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 pdftitle={Equidistribution of Primitive Sextic Torus Packets}}
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\title{Equidistribution of Primitive Sextic Torus Packets}
\author{OpenAI}
\date{October 2026}

\begin{document}
\maketitle

\begin{abstract}
Let $K$ range over totally real sextic fields with no proper
intermediate subfield, and let $D_K$ be the discriminant. For each
$K$, take the complete finite packet of compact diagonal orbits attached
to the ideal classes of $\mathcal O_K$, with all coordinate-sign
translates included and each orbit weighted by its diagonal-orbit
volume. We prove that these packets become equidistributed with respect
to Haar probability on $\SL_6(\Z)\backslash\SL_6(\R)$ as $|D_K|\to\infty$,
with no escape of mass. The obstruction special to composite degree~$6$
is that the measure-classification theorem leaves, as candidates for a
limit, properly embedded block-diagonal orbits of type $3+3$ and
$2+2+2$. The paper excludes these using a conditioning scheme at
intermediate scales: it produces leafwise plaque probabilities inside
each limiting state that retain arithmetic information from the growing
discriminant. A product rule for these probabilities exhibits enough
root-activity to force every homogeneous component to be full.
\end{abstract}

\tableofcontents

\section{Guide for the reader}\label{sec:guide}

This first section explains the main objects and the main theorem
without using any of the machinery that appears later.

\subsection*{Lattices and the space $X_6$}

A \emph{lattice} in $\R^6$ is a discrete grid: the set of all integer
combinations $m_1 v_1+\cdots+m_6 v_6$ of six linearly independent
vectors. Two lattices are considered the same up to scaling of volume,
and by standard linear algebra any unimodular (volume-one) lattice can be
written $\Lambda = \Z^6 g$ for a unique coset $g\in
\SL_6(\Z)\backslash\SL_6(\R)$. This coset space is our universe
\[
 X_6=\SL_6(\Z)\backslash\SL_6(\R),
\]
a quotient of a Lie group. It carries a natural probability measure
$m_6$, the $\SL_6(\R)$-invariant (Haar) probability on $X_6$.

Distributing a random lattice with respect to $m_6$ means choosing its
shape ``uniformly at random.'' Equidistribution of a sequence of
probability measures on $X_6$ is the statement that this uniform choice
is what you get in the limit.

\subsection*{Number fields, ideals, and lattices}

Fix a \emph{totally real number field} $K$ of degree six: a finite
extension $\Q\subset K$ of degree six all of whose archimedean
embeddings $\sigma_1,\dots,\sigma_6\colon K\to\R$ take real values.
Denote by $\mathcal O_K$ the ring of integers. Every nonzero ideal
$I\subset\mathcal O_K$ becomes a lattice in $\R^6$ through the map
$d\mapsto(\sigma_1(d),\dots,\sigma_6(d))$, after rescaling to covolume
one.

The multiplicative units $\mathcal O_K^\times$ act on this lattice by
coordinated rescaling. A fundamental domain for this action is a
bounded region of $A_6$ (the diagonal subgroup of $\SL_6(\R)$), and the
orbit closes into a compact torus. Different nonzero ideals in the same
ordinary ideal class give the same closed $A_6$-orbit up to sign issues,
which we absorb by also letting the lattice be acted on by all
determinant-$\pm1$ sign matrices.

The \emph{sextic packet} $\mathcal P_{K,\sigma}$ is the set of distinct
compact orbits so obtained.
Equipping each orbit with its invariant probability and weighting orbits
by their $A_6$-volume produces a single probability $\mu_{K,\sigma}$ on
$X_6$.

\subsection*{The main theorem}

\begin{theorem}[informal]\label{thm:main-informal}
Let $K_i$ range over totally real sextic fields with no proper
intermediate subfield, with $|D_{K_i}|\to\infty$. For any choice of
embedding ordering, the packet probabilities satisfy
\[
 \mu_{K_i,\sigma_i}\longrightarrow m_6
\]
weakly on $C_c(X_6)$. In particular there is no escape of mass: for every
$\epsilon>0$ there is a compact $C\subset X_6$ with
$\mu_{K_i,\sigma_i}(C)\ge 1-\epsilon$ for all $i$ large.
\end{theorem}

The precise version is Theorem~\ref{thm:main}. The italic words
``no proper intermediate subfield'' (that is, $K$ is \emph{primitive})
play two roles: they make every nonrational integer generate the whole
field, which gives uniform separation in the dual lattice used in the
arithmetic estimates, and they ensure that the fixed-point obstruction to
a proper homogeneous limit---see below---can actually be detected.

\subsection*{Why degree~$6$ is different}

In degree two, Duke's theorem (and its dynamical proof by Einsiedler,
Lindenstrauss, Michel, and Venkatesh) says that quadratic packets
equidistribute. The mechanism is that entropy of the limiting measure
already classifies every limiting $A$-invariant component. In degree
three the same thing works: Einsiedler--Katok--Lindenstrauss prove that
positive entropy forces the component to be Haar---there are no proper
algebraic alternatives.

In degree six, six is composite, so there are block-diagonal closed
subgroups $J\subset\SL_6(\R)$ that contain the whole diagonal $A_6$ and
whose orbits $xJ$ admit finite invariant measure. Standard EKL theory
does not exclude those. The alternatives are
\[
 J \text{ of type } 3+3 \quad\text{or}\quad J \text{ of type } 2+2+2.
\]
The goal of the paper is to rule them out.

\subsection*{How to rule them out: activity of roots}

For $j\ne k$ in $\{1,\dots,6\}$, let $\beta_{jk}(b)=b_k-b_j$ be the
associated root on the diagonal algebra, and let
$U_{jk}=\{1+tE_{jk}:t\in\R\}$ be the one-parameter unipotent subgroup.
On the space of lattices, $U_{jk}$ moves one coordinate entry.

At a limiting (ergodic, invariant) probability $\nu$, the one-dimensional
subgroup $U_{jk}$ acts. We call the root \emph{active} at almost every
$x$ if the conditional measure of $\nu$ along the $U_{jk}$-leaf through
$x$ is not concentrated at a point---that is, if the plaque measure of
$U_{jk}$ at $x$ is supported beyond the identity. Activity is measurable
and transitive under composition of roots.

The key strategy is:
\begin{enumerate}
 \item If every root is active at almost every point of $\nu$, then
 $\nu$ cannot be supported on a proper block-diagonal orbit: a proper
 block-diagonal orbit is fixed pointwise by a nonidentity diagonal, and
 any root crossing two blocks (with unequal diagonal entries) is
 inactive at every such point.
 \item The arithmetic tube estimate (Proposition~\ref{prop:bowen})
 supplies the basic input: two-sided diagonal tubes of a packet have
 exponentially small mass in a range of times $[T_{-,i},T_{+,i}]$
 with $T_{+,i}/T_{-,i}\to\infty$.
 \item An entropy argument on long segments shows that every oriented
 cut is active on a positive proportion of long trajectories.
 \item A conditional-information argument upgrades this to full
 activity at almost every state.
\end{enumerate}

\subsection*{Intermediate-resolution conditioning}

A naive attempt to reason about ``conditional measures along $U_{jk}$
leaves'' fails: the exact conditional measures of a compact packet
along a unipotent orbit are Dirac masses at single points, because the
orbit is closed and the packet measure is analytic and finite. To detect
spread, we must instead condition \emph{before} passing to the limit,
at intermediate resolutions. For a unipotent group $V$ and a plaque
chart $X\supset\mathcal B=\Phi(Q\times B)$, define for each $\mu_i$ and
at depth $l$ the distribution of the plaque coordinate $h$ conditioned
on a small cell in $Q$ of side $\sim 2^{-l}$. As $l$ varies on a broad
range of scales $l_{-,i}\prec l\prec l_{+,i}$, the conditional
probabilities change very little; choosing interior depths and passing
to subsequences produces a limiting state whose data include these
probabilities. The ``plaque rule'' of the paper states that, at a
limiting state, these recorded probabilities glue into a projective,
locally finite measure $m^V_\omega$ on $V$. The measure $m^V_\omega$
encodes exactly the arithmetic that was available at scale $T_{-,i}$
and above; it is much richer than the exact leafwise measure.

\subsection*{What the reader needs}

Sections~\ref{sec:arithmetic}--\ref{sec:homogeneous} contain the
arithmetic and measure-classification foundations. Sections
\ref{sec:conditioning}--\ref{sec:product} build the intermediate
conditional probabilities and the product rule. Sections
\ref{sec:activity}--\ref{sec:combination} prove activity. Section
\ref{sec:assembly} concludes.


\section{Introduction}\label{sec:introduction}

We now state the objects formally. Section~\ref{sec:guide} contains the
same content without the technical notation.

\subsection{Packets and the main theorem}

Let
\[
 X_6=\SL_6(\Z)\backslash\SL_6(\R),\qquad
 A_6=\{\diag(e^{u_1},\dots,e^{u_6}):\textstyle\sum_j u_j=0\}.
\]
Rows act on the right: $xg$ denotes right multiplication of the coset
representative by $g\in\SL_6(\R)$. We denote by $m_6$ the
$\SL_6(\R)$-invariant probability on $X_6$. Haar measure on $A_6$ is
$\dd u_1\cdots\dd u_5$ with $u_6=-\sum_{j=1}^5u_j$.

A degree-six number field $K$ is \emph{primitive} if it has no proper
intermediate field. Fix an ordering $\sigma=(\sigma_1,\dots,\sigma_6)$
of the real embeddings and let $\mathcal O_K$ denote the ring of
integers. For a nonzero fractional ideal $I$, put
\[
 \sigma(I)=\{(\sigma_1(d),\dots,\sigma_6(d)):d\in I\},\qquad
 \Lambda_{I,\sigma}=\covol(\sigma(I))^{-1/6}\sigma(I).
\]
This is a unimodular lattice in $\R^6$, hence determines a point of
$X_6$.

Choose one representative $I$ for each ordinary ideal class. For each
sign matrix $w=\diag(\epsilon_1,\dots,\epsilon_6)$ with
$\epsilon_j\in\{\pm1\}$, form the orbit $\Lambda_{I,\sigma}wA_6$. By
Dirichlet's unit theorem the orbit is compact. Two such orbits (with the
same $I$) coincide as subsets of $X_6$ exactly when the two sign
matrices differ by the signature of a unit; see
Section~\ref{sec:arithmetic}. The \emph{sextic packet}
$\mathcal P_{K,\sigma}$ is the set of distinct compact orbits obtained
this way. Replacing the ideal-class representatives produces the same
packet, so the packet depends only on $K$ and the ordering $\sigma$ of
the embeddings.

For $\mathcal O\in\mathcal P_{K,\sigma}$, let $m_{\mathcal O}$ be the
invariant probability on $\mathcal O$ and $\vol_A(\mathcal O)$ its
$A_6$-volume. The \emph{packet probability} is
\begin{equation}\label{eq:packet-definition}
 \mu_{K,\sigma}=
 \frac{\sum_{\mathcal O\in\mathcal P_{K,\sigma}}
       \vol_A(\mathcal O)\,m_{\mathcal O}}
      {\sum_{\mathcal O\in\mathcal P_{K,\sigma}}
       \vol_A(\mathcal O)}.
\end{equation}

\begin{theorem}\label{thm:main}
Let $K_i$ be primitive totally real sextic fields with
$|D_{K_i}|\to\infty$, and let $\sigma_i$ be any ordering of the
embeddings of $K_i$. Then
\[
 \int_{X_6}f\,\dd\mu_{K_i,\sigma_i}\longrightarrow\int_{X_6}f\,\dd m_6
 \qquad\text{for all }f\in C_c(X_6).
\]
Additionally, there is no escape of mass: for every $\epsilon>0$ there
is a compact $C\subset X_6$ with
$\mu_{K_i,\sigma_i}(C)\ge 1-\epsilon$ for all sufficiently large $i$.
\end{theorem}

\subsection{History and the composite-degree obstruction}

In degree two, quadratic packets reduce to packets of closed geodesics
on the modular surface. Linnik \cite{Linnik1968} proved
equidistribution under an auxiliary splitting condition, which Duke
\cite{Duke1988} removed for fundamental discriminants by analytic
methods. Einsiedler, Lindenstrauss, Michel, and Venkatesh
\cite{ELMV2012} gave a dynamical proof valid for all positive nonsquare
discriminants. In degree three, the same four authors proved
equidistribution of cubic packets \cite{ELMV2011} by combining measure
classification, subconvexity, and local harmonic analysis.

Two of the general tools used here are due to Einsiedler, Katok, and
Lindenstrauss. First, the EKL measure-classification theorem
\cite[Theorem~1.3]{EKL2006}: on $\SL_d(\Z)\backslash\SL_d(\R)$ for
$d\ge3$, any $A$-invariant, $A$-ergodic probability of positive entropy
for some diagonal one-parameter subgroup is the invariant probability
on a closed finite-volume orbit $xJ$ for some closed connected subgroup
$J$ containing $A$. Second, the Einsiedler--Lindenstrauss--Michel--
Venkatesh criterion \cite[Corollary~3.3]{ELMV2009} for positive entropy,
which we use as a black box in Section~\ref{sec:homogeneous}.

The classification does not, in composite degree, reduce the
possibility of limits to Haar. In degree six, there are proper block
subgroups $J\subsetneq G$ that contain the full diagonal $A_6$. Their
orbits $xJ$ carry finite $J$-invariant measure. These are the residuals
of the classification that must be ruled out, using arithmetic that the
measure-classification theorem alone cannot see. Primitivity of $K$ is
what makes that arithmetic available.

Subsequent work has sharpened entropy estimates in higher degree:
Khayutin \cite[Theorem~1.1]{Khayutin2019} treats maximal-order packets
whose Galois closures act two-transitively on the embeddings, and
Wieser--Yang \cite{WieserYang2026} treat quartic packets of maximal type
with quadratic subfields. Neither result covers the degree-six
primitive case here.

The arithmetic estimates adapt the ordinary-vector method of
\cite[Section~2--6]{PrimeDegreePackets} and \cite[Section~3]{QuarticPackets},
with novel uses of Stark's effective zero-free region \cite{Stark1974}
and the short-interval multiplicative-function theorem of Shiu
\cite{Shiu1980} in the form given by Pollack \cite{Pollack2020}.
Primitivity ensures that every nonrational $d\in K$ generates the whole
field, and hence the order $\Z[d]$ has discriminant at least $D_K$.
This yields the uniform separation of rational lattice vectors used in
Proposition~\ref{prop:bowen}.

\subsection{The intermediate-resolution argument}

The core of the paper builds leafwise measures at the limit that retain
information about $\mu_{K_i,\sigma_i}$ at scales that tend to infinity
with the field.

Ordinary-vector estimates prove tightness and give every probability
limit $\mu$ an $O(r^6)$ bound on balls of radius $r$, uniformly when
the centers range over a compact set (Proposition~\ref{prop:ball}).
Since $A_6$ has dimension five, this bound forces positive entropy in
every positive-weight $A_6$-ergodic component
(Proposition~\ref{prop:positive-entropy}). The EKL classification then
makes each such component homogeneous (Proposition~\ref{prop:positive-entropy}).
Every proper homogeneous component is supported on points fixed by a
nonidentity diagonal matrix with algebraic entries. There are only
countably many such matrices, so any such component is confined to a
countable union of smooth pieces (Lemma~\ref{lem:fixed-pieces}).

The arithmetic tube estimate used to detect and exclude those pieces is
Proposition~\ref{prop:bowen}. It bounds the mass of a two-sided tube
\[
 x\bigl(a(tb)Oa(-tb)\cap a(-tb)Oa(tb)\bigr)
\]
in a packet by $e^{-t/3}$, for times
$t\in[T_{-,i},T_{+,i}]$ with $T_{-,i}\to\infty$ and
$T_{+,i}/T_{-,i}\to\infty$. The breadth of this time interval is used
throughout.

For $j\ne k$, let $U_{jk}=\{1+sE_{jk}:s\in\R\}$. Exact conditional
measures of a finite packet along $U_{jk}$ are Dirac masses. To capture
the spread detected by the tube bound, we condition not exact measures
but plaque probabilities at intermediate resolutions: in a local
chart $(q,h)$ for a unipotent group, condition the plaque coordinate
$h$ on a transverse cell of side $2^{-l}$ in $q$. At depths
$l_{-,i}\prec l\prec l_{+,i}$, the packet's conditional probabilities
change negligibly over a wide internal range
(Lemma~\ref{lem:plateau}). The resulting limiting state retains the
plaque probabilities together with the lattice point
(Section~\ref{sec:conditioning}), and the sampling identity
(Lemma~\ref{lem:sampling}) says the sampled plaque coordinate is
distributed according to its own recorded probability.

The product rule (Theorem~\ref{thm:product}) reduces these plaque
probabilities to product factors across commuting root subgroups.
Call a root \emph{active} at a limiting state $\omega$ when
$m^{U_{jk}}_\omega$ is supported beyond the identity
(Section~\ref{sec:activity}). Product structure makes
activity transitive under composed roots
(Proposition~\ref{prop:activity}). Activity is preserved by
diagonal motions (Lemma~\ref{lem:activity-indicators}) and forms a
directed graph that must be strongly connected on cut-crossing
indicators (Section~\ref{sec:combination}).

The entropy argument on long segments
(Proposition~\ref{prop:segment-activity}) uses the tube bound to prove
that crossing-root activity on any long segment of a dominated measure
has positive mean. Composing this with two-edge cycles upgrades the
cut-crossing activity to full activity at almost every state of the
augmented limit (Theorem~\ref{thm:full-activity}). Full activity plus
support on Homogeneous fixed-point pieces gives a contradiction unless
the limiting component is Haar.

\subsection{Conventions}

For the rest of the paper, $n=6$, $G=\SL_n(\R)$, $\Gamma=\SL_n(\Z)$,
$X=\Gamma\backslash G$, and $A=A_n$. Let
$L=\{b\in\R^n:\sum_j b_j=0\}$ be the Lie algebra of $A$ and
$a(b)=\diag(e^{b_1},\dots,e^{b_n})$. The right action is used
throughout: right multiplication by $a(b)$ sends the group element $h$
to $a(-b)ha(b)$. Matrix neighborhoods of the identity and smooth
metrics are interchangeable on fixed compact sets. When a metric on
$X$ is needed it is the quotient of a left-invariant Riemannian metric
on $G$. The notation $B(r)$ denotes a sufficiently small identity
neighborhood of radius $r$. Implicit constants may depend on compact
sets, chosen charts, and $n$, but not on the varying field. All passages
to subsequences preserve the probability limit under consideration.

\section{Arithmetic estimates for the packets}\label{sec:arithmetic}

This section proves the three estimates on which the rest of the paper
relies:
\begin{enumerate}
 \item uniform control of the cusp (Proposition~\ref{prop:cusp});
 \item a dimension bound for limits of packet measures: balls of radius
 $r$ have mass $O(r^6)$ (Proposition~\ref{prop:ball});
 \item exponential decay on two-sided diagonal tubes
 (Proposition~\ref{prop:bowen}).
\end{enumerate}
The first two follow by counting ordinary lattice vectors. The third
counts a vector together with a vector in the dual lattice; their
coordinatewise product lands in the codifferent of $K$, and primitivity
of $K$ supplies the separation needed to count the thin slices that
such products visit.

Throughout, let $K$ be a totally real sextic field with no proper
intermediate field, and let $\sigma$ be an ordering of its real
embeddings. Write
\[
 D=|\Disc(K)|,\quad Q=D^{1/2},\quad
 \kappa=\Res_{s=1}\zeta_K(s),\quad
 d_K(l)=\#\{\mathfrak b\subset\mathcal O_K:N\mathfrak b=l\}.
\]
All ideals counted here are nonzero. For a sequence of fields and
embedding orderings, use the corresponding decorated notation
$D_i,Q_i,\kappa_i,\mu_i$. Unless another dependence is indicated,
constants below are absolute for this fixed degree; in particular, they
do not depend on the field or on the ordering of its embeddings. The
arguments are adapted from the ordinary-vector treatment in
\cite[Section~2--6]{PrimeDegreePackets} and \cite[Section~3]{QuarticPackets},
reproduced in the present normalizations.

\subsection{Volume and unfolding}

We first fix the sign convention in the definition of the packet. The
space $X=\SL_6(\Z)\backslash\SL_6(\R)$ parametrizes ordinary unimodular
lattices. A diagonal sign matrix acts on lattices even when its
determinant is $-1$; the resulting lattice again represents a point of
$X$ (via any positively oriented basis). There is no additional
orientation coordinate, and the packet definition already allows both
determinant signs.

