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% Pointwise Multiple Ergodic Averages for Mixing Transformations
% A self-contained, complete proof, written to be read from front to back.
%
% The file is plain ASCII holding LaTeX source, same convention as the
% original. Every theorem, proposition, and lemma is proved here. Three
% companion results (listed as INPUT A, B, C below) are taken from other
% preprints in this series; they are stated precisely and used as black
% boxes, exactly as in the original.
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% (X,F,mu,T) T is invertible, bimeasurable and mu-preserving.
% mixing mu(A cap T^{-r}B) -> mu(A)mu(B) as |r| -> infty.
% L^infty bounded real measurable functions (complex by linearity).
% L^2, ip, norm the usual Hilbert space of mu-square-integrable functions.
% A_N A_N(f_1,...,f_n)(x) = (1/N) sum_{k=1}^N prod_j f_j(T^{jk}x).
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% The single result proved below is Theorem 0.1: on every mixing system,
% A_N converges almost everywhere to the product of the integrals.
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\begin{document}
\title{Pointwise Multiple Ergodic Averages for Mixing Transformations}
\maketitle
\begin{abstract}
Let $T$ be an invertible mixing probability-preserving transformation
of a probability space $(X,\mathcal F,\mu)$. For every integer $n\ge 2$
and every tuple $f_1,\ldots,f_n\in L^\infty(\mu)$, the consecutive
multiple ergodic averages
\[
A_N(f_1,\ldots,f_n)(x) = \frac1N\sum_{k=1}^N\prod_{j=1}^n f_j(T^{jk}x)
\]
converge $\mu$-almost everywhere to $\prod_{j=1}^n\int_X f_j\dd\mu$ as
$N\to\infty$ through all positive integers. No rate of mixing is
assumed, and $(X,\mathcal F,\mu)$ need not be a standard probability
space.
\end{abstract}
\section{Introduction}
\label{sec:introduction}
\subsection*{The problem and the answer}
\noindent\textbf{Setting.}
$(X,\mathcal F,\mu)$ is a probability space, $T:X\to X$ is invertible,
bimeasurable and measure preserving. $T$ is \emph{mixing} if for every
$A,B\in\mathcal F$
\begin{equation}\label{eq:mixing}
\mu(A\cap T^{-r}B)\longrightarrow\mu(A)\mu(B)\qquad(|r|\to\infty).
\end{equation}
Fix $n\ge 2$ and $f_1,\ldots,f_n\in L^\infty(\mu)$. The object of study
is the sequence of \emph{consecutive multiple ergodic averages}
\[
A_N(f_1,\ldots,f_n)(x)=\frac1N\sum_{k=1}^N\prod_{j=1}^n f_j(T^{jk}x).
\]
These average the integrand along the finite arithmetic progression
$x,Tx,\ldots$ at the $n$ distinct times $k,2k,\ldots,nk$.
\smallskip
\noindent\textbf{Main theorem (Theorem~\ref{thm:main} below).}
If $T$ is mixing, then for $\mu$-almost every $x$
\[
A_N(f_1,\ldots,f_n)(x)\longrightarrow\prod_{j=1}^n\int_X f_j\dd\mu
\]
as $N\to\infty$ through all positive integers. The space $(X,\mathcal F,\mu)$
need not be standard and no rate of mixing is assumed.
The exceptional null set depends on the tuple $(f_1,\ldots,f_n)$ and on
the system, but not on $N$. The theorem says nothing about averages
taken over a different set of slopes, and nothing about uniformity over
functions.
\subsection*{Why this is nontrivial}
The reason the theorem is delicate is the gulf between two very different
notions of convergence.
\begin{itemize}
\item \emph{Norm convergence} asks $A_N\to\int f_1\cdots\int f_n$ in $L^2(\mu)$.
Host--Kra \cite{HostKra2005} (Theorem 1.1) and Ziegler \cite{Ziegler2007}
(Corollary 1.8) proved this for every finite length on every probability-preserving
system.
\item \emph{Pointwise convergence} asks that the same limit holds for
$\mu$-almost every $x$. It is not a corollary of norm convergence:
bounded sequences can converge in $L^2$ and still oscillate on a set of
positive measure. Bourgain's double recurrence theorem
\cite{Bourgain1990} proved $n=2$; higher lengths have only been settled
under structural hypotheses: $K$-systems
\cite{DerrienLesigne1996}, weakly mixing systems whose Pinsker
component has singular spectrum \cite{Assani1998}, and weakly mixing
systems whose pairwise independent self-joinings are independent
\cite{GutmanHuangShaoYe2018}. Huang--Shao--Ye
\cite{HuangShaoYe2019} proved the pointwise statement for every finite
length on ergodic measure-distal systems.
\end{itemize}
Theorem~\ref{thm:main} extends all of these: it covers every mixing
system, every length $n\ge 2$, and does not need ergodicity or standardness
beyond the reductions we make at the start.
The paper continues the fourfold mixing theorem of
\cite{FourfoldCompanion}. Three companion results are used. (A) Ordinary
mixing implies mixing of every finite order, the assertion of Rokhlin's
multiple-mixing problem \cite{MixingCompanion}. (B) An oscillation
estimate for triple averages at distinct integer slopes on any invertible
probability-preserving system \cite{SlopesCompanion}, which yields
pointwise convergence of three-term averages at distinct positive integer
slopes on any system. (C) The Furstenberg--Zimmer structure facts and
relative disjointness between relatively weakly mixing and relatively
distal extensions \cite{FourfoldCompanion}. All three are quoted
verbatim in \S\ref{sec:inputs}. Everything else is proved here.
