\subsection{Global capture and a common invariant exception}
\label{sec:global-statistical-capture}

Choose $h\leq1/16$ and $d<h/8$ as in
Proposition~\ref{prop:borel-small-section-entrance}. In the labeled
section family~\eqref{eq:small-axis-section-family}, let
\[
 \mathcal F=\{(i,y):y\in E\cap F\},\qquad
 \mathcal H=(\{1,2,3\}\times E)\setminus\mathcal F.
\]
The excluded set has finite measure, since
$\mu_{\mathrm{ax}}(\mathcal F)\leq3\mu_h(F)<\infty$.
Every point of $\mathcal H$ has the Borel statistical entrance
obtained by conjugating $W(y)$ by its axis frame.

\begin{lemma}
\label{lem:axis-family-capture}
For $\mu_{\mathrm{ax}}$-almost every point of the labeled section
family with infinitely many future intersections, some nonnegative
time reaches $\mathcal H$.
\end{lemma}

\begin{proof}
The first-return construction of
Section~\ref{sec:measurable-first-returns} applies to this finite
family. To check this explicitly, choose the source and target
labels in each local arrival relation. There are only nine such
pairs. The negative shear bound on every target is uniform, so the
crossings have no finite-time accumulation, just as in
Lemma~\ref{lem:finite-section-crossings}. Their first positive
arrival is attained. The countable event neighborhoods and the
least-time selection therefore give a Borel first return $R$.

The physical section images are disjoint, so orbit uniqueness makes
this labeled return injective. Axis permutations preserve the
constraint volume and flux. On each local branch its weighted
Jacobian $j$ is consequently the same positive-time multiplier
as in~\eqref{eq:arrival-weighted-jacobian}, and $j>1$.
Partitioning the return domain into countably many Borel branch
pieces gives
\[
 \mu_{\mathrm{ax}}(R(B))=\int_B j\,d\mu_{\mathrm{ax}}
\]
for every Borel subset $B$ of that domain.

Apply the capture argument of
Section~\ref{sec:capture-by-expanding-returns} to $\mathcal F$.
The set whose legitimate iterates all remain in $\mathcal F$ is
forward invariant and has finite measure. Strictness of $j>1$
forces its measure to be zero. Its finite backward iterates are
also null by the change of variables formula.
Finally, infinitely many actual intersections are equivalent to
indefinitely continuing first returns: local finiteness orders
the hitting times, and on a bounded interval permits only finitely
many of them. Thus almost every recurrent point reaches the
successful set, possibly at time zero.
\end{proof}

\begin{lemma}
\label{lem:borel-null-saturation}
If $Z\subset\mathcal D_h$ is Borel and has zero three-dimensional
Lebesgue measure, then
$\operatorname{Sat}(\mathcal X_{i,h}(Z))$ is a Borel invariant subset
of $\M$, null on every transverse section and in every regular
four-dimensional chart.
\end{lemma}

\begin{proof}
The domain of the map
\[
 (y,t)\longmapsto\phi_t(\mathcal X_{i,h}(y))
\]
is open, and the map is locally injective. At time zero its derivative
has the three independent section tangents and the complementary
flow direction; at any other existing time the flow transports
this rank-four property. The inverse function theorem in constraint
coordinates gives local injectivity. Cover the domain by countably
many such open sets. On each, the image of its Borel intersection
with $Z\times\R$ is Borel, by the Borel image theorem for continuous
injective maps between standard Borel spaces. Their union is the
full saturation, so it is Borel.

The maximal-flow composition and inverse identities show that
membership in the saturation is unchanged at every existing time.
It lies in $\M$ because the physical phase space is invariant.
Its transverse nullity follows from
Lemma~\ref{lem:transverse-null-target}, applied to the reverse
arrival from an arbitrary transverse section to $\mathcal X_{i,h}(Z)$.
Its four-dimensional nullity is~\eqref{eq:full-orbit-saturation}.
\end{proof}

\begin{theorem}[Global statistical entrance]
\label{thm:common-global-entrance}
For every prescribed $b_*>0$, there are a Borel invariant set
$\Z\subset\M$, containing $\mathcal G$ and null on all transverse
sections and in all regular four-dimensional charts, and a Borel
choice of statistical entrance data~\eqref{eq:statistical-entrance-data}
for every $q\in\M\setminus\Z$, with amplitude $B(q)\leq b_*$.
\end{theorem}

\begin{proof}
For a labeled section point $z=(i,y)$, consider the property that
pairwise product decay along its orbit implies a nonnegative-time
arrival in $\mathcal H$. Theorem~\ref{thm:small-axis-recurrence}
and Lemma~\ref{lem:axis-family-capture} imply this property almost
everywhere on the section family. For each label choose a Borel
Lebesgue-null set $Z_i\subset E$ containing every point where it
fails. Such a hull exists because the points without such an arrival
have outer measure zero; zero flux and zero coordinate volume agree on $\mathcal D_h$,
where $w_h$ is positive. Define
\begin{equation*}
 \Z=\mathcal G\ \cup\ \bigcup_{i=1}^3
               \operatorname{Sat}(\mathcal X_{i,h}(Z_i)).
\end{equation*}
Theorem~\ref{thm:geometric-exception-null} and
Lemma~\ref{lem:borel-null-saturation} give all its stated properties.

If $q\notin\Z$, Theorem~\ref{thm:small-axis-recurrence} gives a
future intersection $\phi_sq=\mathcal X_{i,h}(y)$, $y\in E$.
This point is outside the saturation of $Z_i$, and hence $y\notin Z_i$.
The orbit has product decay by Proposition~\ref{prop:attractor},
and this condition is unchanged by a finite time shift. By the
defining property outside $Z_i$, a later nonnegative time reaches
$\mathcal H$. Thus every $q\notin\Z$ reaches the same fixed
successful target.

It remains to make the entrance choice Borel in $q$. The time-zero
case is retained. For any physical point reaching $\mathcal H$,
its nonnegative hitting times have an attained least element:
in a compact time interval the uniformly transverse target family
has only finitely many crossings. The local implicit arrival
functions cover its spacetime hitting relation by countably many
event neighborhoods. Restrict each branch to the Borel condition
that its endpoint belongs to $\mathcal H$. Taking their infimum,
and the first branch attaining it, gives a Borel least time
$\tau(q)\geq0$ and Borel arrival state $\phi_{\tau(q)}q$.
This is the construction of
Section~\ref{sec:measurable-first-returns} with an arbitrary
physical source and the finite target family.

The labeled section chart is continuous, injective and Borel on
$\{1,2,3\}\times E$; its inverse on its image is Borel.
It therefore recovers a Borel pair $(i(q),y(q))\in\mathcal H$ from
the arrival state. At that point use $W(y(q))$ from
Proposition~\ref{prop:borel-small-section-entrance}. If it is
$(A_0,T_0,B,Y)$, set
\[
 A=A_0A_{i(q)},\qquad T=\tau(q)+T_0.
\]
The product frame is one of the same six frames, and all these
data are Borel. Commutation with the flow gives
\[
 \mathcal I(Y)=A_0\phi_{T_0}\mathcal X_h(y(q))
             =A_0A_{i(q)}\phi_{\tau(q)+T_0}q.
\]
Thus~\eqref{eq:statistical-entrance-data} holds. The total time is
strictly positive because $T_0>0$, even when $\tau=0$.
The chosen initial amplitude is at most $b_*$.
\end{proof}