Let $h_K$ and $R_K$ denote the ordinary class number and regulator, with
$R_K$ the covolume of the absolute logarithms of the units after
deleting the sixth coordinate. Put
\[
 U=\mathcal O_K^\times,\quad
 U^+=\{\epsilon\in U:\sigma_j(\epsilon)>0\,\,(1\le j\le6)\},\quad
 s=[U:U^+].
\]
For a nonnegative Borel $f$ on $\R^6$ and a lattice $\Lambda$, define
$\widehat f(\Lambda)=\sum_{v\in\Lambda\setminus\{0\}}f(v)$. For $p>0$
define a logarithmic fiber integral
\begin{equation}\label{eq:fiber-integral}
 V_f(p)
  =\sum_{\epsilon\in\{\pm1\}^6}\int_{\R^5}
  f\bigl(\epsilon_1e^{u_1},\dots,\epsilon_5e^{u_5},
         \epsilon_6p e^{-u_1-\cdots-u_5}\bigr)
   \,\dd u_1\cdots\dd u_5.
\end{equation}
Thus $V_f(p)$ integrates over the set on which the absolute product of
the six coordinates is $p$, with the Haar normalization for $A$.

\begin{proposition}[Packet volume and vector unfolding]
\label{prop:packet-volume}
Every orbit in $\mathcal P_{K,\sigma}$ has $A$-volume $(s/2)R_K$, and
there are $h_K2^6/s$ distinct orbits. In particular, the total packet
volume is $Q\kappa$. For every nonnegative Borel $f$,
\begin{equation}\label{eq:vector-unfolding}
 \int_X\widehat f\,\dd\mu_{K,\sigma}
   =\frac1{Q\kappa}\sum_{l\ge1}d_K(l)V_f(l/Q),
\end{equation}
with both sides allowed to be infinite.
\end{proposition}

\begin{proof}
A totally real field has only $\pm1$ as roots of unity, so the kernel
of the absolute-logarithmic map on $U$ has order two. Consequently the
image of $U^+$ has index $s/2$ in the image of $U$. The unit theorem
gives covolume $(s/2)R_K$ for the positive-unit logarithms in the
coordinates $u_1,\dots,u_5$.

Suppose a real diagonal map sends $\sigma(I)$ onto $\sigma(J)$ for two
nonzero fractional ideals. Pick $0\ne d\in I$ with image $\sigma(e)$,
$e\in J$. Then all embeddings of $d$ are nonzero, and the diagonal
entries of the map are $\sigma_j(e/d)$, so the map is multiplication by
$e/d\in K^\times$. Hence $J=(e/d)I$. Consequently, different ordinary
ideal classes give disjoint packet orbits. For a fixed ideal $I$, its
diagonal self-automorphisms are its units, because $(I:I)=\mathcal O_K$.
The stabilizer in $A$ is the closure of the image of $U^+$.

Two sign translates lie on the same $A$-orbit exactly when their sign
patterns differ by a unit signature. The signature map on units has
image of cardinality $s$, giving $2^6/s$ orbits per ordinary class and
the stated orbit volume. The analytic class-number formula, in this
normalization, gives $Q\kappa=2^5h_KR_K$; this matches the product of
orbit count and orbit volume.

For unfolding, let $\mathcal F$ be a fundamental domain for the
positive-unit logarithms in the traceless diagonal algebra. Summing over
ordinary class representatives and all sign matrices counts each orbit
$s$ times, so
\begin{equation}\label{eq:packet-integral}
 \int_XF\,\dd\mu_{K,\sigma}
 =\frac1{sQ\kappa}\sum_{[I]}\sum_w
   \int_{\mathcal F}F(\Lambda_{I,\sigma}w a(u))\,\dd u
\end{equation}
for any nonnegative $F$ on $X$, with $\dd u=\dd u_1\cdots\dd u_5$ and
the sixth coordinate the negative sum of the others.

Apply this to $F=\widehat f$. A nonzero $d\in I$ determines the integral
ideal $\mathfrak b=(d)I^{-1}$, and
\[
 \prod_{j=1}^6
 \left|\covol(\sigma(I))^{-1/6}\sigma_j(d)\right|
 =\frac{N\mathfrak b}{Q},
\]
since $\covol(\sigma(I))=QNI$. The ideal $\mathfrak b$ determines
$[I]=[\mathfrak b]^{-1}$ and every integral $\mathfrak b$ occurs. For
fixed $\mathfrak b$, the possible $d$ form a single $U$-orbit, or $s$
orbits under $U^+$. Summing a positive-unit orbit and integrating over
$\mathcal F$ unfolds to the full traceless diagonal algebra.
Multiplication by a representative of $U/U^+$ translates the absolute
logarithms and permutes the signs. After the sum over $w$, each of
these $s$ orbits contributes $V_f(N\mathfrak b/Q)$, and their number
cancels the factor $s$ in \eqref{eq:packet-integral}. The conclusion
follows by nonnegativity.
\end{proof}

\subsection{Uniform counts of ideals}

The normalization by $\kappa$ in \eqref{eq:vector-unfolding} makes it
important to keep this factor in upper bounds for ideal counts. The key
estimate is standard and comes from a zero-free neighborhood of $1$
plus Shiu's short-interval estimate.

\begin{lemma}[Uniform ideal estimates]\label{lem:ideal-estimate}
For all sufficiently large discriminants $D$ in the present family,
\begin{align}
 \kappa&\gg(\log D)^{-1}, \label{eq:residue-lower}\\
 \sum_{p\le D}\frac{d_K(p)}p
   &\le\log(\kappa\log D)+O(1), \label{eq:prime-sum}\\
 \sum_{x-y<l\le x}d_K(l)
   &\ll y\kappa
   \quad\bigl(D^{1/4}\le x\le D,\ x^{1/2}\le y\le x\bigr).
   \label{eq:ideal-interval}
\end{align}
In \eqref{eq:prime-sum}, $p$ ranges over rational primes. Constants and
the discriminant threshold are uniform in $K$.
\end{lemma}

\begin{proof}
Stark's Lemma~3 in \cite{Stark1974} shows that the rectangle
\[
 \Re s\ge1-\frac1{4\log D},\qquad |\Im s|\le\frac1{4\log D}
\]
contains at most one nontrivial zero of $\zeta_K$, and that such a zero
is real and simple. The same paper's Lemma~8 shows that a real zero
$\beta$ with
\[
 1-\frac1{4\cdot 6!\log D}\le\beta<1
\]
would be a zero of a quadratic subfield zeta function
\cite[Lemmas~3 and~8]{Stark1974}. Since $K$ has no quadratic subfield,
there is an absolute $c_0>0$ with
\begin{equation}\label{eq:zero-exclusion}
 |1-\rho|\ge c_0/\log D
\end{equation}
for every nontrivial zero $\rho$ of $\zeta_K$. One may take
$c_0=1/2880$.

Set $s_0=1+1/\log D$. The logarithmic derivative of the completed
Dedekind zeta gives, for real $1<s\le s_0$,
\begin{equation}\label{eq:zeta-derivative}
 S(s):=\sum_\rho\Re\frac1{s-\rho}
   =\frac1{s-1}+\frac12\log D
   +\frac{\zeta'_K(s)}{\zeta_K(s)}+O(1).
\end{equation}
Zeros are counted with multiplicity, and the gamma-factor contributes
$O(1)$ because degree and signature are fixed. The left side is a sum
of nonnegative terms, and $\zeta'_K(s_0)/\zeta_K(s_0)\le0$ by the
Dirichlet series. Hence $S(s_0)\ll\log D$.

The same bound holds uniformly for $1<s\le s_0$. Write
$u=s-1$, $a=1-\Re\rho\ge0$, $v=\Im\rho$. The summand is
$(a+u)/((a+u)^2+v^2)$. For $0<u\le1/\log D$, the numerator is at most
its value at $s_0$, and \eqref{eq:zero-exclusion} bounds
$a^2+v^2\ge c_0^2/(\log D)^2$. Termwise comparison therefore gives the
uniform estimate for $S(s)$ on $(1,s_0]$. Integrating
\eqref{eq:zeta-derivative} after moving $1/(s-1)$ gives
\begin{equation}\label{eq:zeta-residue-comparison}
 \zeta_K(s_0)\asymp\kappa\log D.
\end{equation}
Explicitly, the logarithmic derivative of $(s-1)\zeta_K(s)$ is
$S(s)-\tfrac12\log D+O(1)$, whose integral over this interval is $O(1)$
and whose limit at $s=1$ is $1$. Since $\zeta_K(s_0)\ge1$, this proves
\eqref{eq:residue-lower}.

The Euler product gives $\sum_{p\le D}d_K(p)/p^{s_0}\le\log\zeta_K(s_0)$.
Also $d_K(p)\le6$ and
\[
 \sum_{p\le D}d_K(p)\left(\frac1p-\frac1{p^{s_0}}\right)
 \le\frac6{\log D}\sum_{p\le D}\frac{\log p}{p}\ll1,
\]
by Chebyshev's bound and partial summation. Equation
\eqref{eq:zeta-residue-comparison} then gives \eqref{eq:prime-sum}.

For the interval estimate, we use Shiu's theorem \cite{Shiu1980} in the
form of \cite[Theorem~1.1]{Pollack2020} with empty excluded residue
sets: for $f\ge0$ multiplicative with $f(p^a)\le A_1^a$ and
$f(m)\le A_2(\epsilon)m^\epsilon$, and for each fixed
$0<\beta<1/2$,
\begin{equation}\label{eq:shiu-form}
 \sum_{x-y<m\le x}f(m)
 \ll\frac y{\log x}\exp\left(\sum_{p\le x}\frac{f(p)}p\right)
 \quad(x^\beta<y\le x).
\end{equation}
Constants and thresholds depend only on $\beta,A_1$, and $A_2$. The
function $d_K$ is multiplicative and bounded by $d_6$, since
$d_K(p^a)\le\binom{a+5}{5}\le 6^a$ and $d_K(m)\le d_6(m)\ll_\epsilon
m^\epsilon$; so the hypotheses hold with constants independent of $K$.
Take $\beta=1/3$; the assumption $y\ge x^{1/2}$ lies in range. Since
$x\le D$, equation \eqref{eq:prime-sum} bounds the exponential factor
by $O(\kappa\log D)$, and $x\ge D^{1/4}$ gives
$\log D/\log x\le 4$. This proves \eqref{eq:ideal-interval}.
\end{proof}

\subsection{The cusp and balls in the space of lattices}

For $x\in X$, let $m(x)$ be the length of its shortest nonzero vector
in the sup norm. The sup norm is only used to simplify the fiber
integral.

\begin{proposition}[Uniform cusp bound]\label{prop:cusp}
There exist $C,c>0$ such that, for all sufficiently large $D$,
\[
 \mu_{K,\sigma}\{x:m(x)\le r\}\le Cr^c
 \qquad(0<r<1).
\]
Consequently the packet probabilities form a tight family as
$D\to\infty$, and every subsequential limit is an $A$-invariant
probability measure on $X$.
\end{proposition}

\begin{proof}
Take $f=\one_{[-r,r]^6}$ and put $X_r=Qr^6$. For $0<p\le r^6$, substitute
$t_j=\log r-u_j$ in \eqref{eq:fiber-integral}. The positive-sign region
becomes $t_j\ge0$ and $\sum_{j=1}^5t_j\le\log(r^6/p)$, and
\[
 V_f(p)=\frac{2^6}{5!}\bigl(\log(r^6/p)\bigr)^5
       \quad(0<p\le r^6),
 \qquad V_f(p)=0\quad(p>r^6).
\]
Since the existence of a short vector is bounded by the count of such
vectors, \eqref{eq:vector-unfolding} gives
\begin{equation}\label{eq:cusp-sum}
 \mu_{K,\sigma}\{m\le r\}
 \ll\frac1{Q\kappa}\sum_{l\le X_r}d_K(l)
     \bigl(1+\log(X_r/l)\bigr)^5.
\end{equation}
The sum is empty for $X_r<1$.

Suppose first $r\ge D^{-1/48}$, so $X_r\ge D^{3/8}$. Split the range
$l\ge D^{1/4}$ into dyadic intervals $(2^{-j-1}X_r,2^{-j}X_r]$. Any
such interval with $l\ge D^{1/4}$ has upper endpoint between $D^{1/4}$
and $Q$, and its ideal count is $O(\kappa2^{-j}X_r)$ by
\eqref{eq:ideal-interval} with interval length half its upper endpoint.
The weight in \eqref{eq:cusp-sum} is $O((j+1)^5)$, so the total
contribution of these terms before division by $Q\kappa$ is
\[
 C\kappa X_r\sum_{j\ge0}2^{-j}(j+1)^5\ll\kappa X_r.
\]
For the remainder $l<D^{1/4}$, the divisor estimate
$\sum_{l\le Y}d_6(l)\ll Y(\log(2Y))^5$ gives a bound
$CD^{1/4}(\log D)^{10}$ before normalization, and
\eqref{eq:residue-lower} makes the normalized contribution at most
$CD^{-1/4}(\log D)^{11}\ll r^6$ (since $r^6\ge D^{-1/8}$).

Now let $r<D^{-1/48}$ and $X_r\ge1$. Summing the same divisor estimate
in dyadic blocks gives $\sum_{l\le X_r}d_6(l)(1+\log(X_r/l))^5\ll
X_r(\log(2X_r))^5$. Using $X_r\le Q$ and \eqref{eq:residue-lower},
\eqref{eq:cusp-sum} is at most $Cr^6(\log D)^6$. But in this range
$\log D<48\log(1/r)$, and $r^6(\log(1/r))^6\ll r^3$. The assertion
follows with $c=3$, after increasing $C$.

Mahler's compactness criterion puts $\{x:m(x)\ge r\}$ inside a compact
set for each $r>0$, so tightness follows. Right translation by any
fixed element of $A$ is continuous and preserves compact support, so
invariance passes to the limit.
\end{proof}

Let $B(r)$ denote an open matrix ball about the identity in $G$.

\begin{proposition}[Ordinary balls in a probability limit]
\label{prop:ball}
Suppose $D_i\to\infty$ and $\mu_i$ converges weakly to a probability
$\mu$. For every compact $\Omega\subset X$ there are constants
$C_\Omega,r_\Omega>0$ with
\[
 \mu(xB(r))\le C_\Omega r^6
 \qquad(x\in\Omega,\ 0<r<r_\Omega).
\]
\end{proposition}

\begin{proof}
Pick a nonzero lattice vector at any $x_0\in X$; a full-rank lattice is
not contained in a finite union of proper linear subspaces, so we may
choose such a vector none of whose coordinates is zero. In an injective
local chart at $x_0$, continue this marked vector as $v(x)$; shrink the
chart so its coordinates stay in a bounded interval separated from
zero. A finite number of such charts covers $\Omega$, giving uniform
bounds and uniform derivative bounds.

For $x$ in a chosen chart and $r<r_\Omega$, every lattice in $xB(r)$
has a nonzero vector in the ordinary Euclidean ball
$B_{\R^6}(v(x),C_1r)$. Let $f_{x,r}$ be its indicator. Its support has
absolute coordinate product in an interval $J_{x,r}$ of length at most
$C_2r$, and all these intervals lie in a fixed compact subinterval of
$(0,\infty)$ when $r<r_\Omega$. On this support the first five
logarithmic coordinates each vary by $O(r)$, so
\[
 V_{f_{x,r}}(p)\le C_3r^5\one_{J_{x,r}}(p).
\]
For all sufficiently large $i$, uniformly in the finitely many charts
and in $x\in\Omega$, the ideal norms in $Q_iJ_{x,r}$ can be covered by
an interval with upper endpoint comparable to $Q_i$ and length at most
$C_4Q_ir$. We may take the length comparable to $Q_ir$; it is then at
least the square root of its upper endpoint for large $i$, and the
upper endpoint is between $D_i^{1/4}$ and $D_i$. Hence
\eqref{eq:ideal-interval} and \eqref{eq:vector-unfolding} give
\[
 \mu_i(xB(r))
 \le\int_X\widehat f_{x,r}\,\dd\mu_i
 \le\frac{C_3r^5}{Q_i\kappa_i}
     \sum_{l\in Q_iJ_{x,r}}d_{K_i}(l)
 \le C_5r^6.
\]
The constants are independent of $x$ and $r$; the index threshold may
depend on $r$. Since $xB(r)$ is open, the open-set inequality for weak
convergence gives $\mu(xB(r))\le C_5r^6$.
\end{proof}

\subsection{Trace slices of the codifferent}

For a lattice vector $v$ and dual vector $y$, the coordinatewise product
$v_jy_j$ is nearly preserved under block-diagonal motion. The useful
arithmetic quantity is a partial sum of these products, which is
controlled by the lattice count below.

Let $\mathfrak D_K$ be the different of $K$, so
$\mathfrak D_K^{-1}=\{z\in K:\Tr_{K/\Q}(z\mathcal O_K)\subset\Z\}$.
Give $\R^6$ its Euclidean inner product, set $\mathbf e=(1,\dots,1)$, and
put
\[
 H=\mathbf e^\perp,\qquad \mathcal L_K=\sigma(\mathfrak D_K^{-1})\cap H.
\]

\begin{lemma}[Small cells in a trace slice]\label{lem:trace-slice}
The lattice $\mathcal L_K$ has Euclidean covolume $\sqrt6/Q$ in $H$
and admits a fundamental parallelotope of diameter $O(D^{-1/30})$. Let
$S$ be a nonempty proper subset of $\{1,\dots,6\}$. For each fixed
$M>0$, uniformly for $m\in\Z$ and intervals $J\subset\R$ of length
$\delta>0$,
\begin{equation}\label{eq:trace-slice-count}
 \#\left\{z\in\mathfrak D_K^{-1}:
    \Tr_{K/\Q}z=m,\ |\sigma_j(z)|\le M\ (1\le j\le6),
    \ \sum_{j\in S}\sigma_j(z)\in J\right\}
 \ll_M Q\bigl(\delta+D^{-1/30}\bigr).
\end{equation}
\end{lemma}

\begin{proof}
Choose an integral basis $\alpha_1=1,\alpha_2,\dots,\alpha_6$ of
$\mathcal O_K$. Let $\beta_1,\dots,\beta_6$ be its trace-dual basis,
so $\Tr_{K/\Q}(\alpha_j\beta_k)=\delta_{jk}$. The $\beta_k$ form a
basis of $\mathfrak D_K^{-1}$. In particular
$\Tr_{K/\Q}\beta_k=\delta_{1k}$, so the trace map onto $\Z$ is
surjective and $\sigma(\beta_2),\dots,\sigma(\beta_6)$ form a basis
of $\mathcal L_K$. Orthogonal projection gives, for $2\le j,k\le6$,
\[
 \bigl\langle\operatorname{proj}_H\sigma(\alpha_j),\sigma(\beta_k)
 \bigr\rangle=\delta_{jk},
\]
so the projected vectors are the dual basis in $H$. With
$\operatorname{proj}_H\sigma(\alpha_1)=0$, this gives
\begin{equation}\label{eq:projected-dual}
 \mathcal L_K^*=\operatorname{proj}_H\sigma(\mathcal O_K).
\end{equation}

Take a nonzero vector in the right side of \eqref{eq:projected-dual},
arising from $\alpha\in\mathcal O_K$. The vector is nonzero exactly
when $\alpha\notin\Q$, and then primitivity gives $\Q(\alpha)=K$. The
order $\Z[\alpha]$ has discriminant at least $D$. Differences of
embedding coordinates are unchanged by orthogonal projection, so
\[
 D\le|\Disc(\Z[\alpha])|
   =\prod_{j<k}|\sigma_j(\alpha)-\sigma_k(\alpha)|^2
   \le C\|\operatorname{proj}_H\sigma(\alpha)\|^{30}.
\]
Hence the first successive minimum of $\mathcal L_K^*$ is at least
$cD^{1/30}$, and transference gives
$\lambda_5(\mathcal L_K)\ll\lambda_1(\mathcal L_K^*)^{-1}\ll D^{-1/30}$.
A fixed-dimensional basis bound then produces a basis of
$\mathcal L_K$ with all lengths $O(D^{-1/30})$; see
\cite{Cassels1959}.

The full codifferent has covolume $Q^{-1}$, and its trace planes have
spacing $1/\sqrt6$. Decomposing a basis into five vectors in $H$ and a
vector of trace one gives $Q^{-1}=(1/\sqrt6)\covol_H(\mathcal L_K)$.

