\subsection{Covering the region of dominant shear}
\label{sec:small-section-shear-region}

The statistical construction on a shear layer required at
least one weak wall to be smaller than a prescribed power
of the layer scale. The remaining points have both weak
walls bounded below. Their logarithmic coordinates therefore
occupy a finite rectangle, whose area grows only quadratically
in the logarithm of the scale. This gives a finite excluded
flux on the entire region where shear dominates the weak walls.

For $y=(s,\beta_2,\beta_3)$ write $n_i=e^{-\beta_i}$.
Given $a>0$, $\varepsilon\geq0$ and $P\geq0$, set
\begin{equation}
 \begin{split}
 \mathcal L_a^{\mathrm s}
   &=\{y:a\leq|s|\leq2a,\quad n_2,n_3\leq2\varepsilon a\},\\
 \mathcal U_a^{\mathrm s}
   &=\{y\in\mathcal L_a^{\mathrm s}:n_2>a^P,\ n_3>a^P\}.
 \end{split}
 \label{eq:shear-region-unselected}
\end{equation}
Both sets are Borel. Outside $\mathcal U_a^{\mathrm s}$,
a point of $\mathcal L_a^{\mathrm s}$ satisfies
$\min\{n_2,n_3\}\leq a^P$ and hence belongs to one of
the signed families of Section~\ref{sec:small-section-signed-return}.

\begin{lemma}
\label{lem:shear-unselected-flux}
If $0<h\leq1/8$, $0<a\leq h/8$,
$0\leq\varepsilon\leq1/64$ and $P\geq0$, then
$\mathcal L_a^{\mathrm s}\subset\mathcal D_h$ and
\begin{equation}
 \mu_h(\mathcal U_a^{\mathrm s})
       \leq12P^2a|\log a|^2.
 \label{eq:shear-unselected-flux}
\end{equation}
\end{lemma}

\begin{proof}
The chart radicand and density depend on $s$ only through
$s^2$. Lemma~\ref{lem:small-section-flux-bounds}, with
$\rho=|s|$, therefore gives the chart inclusion and
$w_h\leq3$ on the whole layer.

Here $a\leq1$ and $n_i\leq2\varepsilon a\leq1$.
For a point in $\mathcal U_a^{\mathrm s}$, taking
logarithms of $a^P<n_i\leq1$ gives
\[
 0\leq\beta_i\leq P|\log a|.
\]
Thus
\[
 \mathcal U_a^{\mathrm s}
   \subset[-2a,2a]\times[0,P|\log a|]^2.
\]
This box has Lebesgue volume $4P^2a|\log a|^2$.
Multiplication by the density bound proves
\eqref{eq:shear-unselected-flux}.
\end{proof}

For $0<d\leq h/8$ let $a_j=d2^{-(j+1)}$. The same
lemma and~\eqref{eq:dyadic-logarithmic-sum} imply
\begin{equation}
 \mu_h\left(\bigcup_{j\geq0}\mathcal U_{a_j}^{\mathrm s}\right)
   \leq\sum_{j\geq0}\mu_h(\mathcal U_{a_j}^{\mathrm s})
   \leq132P^2d(1+|\log d|)^2.
 \label{eq:shear-unselected-dyadic-flux}
\end{equation}
The union is Borel.

For fixed statistical parameters, say that $x\in\M$
has a statistical entrance if there are $t>0$, an axis
frame $\mathsf A$, and $(B,Y)\in\mathcal S_{\mathrm{stat}}$
such that
\begin{equation}
 \iota_+(Y)=\operatorname{proj}(\mathsf A\phi_t x).
 \label{eq:statistical-entrance}
\end{equation}
Corollary~\ref{cor:complete-eras-from-survivor} then
provides successive complete eras on this orbit, all six
statistical limits, and entrance times tending to infinity.

Define the full region of dominant shear by
\begin{equation}
 \mathcal B^{\mathrm s}(d,\varepsilon)
   =\{y:0<|s|\leq d,\quad n_2,n_3\leq\varepsilon|s|\}.
 \label{eq:shear-region}
\end{equation}
The half-open intervals $(a_j,2a_j]$ partition $(0,d]$.
Indeed, successive powers of $1/2$ place every $|s|/d\in(0,1]$
in exactly one such interval; their endpoint convention
prevents double counting. Consequently the Borel sets
\begin{equation}
 \mathcal B_j^{\mathrm s}
   =\{y:a_j<|s|\leq2a_j,\quad n_2,n_3\leq\varepsilon|s|\}
 \label{eq:shear-region-annuli}
\end{equation}
partition $\mathcal B^{\mathrm s}(d,\varepsilon)$, and
$\mathcal B_j^{\mathrm s}\subset\mathcal L_{a_j}^{\mathrm s}$.

\begin{theorem}
\label{thm:full-shear-region-statistics}
Fix the statistical parameters and cap of
Theorem~\ref{thm:finite-failure-layer}.
There is $h_*>0$ such that, for $0<h\leq h_*$,
$q>3$, $D>4$ and $P>6D+2$, there is $d_*>0$ such
that every $0<d\leq d_*$ and $0\leq\varepsilon\leq1/128$
admit a Borel set $F_d^{\mathrm s}$ with
\begin{equation}
 \mu_h(F_d^{\mathrm s})
  \leq\bigl[66C_{\log}(K_\sigma,\sigma)+132P^2\bigr]
                      d(1+|\log d|)^2<\infty,
 \label{eq:full-shear-region-flux}
\end{equation}
Every $y\in\mathcal B^{\mathrm s}(d,\varepsilon)\setminus
F_d^{\mathrm s}$ lies in $\mathcal D_h$, and
$\mathcal X_h(y)$ has a statistical entrance with the
chosen parameters and cap.
\end{theorem}

\begin{proof}
Choose the height and scale thresholds to satisfy
Corollary~\ref{cor:dyadic-statistical-failure} and
Lemma~\ref{lem:shear-unselected-flux}, with $d\leq h/8$.
Let $E_d$ be the excluded set of that corollary,
and put
\begin{equation*}
 F_d^{\mathrm s}
       =E_d\cup\bigcup_{j\geq0}\mathcal U_{a_j}^{\mathrm s}.
\end{equation*}
The Borel property and~\eqref{eq:full-shear-region-flux}
follow from~\eqref{eq:dyadic-statistical-failure-flux},
\eqref{eq:shear-unselected-dyadic-flux} and subadditivity.

Take $y$ outside this set and choose its annulus
$\mathcal B_j^{\mathrm s}$. Since
$y\notin\mathcal U_{a_j}^{\mathrm s}$, its wall labels
satisfy $\min\{n_2,n_3\}\leq a_j^P$, and both walls
are at most $2\varepsilon a_j$. Write $\rho=|s|$ and
$\varsigma=\operatorname{sign}s$. Then
\[
 y=(\varsigma\rho,-\log n_2,-\log n_3),\qquad
 \rho\in[a_j,2a_j],\quad (n_2,n_3)\in\Lambda_c(a_j).
\]
The small-section bound puts $y$ in $\mathcal D_h$.
Exclusion from $E_d$, together with the signed return
identity~\eqref{eq:signed-shear-orbit} and its statistical
conclusion, gives~\eqref{eq:statistical-entrance} at the
positive actual return time. The frame is $\mathsf S$
for positive shear and the identity for negative shear.
\end{proof}

The remaining section points can have zero shear or shear
comparable to a weak wall. For these points we vary a weak
wall and the shear together. The resulting radial parameter
retains a positive phase derivative even when the shear vanishes.