before, write $n_i=e^{-\beta_i}$. For $L\geq0$, $a>0$
and $P\geq0$, define
\begin{equation}
 \mathcal U_a^{\mathrm a}
   =\{y:a\leq\max(n_2,n_3)\leq2a,\quad
       |s|\leq L\max(n_2,n_3),\quad a^P<\min(n_2,n_3)\}.
 \label{eq:angular-unselected-region}
\end{equation}
This is a Borel set. Under $h\leq1/8$, $a\leq h/16$
and $2La\leq1/8$, it has flux
\begin{equation}
 \mu_h(\mathcal U_a^{\mathrm a})
               \leq24LP\log2\,a|\log a|.
 \label{eq:angular-unselected-flux}
\end{equation}
To prove this, first suppose $n_2\geq n_3$. Its coordinates lie
in the box
\[
 [-2La,2La]\times[-\log(2a),-\log a]
                       \times[0,P|\log a|].
\]
The first wall interval has logarithmic width $\log2$;
the lower bound on the other wall gives the last interval.
If $n_3\geq n_2$, interchange the two logarithmic coordinates.
Each box has volume $4LP\log2\,a|\log a|$. On
$\mathcal U_a^{\mathrm a}$ both walls are at most $2a\leq h/8$
and $|s|\leq1/8$, so the density estimate in the proof of
Proposition~\ref{prop:angular-failure-flux} gives the factor
three. Summing the two boxes proves
\eqref{eq:angular-unselected-flux}, also when $L=0$.
At the dyadic scales,
\begin{equation}
 \mu_h\left(\bigcup_{j\geq0}\mathcal U_{a_j}^{\mathrm a}\right)
      \leq24LP\log2\,d(|\log d|+2\log2).
 \label{eq:angular-unselected-dyadic-flux}
\end{equation}

The full angular region is
\begin{equation}
 \mathcal B^{\mathrm a}(d,L)
   =\{y:\max(n_2,n_3)\leq d,\quad
                  |s|\leq L\max(n_2,n_3)\}.
 \label{eq:full-angular-region}
\end{equation}
The larger wall is strictly positive, so the half-open
annuli $a_j<\max(n_2,n_3)\leq2a_j$ partition this set.
Outside $\mathcal U_{a_j}^{\mathrm a}$, the smaller wall
is at most $a_j^P$. There are exactly the two representations
needed for the angular families. If $n_2\geq n_3$, set
\[
 \rho=n_2,\qquad \zeta=n_3,\qquad \theta=s/n_2.
\]
Then $y=(\theta\rho,-\log\rho,-\log\zeta)$.
If $n_3\geq n_2$, set
$\rho=n_3$, $\zeta=n_2$, $\theta=-s/n_3$;
then $y=\mathsf s(\theta\rho,-\log\rho,-\log\zeta)$.
In either case $|\theta|\leq L$,
$0<\zeta\leq a_j^P$ and $\rho\in[a_j,2a_j]$.
No division by the shear is used.

\begin{theorem}
\label{thm:full-angular-region-statistics}
There are statistical thresholds $\delta>0$ and, for each
$0<\eta\leq\delta$, $J\geq20$ and
$0<\sigma<1$ with $2J\sigma<1$, a cap
$0<B_{\mathrm{cap}}<1$ and a height $h_*>0$ with the
following property. For $0<h\leq h_*$, $L\geq0$,
$q>3$ and $P>4$, every sufficiently small $d>0$ admits
a Borel set $F_d^{\mathrm a}$ satisfying
\begin{equation}
 \mu_h(F_d^{\mathrm a})\leq
     K_A(L)\,d(1+|\log d|+2\log2),\qquad
 K_A(L)=12L C_1(K_{\mathrm a}(L,\sigma),\sigma)+24LP\log2.
 \label{eq:full-angular-region-flux}
\end{equation}
Every $y\in\mathcal B^{\mathrm a}(d,L)\setminus F_d^{\mathrm a}$
belongs to $\mathcal D_h$, and $\mathcal X_h(y)$ has a
statistical entrance with these parameters and cap.
\end{theorem}

\begin{proof}
Choose the statistical constants from
Theorem~\ref{thm:angular-radial-statistics}. After fixing
$h,L,q,P$, choose $d$ below the common radial threshold,
$h/16$, and $1/[16(L+1)]$. All dyadic scales then satisfy
the conditions of the preceding flux estimates. Put
\[
 F_d^{\mathrm a}=E_d^{\mathrm a}\cup
                         \bigcup_{j\geq0}\mathcal U_{a_j}^{\mathrm a}.
\]
The two unions are Borel. Add
\eqref{eq:angular-dyadic-failure-flux}
and~\eqref{eq:angular-unselected-dyadic-flux}, enlarging
the latter logarithmic factor by one, to obtain
\eqref{eq:full-angular-region-flux}.