The trace-$m$ points form a translate of $\mathcal L_K$. Tile that
affine hyperplane by translates of a fundamental parallelotope,
assigning boundaries measurably. A cell attached to a counted point
lies in the $O(D^{-1/30})$-enlargement of its bounded slice. The
functional $z\mapsto\sum_{j\in S}z_j$ is nonconstant on $H$ (since $S$
is nonempty and proper), so the enlarged slice has five-dimensional
volume $O_M(\delta+D^{-1/30})$, uniformly in its location and in $m$.
Dividing by the cell covolume $\sqrt6/Q$ gives the count.
\end{proof}

\subsection{Exponential decay on diagonal tubes}

For $D_i$ sufficiently large, define
\begin{equation}\label{eq:time-range}
 T_{-,i}=\frac{\log D_i}{(\log\log D_i)^{2/3}},\qquad
 T_{+,i}=\frac{\log D_i}{(\log\log D_i)^{1/3}}.
\end{equation}
Both tend to infinity, their ratio tends to infinity, and
$T_{+,i}=o(\log D_i)$.

\begin{proposition}[Two-sided tube estimate]\label{prop:bowen}
Let $S$ be a nonempty proper subset of $\{1,\dots,6\}$, and let
$b=(b_1,\dots,b_6)\in L$ satisfy
$|b_j-b_k|\ge1$ for $j\in S$, $k\notin S$. For every compact
$\Omega\subset X$ there is an identity neighborhood $O\subset G$ such
that, for all sufficiently large $i$, all $x\in\Omega$, and all
$T_{-,i}\le t\le T_{+,i}$,
\begin{equation}\label{eq:bowen-bound}
 \mu_i\bigl(x(a(tb)Oa(-tb)\cap a(-tb)Oa(tb))\bigr)
   \le e^{-t/3}.
\end{equation}
The neighborhood and threshold can be chosen simultaneously for any
finite collection of pairs $(S,b)$.
\end{proposition}

\begin{proof}
For a lattice $\Lambda$, its Euclidean dual is $\Lambda^*=\{y:\langle
v,y\rangle\in\Z\text{ for all }v\in\Lambda\}$. At each point of
$\Omega$, choose nonzero marked vectors in the lattice and its dual,
both avoiding coordinate hyperplanes, and continue the pair in a local
lattice chart. Finitely many charts cover $\Omega$, and the marked
vectors in these charts lie in fixed boxes separated from those
hyperplanes. Their inner product is an integer constant in each chart,
because under right multiplication by $h$ they transform as
$v\mapsto vh$, $y\mapsto yh^{-\mathsf T}$.

Choose $O$ small enough (in matrix coordinates) that if
$h\in a(tb)Oa(-tb)\cap a(-tb)Oa(tb)$, then $h$ is uniformly close to
the identity and each entry crossing the cut $S\sqcup S^c$ is
$O(e^{-t})$. The two membership conditions respectively bound the
$(j,k)$ entry by a constant times $e^{t(b_j-b_k)}$ and by
$e^{-t(b_j-b_k)}$; their minimum is $O(e^{-t|b_j-b_k|})$. Reduce $O$
further so $h^{-1}$ is uniformly bounded. For the diagonal projection
$P_S$ onto the coordinates in $S$,
\[
 hP_Sh^{-1}-P_S=[h,P_S]h^{-1}=O(e^{-t}).
\]
For the marked pair $v_0,y_0$ at the centre $x$, set $v=v_0h$ and
$y=y_0h^{-\mathsf T}$. These stay in fixed boxes (separated from the
coordinate hyperplanes), and
\begin{align*}
 \sum_{j=1}^6v_jy_j&=\langle v_0,y_0\rangle=:m\in\Z,\\
 \sum_{j\in S}v_jy_j
   &=v_0hP_Sh^{-1}y_0^{\mathsf T}
     =v_0P_Sy_0^{\mathsf T}+O(e^{-t}).
\end{align*}
All constants are uniform in the finite cover. Membership in the tube
therefore forces an ordinary/dual vector pair in the specified bounded
boxes with the trace condition and an $O(e^{-t})$-partial sum.

Count such pairs in the packet. For a representative ideal $I$ and a
packet lattice $c\sigma(I)wa$ with $c=\covol(\sigma(I))^{-1/6}$, its
dual is $c^{-1}\sigma(\mathfrak D_K^{-1}I^{-1})wa^{-1}$. Hence a
nonzero pair comes from
\[
 d\in I\setminus\{0\},\qquad \ell\in\mathfrak D_K^{-1}I^{-1}
 \setminus\{0\}.
\]
The scalar, sign, and diagonal factors cancel in the coordinatewise
product:
\begin{equation}\label{eq:pair-product}
 v_jy_j=\sigma_j(z),\qquad z=d\ell\in\mathfrak D_K^{-1}.
\end{equation}
Thus allowed $z$ lie in a fixed bounded embedding box, have trace $m$,
and have their $S$-sum in an interval of length $O(e^{-t})$.

For a fixed allowed $z$, the ideal $\mathfrak b=(d)I^{-1}$ must divide
$(z)\mathfrak D_K$; indeed
$\mathfrak b\bigl((\ell)I\mathfrak D_K\bigr)=(z)\mathfrak D_K$, and
both factors on the left are integral. Conversely, this divisibility
ensures $z/d\in\mathfrak D_K^{-1}I^{-1}$ whenever $(d)I^{-1}=\mathfrak b$.
For fixed $d,z$ there is exactly one possible $\ell$, namely $z/d$.

Let $f$ be the indicator of a fixed bounded box (or a fixed finite
union of boxes) containing all possible $v$, separated from the
coordinate hyperplanes. Its logarithmic fiber integrals are bounded
uniformly in $p>0$. Repeating the unfolding of
Proposition~\ref{prop:packet-volume} with the constraint
$\mathfrak b\mid(z)\mathfrak D_K$ gives
\begin{equation}\label{eq:pair-unfolding}
 \int_X\#\{\text{admissible pairs with product }z\}\,\dd\mu_{K,\sigma}
 \le\frac1{Q\kappa}\sum_{\mathfrak b\mid(z)\mathfrak D_K}
    V_f(N\mathfrak b/Q)
 \ll\frac1{Q\kappa}\#\{\mathfrak b:\mathfrak b\mid(z)\mathfrak D_K\}.
\end{equation}

The embedding bound gives $N((z)\mathfrak D_K)=D|N_{K/\Q}z|\ll D$.
For an integral ideal of norm $N$, its number of ideal divisors is at
most $d(N)^6$: over each rational prime there are at most six prime
ideals, and each appears with the prime's exponent. The classical
divisor bound then gives, uniformly for allowed $z$,
\begin{equation}\label{eq:ideal-divisors}
 \#\{\mathfrak b:\mathfrak b\mid(z)\mathfrak D_K\}
   \le\exp\left(C\frac{\log D}{\log\log D}\right).
\end{equation}

Lemma~\ref{lem:trace-slice} bounds the number of allowed products by
$O(Q(e^{-t}+D^{-1/30}))$. Since $T_{+,i}/\log D_i\to0$, we have
$D_i^{-1/30}=o(e^{-t})$ uniformly on the time range, so the product
count is $O(Q_ie^{-t})$. Summing \eqref{eq:pair-unfolding} over
these products and using \eqref{eq:ideal-divisors},
\[
 \mu_i\bigl(x(a(tb)Oa(-tb)\cap a(-tb)Oa(tb))\bigr)
 \ll\kappa_i^{-1}\exp\left(-t+C\frac{\log D_i}{\log\log D_i}\right).
\]
Equation \eqref{eq:residue-lower} and $t\ge T_{-,i}$ make the
logarithm of the pre-exponential factor $o(t)$, uniformly on the time
range. Enlarging $D$-threshold makes the last display at most $e^{-t/3}$.
Finitely many cuts and diagonal elements are handled by taking the
smallest neighborhood and largest threshold.
\end{proof}

\section{Homogeneous limits and their fixed points}\label{sec:homogeneous}

The arithmetic estimates of Section~\ref{sec:arithmetic} already imply
that any weak limit of packet probabilities is an $A_6$-invariant
probability. This section shows that any such limit has positive entropy
in every positive-weight ergodic component, and that every positive-entropy
ergodic component is algebraic. The algebraic components are then
classified; the proper ones are the block-diagonal orbits of types
$3+3$ and $2+2+2$. Every such orbit lies in a fixed-point set of a
nonidentity diagonal with algebraic entries.

\subsection{Positive entropy in every positive-weight collection of components}

For an invariant probability $\nu$ and an invertible
measure-preserving map $T$, write $h_\nu(T)$ for its
measure-theoretic entropy. Fix $b\in L$ with distinct coordinates and
put
\[
 \lambda=\min_{j\ne k}|b_j-b_k|>0,\qquad T(x)=xa(b).
\]
The following is an instance of the ordinary-ball argument of
\cite[Proposition~6.1 and Lemma~6.2]{PrimeDegreePackets}. It explains
why a ball estimate for a measure controls the entropy of its ergodic
components without needing to pass that estimate to each individual
component.

\begin{proposition}[Positive entropy from balls]\label{prop:positive-entropy}
Let $\mu$ be an $A$-invariant probability on $X$ satisfying the ball
estimate $(*)$: for every compact $\Omega\subset X$ there are
$C_\Omega,r_\Omega>0$ with
\begin{equation}\label{eq:ordinary-ball-hypothesis}
 \mu(xB(r))\le C_\Omega r^n
 \quad(x\in\Omega,\ 0<r<r_\Omega).
\end{equation}
Then every $A$-invariant probability $\xi\le c\mu$ with $c<\infty$
satisfies $h_\xi(T)\ge\lambda/3$. Consequently almost every
$A$-ergodic component of $\mu$ has positive entropy for $T$.
\end{proposition}

\begin{proof}
Consider the two-sided tube
\[
 \mathcal T_t=a(tb)B(\epsilon)a(-tb)\cap a(-tb)B(\epsilon)a(tb).
\]
If $h\in\mathcal T_t$ and $r=e^{-\lambda t}$, then
$|h_{jj}-1|<\epsilon$ and $|h_{jk}|\le\epsilon r$ for $j\ne k$. In the
determinant expansion every nondiagonal term contains at least two
off-diagonal entries, so $\prod_jh_{jj}=1+O(r^2)$. The matrix
\[
 d(h)=\diag\left(h_{11},\dots,h_{n-1,n-1},
   \Bigl(\prod_{j<n}h_{jj}\Bigr)^{-1}\right)
\]
belongs to a fixed compact $\mathcal D\subset A$ and satisfies
$d(h)^{-1}h\in B(Cr)$. An $r$-mesh of the $(n-1)$-dimensional $\mathcal D$
covers $\mathcal T_t$ by at most $Cr^{-(n-1)}$ sets $d_jB(C'r)$ with
$d_j\in\mathcal D$.

For $x\in\Omega$, all $xd_j$ lie in the compact set
$\Omega\mathcal D$. Distortion of balls under right translation on $X$
is bounded on compact sets, so \eqref{eq:ordinary-ball-hypothesis}
gives $\xi(xd_jB(C'r))\le C'_{\Omega,c}r^n$. Summing over $j$,
\[
 \xi(x\mathcal T_t)\le C_{\Omega,c}e^{-\lambda t}.
\]

The Einsiedler--Lindenstrauss--Michel--Venkatesh entropy criterion
\cite[Corollary~3.3]{ELMV2009} says that if probabilities $\nu_i$
invariant under $a(\R b)$ converge to a probability $\nu$ and $t_i\to\infty$
satisfy
\[
 \nu_i(x[a(t_ib)Oa(-t_ib)\cap a(-t_ib)Oa(t_ib)])\le C_\Omega e^{-2\eta t_i}
 \quad(x\in\Omega)
\]
for every compact $\Omega$ (with $O$ allowed to depend on $\Omega$),
then $h_\nu(T)\ge\eta$. Take $\nu_i=\xi$, $t_i=i$, and $\eta=\lambda/3$.
The displayed hypothesis is satisfied by the previous display.

Write the $A$-ergodic decomposition as $\mu=\int_Z\nu_z\,\dd\tau(z)$. The
map $z\mapsto h_{\nu_z}(T)$ is measurable: for increasing finite Borel
partitions $\mathcal P_k$ generating the Borel sets, it equals
$\sup_k\inf_{N\ge1}\frac1N H_{\nu_z}(\bigvee_{j=0}^{N-1}T^{-j}\mathcal P_k)$.
The entropy integral formula of Rokhlin applies to this decomposition
\cite[Theorem~9.8]{Rokhlin1967}. The component partition is fixed by
$T$, so the entropy of $T$ acting on the quotient is zero, and the
formula remains valid after restricting to any measurable collection of
components.

If the zero-entropy components had positive total weight $w>0$, the
normalized average over that subcollection would satisfy
$\xi\le w^{-1}\mu$ and $h_\xi(T)=0$. This contradicts the first part of
the proposition.
\end{proof}

\subsection{The algebraic alternatives}

We use the EKL measure-classification theorem.

\begin{theorem}[EKL]\label{thm:ekl}
For $n\ge3$, an $A$-invariant $A$-ergodic probability on
$\SL_n(\Z)\backslash\SL_n(\R)$ that has positive entropy for some
diagonal one-parameter subgroup is the invariant probability on a closed
finite-volume orbit $xJ$, where $J$ is a closed connected subgroup
containing $A$.
\end{theorem}

This is \cite[Theorem~1.3]{EKL2006}. Left versus right action differs
by $g\mapsto g^{-1}$ on $\Gamma$, which replaces $a(b)$ by $a(-b)$ and
preserves entropy.

\begin{lemma}[Block groups]\label{lem:block-groups}
Let $J$ be a connected closed subgroup of $G$ containing $A$ and
suppose $xJ$ carries a finite nonzero $J$-invariant measure. Then there
is a partition $\{1,\dots,n\}=B_1\sqcup\cdots\sqcup B_d$ such that $J$
is the identity component of the determinant-one block-diagonal group
for this partition.
\end{lemma}

\begin{proof}
The Lie algebra $\mathfrak j$ contains $L$ and is $\Ad(A)$-invariant.
Since the off-diagonal weight spaces are one-dimensional, $\mathfrak j$
is the direct sum of $L$ and a subset of root lines $\R E_{jk}$. Draw
the edge $j\to k$ when that line occurs. The bracket
$[E_{ij},E_{jk}]=E_{ik}$ makes this graph transitively closed whenever
$i\ne k$.

A Lie group with a lattice is unimodular. Applying the vanishing of its
modular character to $A$ gives
\[
 \sum_{j\to k}(b_j-b_k)=0\qquad(b\in L).
\]
Thus each vertex has equal indegree and outdegree. Every strongly
connected class is complete by transitivity. The graph of these classes
is acyclic. If it had an edge, a source class of an edge-containing
component would have strictly more outgoing than incoming edges within
the original graph, contradicting the sum of the degree balances over
that class. Hence there are no edges between classes, and the resulting
Lie algebra is precisely the block-diagonal one. A connected Lie
subgroup is determined by its Lie algebra.
\end{proof}

The next lemma uses finite volume to identify a rational commutant and
then to exclude its idempotents, forcing a single algebraic unit to fix
the orbit.

\begin{lemma}[Fixed pieces]\label{lem:fixed-pieces}
Every proper orbit $xJ$ arising from Theorem~\ref{thm:ekl} lies in
$\Fix(a)=\{y\in X:ya=y\}$ for some nonidentity $a\in A$ with algebraic
entries. There are only countably many such $a$. For each such $a$, its
fixed set is covered by countably many smooth pieces. Whenever
$a_{jj}\ne a_{kk}$, the direction of $U_{jk}$ is nowhere tangent to
these pieces, and each piece meets any sufficiently small $U_{jk}$
plaque in at most one point.
\end{lemma}

\begin{proof}
Write $x=\Gamma g$ and $\Delta=g^{-1}\Gamma g\cap J$; this is a lattice
in $J$. Use the auxiliary column lattice $\Lambda=g^{-1}\Z^n$ (rational
span $\Lambda_\Q$), which $\Delta$ preserves. We show first that the
real matrix commutant of $\Delta$ equals that of $J$. For any matrix
$C$ commuting with $\Delta$, the map
\[
 \Delta\backslash J\longrightarrow\Mat_n(\R),\qquad
 \Delta j\longmapsto j^{-1}Cj
\]
is well-defined and sends normalized Haar measure to a probability
invariant under conjugation by $A$. Choose a regular $a\in A$. Every
off-diagonal matrix coordinate has a nonzero weight under conjugation,
and recurrence of an invariant probability forces it to vanish almost
surely. Continuity of the map and full support of Haar imply $j^{-1}Cj$
is diagonal for every $j\in J$. Taking $j=1$ shows $C$ is diagonal;
conjugating by a root subgroup inside one of the blocks of
Lemma~\ref{lem:block-groups} forces the corresponding diagonal entries
of $C$ to agree. Hence the commutant of $\Delta$ is the algebra of
block scalars, which is the commutant of $J$.

Define $F=\{C\in\End_\Q(\Lambda_\Q):C\delta=\delta C$ for all
$\delta\in\Delta\}$. In the lattice coordinates supplied by $g$, the
commutation relations are rational linear equations, since
$g\Delta g^{-1}\subset\Gamma$. Finitely many of those equations
determine $F$. So $F$ is a finite-dimensional $\Q$-algebra, and its
real extension is isomorphic to $\R^d$ ($d$ = number of blocks).

Suppose $F$ had a nontrivial idempotent. Its image in real coordinates
selects a nonempty proper union of blocks and hence a subspace $V$
defined over the rational lattice. The group $\Delta$ preserves the
full lattice in $V$ and its inverse does too, so
$\chi(j)=|\det(j|_V)|$ is a character of $J$ trivial on $\Delta$. It is
nontrivial on $A$ since both the selected and the unselected union of
blocks are nonempty. Pushing the finite invariant measure on
$\Delta\backslash J$ forward by $\chi$ would give a finite nonzero
multiplicatively invariant measure on $\R_{>0}$, impossible. Hence
$F$ is a field.

The representation of $F$ on $\Lambda_\Q$ has some dimension $m$ over
$F$. After extending scalars to $\R$, every embedding of $F$ has
multiplicity $m$, so all blocks have size $m$ and $dm=n$. The elements
of $F$ carrying $\Lambda$ into itself form an order: in lattice
coordinates they are $(gFg^{-1})\cap\Mat_n(\Z)$, which is discrete;
clearing denominators gives a full-rank submodule. Thus the multiplier
set is a finitely generated rank-$d$ ring containing $1$. Its Dirichlet
unit group has rank $d-1$.

If $J\ne G$, then $d>1$. Take a unit of infinite order and pass to an
even power so every real embedding is positive. Its action on the
lattice is an automorphism with determinant $N_{F/\Q}(u)^m=1$, so in
real coordinates it is an element $a\in A$. The element $a$ fixes $x$
and is central in $J$ (constant on each block), hence also fixes every
$xj$ with $j\in J$. Its entries are conjugates of an algebraic unit, so
there are countably many possibilities.

Finally, fixing $a$ and lifting to $G$, the fixed-point equation is
$gag^{-1}=\gamma$ for some $\gamma\in\Gamma$. For each $\gamma$, the
nonempty solution set is a coset of the centralizer of $a$; it is
smooth with tangent root directions exactly those satisfying
$a_{jj}=a_{kk}$. Restricting to countably many injective local quotient
charts gives the asserted pieces in $X$. If $a_{jj}\ne a_{kk}$, two
points of one local $U_{jk}$ plaque cannot solve the same lifted
equation: their relative element $1+tE_{jk}$ would commute with $a$,
forcing $t=0$.
\end{proof}

For $n=6$, the equality $dm=6$ leaves as proper finite-volume orbits
only the block sizes $3+3$ and $2+2+2$. The case of $6$ blocks of size
$1$ reduces to $J=A$: right translation of the compact $A$ orbit has
zero entropy, so by Proposition~\ref{prop:positive-entropy} it cannot
occur as a positive-weight component of a packet limit.

By Proposition~\ref{prop:ball}, Proposition~\ref{prop:positive-entropy},
and Theorem~\ref{thm:ekl}, almost every ergodic component of any packet
limit is one of the proper block types or Haar. To complete the proof of
Theorem~\ref{thm:main} it remains to rule out the proper types. This is
the purpose of the rest of the paper.

