\section{The exact flow and the statistical theorem}
\label{sec:flow}

\subsection{The normalized vacuum equations}

We use the normalization of Brehm~\cite[Section~2.2]{Brehm}.
The variable $t$ increases towards the past singularity on the expanding
branch. Write $p=\Sigma_+$ and $s=\Sigma_-$ for the normalized shear
variables, and $N_1,N_2,N_3$ for the curvature variables. Set
\begin{align}
 Q&=N_1^2+N_2^2+N_3^2-2(N_1N_2+N_2N_3+N_3N_1),\notag\\
 S^2&=p^2+s^2,\qquad G=S^2+Q,\notag\\
 \M&=\{(p,s,N_1,N_2,N_3)\in\R^5:G=1,\ N_i>0\}.
 \label{eq:phase-space}
\end{align}
The equations are
\begin{equation}
\begin{aligned}
 p'&=Qp-2N_1^2+N_2^2+N_3^2+N_1N_2-2N_2N_3+N_1N_3,\\
 s'&=Qs+\sqrt3(-N_2^2+N_3^2+N_1N_2-N_1N_3),\\
 N_1'&=(-S^2+2p)N_1,\\
 N_2'&=(-S^2-p+\sqrt3s)N_2,\\
 N_3'&=(-S^2-p-\sqrt3s)N_3.
\end{aligned}
\label{eq:normalized-flow}
\end{equation}
These are the Hubble-normalized equations for vacuum Bianchi IX with
zero cosmological constant. The reduction from the spacetime equations
to this system is described in~\cite[Appendix~A.4]{Brehm}. We denote its
maximal flow by $\phi_t$ and the maximal interval through $q$ by $I(q)$.

\begin{lemma}[Forward existence]
\label{lem:forward}
The flow preserves $\M$. Every $q\in\M$ has $[0,\infty)\subset I(q)$.
For $t\geq0$,
\begin{equation}
 0<N_i(\phi_tq)\leq N_i(q)e^t.
 \label{eq:wall-bound}
\end{equation}
\end{lemma}

\begin{proof}
Differentiating $G$ using~\eqref{eq:normalized-flow} gives $G'=0$.
Each curvature equation has the form $N_i'=\lambda_i N_i$, so
\[
 N_i(t)=N_i(0)\exp\left(\int_0^t\lambda_i(u)\dd u\right)
\]
on the interval of existence. Positivity is preserved. The three rates
can be written
\begin{align*}
 \lambda_1&=1-(p-1)^2-s^2,\\
 \lambda_2&=1-(p+1/2)^2-(s-\sqrt3/2)^2,\\
 \lambda_3&=1-(p+1/2)^2-(s+\sqrt3/2)^2.
\end{align*}
Thus $\lambda_i\leq1$, which proves~\eqref{eq:wall-bound} as long as the
solution exists. On every bounded positive time interval the curvature
variables are bounded. The constraint gives
\[
 p^2+s^2=1-Q\leq1+2(N_1N_2+N_2N_3+N_3N_1),
\]
so the shear variables are bounded as well. A solution of the smooth
polynomial vector field cannot have a finite maximal endpoint while
remaining in a bounded subset of $\R^5$. This proves forward existence.
\end{proof}

The Kasner circle is the zero-curvature boundary
$N_1=N_2=N_3=0$, $p^2+s^2=1$. Its three Taub points are
\[
 T_1=(-1,0),\qquad
 T_2=(1/2,-\sqrt3/2),\qquad
 T_3=(1/2,\sqrt3/2).
\]
When regarded as points of $\R^5$, their three curvature coordinates are
zero. Axis permutations act simultaneously on $(N_1,N_2,N_3)$ and on
the shear components
\[
 \sigma_1=-2p,\qquad \sigma_2=p-\sqrt3s,\qquad
 \sigma_3=p+\sqrt3s.
\]
Two generators, which will also specify the event convention, are
\begin{align}
 \mathsf S(p,s,N_1,N_2,N_3)
   &=(p,-s,N_1,N_3,N_2),\notag\\
 \mathsf C(p,s,N_1,N_2,N_3)
   &=\left(\frac{-p+\sqrt3s}{2},
       \frac{-\sqrt3p-s}{2},N_2,N_3,N_1\right).
 \label{eq:axis-actions}
\end{align}
Both preserve the constraint and commute with the flow, as follows by
substitution in~\eqref{eq:normalized-flow}.