\subsection*{Proof strategy: five steps}
The proof is by contradiction, and it has five parts. Read them once before
diving into details.
\begin{enumerate}
\item[\textbf{(1)}] \emph{Assume failure: a positive witness at variable lengths.}
If Theorem~\ref{thm:main} fails for the tuple $(f_1,\ldots,f_n)$, there is a
smaller index set $J_0\subseteq \{1,\ldots,n\}$, centered real functions $f_i$
bounded by $1$, a set $E_*$ of positive measure and $\delta>0$, such that
$B_L(x)=(1/L)\sum_{l=1}^L\prod_{i\in J_0}f_i(T^{il}x)$ exceeds $2\delta$ for
arbitrarily large integers $L$, at each $x\in E_*$. This is \S\ref{sec:selected}.
\item[\textbf{(2)}] \emph{Turn the failure into a probability law.}
Move to the roof-one suspension $C=X\times[0,1)$. Choose a measurable
selector $N_a$ at each starting point, say $N_a$ takes values in a finite
dyadic grid, and choose it so the trajectory average of the witness exceeds
$\delta$ on $E_*$ for arbitrarily large $a$. Along any free ultrafilter
on $\N$, this defines a probability law $\lambda_Y$ on $Y^I$ for every flow
$Y\to C$. $\lambda_Y$ has fixed marginals, is flow invariant, comes with
small-marginal laws for at most three slots, and is functorial in $Y$.
\item[\textbf{(3)}] \emph{Channels: a book-keeping device for three groups of outputs.}
A quantitative dependence on the original point is captured by three
\emph{groups} of \emph{outputs}, i.e.~factor maps $D_\gamma:Y\to C$. The
two conditions are: the outputs have product spatial law given their phases,
and the original spatial coordinate has product law jointly with any proper
collection of groups. Two facts are needed. Conditioning can be strengthened
to conditioning on $W_Y$, the maximal measure-distal factor. And the
construction $Y\rightsquigarrow\mathsf M_b(Y)$
(each slot gets its own copy of $Y$, scaled by $1/b$) is a channel.
The second fact is where tensor packing enters; it is the technical heart.
\item[\textbf{(4)}] \emph{Minimum-support correlation and saturation.}
Among all nonzero correlations of the form
$\E_{\lambda_Y}[\Phi(P)\prod_{i\in J}h_i(L_i(y_i))]$, over all channels $Y$,
choose one with the minimum number $k$ of slots; among those, maximize the
number $r$ of slots whose role is \emph{output-only}. Then upgrade to a
\emph{saturated} channel: no further extension of the channel can improve the
conditional expectation of any test given the parameters $\sigma(P)$, or the
parameters together with one slot $\sigma(P,y_l)$. A short repackaging turns
the support-$k$ correlation on the saturated channel into a correlation
supported on $k$ slots of the \emph{regrouped} channel
$Z=\mathsf M_b(Y)$, where one slot flips from input to output.
\item[\textbf{(5)}] \emph{Square argument and contradiction.}
On the selected law of $Z$ we have an $I\times I$ array. Row $b$ and column
$b$ both have the old law $\lambda_Y$; they meet at a common corner. Saturation
gives a row--column identity for a bilinear pairing of two conditional
expectations. A Hilbert-space lemma on independent random vectors in $L^2$
of a conditional fibre forces the pairing not to vanish. Packaging it back
as a correlation on $Z$ produces a correlation of support $k$ with $r+1$
output-only slots, contradicting the maximality of $r$.
\end{enumerate}
Only input (B) is used at length 3 on an arbitrary system. Everything for
length $n\ge 4$ is either norm-convergence ($L^2$) or pointwise convergence on
distal systems.
\subsection*{Map of the paper}
\begin{tabular}{ll}
\S\ref{sec:inputs} & standard models, and inputs (A)--(C) stated precisely.\\
\S\ref{sec:selected} & the failure witness, the suspension, the selectors, the laws $\lambda_Y$.\\
\S\ref{sec:channels} & channels, channel conditioning, channel closure by tensor packing.\\
\S\ref{sec:correlations}& minimum support, saturation, the regrouped channel $Z=\mathsf M_b(Y)$.\\
\S\ref{sec:square} & the square argument and the contradiction.\\
\S\ref{sec:packing} & appendix: the tensor packing theorem.
\end{tabular}
\smallskip
A remark on what is and is not elementary. The proof uses genuine results
of ergodic theory (Furstenberg--Zimmer structure, Khintchine-style matrix
estimates, sated-extension constructions). We do not pretend otherwise.
What we have tried to do is to lay out each step so that, on a first reading,
the reader can see \emph{why} it is being done; and, on a second reading, can
check every quantitative estimate.
% The remaining sections follow.
\section{Dynamical inputs and standard models}
\label{sec:inputs}
Fix $n\ge 2$ and put $I=\{1,\ldots,n\}$. We first record three facts
from other papers, and set up the standard-model reduction. The pointwise
input on an arbitrary system is available only for \emph{three} slots;
for any finite number of slots we have only $L^2$ convergence.