\section{Conditional probabilities at intermediate resolutions}
\label{sec:conditioning}

This is the technical core of the paper. We must extract, from each
packet measure $\mu_i$, conditional information along unipotent plaques
that survives the limit. Exact conditional measures give only Dirac
masses because the orbits are compact; instead we condition \emph{small
transverse cells before taking the limit}, at depths that are chosen so
that the packet data change little over a broad range of depths.

\subsection{Charts, patterns, and their extensions}

Throughout, $n=6$, $G=\SL_n(\R)$, $\Gamma=\SL_n(\Z)$, $X=\Gamma\backslash G$,
and $A$ is the full positive diagonal subgroup. Assume we have
$A$-invariant probabilities $\mu_i$ converging weakly to an $A$-invariant
$\mu$, and that the times $T_{-,i},T_{+,i}$ satisfy
$T_{-,i}\to\infty$ and $T_{+,i}/T_{-,i}\to\infty$.

Write $L$ for the traceless diagonal algebra and $a(b)=\exp(b)$.
For $j\ne k$, put
\[
 \beta_{jk}(b)=b_k-b_j,\qquad
 U_{jk}=\{1+tE_{jk}:t\in\R\}.
\]
These signs are chosen for the right action: the displacement $u$ at
$x$ becomes $a(-b)ua(b)$ at $xa(b)$, so its $jk$-entry is multiplied
by $e^{\beta_{jk}(b)}$.

A \emph{pattern} is a set of roots, positive on some element of $L$
and closed under root addition; its \emph{pattern group} $V$ consists
of the unipotent matrices with arbitrary entries in those positions
and zero elsewhere. After reordering coordinates, $V$ is triangular
unipotent. It is normalized by $A$, and there are only finitely many
patterns.

Fix a group $H$ ($V$ or $VA$). A \emph{plaque chart} is an injective
smooth
\[
 \Phi:Q\times B\longrightarrow X,\qquad
 \Phi(q,h)=\psi(q)h
\]
with $Q$ an open box in a complementary transverse manifold and $B$ an
open bounded coordinate box in $H$; it extends smoothly and injectively
to a slightly larger domain. Sets with $q$ fixed are its plaques.

\begin{lemma}[No periods in a pattern group]\label{lem:no-periods}
Let $\nu$ be an $A$-invariant probability on $X$ and let $V$ be a
pattern group. For $\nu$-almost every $x$,
\[
 xv=x,\quad v\in V\quad\Longrightarrow\quad v=1.
\]
At such $x$, the image $xK$ of any prescribed compact $K\subset V$
lies in an embedded plaque with a transverse neighborhood. For any
compact displacement range and any $\epsilon>0$, finitely many inner
plaque charts cover a set of $\nu$-measure at least $1-\epsilon$ and
their outer charts contain that range about every point of the inner
charts.
\end{lemma}

\begin{proof}
Choose $b\in L$ such that $a(-tb)va(tb)\to1$ as $t\to\infty$ for
every $v\in V$. At a point recurrent under $a(b)$, $xv=x$ implies
\[
 xa(kb)\,a(-kb)va(kb)=xa(kb).
\]
A subsequence of the orbit returns to a compact neighborhood of $x$.
The stabilizers of points in that neighborhood contain no nonidentity
element in a fixed identity neighborhood, by discreteness of the lattice
and local quotient charts. The displayed relation is therefore
impossible for $v\ne1$. Poincar\'e recurrence applies almost everywhere.

The orbit map $v\mapsto xv$ is now an injective immersion; its
restriction to a compact set is an embedding. Take a slightly larger
compact domain in $V$ and a complementary transverse slice. Compactness
permits shrinking the slice until its product with the prescribed
domain is injective. A countable subcover and continuity from below
give the finite-cover assertion.
\end{proof}

Fix a countable stock of plaque charts, inner charts, and fixed
subdivisions to which later choices are restricted. It is a basis
wherever local charts are needed, and for pattern groups it carries
the extensions furnished by Lemma~\ref{lem:no-periods}, exhausted by
countable compact displacement ranges. Chart and bin boundaries are
chosen to have $\mu$-null measure: vary coordinate endpoints; at most
countably many widths give a positive-mass boundary.

Two simultaneous usages matter. If $V=WU$ with $U$ a root group, $W$
a normal pattern subgroup, and the patterns disjoint, include charts
\[
 (q,w,u)\longmapsto\psi(q)wu.
\]
Their $V$-, $U$-, and $W$-transverse coordinates are respectively $q$,
$(q,w)$, and $(q,u)$; the last uses normality. Also include charts
$(q,v,a)\mapsto\psi(q)va$ for $VA$ with $V$-transverse coordinate
$(q,a)$. All chart bounds, extensions, bin sizes, and fixed margins
are chosen before $i\to\infty$.

\subsection{A common plateau}

In a chart $\mathcal B=\Phi(Q\times B)$, restrict $\mu_i$ to $\mathcal B$
and partition $Q$ by a shifted dyadic grid of side $2^{-l}$. For
$x=\Phi(q,h)\in\mathcal B$, let $P^{\mathcal B}_{i,l}(x)$ be the
distribution of $h$ conditional on the grid cell of $q$. Zero-mass
columns carry an arbitrary probability. Write $r\prec s$ for
$r_i/s_i\to0$.

\begin{lemma}[A common plateau]\label{lem:plateau}
After passing to a subsequence there are integer depths $l_{-,i}<l_{+,i}$
with
\begin{equation}\label{eq:plateau-room}
 T_-\prec l_-\prec l_+\prec T_+.
\end{equation}
For every fixed chart and every fixed finite plaque-bin partition in the
countable stock, the expected $\ell^1$-distance between the conditional
bin-probability vectors at depths $l_-$ and $l_+$ tends to zero,
averaged over a uniform shift of the nested dyadic grid. The same
conclusion holds for any two intervening depths. The shifts can be
chosen so these conclusions hold for every fixed chart and bin
partition, and so random-grid boundary estimates hold at $l_-,l_+$ and
at any prescribed countable set of interior depths. When coordinate
usages share transverse labels, their grids can share the corresponding
shifts. At interior depths, conditional bin probabilities are unchanged
in expected $\ell^1$-distance by using auxiliary shifted grids with the
same boundary convention.
\end{lemma}

\begin{proof}
Fix a chart, a finite bin partition, and a grid shift. Write $p_l$ for
the conditional bin-probability vector. Its coordinates are bounded
martingales in the increasing transverse grid sigma-fields, so
\[
 \int_{\mathcal B}\|p_v-p_u\|_2^2\,\dd\mu_i
 =\int_{\mathcal B}\|p_v\|_2^2\,\dd\mu_i
  -\int_{\mathcal B}\|p_u\|_2^2\,\dd\mu_i
 \qquad(u\le v).
\]
Summing these over disjoint depth intervals along $\mu_i|_{\mathcal B}$
gives at most $\mu_i(\mathcal B)\le1$. This remains true after averaging
in the shift. Enumerate all chart/bin choices, weight them with positive
summable weights; the weighted sum has the same bounded total variation.

Put $L_i=\log(T_{+,i}/T_{-,i})$. Divide $[0,L_i]$ into $m_i$
consecutive intervals with $m_i\to\infty$ and $m_i=o(L_i)$. Discard
the first and last intervals. A remaining interval has weighted
martingale cost $O(m_i^{-1})$; its endpoints rounded to integers give
$l_-,l_+$. The discarded end intervals and the length
$L_i/m_i\to\infty$ give all three ratios in \eqref{eq:plateau-room}.
Cauchy--Schwarz gives the $\ell^1$ conclusion, and orthogonality of
martingale increments gives it for intervening depths.

For the boundary choice, fix $q$ in a Euclidean transverse coordinate
domain. Under a uniform shift, the probability that $q$ is within
$\rho2^{-l}$ of a depth-$l$ grid wall is $O(\rho)$, uniformly in
$q,l$, and in the sampling measure. Choose $\rho_i\downarrow0$ slowly;
Markov's inequality followed by a diagonal choice over the chart/bin
and depth lists selects shifts satisfying both the conditional-probability
estimates and vanishing boundary mass. Shared labels share the shift
variable; each marginal shift remains uniform.

For the comparison with an auxiliary grid at an interior depth $d$:
outside negligible boundary mass, one of its cells lies inside one
depth-$l_-$ cell and is a union of depth-$l_+$ cells, apart from cells
meeting its walls. Relative collar widths are $O(2^{l_--d})$ and
$O(2^{d-l_+})$, both much smaller than the chosen $\rho_i$. Summing the
martingale bin errors over contained fine cells compares auxiliary to
coarse; discarded mass changes the weighted conditional $\ell^1$ error
by at most twice its total. The same argument compares two selected
grids by sandwiching them between the endpoint grids.
\end{proof}

We fix the following convention. All defining charts, simultaneous
coordinate usages, fixed bin subdivisions, and depth sequences belong
to countable lists; a diagonal selection enforces estimates for each
fixed member but not uniformly over the infinite list at a finite
index. Within $[l_-,l_+]$ any fixed finite number of scales can be
chosen with successive ratios tending to infinity and with room at both
ends (for instance, distinct fixed-fraction interpolations on the
logarithmic depth interval). Auxiliary grids chosen for a single
comparison need not belong to the original list; by
Lemma~\ref{lem:plateau}, their fixed-bin laws are compared to those of
the defining grids. None of these conventions requires absolute
continuity.

\subsection{The augmented probability space}

Choose a defining interior depth for every chart and the shifted grids
of Lemma~\ref{lem:plateau}. For $x$, record in each chart containing
it the conditional plaque probability, together with an isolated dummy
symbol otherwise. Write $Z_i(x)$ for the resulting tuple together with
$x$. Each plaque probability is regarded as a probability on the
compact closure of its coordinate domain. Tightness of $\mu_i$ permits
passage to a subsequence for which $Z_i(x)$ converges in law. The
limiting state is
\[
 \omega=\bigl(x,(P^{\mathcal B})_{\mathcal B}\bigr).
\]
Its $x$-marginal is $\mu$. Fixed-bin probabilities pass to the limit by
the null-boundary convention.

\begin{lemma}[Sampling one plaque probability]\label{lem:sampling}
In any fixed chart, conditional on its transverse coordinate $q$ and
its own limiting probability $P^{\mathcal B}$, the sampled plaque
coordinate $h$ has law $P^{\mathcal B}$. In particular
$h\in\supp P^{\mathcal B}$ almost surely on that chart.
\end{lemma}

\begin{proof}
Let $q_i^0$ be the representative of the cell containing $q$. At the
sequence level, $P^{\mathcal B}_i$ is a function of that cell. For
bounded continuous tests $f$ and $F$,
\[
 \int_{\mathcal B}f(h)F(q_i^0,P^\mathcal B_i)\,\dd\mu_i
 =\int_{\mathcal B}P^\mathcal B_i(f)F(q_i^0,P^\mathcal B_i)\,\dd\mu_i.
\]
On compact inner boxes, replacing $q_i^0$ by $q$ costs an error
tending to zero. Letting $i\to\infty$ and then exhausting the chart by
inner boxes (whose boundary has zero limiting mass) gives the
conditional-law identity. The support assertion follows from a countable
basis for the plaque topology.
\end{proof}

\subsection{Aggregating fine columns into patches}

In an output chart use the two endpoint grids at depths $l_-,l_+$.
Write $E$ for a coarse cell and $e$ for a fine cell contained in $E$.
Fix finitely many plaque bins $J_j$, and let $M_e,N_e$ denote the total
and column masses, $P_E(j)$ the coarse conditional. Set
\[
 \delta_i:=\sum_E\sum_{e\subset E}\sum_j
   |\mu_i(\text{$e$-column},\,h\in J_j)-M_eP_E(j)|.
\]
The plateau estimate says $\delta_i\to0$.

\begin{lemma}[Comparison on trimmed patches]\label{lem:patch-comparison}
In this chart, let $p$ range over pairwise disjoint patches, each lying
transversely in one coarse cell $E(p)$. Suppose a subset of each patch
has the product form
\[
 p^{\mathrm{in}}=
 \bigcup_{e\in I_p}\bigl(e\times B_p\bigr),\qquad
 B_p=\bigcup_{j\in J(p)}J_j,
\]
and the discarded masses $\mu_i(p\setminus p^{\mathrm{in}})$ sum to at
most $\epsilon$. The collections $I_p,J(p)$ may depend on $i,p$, and
the number of patches need not be bounded. If $L_p$ is the plaque-bin
law conditional on $p$, then with errors weighted by $\mu_i(p)$,
\[
 L_p\quad\text{agrees with}\quad
 \frac{\one_{B_p}P_{E(p)}}{P_{E(p)}(B_p)}
\]
to total expected $\ell^1$-error $O(\epsilon)+o(1)$. An arbitrary law
is used if the denominator vanishes. The unnormalized version reads
\begin{equation}\label{eq:patch-unnormalized}
 \sum_p\mu_i(p)
 \left\|P_{E(p)}(B_p)L_p-
             \one_{B_p}P_{E(p)}\right\|_1
 \le O(\epsilon)+o(1).
\end{equation}
If patches and a disjoint refinement into subpatches admit such trims
with the same coarse cell and the same plaque-bin union, their
conditional bin laws agree with the corresponding weighted error.

The assertions also hold for bounded continuous tests up to uniform
continuity error. A test may depend on the sampled point if its norm
and continuity modulus are uniform in the tested plaque variable. At
the output sample, $P_{E(p)}$ may be replaced in fixed-bin expressions
by the recorded data probability, up to error tending to zero.
\end{lemma}

\begin{proof}
For a trimmed patch, sum the errors defining $\delta_i$ over $e\in I_p$
and $j\in J(p)$. The comparison is between the unnormalized bin-mass
vector and $(\sum_{e\in I_p}M_e)\one_{B_p}P_{E(p)}$. Since the trimmed
patches are disjoint, every pair $(e,j)$ occurs in at most one of them,
so the total unnormalized error is at most $\delta_i$. Restoring
discarded mass adds at most $\epsilon$. For two nonnegative finite
vectors $v,w$, \[
 \left\|\frac v{\|v\|_1}-\frac w{\|w\|_1}\right\|_1
 \le\frac{2\|v-w\|_1}{\|v\|_1},
\]
with an arbitrary choice when $\|v\|_1=0$; multiply by
$P_{E(p)}(B_p)\le1$ to compare patches and subpatches via the common
$P_E|_{B_p}$. Approximating continuous tests by bin values gives the
uniform approximation and the sample-dependent-test extension, and the
plateau controls expected fixed-bin distance between the coarse
conditional and the recorded data.
\end{proof}

Arbitrarily small error below is achieved in the order: choose bins and
charts first, then sizes fine enough to reduce error, then let
$i\to\infty$.

\subsection{Projective agreement and diagonal motions}

Center the probability in a chart at its sampled point: if the sample
has group coordinate $h_s$, the group displacement is $h_s^{-1}h$.
Two nonzero measures on a common domain agree \emph{projectively} if
one is a positive scalar multiple of the other there.

\begin{proposition}[Comparison of plaque data]\label{prop:plaque-comparison}
For the augmented limit:
\begin{enumerate}
 \item For every pattern group $V$, its centered probabilities in
 different charts agree projectively on their common plaque sheet near
 the sample. They extend to a measurable nonzero projective locally
 finite measure $m^V_\omega$ on $V$, with $1\in\supp m^V_\omega$.
 \item Fix $b\in L$. In every subsequential joint limit of
 $(Z_i(x),Z_i(xa(b)))$ with $x\sim\mu_i$, the two $V$-measures are
 related by pushforward under $v\mapsto a(-b)va(b)$.
 \item Fix a root $\beta$ and an interior time scale $R$ with room for
 interior depths on both sides. For every fixed $C<\infty$ and every
 sequence $b_i\in\ker\beta$ with $\|b_i\|\le CR_i$, the
 $U_\beta$-measures in every subsequential joint limit of
 $(Z_i(x),Z_i(xa(b_i)))$ agree projectively.
\end{enumerate}
\end{proposition}

\begin{proof}
We prove one local comparison. The motion is either the identity, a
fixed $a(b)$, or $a(b_i)$ with $b_i\in\ker\beta$ and $V=U_\beta$. Write
$c$ for the induced conjugation on $V$; $c$ is fixed in the first two
cases and equals the identity in the third.

The geometry is displayed in group coordinates. For a reparametrized
chart, write its original coordinate as $t\in B_0$ and its group element
$r_q(t)$ with $r_q$ and its inverse smooth on the fixed extension. Given
an input transverse depth $F$ with
\begin{equation}\label{eq:comparison-scales}
 l_-\prec R\prec F\prec l_+,
\end{equation}
input patches are depth-$F$ cells times the full plaque-coordinate
domain; their conditional laws are the depth-$F$ input probabilities.
The depth-$F$ grid may be auxiliary; Lemma~\ref{lem:plateau} compares
it back to the recorded data.

\emph{Fitting into output charts.} By Lemma~\ref{lem:no-periods}, output
inner charts can be chosen whose outer charts accommodate the full
image plaque range with a margin, and a finite such cover omits
arbitrarily small limiting mass. If an image patch meets an inner
chart, the entire image fits in the outer chart for large $i$; this
follows from the transverse diameter estimate below. The map in the two
charts has the form
\[
 (q,h)\longmapsto\bigl(\theta_i(q),\,d_i(q)c(h)\bigr),
\]
with $\theta_i(q)$ independent of $h$ (normality), and $d_i(q)$ a left
translation in $V$. Since both are uniformly controlled,
\begin{equation}\label{eq:transverse-image-width}
 \text{diam}\bigl(\theta_i(\text{input cell})\bigr)
 +\|d_i(q)-d_i(q')\|\le O\bigl(2^{-F}e^{O(R)}\bigr)=o(2^{-l_-}).
\end{equation}
A patch meeting a coarse output wall has its sample in a collar of
relative width $o(1)$ by the grid convention, so its total mass is
$o(1)$. Fine output transverse cells pull back at cost
$O(2^{-l_+}e^{O(R)})=o(2^{-F})$, hence stay within the input cell
except on grid-wall collars of vanishing mass. Fixed plaques admit
common inner $U$-bins (fine output cells for the identity motion,
common inner plaque-bin collection for the diagonal motion) that lie
wholly in the admitted plaque interval up to arbitrarily small sampling
mass. Thus both a patch and its subpatches use the same coarse output
cell and the same plaque-bin union.

\emph{Applying the trim lemma.} Lemma~\ref{lem:patch-comparison} then
gives: the input patch law, converted by the transition map, agrees
with $P_E$ restricted to the admitted bins up to arbitrarily small
weighted error. Taking two continuous displacement tests $f,\phi$
supported inside the input plaque range with margin, cross-multiplication
of the unnormalized comparison gives
\begin{equation}\label{eq:projective-cross-products}
 P_{\mathrm{in}}(f)P_{\mathrm{out}}(\phi\circ c^{-1})
 -P_{\mathrm{in}}(\phi)P_{\mathrm{out}}(f\circ c^{-1})
 \longrightarrow0
\end{equation}
in probability on compared samples. Replacing the depth-$F$ input law
by the recorded probability and the output coarse law by the recorded
output probability uses Lemma~\ref{lem:plateau}; the sample-dependent
centering is allowed by the uniform-test clause of
Lemma~\ref{lem:patch-comparison}.

\emph{Gluing along one plaque.} Applying this with the identity motion,
compare two charts at the sample with a chart extending to contain the
bounded plaque domain of the first. By Lemma~\ref{lem:sampling}, each
probability is positive on every neighborhood of the sample; a
nonnegative continuous test positive at the sample gives positive
integral for both. The cross-product gives projective agreement for
every compactly supported continuous test on the first plaque domain.
To compare two bounded domains, extend to a single chart containing
both, on the same plaque sheet. Exhausting $V$ by compact connected
ranges containing $1$ in their interior produces a coherent
projective family with positive mass near $1$, and hence a locally
finite measure on the union. Local finiteness comes from each compact
set lying in one extension with a finite positive chart mass. Countably
many tests and normalizing bumps give the measurable version.

\emph{Equivariance.} For a fixed diagonal motion, the cross-product
identities give pushforward by $c(v)=a(-b)va(b)$. For the varying
motion in $\ker\beta$, $c$ is the identity, and the estimates are
uniform for $\|b_i\|\le CR_i$: the only growth is $e^{O(R_i)}$,
dominated by \eqref{eq:comparison-scales}. In a paired-state limit the
two base points need not be functions of each other, but the local
identities hold on every paired open margin event, and these events
exhaust probability one because both marginals admit the required
extended charts. Since the argument applies to an arbitrary sequence
$b_i$ under the bound, the last clause follows.
\end{proof}

The measures $m^V_\omega$ retain the sequence information while
possessing the chart-consistency and diagonal-comparison properties of
ordinary plaque measures. The next step is to relate the data for
different pattern groups; that requires a product argument, rather
than a further change of coordinates.