For a point outside $F_d^{\mathrm a}$, choose the annulus
of its larger wall and use the representation above.
The actual angular-return theorem places the corresponding
point in the constraint chart. The chart domain is preserved
by $\mathsf s$, so the same holds in the other wall order.
Exclusion from $E_{a_j}$ gives the actual statistical entrance
by Proposition~\ref{prop:angular-failure-flux}.
\end{proof}

The angular and shear constructions may initially select
different upper amplitude caps. They nevertheless use the
same return branches when their statistical parameters
$\eta,J,\sigma$ agree.

\begin{lemma}
\label{lem:statistical-cap-monotonicity}
Fix $\eta,J,\sigma$. If $B_1\leq B_2$, then the survivor
set and the common statistical set for cap $B_1$ are contained
in their counterparts for cap $B_2$. A common initial pair
$(B,Y)$ has the same finite histories, digits and phases
under either cap. In particular a statistical entrance for
cap $B_1$ remains a statistical entrance for cap $B_2$.
\end{lemma}

\begin{proof}
In the definition of a selected input, the cap appears only
in the inequality $B\leq B_{\mathrm{cap}}$. Increasing it
enlarges the domain of each selected input and leaves the actual
branch and amplitude update unchanged. Induction over a
finite word therefore enlarges every prefix domain. Taking
the union over words of each length and then the intersection
over lengths proves the survivor inclusion.

For a common surviving pair, both words of any given length
belong to the prefix domains for cap $\max(B_1,B_2)$.
Uniqueness of the finite history makes those words equal.
Their successive letters, actual return states and reciprocal
radius phases are consequently equal. In particular their
continued fraction codes are identical. Membership of that
code in the fixed Gauss typical set proves the inclusion
of $\mathcal S_{\mathrm{stat}}$.
The time and axis frame of a statistical entrance do not
change, giving the last assertion.
\end{proof}

Define
\begin{equation}
 \mathcal B(d)=\{y:|s|\leq d,\quad\max(n_2,n_3)\leq d\}.
 \label{eq:whole-small-section}
\end{equation}
This is Borel, and
\begin{equation}
 \mathcal B(d)\subset
       \mathcal B^{\mathrm a}(d,128)
         \cup\mathcal B^{\mathrm s}(d,1/128).
 \label{eq:whole-small-section-cover}
\end{equation}
Indeed, either $|s|\leq128\max(n_2,n_3)$, or
$|s|>128\max(n_2,n_3)>0$ and both weak walls are
at most $|s|/128$.
The two regions meet along their common boundary, as shown in
Figure~\ref{fig:section-coverage}. The distinction is between
directions in the same physical section; it determines which
parameter supplies a nondegenerate phase derivative.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.92\textwidth]{figures/section-coverage.pdf}
\caption{The two regions covering the small section. Write
$r=\max(n_2,n_3)>0$ and project the section coordinates to $(s,128r)$.
Above the two boundary rays, $|s|\leq128r$ gives the angular region
$\mathcal B^{\mathrm a}(d,128)$; below them, $r\leq|s|/128$ gives the
shear region $\mathcal B^{\mathrm s}(d,1/128)$.
Both include the boundary rays. The outer restrictions $|s|\leq d$
and $r\leq d$ are omitted to display the relative sizes.
The arrows illustrate variation of $s$ with both walls fixed and
variation of the larger wall with $s/r$ and the smaller wall fixed.
The projection suppresses the smaller wall and its ordering; the
construction treats both orderings.}
\label{fig:section-coverage}
\end{figure}
\FloatBarrier

\begin{theorem}[Statistical entrances on the whole small section]
\label{thm:whole-small-section-statistics}
There is $0<\delta\leq1/40000$ such that, for every
$0<\eta\leq\delta$, $J\geq20$ and $0<\sigma<1$ with
$2J\sigma<1$, one can choose $0<B_{\mathrm{cap}}<1$
and $h_*>0$ as follows. For every $0<h\leq h_*$,
$q>3$, $D>4$ and $P>6D+2$, there are $d_*>0$
and $C>0$ such that every $0<d\leq d_*$ admits a
Borel set $F_d$ with
\begin{equation}
 \mu_h(F_d)\leq C d(1+|\log d|)^2<\infty.
 \label{eq:whole-small-section-failure-flux}
\end{equation}
Every $y\in\mathcal B(d)\setminus F_d$ lies in
$\mathcal D_h$, and $\mathcal X_h(y)$ has a statistical
entrance for the common parameters and cap.
\end{theorem}

\begin{proof}
Take the minimum of the angular and shear statistical
thresholds. For the chosen $\eta,J,\sigma$, let their
caps be $B_A,B_S$ and set
$B_{\mathrm{cap}}=\max(B_A,B_S)<1$.
Take the smaller height threshold and the smaller scale
threshold from Theorems~\ref{thm:full-angular-region-statistics}
and~\ref{thm:full-shear-region-statistics}, with
$L=128$ and $\varepsilon=1/128$.
The exponent condition implies $P>4$, as required for
the angular part. Put $F_d=F_d^{\mathrm a}\cup F_d^{\mathrm s}$.

Let $K_S=66C_{\log}(K_\sigma,\sigma)+132P^2$ be the
coefficient in~\eqref{eq:full-shear-region-flux}.
Since $\log2\leq1$,
\[
 1+|\log d|+2\log2\leq3(1+|\log d|)^2.
\]
Subadditivity gives~\eqref{eq:whole-small-section-failure-flux}
with $C=3K_A(128)+K_S>0$.
For every point outside $F_d$, the cover
\eqref{eq:whole-small-section-cover} selects an angular
or shear statistical entrance. Lemma~\ref{lem:statistical-cap-monotonicity}
places it in the common cap without changing its orbit
or statistical sequence.
\end{proof}

Theorem~\ref{thm:whole-small-section-statistics} supplies
the finite-flux exceptional set needed for the capture
argument of Section~\ref{sec:capture-by-expanding-returns}.
It covers zero shear and arbitrarily unequal positive weak
walls. We next establish repeated visits in the finitely
many axis frames and transfer the capture conclusion to
arbitrary transverse sections.