\subsection{Reduction to a standard probability space}
Let $(X,\mathcal F,\mu,T)$ be the fixed system, and $f_1,\ldots,f_n$ bounded
real functions. Record all integer translates of the $f_j$ as coordinates
in a single map $\Phi:X\to K^{\Z\times\{1,\ldots,n\}}$, where $K=[-M,M]$ is
a compact real interval containing the ranges of all $f_j$. The pushforward
$\nu=\Phi_*\mu$ is a Borel probability on a compact metrizable space, and $T$
pushes forward to the invertible shift $\sigma$ on $K^{\Z\times I}$, preserving
$\nu$. Since $T$ is mixing, so is $\sigma$.
The original functions become coordinate functions on this factor. A
countable union of translated null sets runs through the change of
representatives, so $\mu$-almost-everywhere convergence of the pushed-forward
averages pulls back to the same statement on $X$. \emph{From now on the
system is standard:} $(X,\mathcal F,\mu,T)$ is a standard Borel probability
space with an invertible measure-preserving transformation.
Throughout the proof, factors and conditional laws are understood modulo null
sets; every function is taken to have a Borel representative. For a pair of
standard probability spaces, conditional expectations of bounded Borel
functions with respect to sub-$\sigma$-algebras admit bounded Borel versions.
\subsection{Input (A): mixing of every finite order}
This is the content of Rokhlin's multiple-mixing problem for one
transformation.
\begin{theorem}[INPUT A: higher-order mixing; \cite[Theorem 1.1]{MixingCompanion}]
\label{thm:input-mixing}
Let $T$ be invertible and mixing on a probability space. For every integer
$s\ge 2$, every $g_1,\ldots,g_s\in L^\infty(\mu)$ and every $s$-tuple of
integers $t_1,\ldots,t_s$,
\[
\int_X\prod_{j=1}^s g_j(T^{t_j}x)\dd\mu(x)\longrightarrow\prod_{j=1}^s\int_X g_j\dd\mu
\]
as $\min_{i\ne j}|t_i-t_j|\to\infty$. For $s=1$ the identity holds at
every single time $t\in\Z$.
\end{theorem}
The cited theorem is stated for sets and for ordered times. Simple-function
approximation, invariance of $\mu$ under $T$ and the finitely many orderings
of the times give the stated formulation. No rate of mixing is asserted or
used.
\subsection{Input (B): smooth triple averages with no mixing}
The second input treats averages of at most three functions, but at distinct
\emph{integer slopes}, and on \emph{any} invertible probability-preserving
system, not just a mixing one.
\begin{definition}[Averaging profile]
An \emph{averaging profile} is a real compactly supported smooth function
$\varphi$ on $\R$ whose mass is one and whose moments of order $1,2,3$ vanish:
\[
\int_\R\varphi(s)\dd s=1,\qquad
\int_\R s^r\varphi(s)\dd s=0\quad(r=1,2,3).
\]
It is \emph{standard} if, for some constant $G\ge 1$ depending on $\varphi$,
\[
\norm{\varphi^{(r)}}_\infty\le (G(r+2))^{G(r+2)}\qquad(r\ge0).
\]
\end{definition}
\begin{proposition}[INPUT B: smooth triple convergence; \cite[Prop 3.5, Lem 3.6, Lem 3.9]{SlopesCompanion}]
\label{prop:input-smooth}
Let $R$ be any invertible probability-preserving transformation of a
standard probability space. Let $b_1,b_2,b_3$ be distinct nonzero integers
with greatest common divisor one, and let $g_1,g_2,g_3$ be bounded real
functions. Then for every standard averaging profile $\varphi$,
\[
V_k^\varphi(y)=\sum_{l\in\Z}2^{-k}\varphi(l/2^k)\prod_{j=1}^3 g_j(R^{b_jl}y)
\]
converges almost everywhere as $k\to\infty$.
Moreover, for each $c\in[1,2]$ and $\epsilon>0$ there exists a standard
averaging profile $\varphi$ with
\[
\norm{\varphi-c^{-1}\mathbf 1_{(0,c]}}_{L^1(\R)}<\epsilon.
\]
\end{proposition}
The second assertion approximates the uniform density on $(0,c]$ in $L^1$
by a smooth profile with the same mass and moments. Neither part uses
mixing or ergodicity.
\subsection{From smooth to ordinary length}
Combining INPUT B with INPUT A gives pointwise convergence for three-slot
averages at distinct positive integer slopes, along all lengths. This is
the only point where any pointwise input for arbitrary systems is used.
\begin{lemma}[Triple pointwise convergence on every system]
\label{lem:triple}
Let $R$ be any invertible probability-preserving transformation of a
standard probability space. For bounded real functions $g_1,\ldots,g_d$
with $d\le 3$, and distinct positive integer slopes $a_1,\ldots,a_d$,
the averages
\[
\frac1N\sum_{l=1}^N\prod_{j=1}^dg_j(R^{a_jl}y)
\]
converge almost everywhere along all positive integer lengths $N$.
\end{lemma}
\begin{proof}
Add constant-one functions and repeat slopes if $d<3$; both additions
multiply the average by a constant and change nothing about its
convergence. Write the resulting slopes as $g\,b_1,g\,b_2,g\,b_3$ where
$g\ge 1$ and $b_1,b_2,b_3$ are distinct positive integers with
$\gcd(b_1,b_2,b_3)=1$.