\section{A product rule for the plaque probabilities}\label{sec:product}

The projective measures constructed in Section~\ref{sec:conditioning}
retain information from conditioning the packet measures at depths
tending to infinity. We now prove a product rule for them, into
one-dimensional root group factors attached to the same augmented state.
That property is what lets root orientations later compose.

Product decompositions for genuine leafwise conditional measures were
developed by Einsiedler and Katok \cite[Proposition~8.3]{EK2003}
\cite[Theorems~8.4--8.5]{EK2005}; see also \cite[Corollary~8.8 and
Theorem~9.8]{EL2010}. Here the underlying measures are the
augmented-state data from Section~\ref{sec:conditioning}, built by
conditioning at intermediate depths and passing to the limit; the cited
leafwise theorems do not directly apply. We prove the product identity
for the data by the chart-comparison machinery of
Section~\ref{sec:conditioning}.

Throughout, use $p\prec q$ for $p_i/q_i\to0$. Every scale used below
lies in the plateau $[l_{-,i},l_{+,i}]$, with room for the displayed
separations and additional separations at both ends.

\begin{theorem}[Product rule]\label{thm:product}
Let $V$ and $W$ be pattern groups, let $U=U_\alpha$ be a one-dimensional
root group, and suppose that multiplication gives a coordinate
decomposition
\[
 V=WU,\qquad W\triangleleft V.
\]
Suppose also that some $a_0\in A$ fixes $U$ and that the automorphism
\begin{equation}\label{eq:contraction-map}
 \Delta(w)=a_0^{-1}wa_0
\end{equation}
strictly contracts $W$. Then, for almost every augmented state $\omega$,
\begin{equation}\label{eq:product-rule}
 m^V_\omega
  \ \propto\
  ((w,u)\longmapsto wu)_*
      \bigl(m^W_\omega\otimes m^U_\omega\bigr).
\end{equation}
The constant of proportionality is positive, and may depend on $\omega$.
\end{theorem}

The case $W=\{1\}$ is immediate. Henceforth $W\ne\{1\}$. The root
constraints defining a contracting logarithm are rational linear
equalities and strict inequalities. Rescale $a_0$ if necessary so that,
in matrix-entry coordinates of $W$,
\begin{equation}\label{eq:integer-contraction}
 \Delta(w)_{jk}=2^{-d_{jk}}w_{jk},
   \qquad d_{jk}\in\N,\ d_{jk}>0.
\end{equation}
Composition of roots adds weights: $d_{j\ell}=d_{jk}+d_{k\ell}$.

The proof has two parts. Contraction of $W$ shows that the $U$
marginal of $m^V_\omega$ is $m^U_\omega$ and independent of the $W$
coordinate (Lemma~\ref{lem:u-factor}). Identifying the remaining $W$
marginal with $m^W_\omega$ uses a deeper argument
(Section~\ref{sec:product-w-id}): a depth-$S$ $U$-cell label contains
only $O(S)$ information, and revealing it can appreciably change the
law of a fixed-length $W$-tile refinement only at a vanishing
proportion of $T$ successive stages when $S\prec T$. Expanding a
selected parent tile then turns this refinement comparison into a
comparison of bounded $W$ tests.

\subsection{The factor fixed by the contraction}

Use the simultaneous chart from
Section~\ref{sec:conditioning},
\[
 (q,w,u)\longmapsto\psi(q)wu,
\]
with transverse coordinates $q$ for $V$, $(q,w)$ for $U$, and $(q,u)$
for $W$. Compatibility transverse grids are used when the same $q$
occurs.

\begin{lemma}\label{lem:u-factor}
In this chart, the limiting $V$-plaque probability is a product in
coordinates $w,u$, and its $u$-marginal equals the $U$-plaque
probability at the sample. Consequently there is a projective locally
finite measure $\nu^W_\omega$ on $W$ with
\begin{equation}\label{eq:provisional-product}
 m^V_\omega
  \ \propto\
  ((w,u)\longmapsto wu)_*
      (\nu^W_\omega\otimes m^U_\omega).
\end{equation}
\end{lemma}

\begin{proof}
Choose interior scales $l_-\prec T\prec F\prec l_+$.
The diagonal motion $a_0^{\lfloor T\rfloor}$ acts on a patch with
$q$-cell $Q_F$ and full $w,u$ ranges by contracting $w$ at rate
$e^{-cT}$ while fixing the $u$ range.

Fix a loss $\epsilon>0$. A finite cover of output $\mu_i$-mass by
inner $U$-charts exists with plaque extensions containing all image
patches, by Lemma~\ref{lem:no-periods}; the loss is at most
$\epsilon$. An input patch whose image meets an inner output chart
lies in its outer chart for sufficiently large $i$: its $W$-range
shrinks by $O(e^{-cT})$ and its $q$-range variation, bounded by
\[
 O(2^{-F}e^{CT}),
\]
is negligible compared to $2^{-l_-}$. The transverse image of a patch
in the output $U$-chart has diameter
\[
 O(e^{-cT}+2^{-F}e^{CT})=o(2^{-l_-}).
\]
The transverse projection is independent of $u$ (normality of $W$),
and its restriction to the input $(q,w)$-range is injective. In the
output $u$-coordinate the input interval is translated by a parameter
varying by $o(1)$ across a patch. Therefore a patch and its subpatches
uses a common coarse output cell and a common plaque-bin union after
trimming.

A patch meeting a coarse output wall has its sample in a collar of
relative width $o(1)$ by the grid convention, so its mass $o(1)$.
Fine output transverse cells pull back at cost $O(e^{CT}\cdot2^{-l_+})$,
negligible compared with the input mesh; they stay inside the prescribed
input label except on collars of $o(1)$ mass. Fixed output plaque bins
lie wholly in the admitted plaque interval (up to arbitrarily small
sampling mass) after removing fixed input chart-boundary collars.
Thus Lemma~\ref{lem:patch-comparison} applies to both patches and
subpatches with a common coarse output cell and plaque-bin union, and
gives, for every continuous $f$ on the input $u$-interval,
\begin{equation}\label{eq:u-independence-prelimit}
 \int
 \left|
  \E_{\mu_i}(f(u)\mid Q_F,W_F)
   -\E_{\mu_i}(f(u)\mid Q_F)
 \right|\,\dd\mu_i
 \longrightarrow0.
\end{equation}
Reducing bin and collar errors arbitrarily and then cover loss
arbitrarily, and letting $i\to\infty$, gives the limit zero; the
common translation of $u$ costs nothing, because its variation tends
to zero and $f$ is uniformly continuous.

The plateau comparison replaces these depths by the recorded depths
of the augmented data. Independence follows by replacing bounded
continuous $g(w)$ with its bin values on $W_F$, using conditional
expectation and
\begin{equation}\label{eq:product-independence-bound}
 \begin{aligned}
 &\int\left|
  \E(f(u)g(w)\mid Q_F)-\E(f(u)\mid Q_F)\E(g(w)\mid Q_F)
  \right|\,\dd\mu_i\\
 &\ \le
  \|g\|_\infty\int\left|
   \E(f(u)\mid Q_F,W_F)-\E(f(u)\mid Q_F)
   \right|\,\dd\mu_i
  +2\|f\|_\infty\omega_g(C2^{-F}),
 \end{aligned}
\end{equation}
which tends to zero. Hence $P^V(fg)=P^V(f)P^V(g)$ for bounded
continuous $f,g$. In displacement coordinates from a sample with
group coordinates $(w_s,u_s)$, the group element leading to $wu$ has
$(W,U)$-coordinates
\begin{equation}\label{eq:product-displacement}
 \bigl(\Ad_{u_s^{-1}}(w_s^{-1}w),\ u_s^{-1}u\bigr),
\end{equation}
so the product also holds in displacement coordinates. Projective
chart comparison (Proposition~\ref{prop:plaque-comparison}) then
identifies $W$-marginals across charts, giving a projective
$\nu^W_\omega$ and \eqref{eq:provisional-product}.
\end{proof}

\subsection{Nested tiles and the information cost}
\label{sec:product-tiles}

To identify $\nu^W_\omega=m^W_\omega$ projectively, we compare small
$W$ pieces before a reverse motion $a_0^{-(k+m_0)}$ with bounded $W$
ranges after it. We first construct a subdivision of $W$ compatible
with $\Delta$.

Let $\Gamma_W$ be the lattice of pattern matrices with integral
entries; Haar on $W$ is Lebesgue in its entries, so covolume $(\Gamma_W)=1$.

\begin{lemma}[Tiles adapted to the contraction]\label{lem:tiles}
There is a compact $\mathcal F\subset W$ of Haar measure one and
Haar-null boundary such that the translates $\gamma\mathcal F$,
$\gamma\in\Gamma_W$, tile $W$ with disjoint interiors, and
\[
 \mathcal F=\bigcup_{d\in\mathcal D}\Delta(d)\Delta(\mathcal F),\qquad
 \mathcal D=\{\gamma\in\Gamma_W:0\le\gamma_{jk}<2^{d_{jk}}
             \ \text{for each entry }jk\},
\]
with the union disjoint off boundaries. For every $k\ge0$, the tiles
$\gamma\Delta^k\mathcal F$, $\gamma\in\Delta^k\Gamma_W$, form a
subdivision, nested in $k$. They may be shifted by a common left
translation $g$. If $g$ has Haar probability on $W/\Gamma_W$, then for
each fixed $w$, the coordinate of $w$ in its shifted level-$k$ tile
has uniform Haar distribution on $\mathcal F$ after rescaling by
$\Delta^{-k}$; in particular the probability of lying in a
$\rho$-collar of $\partial\mathcal F$ tends to zero as $\rho\downarrow0$,
uniformly in $k,w$.
\end{lemma}

\begin{proof}
For a pattern entry $\alpha$, let $d_\alpha$ be its weight and
$B_\alpha=2^{d_\alpha}$. Let
\[
 \mathcal F=\left\{\prod_{\nu=1}^{\infty}\Delta^\nu(d_\nu):
   \ d_\nu\in\mathcal D\right\}
\]
with factors ordered by $\nu$. Contracting and using the finite
nilpotent formulas, the products converge uniformly (each entry
expands as a convergent series with geometrically decaying weights),
so $\mathcal F$ is compact.

Order entries by increasing root height in upper-triangular form. In
an entry $\alpha$, the coordinate in the expansion of an element of
$\mathcal F$ is
\[
 \sum_{\nu\ge1}B_\alpha^{-\nu}(d_\nu)_\alpha+C_\alpha,
\]
where $C_\alpha$ depends only on digits in lower-height entries. At
height one, this is the base-$B_\alpha$ expansion of a number in
$[0,1]$; inductively, once lower-height digits are specified away from
their countable ambiguity sets, the possible $\alpha$ coordinates form
$[C_\alpha,C_\alpha+1]$ with a unique expansion.
Left multiplication by $\gamma\in\Gamma_W$ has the same triangular
form: the $\alpha$ coordinate changes by its integral $\alpha$-entry
plus a function of lower-height entries, and reduction by these a.e.
unique expansions places each point in exactly one translate.
Fubini at each finite entry proves a.e. covering and Haar measure one.
Local finiteness follows from compactness of $\mathcal F$ and
discreteness of $\Gamma_W$; the union is closed and covers a.e.,
hence covers $W$.

Splitting the first digit gives
\[
 \mathcal F=\bigcup_{d\in\mathcal D}\Delta(d)\Delta(\mathcal F),
\]
and triangular reduction shows $\mathcal D$ represents right cosets of
$\Delta^{-1}\Gamma_W$ in $\Gamma_W$, so $\Delta\Gamma_W$ is a disjoint
union of $\Gamma_W\Delta(d)$, $d\in\mathcal D$. Iteration gives the
level-$k$ subdivision and nesting.

Baire category gives nonempty interior to $\mathcal F$; iterated
descendants give interior and vanishing diameter, so
$\overline{\mathrm{int}\,\mathcal F}=\mathcal F$. Interior intersections
of two translates would have positive volume, contradicting a.e.
uniqueness; so boundaries are contained in the null pairwise overlaps.

For the law of a shifted coordinate: since weights are positive
integers, $\Gamma_W\subset\Delta^k\Gamma_W$, and
$W/\Gamma_W\to W/\Delta^k\Gamma_W$ pushes Haar probability to Haar
probability. Inversion and right multiplication by $w$ give invariant
volume on the left quotient, and the representative of $g^{-1}w$ in
$\Delta^k\mathcal F$ has normalized Haar distribution on that tile;
rescaling gives Haar on $\mathcal F$. This is exactly the tile coordinate
of $w$ in $g\gamma\Delta^k\mathcal F$. Fubini for any sampling
probability gives the collar consequence.
\end{proof}

For finite labels write $H$ for Shannon entropy with natural logarithms
and $I(Y;Z\mid X)=H(Y\mid X)-H(Y\mid X,Z)$ for conditional mutual
information.

\begin{lemma}[Information in a transverse label]\label{lem:stage-information}
Fix a chart $(q,w,u)$. For integers $F,S\ge1$, let $Q_F$ be the
depth-$F$ $q$-cell label and $U_S$ the depth-$S$ $u$-cell label. Let
$K_k$ be the level-$k$ tile label of $w$; shifts may be fixed or
averaged. For every pair of integers $T\ge j\ge1$,
\begin{equation}\label{eq:stage-information}
 \sum_{k=0}^{T-j}I_{\mu_i}(K_{k+j};U_S\mid Q_F,K_k)\le Cj(1+S),
\end{equation}
with $C$ depending only on the fixed coordinate domain, not on
$i,F,S,T,j$, the sampling measure, or the tile shift. Consequently,
along any sequence $T\to\infty$, $S/T\to0$, for every fixed $j$ the
average over $T/2\le k\le3T/4$ of the $\ell^1$-error between
$\mathcal L(K_{k+j}\mid Q_F,K_k,U_S)$ and
$\mathcal L(K_{k+j}\mid Q_F,K_k)$ tends to zero. The same holds after
averaging in the shift.
\end{lemma}

\begin{proof}
The $u$-cells meeting a fixed bounded interval are at most $O(2^S)$
many, so $H(U_S)\le C(1+S)$. Because tile labels are nested, the chain
rule gives
\[
 \sum_{k=0}^{L-1}I(K_{k+1};U_S\mid Q_F,K_k)
 =I(K_L;U_S\mid Q_F,K_0)\le H(U_S).
\]
For a jump of length $j$, expand its mutual information into these
one-stage increments; each increment occurs at most $j$ times,
proving \eqref{eq:stage-information}. Pinsker's inequality and
Cauchy--Schwarz bound the average conditional $\ell^1$-error by
$C\sqrt{j(1+S)/T}$. The entropy bound is shift-invariant.
\end{proof}

\subsection{Identification of the contracted factor}
\label{sec:product-w-id}

Choose interior scales
\begin{equation}\label{eq:w-factor-scales}
 l_-\prec S\prec T\prec F\prec l_+.
\end{equation}
Fix nonnegative $f,\phi\in C_c(W)$, bounded by one, with $\phi(1)>0$.
It suffices to show
\begin{equation}\label{eq:w-ratio-target}
 \frac{\nu^W_\omega(f)}{\nu^W_\omega(\phi)}
 =
 \frac{m^W_\omega(f)}{m^W_\omega(\phi)}
\end{equation}
almost surely; identity support gives positive denominators, and
countably many tests then give projective equality.

The comparison is made at the output of
\begin{equation}\label{eq:reverse-motion}
 a_0^{-(k+m_0)},\qquad T/2\le k\le3T/4,
\end{equation}
with $m_0$ a fixed positive integer. A level-$k$ input tile becomes
the fixed output shape $\Delta^{-m_0}\mathcal F$; the level-$(k+j)$
tiles, with $j$ fixed after $m_0$, resolve bounded $W$ tests on that
shape.

Inside one input chart, the parent patch has label $p=(Q_F,K_k)$
(whole $u$-interval available); the subpatch has label
$p'=(Q_F,K_k,U_S)$. By Lemma~\ref{lem:stage-information},
\begin{equation}\label{eq:w-two-tile-laws}
 \mathcal L_{\mu_i}(K_{k+j}\mid Q_F,K_k)\ \text{and}\
 \mathcal L_{\mu_i}(K_{k+j}\mid Q_F,K_k,U_S)
\end{equation}
agree in expected $\ell^1$-distance for suitable shifts and stages.
At output, the parent tile has fixed shape
$\Delta^{-m_0}\mathcal F$; descendants have shape
$\Delta^{j-m_0}\mathcal F$; increasing the fixed $j$ makes them as
small as prescribed.

\emph{Coverage and geometry.} Let $\epsilon>0$. Choose finitely many
input charts with inner boxes and fixed margins covering all but a
small multiple of $\epsilon$. Choose whole level-$k$ $w$-tiles as
patch and subpatch. Tiles meeting inner boxes lie inside outer input
$w$-ranges for large $i$ (diameters tend to zero). Pick a
tile-coordinate collar thin enough that its expected mass is much
less than the allowed coverage loss; Lemma~\ref{lem:tiles} supplies
this uniformly in $k$. Outside this collar the sample has a fixed
positive margin in $\mathcal F$; choose $m_0$ so that every bounded
displacement in a fixed compact containing
$\supp f\cup\supp\phi\cup\{1\}$ lies, with margin, in the output tile
about that sample. Indeed, pulling back to the input tile coordinate
contracts by $\Delta^{m_0}$ up to the bounded conjugation
$u_0(\cdot)u_0^{-1}$.

Take finitely many output charts of the same simultaneous form, with
inner boxes and $V$-extensions large enough for the images of these
patches. For fixed $m_0$, all displacement ranges from a sample to
points in its output patch lie in a fixed compact ($W$-tile origin,
fixed $U$-base, fixed shape $\Delta^{-m_0}\mathcal F$); the $q$
variation after \eqref{eq:reverse-motion} is $O(2^{-F}e^{CT})$.
A patch meeting an output inner box fits in its extended chart with
a margin. Assign patches to finitely many output charts.

These choices fix charts, tile extent, and test-range margins but
\emph{not} bin-comparison accuracy. Later refinements (finer tile
exclusion, finer output bins, larger fixed $j$) improve errors without
changing coverage. Averaging in shifts and in $k\in[T/2,3T/4]$
simultaneously with the information estimate and applying Markov's
inequality picks a common shift and $k=k_i$ that satisfy both coverage
and comparison-error bounds. The scales satisfy
\eqref{eq:w-factor-scales}.

\emph{Output comparisons.} Write an assigned output chart as
$(q',w',u')$. Left multiplication by $w_0u_0$ sends $wu$ to
\begin{equation}\label{eq:separate-left-translation}
 (w_0\Ad_{u_0}(w))\,(u_0u).
\end{equation}
Thus the two plaque coordinates change separately and the parameters
of this bounded left multiplication vary by $O(2^{-F}e^{CT})$ across
the input $q$-cell. The $V$-transverse diameter in $q'$ is bounded by
the same quantity, which is smaller than $2^{-CS}$ for every fixed $C$
once $i$ is large. For a subpatch, the $W$-transverse coordinates
$(q',u')$ have diameter $O(2^{-S}+2^{-F}e^{CT})=o(2^{-l_-})$. Discard
patches or subpatch comparisons whose sample lies in a coarse output
transverse wall collar; total discarded mass $o(1)$.

Apply Lemma~\ref{lem:patch-comparison} to $V$-data on patches and to
$W$-data on subpatches, with common coarse output cell and common
plaque-bin union: for a patch collections of $w'$ and $u'$ bins in
product form; for each subpatch the same inner collection of $w'$
bins; both retained over the entire corresponding transverse
projection, by \eqref{eq:separate-left-translation} below. Inner
products lose arbitrarily small sampling mass (transverse maps are
injective on a patch or subpatch, extended charts describe one plaque
sheet, and the pullback cost $e^{CT}$ stays inside input labels
because $F,S\prec l_+$). Outside the collars, the sample has a definite
margin in admitted $w'$ and $u'$ ranges, so sufficiently fine fixed
plaque bins are admitted.

\emph{Comparison of the $W$-tests.} At a sampled output point, let
$P_V,P_W$ denote its finite-$i$ chart probabilities. Interpret
$f,\phi$ in the output $w'$ coordinate using the displacement formula
\eqref{eq:product-displacement}; their norms and continuity moduli
are uniform on the finite chart ranges. Let $B_u$ denote the union of
inner $u'$ bins for the patch. We claim that, on the covered samples,
\begin{equation}\label{eq:w-cross-product}
 P_V(f\,1_{B_u})P_W(\phi)
  -P_V(\phi\,1_{B_u})P_W(f)
\end{equation}
has arbitrarily small expected absolute value as $i\to\infty$.