The first Taub subspace is
$\mathcal T_1=\{q\in\M:s=0,\ N_2=N_3\}$; the other two are obtained by
cyclic permutation. Put $\mathcal T=\bigcup_{i=1}^3\mathcal T_i$ and
\[
 \delta_i=2\sqrt{N_jN_k},\qquad \{i,j,k\}=\{1,2,3\}.
\]

\begin{proposition}[Decay of the curvature products]
\label{prop:attractor}
For every $q\in\M\setminus\mathcal T$ and each $i=1,2,3$,
\begin{equation}
 \delta_i(\phi_tq)\longrightarrow0\qquad(t\longrightarrow\infty).
 \label{eq:attractor-input}
\end{equation}
In particular, $N_j(\phi_tq)N_k(\phi_tq)\to0$ whenever $j\ne k$.
\end{proposition}

\begin{proof}
Brehm's global IX classification~\cite[Theorem~3, p.~45]{Brehm}
places every initial point outside $\mathcal T$ in the attracting class.
After a finite time, his local attractor theorem applies. Equation~(6.2c)
of that theorem~\cite[Theorem~2, p.~44]{Brehm} is
\eqref{eq:attractor-input}. A finite shift of time does not alter this
limit. The last assertion follows from $\delta_i^2=4N_jN_k$.
\end{proof}

\subsection{Coordinates for a curvature core}

In an ordered axis frame with $A=1+p>0$, define
\begin{equation}
 M=\frac{\sqrt3s}{A},\qquad
 (U,V,W)=\frac{\sqrt3}{A}(N_1,N_2,N_3),\qquad
 \dd\tau=A\dd t.
 \label{eq:projective-coordinates}
\end{equation}
The variable $\tau$ is used for local estimates. All arrival times in the
main theorem refer to $t$. The constraint yields the inverse formula
\begin{equation}
 \begin{split}
 D&=3+M^2+U^2+(V-W)^2-2U(V+W),\qquad A=\frac6D,\\
 (p,s,N_1,N_2,N_3)&=
 \left(A-1,\frac{AM}{\sqrt3},\frac{AU}{\sqrt3},
       \frac{AV}{\sqrt3},\frac{AW}{\sqrt3}\right).
 \end{split}
 \label{eq:projective-inverse}
\end{equation}
The inverse has $G=1$ whenever $D\ne0$, and $D>0$ corresponds to $A>0$.
For example, substituting~\eqref{eq:projective-coordinates} into $D$
gives $D=6/(1+p)$ directly from $G=1$.

Write $C=U(V+W-U)$. Quotient differentiation in
\eqref{eq:normalized-flow}, followed by division by $A$, gives
\begin{equation}
\begin{aligned}
 \dot M&=-V^2+W^2+U(V-W)-MC,\\
 \dot U&=\tfrac12\bigl(1-M^2-(V-W)^2+U^2\bigr)U,\\
 \dot V&=(M-1-C)V,\\
 \dot W&=(-M-1-C)W.
\end{aligned}
\label{eq:projective-flow}
\end{equation}
Dots denote $\tau$ derivatives throughout the paper. These identities use the constraint in
the curvature equations; the equation for $M$ follows without it.
The time change is increasing on the coordinate domain, so it preserves
the order and crossing direction of its events.

Two identities explain the usefulness of these coordinates. If
$E=(M-1)^2+V^2$, then
\begin{equation}
 \begin{aligned}
  \dot E&=-2CE+2(M-1)(U-W)^2,\\
  \frac{d}{d\tau}(VW)&=-2(1+C)VW.
 \end{aligned}
 \label{eq:energy-identities}
\end{equation}
They follow by differentiating with~\eqref{eq:projective-flow}. On the
single-wall boundary $U=W=0$, the first identity makes $E$ constant.
For positive small $U,W$, it identifies the terms that change the core
radius. The later estimates retain these terms and control their
accumulated effect through an entire era.