By scaling we may assume $|g_j|\le1$. Put $S=R^g$ and
\[
F_l=\prod_{j=1}^3 g_j(S^{b_j l}y).
\]
Then $|F_l|\le 1$. Let $N_k=\lfloor c2^k\rfloor$ for $c\in[1,2]$.
\emph{The smooth-to-ordinary step.}
For $\varphi$ a profile, we have
\[
V_k^\varphi(y)=\sum_{l\in\Z}2^{-k}\varphi(l/2^k)F_l.
\]
For $h_c=c^{-1}\mathbf 1_{(0,c]}$ set
\[
W_k(y)=\sum_{l\in\Z}2^{-k}h_c(l/2^k)F_l.
\]
Then $W_k=(1/N_k)\sum_{l=1}^{N_k}F_l+O(1/(c2^k))$ because $h_c$ is the
normalized indicator of $(0,c]$, using $\lfloor c2^k\rfloor=N_k$ and the
error on the last $c2^k-N_k$ terms being at most $1/(c2^k)$. Also
\[
|V_k^\varphi(y)-W_k(y)|\le 2^{-k}\sum_{l\in\Z}|h_c(l/2^k)-\varphi(l/2^k)|.
\]
By Riemann sums the right side tends to $\norm{h_c-\varphi}_1$ uniformly in
$y$. Now pick a sequence of profiles $\varphi^{(m)}$ with
$\norm{h_c-\varphi^{(m)}}_1\to0$. On a common conull set (intersection
of the conull sets supplied by Proposition~\ref{prop:input-smooth} at each
$m$) all $V_k^{\varphi^{(m)}}(y)$ converge. For $m\to\infty$ the above
display and the triangle inequality show that $W_k(y)$ is Cauchy. Since
$W_k$ and the ordinary average differ by $O(2^{-k})$, the ordinary averages
along $N_k=\lfloor c2^k\rfloor$ form a Cauchy sequence for $\mu$-almost
every $y$. This holds for every $c$ in a countable dense subset of
$[1,2]$.
\emph{All lengths, not just dyadic.}
For any bounded sequence $|H_l|\le1$ and positive integers $M\le N$,
\[
\left|\frac1M\sum_{l=1}^M H_l-\frac1N\sum_{l=1}^N H_l\right|\le\frac{2|N-M|}{\max(N,M)}.
\]
For $2^k\le N<2^{k+1}$ pick a finite rational grid $c\in[1,2]$ with mesh
less than some $\eta$, and compare $N$ to the sequence values at
$\lfloor c2^k\rfloor$. The error is at most $O(\eta)$. Letting $\eta\to0$
on a countable union of scales preserves almost everywhere convergence.
Therefore the average along every complex of lengths converges almost
everywhere.
\emph{Identifying the limit.}
Lemma~\ref{lem:relative-norm} applied over the trivial factor shows that
the ordinary three-slot averages converge to the product of the means in
$L^2$. If their almost-everywhere limit exists on a positive-measure set,
it must equal that $L^2$ limit, so all the a.e. limits we produce at
different $c$ agree. This identification is not needed below; only
existence of the a.e. limit is.
\end{proof}
\subsection{Input (C): Furstenberg--Zimmer structure facts}
The third input packages the standard relative ergodic theory. Recall an
extension $Y\to B$ of an invertible system $R$ is \emph{relatively weakly
mixing} if for bounded $f,g$ with $\E[f\mid B]=0$,
\begin{equation}\label{eq:rwm}
\frac1H\sum_{h=1}^H\norm{\E[\bar f\,(g\circ R^h)\mid B]}_2^2\longrightarrow0.
\end{equation}
It is \emph{relatively compact} if the $L^\infty(B)$-modules spanned by
finite sums $\sum_j a_jF_j$ ($a_j\in L^\infty(B)$, $F_j\in L^2$ fixed) are
dense. A \emph{relatively distal} extension is a tower of relatively
compact extensions and inverse limits. A system is \emph{distal} if it is
distal over the trivial factor. A \emph{joining} of two systems is an
invariant probability on the product with the prescribed marginals.
\begin{proposition}[INPUT C: relative structure; \cite[Prop 2.2]{FourfoldCompanion}]
\label{prop:input-structure}
The following hold for standard probability-preserving systems, without
an ergodicity hypothesis.
\begin{enumerate}
\item Every system $Y$ has a maximal measure-distal factor $W_Y$, and the
extension $Y\to W_Y$ is relatively weakly mixing. Distal factors of any
extension pull into $W_Y$. Automorphisms commuting with $R$ preserve $W_Y$.
\item Factors and joinings of distal systems are distal. A distal system is
relatively distal over each factor. Distality is equivalent for $R$ and any
nonzero integer power.
\item Relative weak mixing passes to nonzero integer powers. The
conditional product of relatively weakly mixing fibre laws over an invariant
joining of the base is relatively weakly mixing over that joining.
\item A relatively weakly mixing extension and a relatively distal extension
of the same base are relatively disjoint: every invariant joining identifying
their bases equals the conditional product over that base.
\item For a distal transformation $R$, every $d\ge 1$ and every tuple of
bounded $g_1,\ldots,g_d$, the averages
\[
\frac1N\sum_{l=1}^N\prod_{j=1}^dg_j(R^{jl}y)
\]
converge almost everywhere.