Write $A_p$ and $A_{p'}$ for the output plaque-coordinate laws when
sampling the input patch $p$ and its subpatch $p'$. Let
$e_f,e_\phi,e'_f,e'_\phi$ be the absolute errors in the four
comparisons
\[
 c_pA_p(f)\simeq P_V(f1_{B_u}),\quad
 c_pA_p(\phi)\simeq P_V(\phi 1_{B_u}),\quad
 c_{p'}A_{p'}(f)\simeq P_W(f),\quad
 c_{p'}A_{p'}(\phi)\simeq P_W(\phi),
\]
with each inequality an absolute error (after weighting by the patch
mass) that is made arbitrarily small. Then $c_p,c_{p'}\in[0,1]$. At
input, Lemma~\ref{lem:stage-information} compares the level-$(k+j)$
tile law in $p$ and $p'$; at output these finer tiles have relative
shape $\Delta^{j-m_0}\mathcal F$, whose diameter $\sigma_{i,j}$ can be
made as small as prescribed for fixed $m_0$ and $j$ large. The
conversion to $w'$ coordinates has uniformly bounded distortion and
is independent of $u$, by \eqref{eq:separate-left-translation}. Let
$\eta_i(x)$ be the $\ell^1$-distance between the two conditional
level-$(k+j)$ tile laws; it satisfies
$\E\,\eta_i\to0$ by Lemma~\ref{lem:stage-information}, and for
$h\in\{f,\phi\}$ with common continuity modulus $\omega_h$,
\begin{equation}\label{eq:tile-test-comparison}
 |A_p(h)-A_{p'}(h)|\le\eta_i(x)+2\omega_h(\sigma_{i,j}).
\end{equation}

Multiply the four mass comparisons and cross-multiply without dividing
by $c_p,c_{p'}$; the discrepancy is bounded by
\[
 e_f(x)+e_\phi(x)+e'_f(x)+e'_\phi(x)
  +2\eta_i(x)+2\omega_f(\sigma_{i,j})+2\omega_\phi(\sigma_{i,j}).
\]
Integrating bounds the first four terms by the total trim error, the
fifth by the stage-information error, and the last two by their uniform
moduli. Choose trim accuracy and fixed bins first, then $j$ large,
then let $i\to\infty$.

\emph{Removing the $U$-restriction.} By Lemma~\ref{lem:u-factor}, the
limiting $V$ data has independent $w',u'$ coordinates, so
$P_V(f1_{B_u})-P_V(f)P_V(B_u)\to0$ in probability on our charts, and
the same with $\phi$. Since $B_u$ comes from one fixed finite
subdivision and each bin boundary has zero limiting mass, the
convergence is simultaneous over the whole finite bin collection;
continuous displacement tests vary with the sample, and output sampling
has law $\mu_i$, so the same holds at output. Hence
\eqref{eq:w-cross-product} becomes
\begin{equation}\label{eq:w-cross-product-reduced}
 P_V(B_u)\bigl(P_V(f)P_W(\phi)-P_V(\phi)P_W(f)\bigr)\simeq0.
\end{equation}

\emph{Denominators and projective limit.} On each fixed covered chart,
$B_u$ contains a fixed $U$-neighborhood of the sample; choose a
continuous bump supported there, positive at the sampled $u'$.
Its limit $V$-integral $P_V(B_u)$ is positive, and identity support
gives $P_V(\phi)>0$ and $P_W(\phi)>0$. Hence outside a further small
exceptional probability there is $\kappa>0$ with
\begin{equation}\label{eq:w-denominators}
 P_V(B_u)\ge\kappa,\quad P_V(\phi)\ge\kappa,\quad P_W(\phi)\ge\kappa.
\end{equation}
Fix coverage, $m_0$, output charts, and test-range margins first; then
$\kappa$; then refine bin approximations and tile collars and enlarge
$j$ to make
$|\eqref{eq:w-cross-product-reduced}|$ small compared to $\kappa^3$.
Then dividing gives
\[
 \frac{P_V(f)}{P_V(\phi)}=\frac{P_W(f)}{P_W(\phi)}+o(1)
\]
outside an arbitrarily small exceptional probability. Precisely, if
$\mathcal G_i$ is the covered set on which
\eqref{eq:w-denominators} holds and $D_i(x)$ is the left side of
\eqref{eq:w-cross-product-reduced}, then for every $\delta>0$,
\[
 \mu_i\Bigl(\mathcal G_i\cap
 \bigl\{\bigl|\tfrac{P_V(f)}{P_V(\phi)}
             -\tfrac{P_W(f)}{P_W(\phi)}\bigr|>\delta\bigr\}\Bigr)
 \le\frac1{\delta\kappa^3}\int_{\mathcal G_i}|D_i|\,\dd\mu_i.
\]

Suppose for contradiction the desired ratio equality failed at the
projective limit. A countable choice of tests and charts gives a
reference output chart and a positive-probability set on which the two
ratios differ by more than some fixed positive number; restrict to
a set with positive denominators and inner test-support margins, so
that both $\mu$-probability and finite-$i$ ratios admit lower bounds.
Applying the coverage construction with exception probability far
smaller than this discrepancy probability and using
Proposition~\ref{prop:plaque-comparison} and Lemma~\ref{lem:u-factor}
identifies the two limiting ratios on the samples covered by both the
reference chart and an assigned output chart, contradicting persistence
of the discrepancy. Hence \eqref{eq:w-ratio-target} holds, and
$\nu^W_\omega\propto m^W_\omega$ almost surely. With
\eqref{eq:provisional-product}, the proof of Theorem~\ref{thm:product}
is complete.

\section{Active roots and their support}\label{sec:activity}

The product rule of Section~\ref{sec:product} organizes the plaque
measures into a directed graph at each augmented state. This section
shows that the arrows compose, describes the support of the
positive-root plaques, and realizes the graph by measurable indicators
on the original spaces. The resulting indicators will be used in the
entropy argument of Section~\ref{sec:segment-entropy} and in the
combination argument of Section~\ref{sec:combination}.

Write $\mathsf P$ for the law of the limiting augmented state
$\omega=(x,(P^\mathcal B)_\mathcal B)$ from Section~\ref{sec:conditioning}.
For $j\ne k$ define
\[
 a_{jk}(\omega)
  =\one_{\{m^{U_{jk}}_\omega\text{ is not supported on }\{1\}\}}.
\]
We say the root $jk$ is \emph{active} at $\omega$ when $a_{jk}(\omega)=1$.
This definition does not depend on the representative of the
projective measure: after normalizing by a positive integral against a
bump near the identity, activity is detected by countably many
nonnegative continuous tests supported away from the identity. All
statements below are made on a common set of full $\mathsf P$-measure
on which the comparison and product rules hold.

\begin{proposition}[Composition and positive-root support]
\label{prop:activity}
Almost surely, activity is transitively closed:
\[
 a_{pq}(\omega)a_{qr}(\omega)\bigl(1-a_{pr}(\omega)\bigr)=0
 \qquad(p,q,r\text{ distinct}).
\]
For a regular $b\in L$, let $V$ be the pattern group of all
$b$-positive roots, and let $V_\omega^+$ be the subgroup generated by
the active $b$-positive root groups. These active roots form a pattern,
and
\[
 \supp m^V_\omega\subset V_\omega^+.
\]
Both conclusions hold simultaneously for every regular $b$.
\end{proposition}

\begin{proof}
We first show that Theorem~\ref{thm:product} factors the measure of any
pattern group into its one-dimensional root measures, in a suitable
order. The cone generated by its roots is pointed. Choose a root
$\alpha$ on an exposed extremal ray. No other root in the pattern is
proportional to $\alpha$, so a diagonal logarithm can vanish on
$\alpha$ and be strictly negative on every other root. Removing
$\alpha$ leaves a normal pattern subgroup $W$: a sum of two roots in
the pattern cannot equal $\alpha$, since extremality would force both
summands onto its ray. Thus Theorem~\ref{thm:product} applies with
$V=WU_\alpha$. Iterate, allowing $W=\{1\}$ at the last step.

For transitivity, fix distinct $p,q,r$ and consider the three-root
Heisenberg group generated by $U_{pq},U_{qr},U_{pr}$. If $a_{pr}=0$,
factor first with $U_{pq}$ as the final factor and then with $U_{qr}$
as the final factor; both give the same projective measure. Write
$u_{jk}(t)=1+tE_{jk}$. Since $U_{pr}$ is a point mass, the first
description is supported on matrices
\[
 u_{qr}(t)u_{pq}(s)=1+sE_{pq}+tE_{qr},
\]
whereas the second is the pushforward of the product of the two root
measures under
\[
 u_{pq}(s)u_{qr}(t)=1+sE_{pq}+tE_{qr}+stE_{pr}.
\]
If both roots were active, each root measure would give positive mass
to some compact parameter set away from zero; their product would give
positive mass to matrices with nonzero $pr$-entry, contradicting the
first description. The projective identities hold on compact parameter
ranges, so all masses are finite. Thus transitivity holds.

For the support assertion, iterate the successive factorization.
Every inactive root factor is a point mass at the identity; each
remaining factor lies on its own active root group. The product lies
in the subgroup generated by the active roots. Transitivity shows
exactly that the active positive roots are closed under root addition,
so that subgroup is a pattern group and is closed. There are finitely
many positive-root patterns, so the simultaneous assertion holds.
\end{proof}

The entropy argument uses plaques enlarged by the diagonal group.
The next lemma describes the support of those enlarged plaques at the
sampled state.

\begin{lemma}[Support on plaques enlarged by $A$]\label{lem:va-support}
With $V$ and $V_\omega^+$ as in Proposition~\ref{prop:activity}, the
limiting $VA$-plaque probability, written in displacement coordinates
from its sample, lies in $V_\omega^+A$ within its chart.
\end{lemma}

\begin{proof}
Use a product box with coordinates $\psi(q)va$ and product coordinate
domains. Local $A$-invariance gives, for each $i$, a product
decomposition
\[
 d\mu_i(q,v,a)=d\nu_i(q,v)\,da
\]
inside the box, where $da$ is Haar on its coordinate box, absorbed into
$\nu_i$. For $VA$-plaques the transverse coordinate is $q$; for
$V$-plaques it is $(q,a)$. Use a common $q$ grid, and supplement it by
an $a$ grid for the second. Conditional on a $q$-cell, the $VA$
probability is the product of a probability on $v$ and normalized Haar
on $a$; conditioning also on an $a$-cell leaves the $v$ probability
unchanged. These are exact identities before the limit, and the
interior-depth comparisons allow compatible grids in the defining
data. The identities pass to the joint limit by continuous tests and
the null-boundary convention.

At a sample $x=\psi(q_s)v_sa_s$ the $V$-displacements obtained by
varying $v$ at fixed $a_s$ are $a_s^{-1}v_s^{-1}va_s$. By
Proposition~\ref{prop:activity}, their probability lies in
$V_\omega^+$. A general $VA$-displacement is
\[
 a_s^{-1}v_s^{-1}va
 =(a_s^{-1}v_s^{-1}va_s)(a_s^{-1}a),
\]
which belongs to $V_\omega^+A$ almost surely. The argument holds on
inner chart boxes; exhausting them gives the statement on the whole
plaque box, whose boundary has zero probability.
\end{proof}

\begin{lemma}[Indicators along the sequence]\label{lem:activity-indicators}
Fix an interior scale $R_i$ in the long-wall comparison of
Proposition~\ref{prop:plaque-comparison}, ratio-separated from
interior depths on both sides. There are measurable
$g_{jk,i}:X\to\{0,1\}$ with
\[
 \operatorname{Law}_{\mu_i}(Z_i(x),(g_{jk,i}(x))_{j\ne k})
 \ \Longrightarrow\
 \operatorname{Law}_{\mathsf P}(\omega,(a_{jk}(\omega))_{j\ne k}).
\]
The $g$ satisfy asymptotic transitivity:
\[
 \int_X g_{pq,i}(x)g_{qr,i}(x)\bigl(1-g_{pr,i}(x)\bigr)\,\dd\mu_i(x)
   =o(1)
 \quad(p,q,r\text{ distinct}),
\]
and, for every fixed $C<\infty$,
\begin{equation}\label{eq:wall-invariance}
 \sup_{\substack{b\in\ker\beta_{jk}\\ \|b\|\le CR_i}}
 \int_X|g_{jk,i}(xa(b))-g_{jk,i}(x)|\,\dd\mu_i(x)\longrightarrow0.
\end{equation}
\end{lemma}

\begin{proof}
Let $\mathcal S$ be the separable metric state space used for $Z_i$,
and put $A_{jk}=\{\omega:a_{jk}(\omega)=1\}$. Choose
$\epsilon_r\downarrow0$. For each root and each $r$ there is a
$\mathsf P$-continuity set $D_{jk,r}\subset\mathcal S$ with
$\mathsf P(D_{jk,r}\triangle A_{jk})<\epsilon_r$. For completeness,
use countably many continuous coordinate tests generating the Borel
$\sigma$-algebra of $\mathcal S$, and for each test choose a countable
dense family of thresholds outside the atoms of its distribution.
Finite Boolean combinations of these threshold sets have null boundary
and approximate any measurable set in $\mathsf P$-measure.

For fixed $r$, set $g^{(r)}_{jk,i}(x)=\one_{D_{jk,r}}(Z_i(x))$. Weak
convergence of the states and null boundaries give joint convergence
of states and these finitely many indicators. Since the exact
indicators are transitive, the limiting transitivity defect for any
triple is at most $3\epsilon_r$.

We next prove the uniform wall estimate at this fixed approximation
level. Fix a root, $C$, and any sequence $b_i\in\ker\beta_{jk}$ with
$\|b_i\|\le CR_i$. Both marginals of $(Z_i(x),Z_i(xa(b_i)))$ have the
ordinary state law (since $\mu_i$ is $A$-invariant); consequently
these paired laws are tight. In every subsequential limit the two
marginals are $\mathsf P$, and Proposition~\ref{prop:plaque-comparison}
identifies the two projective $U_{jk}$ measures, so their exact
activity indicators are equal. Applying the continuity-set
approximation separately to the two marginals gives, in each such
limit,
\[
 \E\bigl|\one_{D_{jk,r}}(\omega)-\one_{D_{jk,r}}(\omega')\bigr|
 \le 2\epsilon_r.
\]
The boundary of the tested event is contained in the union of the two
marginal boundaries, so the corresponding prelimit expectations
converge along this subsequence. Since $b_i$ was arbitrary, a
sequence chosen within vanishing error of the supremum gives
\[
 \limsup_i\ \sup_{\substack{b\in\ker\beta_{jk}\\\|b\|\le CR_i}}
 \int_X\bigl|g^{(r)}_{jk,i}(xa(b))-g^{(r)}_{jk,i}(x)\bigr|\,\dd\mu_i(x)
 \le 2\epsilon_r.
\]
The limit of paired states is used even when $b_i$ diverges.

Finally choose $r=r(i)\to\infty$ slowly enough. At stage $r$, impose
the preceding estimates for all roots, all triples, and all integers
$1\le C\le r$, together with the first $r$ tests of a
convergence-determining family for the joint state and indicator law.
The limiting laws at fixed $r$ differ from the exact activity law by
at most the sum of the finitely many approximation errors. A diagonal
choice therefore gives joint convergence, vanishing transitivity
defects, and \eqref{eq:wall-invariance} for
$g_{jk,i}=g^{(r(i))}_{jk,i}$.
\end{proof}

\begin{lemma}[Inactivity on pieces with countable plaque sections]
\label{lem:transverse-inactive}
Let $S\subset X$ be Borel, with each local $U_{jk}$ plaque meeting $S$
in at most countably many points. Then
\[
 a_{jk}(\omega)=0
 \quad\text{for $\mathsf P$-almost every state with }x\in S.
\]
In particular, this applies to the fixed-point pieces of
Lemma~\ref{lem:fixed-pieces} for roots crossing distinct eigenvalues;
each such piece meets a sufficiently small root plaque in at most one
point. It also applies to countable unions of these pieces.
\end{lemma}

\begin{proof}
Work first in a $U_{jk}$ flow box with coordinates $(q,h)$ and plaque
probability $P^\mathcal B$. For each $q$, the section
$S_q=\{h:\psi(q)h\in S\}$ is countable. The sampling identity for this
single chart states that the conditional law of the sampled $h$ given
$(q,P^\mathcal B)$ is $P^\mathcal B$. A probability measure assigns
mass zero to those points of a countable set which are not its atoms,
so
\[
 \mathsf P\{x\in S\cap\mathcal B,\ P^\mathcal B(\{h\})=0\}=0.
\]
No conditioning on the other plaque probabilities is used. Projective
agreement in the chart identifies an atom at the sampled $h$ with an
atom at the identity of $m^{U_{jk}}_\omega$. A countable flow-box cover
therefore shows that this root measure has an atom at the identity
almost surely over $S$.

It remains to show that, under the full state law, an atom at the
identity forces the entire root measure to be supported there. Identify
$U_{jk}$ with $\R$ by $t\mapsto 1+tE_{jk}$. Choose $b\in L$ with
$\lambda=e^{\beta_{jk}(b)}>1$. Fixed-motion equivariance in
Proposition~\ref{prop:plaque-comparison}, together with $A$-invariance
of the prelimit sampling law, says that the projective law of
$m^{U_{jk}}_\omega$ is invariant under the pushforward $t\mapsto
\lambda t$.

The atom event is invariant under dilation. If it had positive
probability, condition the projective-measure law on it and normalize
each measure by its atom: $\nu=m/m(\{0\})$. This is a measurable
normalization, $\nu(\{0\})=1$, and its law is dilation-invariant. For
every compact $K$ disjoint from zero,
\[
 (t\mapsto\lambda^nt)_*\nu(K)
       =\nu(\lambda^{-n}K)\longrightarrow0,
\]
as $\lambda^{-n}K$ lies in a shrinking punctured neighborhood of zero,
whose mass tends to zero by local finiteness. Dilation invariance makes
each quantity in the display have the same distribution as
$\nu(K)$, so almost-sure convergence to zero forces $\nu(K)=0$ almost
surely. A countable family of intervals covers $\R\setminus\{0\}$,
giving $\nu=\delta_0$ almost surely. Atoms of probability zero are
harmless. Together with the first part this proves the lemma.
\end{proof}

\section{Activity along long diagonal segments}\label{sec:segment-entropy}

Root activity was defined at a single augmented state. This section
shows that activity cannot vanish on a cut-crossing family along a long
diagonal segment for a positive proportion of starting points. This is
the entropy input for the combination argument of
Section~\ref{sec:combination}.

If $\mu_i(E_i)\ge p>0$, the normalized restriction
$\xi_i=\mu_i|_{E_i}/\mu_i(E_i)$ satisfies $\xi_i\le p^{-1}\mu_i$.
Proving the desired bound for every such dominated law excludes the
possibility of a positive-mass exceptional set.

An \emph{oriented cut} is a nonempty proper subset
$J\subset\{1,\dots,6\}$; its roots point from $J$ to the complement:
\[
 \mathcal E_J=\{\beta_{jk}:j\in J,\ k\notin J\}.
\]
The complementary subset gives the opposite orientation. Fix a small
$\delta>0$. For each $J$ choose regular $b_J\in L$ with
\begin{equation}\label{eq:cut-direction}
 \beta(b_J)\ge1\quad(\beta\in\mathcal E_J),\qquad
 |(b_J)_k-(b_J)_j|\le\delta\quad(j,k\text{ on the same side}).
\end{equation}
Such choices exist in a fixed bounded subset of $L$: place the two
sides at distinct levels, perturb within each side, and center at the
average. There are only finitely many cuts. Write $\eta=1/3$ for the
exponential constant of Proposition~\ref{prop:bowen}; it is independent
of $\delta$. Write $g_{\beta,i}=g_{jk,i}$ when $\beta=\beta_{jk}$
and put $A_{J,i}(x)=\sum_{\beta\in\mathcal E_J}g_{\beta,i}(x)$. Choose
the common interior scale $R_i$ as in \eqref{eq:wall-invariance},
leaving separated interior depths both below and above it, so that
$T_{-,i}\prec R_i\prec T_{+,i}$.

\begin{proposition}[Activity on every long segment]
\label{prop:segment-activity}
For $\delta$ sufficiently small, we have: for every oriented cut $J$,
every $C<\infty$, every $\tau>0$, and every sequence of probabilities
$\xi_i$ on $X$ with $\xi_i\le C\mu_i$,
\begin{equation}\label{eq:segment-activity}
 \liminf_{i\to\infty}\frac1{R_i}
 \int_0^{\tau R_i}\int_X
 A_{J,i}(xa(tb_J))\,\dd\xi_i(x)\,\dd t>0.
\end{equation}
The same assertion holds after any subsequence.
\end{proposition}

The proof compares two entropy bounds for orbit names. The tube
estimate gives a positive entropy cost per unit time. If the crossing
roots are inactive, the plaque probabilities vary only in directions
of expansion rate $\le\delta$, so their names have much smaller coding
cost. This use of arithmetic separation to bound entropy belongs to the
method of \cite{ELMV2009}; here the intermediate conditioning and the
dominated starting probabilities require the additional details below.
We use Shannon entropy with natural logarithms, including chain rule
and the elementary finite-alphabet error bound; see
\cite[Chapter~2]{CoverThomas2006}.

