\subsection{Measurable statistical entrances on the whole section}
\label{sec:measurable-statistical-entrances}

The local constructions give actual statistical entrances together
with their times and axis frames. We now choose these data on the
whole small section. Two details are useful: the initial amplitude
can be made smaller than any prescribed positive bound, and all
successful points use one Borel choice of entrance data.

\begin{lemma}[Restriction of the amplitude cap]
\label{lem:survivor-cap-restriction}
Suppose $(B,Y)$ is a survivor with cap $B_0$, and
$0<B\leq B_1\leq1$. Then it is a survivor with cap $B_1$, with
identical digits, states and phases. In particular, a statistical
survivor remains statistical after this change of cap.
\end{lemma}

\begin{proof}
The selected entrance and return conditions
\eqref{eq:era-selected-entrance}--\eqref{eq:era-selected-return}
depend on the cap only through the upper bound on the amplitude.
At the end of a history of length $n$ that amplitude is
$B^{(4/3)^n}\leq B\leq B_1$. Thus every existing history for cap
$B_0$ is also a history for cap $B_1$, with the same return states.
This proves survival. Uniqueness of selected histories identifies
the digits and phases. Equivalently, compare both histories under
the larger cap, as in Lemma~\ref{lem:statistical-cap-monotonicity}.
\end{proof}

Fix the era parameters in their allowed range. A statistical entrance
datum at a physical state $q$ is a tuple
\begin{equation}
 (A,T,B,Y),\qquad T>0,\quad
 (B,Y)\in\mathcal S_{\mathrm{stat}},\qquad
 \iota_+(Y)=\operatorname{proj}(A\phi_Tq),
 \label{eq:statistical-entrance-data}
\end{equation}
where $A$ is one of the six ordered frames.

\begin{proposition}
\label{prop:borel-small-section-entrance}
There are $B_{\mathrm{cap}}\in(0,1)$ and $h_*>0$ such that, for
each $0<h\leq h_*$ and each $b_*>0$, one can choose $d>0$ with
$d<h/8$, a Borel set $F\subset\mathcal B(d)$ with $\mu_h(F)<\infty$,
and a Borel function $W$ assigning statistical entrance data to every
$\mathcal X_h(y)$ with $y\in\mathcal B(d)\setminus F$.
Its amplitude satisfies $B(y)\leq b_*$.
\end{proposition}

\begin{proof}
Fix the exponents $q=50$, $D=90$, $P=550$ in the small-section
constructions. They satisfy $q>3$, $D>4$, $P>6D+2$.
Take $\varepsilon=1/128$ and $L=128$, and choose a common cap
for the shear and angular statistical sets as in
Theorem~\ref{thm:whole-small-section-statistics}.

We specify a countable partition on which the actual entrance
formulas can be used. In the open section
\[
 \mathcal B^\circ(d)=\{y:|s|<d,\ \max(n_2,n_3)<d\},
 \qquad n_j=e^{-\beta_j},
\]
use shear coordinates when $\max(n_2,n_3)\leq\varepsilon|s|$,
and angular coordinates otherwise. The radial variable is
$r=|s|$ in the first case and $r=\max(n_2,n_3)$ in the second.
Discard the countable set of radial levels
$r=d2^{-j-1}$, and use the unique open interval
$(a,2a)$ with $a=d2^{-j-1}$ containing $r$.
In the shear part the sign of $s$ distinguishes two pieces.
In the angular part take the order $n_2\geq n_3$ on one piece and
$n_3>n_2$ on the other. Their parameters are respectively
\[
 (r,\theta,\zeta)=(n_2,s/n_2,n_3),\qquad
 (r,\theta,\zeta)=(n_3,-s/n_3,n_2).
\]
Here $|\theta|<L$. The negative shear piece uses the reflected
positive shear family, including the exchange of its two wall labels.
These are disjoint Borel pieces with explicit Borel inverses.

On each piece exclude points where the smaller-wall selection
condition $\min(n_2,n_3)\leq a^P$ fails, or where the output
pair $(B,Y)$ is outside the common statistical set. These conditions
are Borel by the joint regularity and measurability proved for the
two families. An excluded point satisfying the selection condition belongs to the
corresponding radial excluded set estimated in
Sections~\ref{sec:small-section-shear-region}
and~\ref{sec:whole-small-section}; an unselected point belongs to
the finite logarithmic regions estimated there. Cap monotonicity
ensures the same containment when the common cap is larger than
the cap used for either estimate.

Add the discarded radial levels and the outer boundary
$\mathcal B(d)\setminus\mathcal B^\circ(d)$ to $F$.
These sets are null in logarithmic coordinates. Each radial level
is contained in the union of the four coordinate levels
$s=\pm a$, $n_2=a$, $n_3=a$; the outer boundary is contained in
the analogous levels with $a=d$. Thus they also have zero $\mu_h$
measure. The regional bounds give
\[
 \mu_h(F)\leq C_h d(1+|\log d|)^2<\infty.
\]
Decrease $d$ so that the whole section lies in $\mathcal D_h$ and
all local constructions apply.

On a successful shear piece the initial amplitude bound is
$800n_2n_3/a^4\leq800a$. On an angular piece it is
$6400M_L\zeta/a^3\leq51200(L+1)a$, with $M_L=8(L+1)$.
Since $a\leq d$, decreasing $d$ once more makes both bounds at
most $b_*$. This does not change the common cap.

The entrance time, state and amplitude are jointly Borel on each
selected parameter domain; the frame takes one of finitely many
values. Compose them with the explicit piece inverses and use the
corresponding formula on each successful piece. A countable Borel
partition makes the resulting function $W$ Borel. Extend it by a
fixed tuple elsewhere. The orbit identities for both shear
signs and both wall orders give~\eqref{eq:statistical-entrance-data}
at every successful point.
\end{proof}

The partition gives a Borel choice of entrance and preserves the
return identities and measure estimates on each piece.