\end{enumerate}
For a flow $Y$ (a jointly Borel action of $\R$), the maximal distal factor of
time one is preserved by the flow, and its action at every nonzero rational
time is distal.
\end{proposition}
The cited results include the nonergodic form of the Furstenberg--Zimmer
structure theorem \cite{Zimmer1976,Jamneshan2023} and the pointwise theorem
for distal systems \cite[Theorem C]{HuangShaoYe2019}; see
\cite[Prop 2.2]{FourfoldCompanion} for the proof in the nonergodic setup. In
the flow assertion, invariance of $W_Y$ under every flow time is immediate
because each time commutes with time one. If $r=p/q\ne 0$ is rational, the
$q$-th power of $S_Y^r$ equals $S_Y^p$, which is distal; hence $S_Y^r$ is
distal on $W_Y$.
\subsection{Norm convergence on relatively weakly mixing extensions}
The following is the standardised version of $L^2$ convergence we use
throughout.
\begin{lemma}[Relative $L^2$ convergence]
\label{lem:relative-norm}
Suppose $Y\to B$ is a relatively weakly mixing extension of $R$. For
distinct nonzero integers $a_1,\ldots,a_d$ and bounded real $G_1,\ldots,G_d$,
\[
\left\|\frac1N\sum_{l=1}^N\left(\prod_{i=1}^dG_i\circ R^{a_il}-\prod_{i=1}^d\E[G_i\mid B]\circ R^{a_il}\right)\right\|_2\longrightarrow0.
\]
\end{lemma}
\begin{proof}
By telescoping and scaling, it suffices to show the average of
$v_l=\prod_i G_i\circ R^{a_il}$ tends to zero in $L^2$ when $|G_i|\le1$ for
all $i$ and one of them, say $G_s$, is centered over $B$.
Induction on $d$. For fixed $h\ge1$,
\[
F_{i,h}=\overline{G_i}\,(G_i\circ R^{a_ih}),
\]
so by invariance
\[
\ip{v_l}{v_{l+h}}=\int_Y F_{s,h}\prod_{i\ne s}F_{i,h}\circ R^{(a_i-a_s)l}\dd m_Y.
\]
The induction hypothesis replaces the average over $l$ of the product over
$i\ne s$ by its conditional expectation over $B$. The remaining product is
$B$-measurable and bounded by one; hence
\[
\limsup_{N\to\infty}\left|\frac1N\sum_{l=1}^N\ip{v_l}{v_{l+h}}\right|\le\norm{\E[F_{s,h}\mid B]}_1.
\]
For $d=1$ this bound holds directly; for $d>1$ it is the induction output.
The Hilbert-space van der Corput inequality bounds the size of the limit in
$L^2$ by a constant times
\[
\frac1H+\frac1H\sum_{h=1}^H\norm{\E[F_{s,h}\mid B]}_1.
\]
Relative weak mixing of $B\subseteq Y$ over $R^{a_s}$ (Assertion~3 of
Proposition~\ref{prop:input-structure}) makes each $\norm{\E[F_{s,h}\mid B]}_1$
small in average, and Cauchy--Schwarz (since $F_{s,h}$ is bounded) makes the
whole expression tend to zero as $H\to\infty$. This completes the induction.
\end{proof}
Inserting constant-one functions in omitted positions, and using the
pointwise theorem for distal systems (Assertion~5 of
Proposition~\ref{prop:input-structure}) to get an $L^2$ limit of the
distal-factor averages, gives an $L^2$ limit for arbitrary bounded tuples
at distinct positive integer slopes on any standard system. On the
original mixing system, the trivial factor already satisfies
\eqref{eq:rwm}, and the limit is the product of the means.
\subsection{A first corollary for the original system}
Before we start the proof, we note the easy consequence we will use to
\emph{produce} the failure witness in \S\ref{sec:selected}.
\begin{lemma}\label{lem:norm-vanishes}
Suppose $T$ is mixing. For any $f_1,\ldots,f_n\in L^\infty(\mu)$ with
$\int f_i\dd\mu=0$ for some $i$, the averages $A_N(f_1,\ldots,f_n)$ tend to
zero in $L^2(\mu)$ as $N\to\infty$.
\end{lemma}
\begin{proof}
Apply Lemma~\ref{lem:relative-norm} with $B$ the trivial factor, the
identity transformation $R=T$, and $a_i=i$. Because $T$ is mixing, the
trivial factor is a relatively weakly mixing extension. Then the product
$\prod_i\E[f_i\mid B]=\prod_i\int f_i=0$, and the lemma follows.
\end{proof}
\section{The failure witness and selected laws}
\label{sec:selected}
Assume, toward a contradiction, that Theorem~\ref{thm:main} fails for the
fixed slopes $I=\{1,\ldots,n\}$ and some $f_1,\ldots,f_n\in L^\infty(\mu)$.
We extract a positive witness, move to the roof-one suspension $C$, define
measurable averaging lengths $N_a$, and along a free ultrafilter obtain a
probability law $\lambda_Y$ on $Y^I$ for every flow $Y$ over $C$. The
construction follows \cite[Sections 3--6]{FourfoldCompanion}, with the
extension and conditioning facts spelled out for every $n$.