\subsection{Partitions and the cusp}

For a countable partition $\mathcal P$ and probability $\nu$, write
$H_\nu(\mathcal P)=-\sum_{P\in\mathcal P}\nu(P)\log\nu(P)$, $0\log0=0$.
Fix a regular $b\in L$ and $T(x)=xa(b)$. Write
$\mathcal P_{[r,s]}=\bigvee_{t=r}^sT^{-t}\mathcal P$ for the name on
integer times $r,\dots,s$. Let $V$ be the pattern group of $b$-positive
roots.

\begin{lemma}[A partition for consistent orbit names]
\label{lem:entropy-partition}
Given a sufficiently small identity neighborhood $O\subset G$, there
is a countable Borel partition $\mathcal P$ of $X$ with:
\begin{enumerate}
 \item every atom has $\mu$-null boundary and lies in an inner part of
 a relatively compact $VA$-plaque chart from the fixed stock, extended
 beyond the atom with margin;
 \item if $x,y$ have the same symbols at times $0,\dots,N$, there are
 relative lifts $h_t\in O$ with
 \[
 y a(tb)=xa(tb)h_t,\qquad h_{t+1}=a(-b)h_ta(b), \quad 0\le t\le N;
 \]
 \item for every fixed $C$, $H_\nu(\mathcal P)$ is bounded uniformly
 in $i$ and $\nu\le C\mu_i$, and there are finite coarsenings
 $\mathcal P^{(q)}$ (keep finitely many atoms, merge the rest) with
 \begin{equation}\label{eq:partition-entropy-tails}
  \lim_{q\to\infty}\limsup_{i\to\infty}
  \sup_{\nu\le C\mu_i}
  H_\nu(\mathcal P\mid\mathcal P^{(q)})=0.
 \end{equation}
\end{enumerate}
\end{lemma}

\begin{proof}
Let $m(x)$ be the shortest-vector length. Divide $X$ into layers with
$m(x)\asymp e^{-k}$ (integers $k\ge0$), the first layer also containing
the thick part. Layer boundaries can be chosen $\mu$-null. Successive
minima and lattice basis reduction give a representative matrix $g$ in
the $k$-th layer with
\[
 \|g\|+\|g^{-1}\|\le e^{C_1(1+k)}.
\]
Consequently the injectivity radius for relative group coordinates in
that layer is at least $e^{-C_2(1+k)}$: a nonidentity integral $\gamma$
giving a smaller return would, after conjugation by $g$, have
$\|\gamma-1\|<1$, impossible.

Fix a quantitative margin condition. Let $d_G$ be the left-invariant
Riemannian distance on $G$. In a fixed identity neighborhood of $G$
take a smooth retraction $\pi$ onto $VA$, with $\pi(h)=h$ on $VA$ and
\[
 d_G(h,\pi(h))\le C_\pi\dist_G(h,VA),\qquad
 d_G(1,\pi(h))\le C_\pi d_G(1,h).
\]
After shrinking the neighborhood, $1$ and $\pi(h)$ are joined inside
$VA$ by a path of length $\le C_\pi d_G(1,h)$.

In the $k$-th layer, take outer $VA$ product charts of radius
$r_k=e^{-C_3(1+k)}$ with $C_3$ large enough for injectivity, and inner
boxes whose relative-coordinate distance from the outer boundary is at
least a fixed multiple of $r_k$. Use translates of one fixed smooth
group-coordinate chart, so local product and metric constants are
uniform. Cover the layer by smaller symbol boxes inside these inner
boxes, with relative-group diameter at most $c_*r_k$, $c_*>0$ small
compared with both the inner-to-outer margin and the radius of
$\pi$. Then whenever $x,y=xh$ lie in one symbol box, the path from
$x$ to $x\pi(h)$ stays inside its outer chart, so $x\pi(h)$ is on the
same chart plaque as $x$.

The number of symbol boxes is at most $e^{C_4(1+k)}$: subdivide
bounded-matrix representatives at an exponentially small mesh, refine
by a fixed factor when necessary. Passing from cover to partition by
successive differences does not increase pieces or diameters, and
choosing chart centers within a small fixed fraction of the covering
radius preserves inner-chart margins. Chart and cell boundaries are
$\mu$-null: vary centers and widths, avoiding at most countably many
widths of positive boundary mass. Each atom stays inside a chart with
a margin, though it need not be a coordinate rectangle.

The shortest-vector length changes by at most a fixed factor under
$T$, so we can choose boxes so small that both a relative lift in one
symbol and its $a(b)$-conjugate lie in an injective neighborhood at
the next symbol. Uniqueness in that neighborhood identifies the latter
lift with the lift chosen there, and induction gives assertion~2.
Decreasing box sizes by a common factor ensures every lift lies in
the prescribed $O$.

By Proposition~\ref{prop:cusp}, every $\nu\le C\mu_i$ assigns layer
$k$ mass at most $C_5e^{-ck}$, uniformly in $i$. If that mass is
$p_k$ and the layer has at most $M_k$ symbols, their entropy is at
most $p_k\log M_k-p_k\log p_k$. With $\log M_k=O(1+k)$ and
$p_k\le C_5e^{-ck}$, the sum converges uniformly and the tail tends
to zero. Keeping all symbols of the first finitely many layers proves
both entropy assertions.
\end{proof}

\subsection{What an inactive cut implies inside a cell}

Fix $J$, put $b=b_J$, and let $V_s$ be the pattern of positive roots
with both endpoints on the same side of $J$. Set $H_s=V_sA$. The
cross-root group $Z_J$ is normal in $VA$ with unique factorization
$h=zd$, $z\in Z_J$, $d\in H_s$. In a product chart $x=\psi(q)zd$, the
$H_s$ plaques are exactly the sets with $(q,z)$ fixed. By
Lemma~\ref{lem:va-support}, at a limiting state whose crossing-root
indicators all vanish, the $VA$ probability is supported on the $H_s$
plaque through the sampled point. The next lemma records the required
change of measure.

\begin{lemma}\label{lem:inactive-cell-concentration}
Fix finitely many atoms $S$ of $\mathcal P$, with their $VA$ charts,
and an interior transverse depth $l_i\prec R_i$. Suppose
$\nu_i\le C\mu_i$ satisfy $\int A_{J,i}\,\dd\nu_i\to0$. Condition on
one symbol $S$ and on its transverse $q$ cell, and draw two independent
points from the resulting conditional probability of $\nu_i$. Then
their $z$-coordinates differ by a quantity tending to zero in
probability, averaged with the conditional probabilities.

The conclusion also holds on average over any finite list of such
probabilities, with probability weights, if the weighted mean of
$\int A_{J,i}\,\dd\nu_i$ tends to zero.
\end{lemma}

\begin{proof}
First sample $x$ under $\nu_i$ and draw the second plaque coordinate
from the $\mu_i$-probability conditioned only on its transverse cell in
the whole chart. The depth comparison allows this conditional to be
replaced in the limit by the chart probability of the state data. Any
joint limit with $x$ sampled under $\nu_i$ is dominated by $C$ times
the state law, so its crossing-root indicators vanish almost surely.
Lemma~\ref{lem:va-support} puts the second point on the same $H_s$
plaque as the sample, with $z$-coordinate equal to the sampled $z$.
Since the transverse mesh tends to zero, continuous tests on the fixed
compact chart boxes show that the prelimit $z$-difference tends to
zero in probability. No conditioning on the other plaque probabilities
is used.

To change the second draw to the $\nu_i$-conditional, fix
$\epsilon>0$ and discard events $E=S\cap\{q\text{ in a given cell}\}$
with
\[
 \nu_i(E)<\epsilon\mu_i(\{q\text{ in that cell}\}\cap\mathcal B).
\]
Their total $\nu_i$-mass is at most $\epsilon$ times the number of
symbols. On each remaining event the conditional $\nu_i$-law has
density at most $C/\epsilon$ with respect to the $\mu_i$-chart-cell
law, so the preceding convergence remains for the second draw. Let
$i\to\infty$ and then $\epsilon\to0$.

For the averaged statement, note a uniform sequential consequence:
no sequence of dominated probabilities can have indicator expectation
tending to zero while the asserted pair error stays bounded away from
zero. Let $a_i\to0$ be the weighted mean; discard members with
expectation $>\sqrt{a_i+i^{-1}}$; their total weight goes to zero by
Markov. On the remaining members the pair errors go to zero by the
sequential consequence.
\end{proof}

\subsection{The two entropy bounds}

\begin{proof}[Proof of Proposition~\ref{prop:segment-activity}]
Suppose that along a subsequence the expression in
\eqref{eq:segment-activity} goes to zero. Choose even integers
$N_i\sim\tau R_i$ with $N_i+2<\tau R_i$. Averaging in a starting shift
$s\in[0,1]$ lets us replace $\xi_i$ by its translate through some
$s_i\in[0,1]$ so that
\begin{equation}\label{eq:vanishing-discrete-activity}
 \frac1{N_i+1}\sum_{t=0}^{N_i}
 \int A_{J,i}(T^tx)\,\dd\xi_i(x)\longrightarrow0.
\end{equation}
All translates still satisfy the domination bound. Hence every time-$t$
distribution $\nu_{i,t}=(T^t)_*\xi_i$ does too.

Choose a compact $K$ with $\nu_{i,N_i/2}(K)\ge1-\epsilon_0$ for all
large $i$, using the uniform cusp bound and domination. Choose $O$ in
Lemma~\ref{lem:entropy-partition} small enough for
Proposition~\ref{prop:bowen} on $K$. If a name atom
$B\in\mathcal P_{[0,N_i]}$ meets $T^{-N_i/2}K$, consistency of relative
lifts puts every midpoint of $B$ in the tube about its reference
midpoint over time $N_i/2$. Since $T_{-,i}\prec R_i\prec T_{+,i}$,
that time lies in the arithmetic range. Hence
\[
 \xi_i(B)\le Ce^{-\eta N_i/2}
\]
for those atoms, whose total $\xi_i$-mass is at least
$1-\epsilon_0$. Integrating the negative logarithm of the atom mass
gives, after decreasing $\epsilon_0$ and taking $i$ large,
\begin{equation}\label{eq:name-entropy-lower}
 \frac1{N_i}H_{\xi_i}(\mathcal P_{[0,N_i]})\ge\frac\eta3.
\end{equation}

We next derive the incompatible upper bound. Fix a small target
$\varepsilon>0$. First pick a finite coarsening $\mathcal P^0$ so that
\eqref{eq:partition-entropy-tails} is as small as required below; let
$q_0$ be its alphabet size. Choose also a finite collection
$\mathcal S$ of original symbols whose complement has arbitrarily
small probability under every $\nu\le C\mu_i$.

The boundary of $\mathcal P^0$ is a finite union of compact $\mu$-null
sets. Choose accuracy $\rho>0$ so the $2\rho$-neighborhood of the
boundary has arbitrarily small probability under every
$\nu\le C\mu_i$ for large $i$. To check uniformly: take decreasing
closed neighborhoods of the boundary; their $\mu$-masses tend to
zero, and the weak-convergence upper bound applies to each fixed
neighborhood. Incorrect symbols can then occur only near the boundary
or on the exceptions below.

Choose transverse grids in the charts for $\mathcal S$ at interior
depth $l_i\prec R_i$. Let $M$ be a large integer, fixed while
$i\to\infty$ and chosen after the charts and $\rho$. The concentration
tolerances below may depend on $M$. Split the name into blocks of $M$
future symbols given the past through the block start. An initial
segment of length $t_i=o(N_i)$ and a final remainder of length $<M$
cost $o(N_i)$ entropy by the uniform single-symbol bound.

We now bound the extra cost of the transverse label at every block.
In a past-name atom at time $t$, the negative root entries of the
relative lift are $O(e^{-c_bt})$ with
$c_b=\min_{\beta(b)>0}\beta(b)>0$. Triangular coordinates in a small
group neighborhood put the lift at distance $O(e^{-c_bt})$ from $VA$.
Let $x,y=xh$ be two current points in one past-name atom and use the
retraction $\pi$. The diameter condition ensures $x\pi(h)$ lies on the
same plaque of the current outer chart as $x$, so $q(x\pi(h))=q(x)$.
Lipschitz continuity of $q$ on that chart gives
\[
 |q(y)-q(x)|
 \le Cd_G(h,\pi(h))
 \le C'\dist_G(h,VA)
 \le C''e^{-c_bt},
\]
with constants uniform in the finite collection of current charts.
Choose $t_i$ a sufficiently large fixed multiple of $l_i$; then
$e^{-c_bt}<2^{-l_i}$ for $t\ge t_i$, and $t_i=o(N_i)$. Consequently
only $B_0$ neighboring transverse cells are possible. Given the past,
their label has entropy at most $\log B_0$. Use a single dummy label
for current symbols outside $\mathcal S$.

Let $\mathcal Q_i$ record this cell label in the current symbol and a
dummy label outside $\mathcal S$, and write
$\mathcal F_t=\mathcal P_{[0,t]}$. After revealing the cell label we
may discard the rest of the past in upper-bounding entropy of the next
coarsened block; by the chain rule and monotonicity under conditioning,
\begin{equation}\label{eq:coarse-block-entropy}
 \begin{aligned}
 &H_{\xi_i}\bigl((\mathcal P^0)_{[t+1,t+M]}\mid\mathcal F_t\bigr)\\
 &\le H_{\xi_i}\bigl(T^{-t}\mathcal Q_i\mid\mathcal F_t\bigr)
   +H_{\nu_{i,t}}\bigl((\mathcal P^0)_{[1,M]}\mid\mathcal P\vee\mathcal Q_i\bigr)\\
 &\le\log B_0+
   H_{\nu_{i,t}}\bigl((\mathcal P^0)_{[1,M]}\mid\mathcal P\vee\mathcal Q_i\bigr).
 \end{aligned}
\end{equation}
The last conditioning is just the current symbol and its cell. By
\eqref{eq:vanishing-discrete-activity}, the mean of crossing-root
activity at block starts goes to zero. The averaged form of
Lemma~\ref{lem:inactive-cell-concentration} applies to their
distributions. For each symbol-cell event, choose a $z$-center so
that, up to probability going to zero on average, $z$ lies within any
prescribed fixed tolerance of that center.

The $d$-coordinates lie in a fixed compact part of $H_s$ within each
chart. Every root in $H_s$ has expansion rate at most $\delta$, and
$A$ commutes with the motion. A mesh in these coordinates with accuracy
a fixed multiple of $\rho e^{-\delta M}$ predicts the trajectory over
$M$ steps to distance accuracy $\rho$, provided $z$ and the transverse
coordinates are sufficiently close to their centers. The number of mesh
points is at most
\begin{equation}\label{eq:slow-direction-mesh}
 C_0\exp(C_6\delta M).
\end{equation}
Here $C_6$ depends only on the dimension, and $C_0$ depends on the
fixed charts and $\rho$ but not on $M$. This follows from finitely many
smooth coordinate meshes on the compact $d$ ranges: conjugation
multiplies every entry of a group-relative error by at most
$e^{\delta M}$.

The cross-root center tolerance may need to be exponentially small in
$M$, because cross-root directions expand faster. This costs no new
mesh labels: $M$ is fixed before applying the concentration lemma, so
any positive tolerance is allowed. The transverse width also goes to
zero for fixed $M$. Thus $C_0$ in \eqref{eq:slow-direction-mesh}
remains independent of $M$.

Encode the next $M$ symbols of $\mathcal P^0$ by a mesh label, with a
single additional label for an omitted chart or an outlier, and then by
corrections to the predicted symbols. Let $e_{i,t}$ be the average
probability of incorrect prediction over these $M$ positions, with
$h_2(e)=-e\log e-(1-e)\log(1-e)$ and the usual conventions. Then
\begin{equation}\label{eq:mesh-correction-entropy}
 \begin{aligned}
 &H_{\nu_{i,t}}\bigl((\mathcal P^0)_{[1,M]}\mid\mathcal P\vee\mathcal Q_i\bigr)\\
 &\le
 \log\bigl(C_0e^{C_6\delta M}+1\bigr)
 +M\bigl(h_2(e_{i,t})+e_{i,t}\log q_0\bigr).
 \end{aligned}
\end{equation}
For completeness, reveal the error indicator at each position, and on
an error reveal the actual symbol; subadditivity and concavity of
$h_2$ give this bound also when error probabilities differ across
positions and blocks. Chart omissions and boundary neighborhoods were
chosen with arbitrarily small probability; concentration errors tend
to zero on average. So the mean of $e_{i,t}$ can be made as small as
required.

Finally, the original block is recovered from its coarsened block at
\begin{equation}\label{eq:restore-block-entropy}
 \begin{aligned}
 &H_{\xi_i}\bigl(\mathcal P_{[t+1,t+M]}\mid\mathcal F_t\bigr)\\
 &\le
 H_{\xi_i}\bigl((\mathcal P^0)_{[t+1,t+M]}\mid\mathcal F_t\bigr)
 +\sum_{r=1}^M H_{\nu_{i,t+r}}(\mathcal P\mid\mathcal P^0).
 \end{aligned}
\end{equation}
Indeed, conditional on the coarsened block, revealing each original
symbol costs at most its one-step conditional entropy given its
coarsened symbol. Each summand is controlled by
\eqref{eq:partition-entropy-tails}.

