\subsection{Taub recurrence in transverse measure} \label{sec:taub-recurrence-in-measure} The late core entrance turns the null set of uniformly avoiding core states into a statement on every transverse section. We first explain the measure transfer, since an arbitrary union over arrival times would not preserve measure zero. \begin{lemma} \label{lem:transverse-null-target} Let $C:U\to\M$ be a $C^1$ transverse chart, with $U\subset\R^3$ open. Let $Z$ be a null set in the coordinates of a core or fixed-wall section, in any ordered frame. The set of $y\in U$ whose orbit meets $Z$ at a transverse arrival has outer Lebesgue measure zero. The arrival time may have either sign, provided it belongs to the maximal interval of the orbit. \end{lemma} \begin{proof} At any such pair $(y_0,t_0)$, transversality at the target and the implicit function theorem give a local arrival time $t(y)$ and a $C^1$ map $R(y)$ to target coordinates. It is a local diffeomorphism. Indeed, differentiation of the actual arrival gives \[ D(C_{\mathrm{target}}\circ R)[v] =D\phi_{t(y)}(C(y))DC[v]+F(\phi_{t(y)}C(y))Dt[v]. \] Contracting the constraint volume with $F$ removes the second term. The remaining flux is a nonzero multiple of the initial transverse flux, by~\eqref{eq:section-flux-transport}. Thus $\det DR\ne0$, as in Proposition~\ref{prop:local-arrival-measure}. A local $C^1$ inverse sends null sets to null sets, by covering its domain by countably many compact subsets on which it is Lipschitz. Hence $R^{-1}Z$ is locally null. Cover the set of arrival pairs in $U\times\R$ by these open neighborhoods. Second countability gives a countable subcover. In each neighborhood the source coordinates of all relevant pairs belong to the corresponding null set $R^{-1}Z$. Their countable union contains every source point under consideration. \end{proof} We will also use the four-dimensional version of this observation. If $C:U\subset\R^3\to\M$ is $C^1$ and $Z\subset U$ is null, then \begin{equation} \operatorname{Sat}(C(Z)) =\{\phi_t(C(z)):z\in Z,\ t\in I(C(z))\} \label{eq:full-orbit-saturation} \end{equation} is null in every regular four-dimensional chart of $\M$. The relevant source points lie in $Z\times\R$, a four-dimensional null set by Tonelli. The orbit map is $C^1$ on its open domain. For any linear $P:\R^5\to\R^4$, its composition with $P$ therefore sends this null set to a null set, using a countable local Lipschitz cover. If $C_4$ is a regular four-dimensional chart, choose $P$ to be a left inverse of $DC_4$ at a given point. Then $P\circ C_4$ is a local diffeomorphism, so the chart preimage of the saturation is locally null. A countable cover proves the assertion. \begin{proposition} \label{prop:eventual-avoidance-null} Fix $\epsilon>0$. The set of $q\in\M$ satisfying \eqref{eq:pair-product-decay} and eventually remaining at distance at least $\epsilon$ from all Taub points is null on every $C^1$ transverse section and in every regular four-dimensional chart. \end{proposition} \begin{proof} Choose $B>0$ so that Proposition~\ref{prop:taub-avoiding-core-null} applies. Proposition~\ref{prop:late-small-compact-core} shows that each orbit in question reaches, in one of the six frames, a core whose phase coordinates $(x,u,w)$ belong to $\mathcal A_{\epsilon,B}$. Its arrival is transverse. The change from these coordinates to physical core coordinates is $C^1$ on $00$. Hence their images in any core section coordinates form a null set. Apply Lemma~\ref{lem:transverse-null-target} to each of the six frames. Their finite union proves the assertion on arbitrary transverse charts. For four-dimensional charts use \eqref{eq:full-orbit-saturation} for the same six null sources. The inverse flow from the core entrance recovers the original point, so these saturations contain the whole set in the proposition. \end{proof} Let $\mathcal G$ consist of the Taub subspaces together with the eventually uniformly avoiding trajectories: \begin{equation} \begin{split} \mathcal G=\mathcal T\ \cup\ \{q\in\M:\;&\text{there exist } \epsilon>0,\ R\geq0\text{ such that}\\[-2pt] &\dist(\phi_tq,T_i)\geq\epsilon \text{ for every }t\geq R\text{ and }i\}. \end{split} \label{eq:geometric-exception} \end{equation} \begin{theorem} \label{thm:geometric-exception-null} The set $\mathcal G$ is Borel and invariant throughout every maximal existence interval. It is null on every $C^1$ transverse section and in every regular four-dimensional chart. Every $q\in\M\setminus\mathcal G$ satisfies~\eqref{eq:pair-product-decay}, lies outside all Taub subspaces, and has arbitrarily late visits to arbitrarily small neighborhoods of the Taub points. \end{theorem} \begin{proof} The Taub subspaces are closed relative to $\M$ and invariant. For the first subspace, this also follows from $\mathcal T_1=\{q:\mathsf S q=q\}$: commutation with the flow and uniqueness show that the fixed point condition holds at one time if and only if it holds at every existing time. The second set in~\eqref{eq:geometric-exception} equals \begin{equation*} \bigcup_{n,m\in\N}\ \bigcap_{t\geq m}\ \bigcap_{i=1}^3 \{q\in\M:\dist(\phi_tq,T_i)\geq1/(n+1)\}. \end{equation*} For fixed $n,m$, the intersection is closed relative to $\M$, because every positive-time flow map is continuous there. Only the two outer unions need be countable. This proves Borel measurability. The eventual condition is unchanged by any fixed shift of time within a maximal interval, proving invariance. We verify nullity of $\mathcal T_1$ on a transverse chart $C$. At a point of this subspace, $s=0$, $N_2=N_3$ and $s'=0$. If $D(s\circ C)$ vanished, every vector in the determinant defining the transverse flux, including $q/2$ and $F(q)$, would have zero $s$ component. The flux would vanish, a contradiction. Thus each such point lies on a regular level of the scalar function $s\circ C$. The implicit function theorem and a countable cover make its zero set null in three dimensions. Repeat in the other two frames. For four-dimensional nullity, parametrize the ambient first Taub subspace by $(p,n,b)\mapsto(p,0,n,b,b)$. After any linear projection to $\R^4$, its image is the smooth image of a null coordinate hyperplane in $\R^4$. The same local inverse argument just used for~\eqref{eq:full-orbit-saturation} proves nullity in every regular chart. Axis permutations give the other two subspaces. Outside $\mathcal T$, Proposition~\ref{prop:attractor} supplies \eqref{eq:pair-product-decay}. For each $n$, apply Proposition~\ref{prop:eventual-avoidance-null} with $\epsilon=1/(n+1)$. The countable union covers the eventually avoiding part of $\mathcal G$ outside $\mathcal T$, proving nullity. Finally, negating eventual avoidance says precisely that for every $\epsilon>0$ and every $R$ there are $t\geq R$ and $i$ with $\dist(\phi_tq,T_i)<\epsilon$. \end{proof} Recurrence near Taub follows from product decay, the actual core construction, and the exclusion of bounded digit survivors.