\subsection{A positive witness at selected lengths}
Expanding each $f_j$ into its mean and centered part, and each centered part
into real and imaginary parts, we find a nonempty $J_0\subseteq I$ and real
functions $f_i$, $i\in J_0$, with
\[
\int_Xf_i\,\dd\mu=0,\qquad \norm{f_i}_\infty\le 1,
\qquad
B_L(x)=\frac1L\sum_{l=1}^L\prod_{i\in J_0}f_i(T^{il}x)
\]
such that $B_L$ does not tend to zero on a set of positive measure. Here and
below $B_L$ has integer subscript. Lemma~\ref{lem:norm-vanishes} gives
$B_L\to0$ in $L^2(\mu)$, so the failure is genuinely pointwise: $B_L$
oscillates on a set of positive measure, and its limit is $0$ in $L^2$.
After changing the sign of one $f_i$, there exist $\delta>0$ and a
measurable set $E_*\subseteq X$ of positive measure with
\begin{equation}\label{eq:failure-witness}
B_L(x)>2\delta\quad\text{for arbitrarily large }L,\qquad
x\in E_*.
\end{equation}
\subsection{The suspension and the selectors}
Set
\begin{equation}\label{eq:suspension}
C=X\times[0,1),\qquad m_C=\mu\otimes\dd u,
\qquad
S_C^t(x,u)=\bigl(T^{\lfloor u+t\rfloor}x,\{u+t\}\bigr),
\end{equation}
and write $\xi(x,u)=x$, $\upsilon(x,u)=u$, viewing $\upsilon$ also as a
coordinate on $\R/\Z$. Then $(C,m_C,S_C)$ is a measure-preserving flow.
\begin{lemma}[Selected positive witness]
\label{lem:selected-witness}
There are a finite set $F\subseteq[1,2]$ with $1\in F$, and measurable
functions $N_a:C\to[2^a,\infty)$, $a\in\N$, each independent of the
phase coordinate, with
\begin{equation}\label{eq:selector-grid}
N_a(x,u)\in\{c2^k:c\in F,\ k\ge a\},\qquad
N_a(x,u)=2^a\ (x\notin E_*),
\end{equation}
and such that $B_{\lfloor N_a(x,u)\rfloor}(x)>\delta$ on $E_*$. Moreover
there is a continuous nonnegative function $\Psi$ on $(\R/\Z)^I$ with
\begin{equation}\label{eq:selected-positive-limit}
\liminf_{a\to\infty}\int_C\frac1{N_a(z)}
\int_0^{N_a(z)}
\Psi\bigl((\upsilon(S_C^{it}z))_{i\in I}\bigr)
\prod_{i\in J_0}f_i\bigl(\xi(S_C^{it}z)\bigr)\dd t\dd m_C(z)>0.
\end{equation}
\end{lemma}
\begin{proof}
\emph{Length-change.}
For a bounded sequence $|H_l|\le1$ and positive integers $L\le M$,
\[
\left|\frac1L\sum_{l=1}^LH_l-\frac1M\sum_{l=1}^MH_l\right|
\le\frac{2|M-L|}{\max(M,L)}.
\]
Pick a finite grid $F\subseteq[1,2]$ containing $1$, so fine that for every
integer $L$ there exist $c\in F$ and $k=\lfloor\log_2L\rfloor$ with
$|L-c2^k|<\eta L$; choose $\eta$ so small that the length-change bound
stays below $\delta$. Then for every $x\in E_*$ and every $L$ at which
$B_L(x)>2\delta$, the corresponding dyadic average
$B_{\lfloor c2^k\rfloor}(x)$ exceeds $\delta$.
\emph{The measurable choice.}
Fix an enumeration of the countable set
$\{(c,k):c\in F,\,k\in\N\}$. For $a\in\N$ and $x\in E_*$, let $(c,k)$ be
the first pair in this enumeration with $k\ge a$ and
$B_{\lfloor c2^k\rfloor}(x)>\delta$. On $E_*$ we set $N_a(x,u)=c2^k$;
on $X\setminus E_*$ we set $N_a=2^a$. As a map $C\to[2^a,\infty)$ this is
measurable and independent of $u$. By construction
$B_{\lfloor N_a(x,u)\rfloor}(x)>\delta$ on $E_*$.
\emph{The phase witness.}
Choose continuous nonnegative circle functions $\alpha,\beta$, not
identically zero, supported inside $(1/4,1/2)$ and $(0,1/(4n))$
respectively. Put
\begin{equation}\label{eq:phase-witness}
\Psi(v)=\alpha(2v_1-v_2)\beta(v_2-v_1),
\qquad v=(v_i)_{i\in I}\in(\R/\Z)^I.
\end{equation}
For $z=(x,u)$ and $t=l+s$ with $l\in\Z$, $0\le s<1$, the phase of slot $i$
is $\upsilon(S_C^{it}z)=\{u+is\}$. The support of $\Psi$ then forces
$0<u+is<1$ for every $i\in I$. In particular, on the support of $\Psi$,
$\xi(S_C^{it}z)=T^{il}x$ for every $i$.
Summing complete unit intervals of length $1$ in $t$ and using the support
condition, we obtain, uniformly in $z$ and $L\ge2$,
\begin{align}
&\frac1L\int_0^L
\Psi\bigl((\upsilon(S_C^{it}z))_i\bigr)
\prod_{i\in J_0}f_i\bigl(\xi(S_C^{it}z)\bigr)\dd t
\notag\\
&\hspace{18mm}
=\alpha(u)\left(\int_0^1\beta(s)\dd s\right)B_{\lfloor L\rfloor}(x)
+O(L^{-1}).