We write $\Dt=d/dt$ for differentiation in the original normalized
time and use $\delta$ for tangent variations. The local estimates also
use an increasing angle or a growing wall as an independent variable.
The derivative conventions are summarized here:
\begin{center}
\begin{tabular}{@{}ll@{}}
Independent variable & Derivative convention\\[3pt]
Normalized time $t$ & $\Dt$ or $d/dt$; primes in~\eqref{eq:normalized-flow}\\
Projective time $\tau$ & An overdot\\
Core angle or local wall clock & A prime after the clock is specified\\
Radial family parameter $\rho$ & A prime along the specified family
\end{tabular}
\end{center}
\subsection{Events and complete eras}
\label{subsec:events}

We now define the observations whose statistics are studied. A core starts
at an upward crossing of $V=1$ and ends at its first subsequent return to
$V=1$, with downward crossing there. Between these two events $V>1$ and
the projective coordinate domain remains valid. At the exit set
\[
 R=\sqrt{(M-1)^2+1}.
\]
There are two kinds of connection to the next core.
If $R>4$, change the ordered frame by $\mathsf S$; in projective
coordinates this replaces $(M,U,V,W)$ by $(-M,U,W,V)$.
If $2<R<4$, change the frame by $\mathsf C$ in
\eqref{eq:axis-actions}. In either case the next entrance is the first
subsequent $V=1$ event in the new frame, and its crossing is upward.
We call the first connection a reflection and the second a terminal
connection.

\begin{definition}[Complete era]
\label{def:era}
A complete era begins at the core entrance following a terminal
connection. It consists of successive cores joined by reflections,
up to and including the first core followed by a terminal connection.
Its length $K$ is the number of cores it contains. At its initial
entrance define
\begin{equation}
 \kappa=\sqrt{(M-1)^2+1},\qquad
 \widehat u=\kappa/2,\qquad x=2/\kappa.
 \label{eq:era-observables}
\end{equation}
\end{definition}

The coordinates $(\kappa,u,w)$ at an entrance record its radius and
its two small walls. Here $u$ and $w$ are the entrance values of $U$
and $W$, while the era parameter is $\widehat u=\kappa/2$.
The statistics concern $K$ and $x=1/\widehat u$. The wall amplitudes
$u,w$ control the errors in their return law. All these radii and
phases are dimensionless coordinates of the normalized state.

The existence assertion below includes every event in this definition.
The constructed tail avoids $R=2,4$, and its first era has a preceding
terminal connection on the same trajectory. At each constructed entrance the
projective state has the form
\begin{equation}
  (M,U,V,W)=\left(1+\sqrt{\kappa^2-1},u,1,w\right),
  \qquad \kappa>2,\quad u,w>0.
  \label{eq:core-entry}
\end{equation}
The identity $K=\lfloor\widehat u\rfloor$ will be a consequence of the
return estimates; $K$ is defined by counting events.
Figure~\ref{fig:era-events} shows the ordering of these events.

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{figures/era-events.pdf}
\caption{An era of three cores and the beginning of the next era.
Each schematic profile uses its own ordered frame, with entrance and
exit marked at $V=1$. The connections use $\mathsf S$ for $R>4$ and
$\mathsf C$ for $2<R<4$. A terminal connection precedes $t_m$.
The bracket counts the cores of the era, and $t_{m+1}$ marks the
following era entrance. Spacing along $t$ indicates event order.}
\label{fig:era-events}
\end{figure}
\FloatBarrier

\subsection{The statistical law}

A smooth transverse section is a three-dimensional smooth submanifold
of $\M$ whose tangent space is complementary to the flow direction.
A set is null on such a section if its intersection with the section
has measure zero for every smooth positive density. Equivalently, its
inverse image has three-dimensional Lebesgue measure zero in every
local section chart. We use the corresponding coordinate definition
for null sets in the four-dimensional manifold $\M$.