Sum \eqref{eq:restore-block-entropy} over blocks and use
\eqref{eq:coarse-block-entropy} and \eqref{eq:mesh-correction-entropy}.
The chain rule and negligible initial and final pieces then give
\begin{equation}\label{eq:name-entropy-upper}
 \limsup_{i\to\infty}\frac1{N_i}
 H_{\xi_i}(\mathcal P_{[0,N_i]})
 \le C_6\delta+\varepsilon+
 \frac{\log B_0+\log(C_0+1)}M.
\end{equation}
Choose $\delta$ so $C_6\delta<\eta/12$, truncation, charts, and
distance accuracy so $\varepsilon<\eta/12$, and $M$ so
$(\log B_0+\log(C_0+1))/M<\eta/12$. Letting $i\to\infty$ contradicts
\eqref{eq:name-entropy-lower}. This proves the proposition, including
its subsequence formulation.
\end{proof}

\section{Combining the root orientations}\label{sec:combination}

The previous section produced activity \emph{somewhere} on every long
segment across each oriented cut. This section shows that actually
every root is active at almost every state of the augmented limit.
The bridge is that, on the scale $R_i$, the plaque activity of a root
$jk$ is nearly a function of the value of the root coordinate alone.
Transitivity then becomes an approximate addition rule on real
functions, and two-edge cycles force the inactive sets to have measure
zero.

\begin{theorem}[Full activity]\label{thm:full-activity}
For every ordered pair $j\ne k$ in $\{1,\dots,6\}$,
\begin{equation}\label{eq:full-activity}
 \int_X g_{jk,i}\,\dd\mu_i\longrightarrow1.
\end{equation}
Consequently every root is active almost surely in the limiting state
law.
\end{theorem}

\subsection{Functions of a root coordinate}

For each root $\beta$, fix $b_\beta\in L$ with $\beta(b_\beta)=1$. If
$x$ is sampled with law $\mu_i$, define
\begin{equation}\label{eq:root-line-functions}
 G_{\beta,i}^{x}(s)=g_{\beta,i}\bigl(xa(R_isb_\beta)\bigr),
 \qquad s\in\R.
\end{equation}
These take values in $\{0,1\}$. Regard them as elements of the compact
metrizable space
\[
 \mathcal B=\{f\in L^\infty_{\mathrm{loc}}(\R):0\le f\le1\}
\]
with weak-* convergence on every bounded interval (metrized by countably
many $L^1$ test functions on each $[-q,q]$). The finite product
$\mathcal B^{30}$ is also compact. After a subsequence, the laws of the
arrays in \eqref{eq:root-line-functions} converge to a law of functions
$(F_\beta)_\beta$. Write $G_{jk,i}^x=G_{\beta_{jk},i}^x$ and
$F_{jk}=F_{\beta_{jk}}$.

\begin{lemma}\label{lem:root-line-laws}
For any such convergent subsequence:
\begin{enumerate}
 \item almost surely, for every $v_0\in L$ with rational coordinates,
 every oriented cut $J$, and every positive rational $\tau$,
 \begin{equation}\label{eq:positive-cut-integral}
  \sum_{\beta\in\mathcal E_J}\int_0^\tau
  F_\beta\bigl(\beta(v_0+tb_J)\bigr)\,\dd t>0;
 \end{equation}
 \item for distinct $j,k,\ell$ and every bounded rectangle
 $B\subset\R^2$, the nonnegative defects satisfy
 \begin{equation}\label{eq:line-transitivity-defect}
  \E_{x\sim\mu_i}\int_B G_{jk,i}^{x}(s)G_{k\ell,i}^{x}(t)
   \bigl(1-G_{j\ell,i}^{x}(s+t)\bigr)\,\dd s\,\dd t\longrightarrow0.
 \end{equation}
\end{enumerate}
\end{lemma}

\begin{proof}
For $v$ in a fixed bounded subset of $L$, the vector $v-\beta(v)b_\beta$
belongs to $\ker\beta$ and remains bounded. Wall invariance
\eqref{eq:wall-invariance} and invariance of $\mu_i$ under the common
translation give
\begin{equation}\label{eq:common-translate-comparison}
 \sup_{v\text{ in a fixed bounded set}}
 \int_X\bigl|
 G_{\beta,i}^{x}\bigl(\beta(v)\bigr)
 -g_{\beta,i}\bigl(xa(R_iv)\bigr)
 \bigr|\,\dd\mu_i(x)\longrightarrow0.
\end{equation}

Fix the parameters in assertion~1. The left side of
\eqref{eq:positive-cut-integral} is continuous in the weak-* array,
because $\beta(b_J)\ne0$ for every crossing root. By
\eqref{eq:common-translate-comparison}, it differs in $L^1(\mu_i)$ from
\[
 Y_i(x)=\sum_{\beta\in\mathcal E_J}\int_0^\tau
 g_{\beta,i}\bigl(xa(R_i(v_0+tb_J))\bigr)\,\dd t
\]
by $o(1)$. If the limiting observable were zero on a set of probability
$p>0$, convergence in law and nonnegativity would allow thresholds
$\varepsilon_i\downarrow0$ and events $E_i$ with
$\mu_i(E_i)\ge p/2$ and $Y_i\le\varepsilon_i$ on $E_i$. The normalized
restrictions $\mu_i|_{E_i}/\mu_i(E_i)$, translated by $a(R_iv_0)$, then
satisfy $\xi_i\le(2/p)\mu_i$ and
\[
 \frac1{R_i}\int_0^{\tau R_i}\int_X
 A_{J,i}(xa(ub_J))\,\dd\xi_i(x)\,\dd u\longrightarrow0,
\]
contradicting Proposition~\ref{prop:segment-activity}. The parameters
in~1 range over a countable set, so the assertion holds simultaneously
almost surely.

For assertion~2, $\beta_{jk}$ and $\beta_{k\ell}$ are linearly
independent: choose linear $v(s,t)$ with
$\beta_{jk}(v)=s$, $\beta_{k\ell}(v)=t$, hence
$\beta_{j\ell}(v)=s+t$. On a bounded rectangle,
\eqref{eq:common-translate-comparison} replaces all three functions in
the defect by their indicators at the common point $xa(R_iv(s,t))$ with
total expected error $o(1)$. The expected defect there is $o(1)$ by
the approximate transitivity of Lemma~\ref{lem:activity-indicators};
its distribution is unchanged by the translation. Integrating over the
rectangle proves the claim.
\end{proof}

We pass transitivity through weak-* convergence by selecting
deterministic arrays whose defects themselves tend to zero.

\begin{lemma}\label{lem:deterministic-root-arrays}
If \eqref{eq:full-activity} fails for some root, then there is a
subsequence and a deterministic choice of arrays $(G_{\beta,i})_\beta$,
each taken from \eqref{eq:root-line-functions}, with:
\begin{enumerate}
 \item $G_{\beta,i}\to F_\beta$ weak-* on bounded intervals, and the
 deterministic limits satisfy \eqref{eq:positive-cut-integral} for its
 stated parameters;
 \item every defect in \eqref{eq:line-transitivity-defect}, without
 its expectation, tends to zero on every bounded rectangle;
 \item for some $\beta_0$ and some bounded interval $I$ of positive
 length,
 \begin{equation}\label{eq:limiting-deficiency}
  \int_I(1-F_{\beta_0}(s))\,\dd s>0.
 \end{equation}
\end{enumerate}
\end{lemma}

\begin{proof}
Pass to a subsequence on which
$\int_X(1-g_{\beta_0,i})\,\dd\mu_i\ge\varepsilon$ for some fixed
$\varepsilon>0$. Invariance of $\mu_i$ and Fubini give, for any fixed
bounded interval $I$,
\[
 \E\int_I(1-G_{\beta_0,i}^{x}(s))\,\dd s
 =|I|\int_X(1-g_{\beta_0,i})\,\dd\mu_i\ge\varepsilon|I|.
\]
The integral is bounded weak-* continuous, so the same inequality holds
for the limiting law. Choose a point $F$ in its support with positive
deficiency that satisfies assertion~1 of Lemma~\ref{lem:root-line-laws}.

Fix an integer $q$. Take a neighborhood $U_q$ of $F$ of diameter
less than $1/q$. It has positive limiting probability, hence the
prelimit probabilities of $U_q$ are bounded below at large indices.
There are finitely many root compositions, so by
\eqref{eq:line-transitivity-defect} the sum of their defects on
$[-q,q]^2$ has expectation tending to zero. Choose an index so large
that, by Markov's inequality, the probability that this sum exceeds
$1/q$ is less than half the probability of $U_q$. Then some sampled
array in $U_q$ has total defect at most $1/q$. Choose these indices
increasing. They converge to $F$ and have vanishing defects on every
bounded rectangle by nonnegativity and containment in $[-q,q]^2$.
\end{proof}

\subsection{From cut activity to every interval}

From now on fix deterministic arrays as in
Lemma~\ref{lem:deterministic-root-arrays}. Their weak-* limits may take
values in $[0,1]$.

\begin{lemma}\label{lem:positive-root-intervals}
For every root $\beta$ and every nonempty open interval $I\subset\R$,
\[
 \int_IF_\beta(s)\,\dd s>0.
\]
\end{lemma}

\begin{proof}
Fix $\beta=\beta_{jk}$ and a bounded nonempty open interval $I$.
Choose rational $v_0\in L$ with $\beta_{jk}(v_0)\in I$, and $r>0$
small enough that
$[\beta_{jk}(v_0)-6r,\ \beta_{jk}(v_0)+6r]\subset I$.
For every root $\alpha$, put
$I_\alpha=(\alpha(v_0)-r,\alpha(v_0)+r)$. Draw a directed edge $p\to q$
when $\int_{I_{\beta_{pq}}}F_{pq}>0$.

This graph is strongly connected. Indeed, for any nonempty proper
subset $J$, choose positive rational $\tau$ small enough that
$\alpha(v_0+tb_J)\in I_\alpha$ for every $\alpha\in\mathcal E_J$ and
$0\le t\le\tau$. Then \eqref{eq:positive-cut-integral} supplies an edge
from $J$ to its complement. A finite directed graph with an outgoing
edge across every nontrivial cut is strongly connected: otherwise the
set of vertices reachable from some vertex is a nonempty proper set
with no outgoing edge.

Take a simple directed path $j=p_0,\dots,p_m=k$ with $m\le5$. Write
$\alpha_h=\beta_{p_{h-1}p_h}$ and $I_h=I_{\alpha_h}$. Weak-*
convergence gives positive lower bounds, at large $i$, on
\[
 \int_{I_h}G_{\alpha_h,i}(s)\,\dd s,\qquad 1\le h\le m.
\]
So the product of these integrals is bounded below by a positive
constant. Interpret it as the measure of tuples
$(s_1,\dots,s_m)\in\prod_h I_h$ at which all the path labels equal one.

Except for a set of tuple measure tending to zero, successive
composition of these labels gives
\[
 G_{p_0p_h,i}(s_1+\cdots+s_h)=1,\qquad 1\le h\le m.
\]
At step $h$, the failure of this implication, given all earlier,
is bounded by the transitivity defect for $p_0,p_{h-1},p_h$; the
path is simple, so these are distinct. Integration in
$s_1,\dots,s_{h-1}$ pushes Lebesgue measure on bounded intervals to a
bounded-density measure in their sum; multiplying by earlier
indicator restrictions only decreases the measure. Defect integrals on
a fixed bounded rectangle therefore bound the failure measure by a
fixed multiple of $o(1)$; summing gives the assertion. The $m=1$ case
needs no composition.

The final sum $s_1+\cdots+s_m$ lies in $I$, because
$\alpha_1(v_0)+\cdots+\alpha_m(v_0)=\beta_{jk}(v_0)$. Its pushforward
density is bounded above by a fixed constant, so a positive measure of
tuples has final label one. Hence
$\liminf_i\int_IG_{jk,i}>0$, and weak-* convergence proves the lemma.
\end{proof}

We have found positive activity in every interval for every root. To
eliminate a positive inactive set, translate an active set by sums of
parameters from two opposite auxiliary roots. Two-edge cycles close at
the original root with the sum of the two parameters; interval
positivity gives parameter sets of positive measure on both sides of
zero. Mixing normalized Lebesgue measures gives bounded densities of
mean zero, bounded support, and separated variance. The sum of two
independent parameters describes one cycle increment. Fixing a
bounded interval $I$, one needs a fixed number of independent cycle
increments to have density bounded below on $I-I$, so every part of
the active set interacts with every part of the inactive set inside
$I$. The next lemma provides that bound.

\subsection{A uniform convolution estimate}

\begin{lemma}\label{lem:convolution}
Fix $B,M,v,L>0$. For any probability density $p$ on $\R$ with
\[
 \supp p\subset[-B,B],\quad
 0\le p\le M,\quad
 \int xp(x)\,\dd x=0,\quad
 \int x^2p(x)\,\dd x\ge v,
\]
there are an integer $m\ge2$ and a $c>0$ (depending only on
$B,M,v,L$) with
\[
 p^{*m}(x)\ge c\qquad(|x|\le L).
\]
The inequality uses the continuous representative of the convolution.
\end{lemma}

\begin{proof}
If the family is empty there is nothing to prove. Otherwise write
$\sigma_p^2=\int x^2p(x)\,\dd x\in[v,B^2]$ and
$\phi_p(t)=\int e^{itx}p(x)\,\dd x$. Bounded support and zero mean give
\[
 \phi_p(t)=1-\frac12\sigma_p^2t^2+O(B^3|t|^3).
\]
Applying the same expansion to $|\phi_p(t)|^2$, we get, for some
$\delta>0$ depending only on $B,v$,
\begin{equation}\label{eq:small-frequency-domination}
 |\phi_p(t)|\le e^{-vt^2/4}\qquad(|t|\le\delta).
\end{equation}

There is also a uniform gap below one for $|t|\ge\delta$. For
$|t|\ge\delta$, an angle $\theta$, and $0<a<\pi$, the set
\[
 \{x\in[-B,B]:\operatorname{dist}(tx-\theta,2\pi\Z)<a\}
\]
has Lebesgue measure at most $a(2B/\pi+4/\delta)$ (each full period
contributes fraction $a/\pi$; at most two incomplete periods at the
ends). Choose $a$ so small that $M$ times this bound is at most
$1/2$. With $\theta=\arg\phi_p(t)$, at least half of the probability
lies outside this arc; taking the real part after rotating by
$e^{-i\theta}$ gives
\begin{equation}\label{eq:large-frequency-gap}
 \sup_p\sup_{|t|\ge\delta}|\phi_p(t)|\le(1+\cos a)/2=:\rho<1.
\end{equation}

Plancherel gives
$\int_\R|\phi_p(t)|^2\,\dd t=2\pi\int_\R p(x)^2\,\dd x\le2\pi M$.
For $m\ge2$, $\phi_p^m$ is integrable and Fourier inversion defines a
continuous representative of $p^{*m}$. We claim
\begin{equation}\label{eq:uniform-local-clt}
 \sup_p\sup_{x\in\R}\Bigl|
 \sqrt m\,p^{*m}(x)
 -\frac1{\sqrt{2\pi}\,\sigma_p}\exp
 \Bigl(-\frac{x^2}{2m\sigma_p^2}\Bigr)\Bigr|\longrightarrow0.
\end{equation}
Change variables $u=t\sqrt m$ in the inversion formula; the difference
in \eqref{eq:uniform-local-clt} is at most $1/(2\pi)$ times
\[
 \int_\R\bigl|
 \phi_p(u/\sqrt m)^m-e^{-\sigma_p^2u^2/2}\bigr|\,\dd u.
\]
On each fixed bounded $u$-interval, the local expansion proves uniform
convergence of the integrand to zero. On $|u|\le\delta\sqrt m$,
\eqref{eq:small-frequency-domination} bounds the first term by
$e^{-vu^2/4}$, while the Gaussian is at most $e^{-vu^2/2}$. On
$|u|>\delta\sqrt m$, \eqref{eq:large-frequency-gap} and Plancherel
give $\int_{|u|>\delta\sqrt m}|\phi_p(u/\sqrt m)|^m\,\dd u\le
2\pi M\sqrt m\,\rho^{m-2}\to0$, and the Gaussian tail goes to zero as
well.

For $|x|\le L$ and large $m$, the Gaussian term in
\eqref{eq:uniform-local-clt} is at least $1/(2\sqrt{2\pi}B)$, uniformly
in $p$. Increase $m$ so the error is at most $1/(4\sqrt{2\pi}B)$.
Then $p^{*m}(x)\ge 1/(4\sqrt{2\pi}B\sqrt m)$ for $|x|\le L$.
\end{proof}

\subsection{Cycles remove the inactive sets}

\begin{proof}[Proof of Theorem~\ref{thm:full-activity}]
Suppose the theorem fails and take the deterministic arrays and
limits from Lemma~\ref{lem:deterministic-root-arrays}. Fix an edge
$j\to k$ and pick $\ell$ distinct from $j,k$.
Lemma~\ref{lem:positive-root-intervals} gives, for the roots $\ell j$
and $j\ell$, positive limiting integral on each of the intervals
$(-2,-1)$ and $(1,2)$. So at large $i$, each of the four sets where
the relevant $G$ equals one has length bounded below.

For each auxiliary root, normalize Lebesgue measure on its two sets to
probability densities $p_i^-,p_i^+$. Their means lie in $[-2,-1]$ and
$[1,2]$. There is a convex combination
$p_i=\lambda_ip_i^-+(1-\lambda_i)p_i^+$ with zero mean; the coefficients
lie in $[1/3,2/3]$. Then $p_i$ has support in $[-2,2]$, a common upper
bound, and variance at least one. Call the density for $\ell j$
$p_i$ and for $j\ell$ $q_i$.

Starting from $G_{jk,i}(t)=1$, composing with parameter $s$ in the
support of $p_i$ gives the edge $\ell k$, then with parameter $s'$ in
the support of $q_i$ gives back $jk$:
\[
 \ell j+jk=\ell k,\qquad j\ell+\ell k=jk.
\]
Both are root compositions with three distinct vertices. Their defect
estimates give, for every fixed bounded interval $I$,
\begin{equation}\label{eq:one-cycle-error}
 \int_I\int_\R\int_\R
 G_{jk,i}(t)\bigl(1-G_{jk,i}(t+s+s')\bigr)
 p_i(s)q_i(s')\,\dd s\,\dd s'\,\dd t\longrightarrow0.
\end{equation}
Indeed, off null parameter sets both auxiliary labels are one; the
indicator is then bounded by the two composition defects. For the
first defect, integrate out $s'$ and use the uniform density bound on
$p_i$. For the second, $t+s$ has bounded density on a bounded interval
when $t\in I$ (Lebesgue) and $s\sim p_i$; use the bound on $q_i$ and
the defect estimate for $(j\ell,\ell k)$.

Set $r_i=p_i*q_i$. These cycle-increment densities have support in
$[-4,4]$, common density bound, mean zero, and variance at least two.
Iterate \eqref{eq:one-cycle-error} any fixed number $m$ of times to
obtain
\begin{equation}\label{eq:repeated-cycle-error}
 \int_I\int_\R
 G_{jk,i}(t)\bigl(1-G_{jk,i}(t+u)\bigr)
 r_i^{*m}(u)\,\dd u\,\dd t\longrightarrow0.
\end{equation}
Along $m$ increments, a path from label one to label zero has a first
transition from one to zero; bound the indicator by the sum over these
transitions. Intermediate positions lie in $I+[-4m,4m]$; after
integrating earlier increments, their distributions have densities
bounded relative to Lebesgue on that larger interval, because $t$ is
integrated over $I$ and all increments have probability laws. Each
term is bounded by the one-cycle error on the larger interval, giving
\eqref{eq:repeated-cycle-error}.

Fix a bounded interval $I$ of positive length. Apply
Lemma~\ref{lem:convolution} to $r_i$ with $L$ large enough that
$I-I\subset[-L,L]$; get a fixed power $m$ and a lower bound $c>0$.
In \eqref{eq:repeated-cycle-error} keep only the pairs $t,t+u\in I$:
\[
 c\Bigl(\int_IG_{jk,i}(t)\,\dd t\Bigr)
  \Bigl(\int_I(1-G_{jk,i}(t))\,\dd t\Bigr)\longrightarrow0.
\]
The first factor has positive lower limit by
Lemma~\ref{lem:positive-root-intervals}. Hence
$\int_I(1-G_{jk,i})\to0$ for every such $I$. By weak-* convergence,
$F_{jk}=1$ almost everywhere. Applying this to every root contradicts
the deficiency \eqref{eq:limiting-deficiency}; thus
\eqref{eq:full-activity} holds.

By Lemma~\ref{lem:activity-indicators}, the sequence indicators
converge jointly in law to the activity indicators of the limiting
state. Since each has expectation tending to one, every limiting root
indicator is almost surely one. There are thirty roots; the conclusion
holds for all of them simultaneously.
\end{proof}

\section{Proof of the equidistribution theorem}\label{sec:assembly}

We now return to the packet sequence of Theorem~\ref{thm:main}.
Proposition~\ref{prop:cusp} and Mahler's compactness criterion give
tightness. Take any subsequence with a weak probability limit $\mu$;
it is $A$-invariant, and Proposition~\ref{prop:ball} gives the
ordinary-ball estimate required by
Proposition~\ref{prop:positive-entropy}. Theorem~\ref{thm:ekl}
therefore makes almost every $A$-ergodic component of $\mu$
homogeneous.

Apply the intermediate-conditioning construction to this subsequence,
passing further to a subsequence wherever necessary. The augmented
limiting state projects to $\mu$ on $X$. Proposition~\ref{prop:bowen}
supplies the tube estimate throughout the required time interval, and
Theorem~\ref{thm:full-activity} makes every root active almost surely
in this state.

If proper homogeneous components had positive total weight,
Lemma~\ref{lem:fixed-pieces} would place positive $\mu$-mass on the
countable union of their fixed-point pieces. At least one piece would
then have positive mass. Its fixing diagonal is nonidentity, so it has
two unequal diagonal entries, and the corresponding root plaques meet
that piece in at most countably many points. By
Lemma~\ref{lem:transverse-inactive}, that root is inactive almost surely
on states projecting to the piece, a contradiction with the previous
paragraph. Thus almost every component is the $G$-invariant
probability, and $\mu=m_6$.

Every weakly convergent subsequence supplied by tightness has the same
limit. It follows that the original sequence converges to $m_6$ on
$C_c(X_6)$. Proposition~\ref{prop:cusp} already gives the compact sets
required for the nonescape assertion. The estimates are uniform in the
embedding ordering, so the conclusion holds for the arbitrary choices
in Theorem~\ref{thm:main}.

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\end{thebibliography}

\end{document}