\label{eq:suspension-witness}
\end{align}
The error covers the incomplete last unit interval, the change of index
summation from $0,\ldots,\lfloor L\rfloor-1$ to $1,\ldots,\lfloor L\rfloor$,
and the normalisation.
\emph{Integrating.}
On $E_*$, with $L=N_a(z)$, the integral of the main term in
\eqref{eq:suspension-witness} is at least
\[
\delta\mu(E_*)\left(\int_0^1\alpha(u)\dd u\right)\left(\int_0^1\beta(s)\dd s\right)>0,
\]
since $B_{\lfloor N_a(z)\rfloor}>\delta$ on $E_*$. On $X\setminus E_*$, the
function $N_a\equiv 2^a$ is independent of $u$, and $B_{2^a}\to0$ in $L^2(\mu)$
by Lemma~\ref{lem:norm-vanishes}; hence the contribution of this region
tends to zero. The uniform error is $O(2^{-a})$. Taking $\liminf$ over
$a$ completes the proof.
\end{proof}
\subsection{Selected laws on every flow over $C$}
Fix the selectors and a free ultrafilter $\mathcal U$ on $\N$.
\begin{definition}[Flow over $C$; extensions]
A \emph{flow over $C$} is a standard probability space $(Y,m_Y)$ with a
jointly Borel probability-preserving action $S_Y^t$, $t\in\R$, and a
probability-preserving factor map $\pi_Y:Y\to C$ with $\pi_YS_Y^t=S_C^t\pi_Y$
mod $m_Y$ for each fixed $t$. An \emph{extension} $\rho:Y'\to Y$ of such
flows is a probability-preserving factor map, equivariant modulo null sets,
with $\pi_{Y'}=\pi_Y\circ\rho$ modulo null sets. We do not require a common
conull set on which every real-time factor identity holds; joint
measurability and Fubini justify their uses in time integrals.
\end{definition}
\begin{proposition}[Selected joining laws]
\label{prop:selected-law}
Every flow $Y$ over $C$ has a unique probability $\lambda_Y$ on $Y^I$
satisfying, for all bounded measurable $G_i:Y\to\R$,
\begin{equation}\label{eq:selected-law}
\int_{Y^I}\prod_{i\in I}G_i(y_i)\dd\lambda_Y
=\lim_{a\to\mathcal U}\int_Y\frac1{N_a(\pi_Yy)}
\int_0^{N_a(\pi_Yy)}\prod_{i\in I}G_i(S_Y^{it}y)\dd t\dd m_Y(y).
\end{equation}
Its marginals are $m_Y$, and $\lambda_Y$ is invariant under
$(y_i)_{i\in I}\mapsto(S_Y^{it}y_i)_{i\in I}$ for every $t\in\R$. If
$\rho:Y'\to Y$ is an extension over $C$, then $(\rho^I)_*\lambda_{Y'}=\lambda_Y$.
Countable inverse limits exist in this category. The selected law at the
limit projects to the selected law at each stage. Pullbacks of the stage
$\sigma$-algebras have dense union in $L^2(\lambda_Y)$ at the limit.
\end{proposition}
\begin{proof}
\emph{Construction at each $a$.}
For each $a$, the expression before the ultralimit in \eqref{eq:selected-law}
defines a probability measure $\lambda_{Y,a}$ on $Y^I$, since $N_a\circ\pi_Y$
is measurable, $y\mapsto S_Y^{it}y$ is jointly Borel, and the integrand is
bounded.
\emph{Marginals.}
We show $\lambda_{Y,a}\to m_Y$ on marginals as $a\to\infty$ along every
bounded $G:Y\to\R$. By Birkhoff applied to the auxiliary function
$y\mapsto\int_0^1G(S_Y^{is}y)\dd s$ and to $S_Y^i$, the continuous-time
averages $(1/L)\int_0^LG(S_Y^{it}y)\dd t$ converge almost everywhere as
$L\to\infty$, and their a.e. limit $\bar G$ satisfies
$\int_Y\bar G\dd m_Y=\int_YG\dd m_Y$ since every deterministic average has
this integral. As $N_a\ge 2^a\to\infty$, bounded convergence gives
$\int_Y(1/N_a)\int_0^{N_a}G(S_Y^{it}y)\dd t\dd m_Y(y)\to\int_YG\dd m_Y$.
\emph{Ultralimit.}
Embed $Y$ into a compact metrizable $K$ (as a Borel subset with $m_Y$
extended by zero). Along the ultrafilter $\mathcal U$,
$\lambda_{Y,a}$ against continuous functions on $K^I$ converges to a
positive normalised functional, hence a probability $K^I$. Its marginals
are $m_Y$, so it is concentrated on $Y^I$. Denote it $\lambda_Y$.
\emph{Formula for all bounded tests.}
For bounded $G_i:Y\to\R$ with $|G_i|\le1$, approximate each by continuous
functions on $K$ with the same bound in $L^1(m_Y)$ to within $\epsilon$.