Let $\nu$ be the probability measure on $(0,1)$ given by
\begin{equation}
 \dd\nu(x)=\frac{\dd x}{\log2\,(1+x)}.
 \label{eq:gauss-measure}
\end{equation}
For a positive integer $k$ put $g_k(x)=(k+x)^{-1}$. For a finite word
$\mathbf b=(b_0,\ldots,b_{r-1})$ of positive integers, its Gauss cylinder
is the open interval
\[
 I_{\mathbf b}=g_{b_0}\circ\cdots\circ g_{b_{r-1}}((0,1)).
\]
On irrational points this is precisely the set with the indicated
initial continued fraction digits. The rational points have zero
$\nu$ measure.

\begin{samepage}
\begin{theorem}[Era statistics]
\label{thm:main}
There is a set $\Z\subset\M$, null in every regular coordinate chart
and on every smooth transverse section, such that each
$q\in\M\setminus\Z$ admits an infinite sequence of successive complete
eras. Their entrance times $t_m(q)$ are positive, strictly increasing,
and tend to infinity. Writing $K_m(q)$ and $x_m(q)$ for their lengths
and phases, at each entrance we have
\begin{equation}
 K_m=\lfloor\widehat u_m\rfloor,\qquad
 \widehat u_m>1,\qquad x_m=1/\widehat u_m\in(0,1).
 \label{eq:count-floor}
\end{equation}
For every such $q$, the following limits hold for all
$f\in C([0,1])$, all finite positive words $\mathbf b$, and all positive
integers $k,R$:
\begin{align}
 \frac1n\sum_{m=0}^{n-1} f(x_m)
   &\longrightarrow\int_0^1 f(x)\dd\nu(x),\label{eq:phase-law}\\
 \frac1n\#\{m<n:(K_m,\ldots,K_{m+r-1})=\mathbf b\}
   &\longrightarrow\nu(I_{\mathbf b}),
   \label{eq:block-law}\\
 \frac1n\#\{m<n:K_m=k\}
   &\longrightarrow\frac1{\log2}
       \log\frac{(k+1)^2}{k(k+2)},\label{eq:digit-law}\\
 \frac1n\#\{m<n:K_m\geq k\}
   &\longrightarrow\frac{\log(1+1/k)}{\log2},\label{eq:tail-law}\\
 \frac1n\sum_{m=0}^{n-1}\min(K_m,R)
   &\longrightarrow\frac{\log(R+1)}{\log2},\label{eq:truncated-law}\\
 \frac1n\sum_{m=0}^{n-1}K_m&\longrightarrow+\infty.
   \label{eq:infinite-mean}
\end{align}
The set $\Z$ can be chosen Borel, containing $\mathcal T$, and invariant
throughout each maximal existence interval:
\[
  \phi_tq\in\Z\quad\Longleftrightarrow\quad q\in\Z,
  \qquad q\in\M,\quad t\in I(q).
\]
The ordered frames $a_m(q)$ and entrance times $t_m(q)$ can be chosen
as Borel functions on $\M\setminus\Z$. Their corresponding lengths
$K_m(q)$ and phases $x_m(q)$ are also Borel functions.
This one exceptional set applies to all sections, densities, observation
functions, words, and thresholds above.
\end{theorem}
\end{samepage}

The theorem samples eras with equal weight. The divergent mean
in~\eqref{eq:infinite-mean} counts cores per era.
The invariant exceptional set also permits every finite change of the
time origin within $I(q)$.

\begin{corollary}[Two observable consequences]
\label{cor:consequences}
For each trajectory and era sequence in Theorem~\ref{thm:main}, the
frequency of adjacent eras each containing one core is
$\log(10/9)/\log2$. If $p_{\geq k}$ denotes the limit
in~\eqref{eq:tail-law}, then
\begin{equation}
 \lim_{k\to\infty}k p_{\geq k}=\frac1{\log2}.
 \label{eq:tail-coefficient}
\end{equation}
\end{corollary}

\begin{proof}
The interval $I_{(1,1)}$ is $(1/2,2/3)$, so
\[
 \nu(I_{(1,1)})=\frac{\log(5/3)-\log(3/2)}{\log2}
 =\frac{\log(10/9)}{\log2}.
\]
For the second assertion use
\[
 \frac1{k+1}\leq\log(1+1/k)\leq\frac1k
\]
and squeeze $kp_{\geq k}$ between $k/((k+1)\log2)$ and $1/\log2$.
\end{proof}

The adjacent unit-era frequency differs from the square of the
single-era frequency in~\eqref{eq:digit-law}; the joint law therefore
records dependence between consecutive lengths. Long eras have a tail
of order $k^{-1}$. They become rarer as the threshold increases, yet
their contribution makes the arithmetic mean diverge. The frequency
limit is taken at each fixed threshold before $k$ tends to infinity.