\subsection{Proof of the statistical theorem}
\label{sec:proof-statistical-theorem}

The whole-section estimate supplies finite excluded flux, recurrence
supplies repeated visits, and strict expansion gives capture.
Theorem~\ref{thm:common-global-entrance} has combined these results into
a measurable statistical entrance of arbitrarily small amplitude.
We now join its selected returns into one infinite sequence of complete
eras and transfer their statistics to the original trajectory.

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Choose $\eta>0$ below the finitely many thresholds in the era,
section and entrance results, with $\eta\leq1/40000$. Set
\[
 J=45,\qquad \sigma=1/400.
\]
These satisfy $J\geq20$ and $2J\sigma<1$.
Choose a positive amplitude cap $b_*<1$ below the thresholds for
the Borel selected return, the actual event itineraries, the
ordinary return-time functions and the absence of finite-time
accumulation in Section~\ref{sec:era-original-time}.
Theorem~\ref{thm:common-global-entrance} gives one invariant null
set $\Z$ and Borel entrance data $(A,T,B,Y)$ on its complement,
with $B\leq b_*$.

The statistical survivor in those data was obtained with the cap
used by the section construction. Apply
Lemma~\ref{lem:survivor-cap-restriction} to replace that cap by $b_*$.
Its digits, states and phases are unchanged. The Borel map
$q\mapsto(B(q),Y(q))$ takes values in this one Borel survivor set.
Proposition~\ref{prop:era-state-coding} therefore gives Borel
digits $a_n(q)$, states $Y_n(q)$ and phases $x_n(q)$.

Starting at the positive original time $T(q)$ in the frame $A(q)$,
Propositions~\ref{prop:era-survivor-physical-orbit}
and~\ref{prop:era-time-frame-borel} assemble these states into one
actual orbit with Borel frames and entrance times. Every selected
return consists of its ordered cores, reflections and terminal
connection, by Proposition~\ref{prop:era-original-events}.
The times are positive and strictly increasing, and they tend to
infinity by Theorem~\ref{thm:era-no-accumulation}.

Delete the first selected era. Each remaining era then has its
preceding terminal connection on this same trajectory, so it is
complete in the sense of Definition~\ref{def:era}.
Corollary~\ref{cor:complete-eras-from-survivor} gives the required
sequences by setting $K_m=a_{m+1}$ and using the corresponding
shifted states, frames and times. At every entrance the state is
$\iota_+(Y_{m+1})$, and its radius is the first coordinate of
$Y_{m+1}$. The selected entrance inequalities give
\[
 K_m=\left\lfloor\frac{Y_{m+1,0}}2\right\rfloor,
 \qquad Y_{m+1,0}>2,
 \qquad x_m=\frac2{Y_{m+1,0}}\in(0,1).
\]
These are precisely~\eqref{eq:count-floor} and the physical
observables~\eqref{eq:era-observables}.

The statistical entrance lies in the common statistical set, so
its selected digits and phases satisfy all six limits in the
theorem. A fixed shift preserves those limits, including the
divergent nonnegative mean, by
Lemma~\ref{lem:era-statistics-finite-shift}. The same limits thus
hold for the complete eras just chosen.

Finally, $\Z$ contains $\mathcal T$ because it contains $\mathcal G$.
Its invariance and its nullity in every transverse and
four-dimensional chart are already part of
Theorem~\ref{thm:common-global-entrance}. Every measure obtained by
putting a nonnegative density on coordinate Lebesgue measure is
absolutely continuous with respect to it. Hence the same $\Z$ is
null for every smooth positive density in the statement. The
statistical set was chosen using all finite cylinders and the
single common Gauss-typical set, before the initial section or
observation function was specified. This proves all assertions on the
same complement of $\Z$.
\end{proof}

\section{Consequences and further questions}
\label{sec:concluding-discussion}

The Gauss law describes a sequence of events on the exact vacuum
Einstein trajectory. In particular, the long-era tail and the
logarithmically growing truncated mean count actual curvature
cores. Almost every physical initial condition, in transverse
or four-dimensional measure, admits such an era tail.
Short eras carry a substantial fraction of the frequency, while
the rare long eras prevent the arithmetic mean from being finite.

The joint distributions describe correlations along each typical orbit.
Corollary~\ref{cor:consequences} gives the adjacent unit-era frequency
$\log(10/9)/\log2$, while the general cylinder formula determines
every finite block. The same scalar distributions occur in the
billiard description~\cite{DL}; the present conclusion obtains
them for the exact normalized flow with a common invariant
exceptional set. The conclusion applies to every initial distribution
that is absolutely continuous in local coordinates.

Two parts of the proof account for its range near the Taub points.
Within an era, phase errors are summed against the loss of transverse
amplitude, so the error bound does not grow with the number of cores.
On the small section, shear and angular families cover complementary
directions, including zero shear and arbitrarily unequal weak walls.
Their remaining excluded set has finite flux. Strict expansion of
that flux and actual recurrence then turn a local statistical
construction into a full-measure statement.

The nearest further questions concern duration and fluctuations:
statistics weighted by the time spent in cores and connections,
and rates of convergence of the empirical frequencies.

The full axis sequence retains information discarded by the scalar
era phase. Its joint statistics, together with extensions to matter
models or nonzero cosmological constant, provide further directions
for research. The exact event construction supplies observables for
these questions.