\subsection{Repeated visits to the small section family}
\label{sec:recurrent-axis-sections}

We use three copies of the small section, one for each choice of
dominant wall. This finite family accommodates the Taub point visited
by an orbit without changing the section height or its transverse
size along that orbit.

For $i=1,2,3$, let $A_i$ be the axis permutation interchanging axis
$1$ with axis $i$, with $A_1$ the identity. These are among the six
ordered frames and are involutions. Define
\begin{equation}
 \mathcal X_{i,h}(y)=A_i^{-1}\mathcal X_h(y),\qquad
 E=\mathcal B(d)\cap\mathcal D_h,
 \qquad
 \Sigma_{h,d}=\bigcup_{i=1}^3\mathcal X_{i,h}(E),
 \label{eq:small-axis-section-family}
\end{equation}
where $\mathcal B(d)$ is~\eqref{eq:whole-small-section}.
We retain the label $i$ when using section coordinates, and give the
disjoint coordinate space $\{1,2,3\}\times E$ the measure
\begin{equation}
 \mu_{\mathrm{ax}}(H)=\sum_{i=1}^3\mu_h(H_i),\qquad
 H_i=\{y:(i,y)\in H\}.
 \label{eq:axis-family-measure}
\end{equation}
For $d<h/8$ the physical images are disjoint: in the $i$th image
the $i$th wall is $h$, while both other walls are less than $h$.
Each individual chart is injective, so the labeled chart is injective
as well.

\begin{theorem}[Recurrence of the section family]
\label{thm:small-axis-recurrence}
Let $0<h\leq1/16$ and $0<d<h/8$. Every
$q\in\M\setminus\mathcal G$ meets $\Sigma_{h,d}$ at arbitrarily
late times. More precisely, there is a fixed $i$ such that, for every
$R$ and $\zeta>0$, some $t>\max(R,0)$ has
\[
 \phi_tq=\mathcal X_{i,h}(s,\beta_2,\beta_3),\qquad
 |s|<d,\quad e^{-\beta_2},e^{-\beta_3}<d,
\]
and the squared transverse Taub radius in this frame is less than
$\zeta$. In particular, for $\mu_{\mathrm{ax}}$-almost every point
of the labeled section family, the original flow has infinitely
many future intersections with this family.
\end{theorem}

\begin{proof}
By Theorem~\ref{thm:geometric-exception-null}, the orbit has pairwise
product decay and arbitrarily late visits to the union of arbitrarily
small Taub neighborhoods. At least one fixed Taub point has this
property. Otherwise, for each of the three points one could choose
a radius and a starting time after which its neighborhood is never
visited. The minimum of the radii and the maximum of the starting
times would contradict the visits to the union.

Use $A_i$ to send this fixed Taub point to $T_1$. It preserves the
Euclidean metric, permutes the wall products, and commutes with the
flow. The initial transverse radius in this frame is positive,
since $q\notin\mathcal T$. Corollary~\ref{cor:late-fixed-axis-crossings}
gives the asserted crossings in that frame. Their negative shear
coordinate $p$ and the constraint reconstruct exactly the negative
square root in $\mathcal X_h$. Taking
$\beta_j=-\log N_j$ therefore gives the claimed chart coordinates.

Each $\mathcal X_{i,h}$ is a smooth transverse chart on
$\mathcal D_h$. The pullback of $\mathcal G$ is null by
Theorem~\ref{thm:geometric-exception-null}, and remains null for
$\mu_h$, whose density is positive and finite there.
Outside these three null sets, the first part gives unbounded
future hitting times and hence infinitely many intersections.
\end{proof}

This supplies the recurrence required by the capture argument.
The remaining step is to apply capture on the labeled section family,
choose its successful core entrances measurably, and remove one
invariant null set for all initial sections at once.