The difference of the two product integrals \eqref{eq:selected-law} at stage
$a$ is bounded by $\sum_i\epsilon$. Under $\lambda_Y$ these errors are
$L^1$-errors, and under $\lambda_{Y,a}$ they converge to those errors by
marginal convergence. Letting $\epsilon\to0$ gives the formula.
Product tests determine a unique probability on $Y^I$, giving uniqueness.
All identities depend only on $G_i$ up to a.e. equivalence.
\emph{Invariance.}
Fix $t_0\in\R$ and shift the time interval in \eqref{eq:selected-law} by
$t_0$. The selector $N_a\circ\pi_Y$ does not change under
$S_Y^{t_0}$. For $|G_i|\le1$ the error is at most $2|t_0|2^{-a}$,
proving invariance under $S_Y^{t_0}$ in each slot.
\emph{Functoriality.}
Substitute $G_i\circ\rho$ for $G_i$ in the formula and use $\pi_{Y'}=\pi_Y\rho$
plus Fubini and equivariance. The right-hand side of \eqref{eq:selected-law}
is identical for $Y$ and $Y'$.
\emph{Inverse limits.}
For $Y_0\leftarrow Y_1\leftarrow\cdots$, let $Y_\infty$ be the full
countable product of the underlying standard spaces with the consistent
inverse-limit measure and coordinatewise flow. Joint measurability is
automatic for coordinatewise actions of $\R$ on a countable product. The
projections are measurable, $S_{Y_\infty}^t$ preserves $m_{Y_\infty}$
because every cylinder has the same measure as its preimage. Stages are
factor maps mod null sets; projection through $Y_0$ gives the factor
$Y_\infty\to C$. Functoriality gives the claimed projections of stage
selected laws. The stage $\sigma$-algebras generate the product
$\sigma$-algebra; pullbacks are increasing and generate $L^2(\lambda_Y)$,
by the usual $L^2$ approximation by increasing $\sigma$-algebras.
\end{proof}
\subsection{Distal parameters and small marginals}
For a flow $Y$ over $C$, let $\omega_Y:Y\to W_Y$ be its maximal measure-distal
factor for time one, and $\nu_Y=(\omega_Y)_*m_Y$. Write $m_{Y,w}$ for
its conditional measures, so $m_Y=\int_{W_Y}m_{Y,w}\dd\nu_Y(w)$. Under
$\lambda_Y$ put
\begin{equation}\label{eq:distal-parameters}
P_i=\omega_Y(y_i),\qquad P=(P_i)_{i\in I}.
\end{equation}
By Assertion 1 of Proposition~\ref{prop:input-structure}, $W_Y$ is preserved
by every flow time. Under an extension $\rho:Y'\to Y$, the pullback of $W_Y$
is a distal factor of $Y'$ for time one, hence is contained in $W_{Y'}$.
Thus $P$ on $Y^I$ is a measurable function of the corresponding array on
$(Y')^I$.
\begin{proposition}[Parameter and small-marginal laws]
\label{prop:selected-parameters}
For every flow $Y$ over $C$ and every bounded measurable $H_i$ on $W_Y$,
\begin{equation}\label{eq:deterministic-parameters}
\E_{\lambda_Y}\prod_{i\in I}H_i(P_i)
=\lim_{L\to\infty}\int_Y\frac1L\int_0^L
\prod_{i\in I}H_i(\omega_Y(S_Y^{it}y))\dd t\dd m_Y(y).
\end{equation}
For every nonempty $J\subseteq I$ with $|J|\le 3$,
\begin{equation}\label{eq:small-conditional-law}
\law_{\lambda_Y}\bigl((y_i)_{i\in J}\mid P\bigr)
=\bigotimes_{i\in J}m_{Y,P_i}.
\end{equation}
\end{proposition}
\begin{proof}
\emph{From discrete to continuous time.}
Write $t=l+s$ with $l\ge0$ integer and $0\le s<1$. For each fixed $s$, a
continuous-time product becomes the discrete-time product at slopes $i$ with
slot functions $G_i\circ S_Y^{is}$, evaluated at times $l\ge0$. If the
discrete averages converge almost everywhere, Fubini and bounded convergence
give that the $s$-average of them does. Starting the discrete average at
$l=0$ rather than $l=1$, and dropping an incomplete final unit interval,
contributes an error that tends to zero. The same argument applied to real
$L\ge2$ in place of dyadic $N$ shows convergence along all real lengths.
\emph{Deterministic parameters.}
Apply this reduction to the functions $H_i\circ\omega_Y$, which are fixed by
all flow times (Assertion 1 of Proposition~\ref{prop:input-structure});
pointwise convergence on distal systems follows from Assertion 5. Since
$N_a\to\infty$, bounded convergence with \eqref{eq:selected-law} gives
\eqref{eq:deterministic-parameters}.
\emph{Small marginals off the parameter.}
The extension $Y\to W_Y$ is relatively weakly mixing. For $|J|\le3$,
Lemma~\ref{lem:triple} applied to the discrete averages at slopes $i\in J$
shows that the $J$-marginal attains the integrated deterministic-length law.
By Lemma~\ref{lem:relative-norm}, the difference of the deterministic
average $\prod_{i\in J}G_i(y_i)$ and the same product with each $G_i$
replaced by $\E[G_i\mid W_Y]$ tends to zero in $L^2$. Integrating in $s$
gives
\begin{equation}\label{eq:small-replacement}
\E_{\lambda_Y}\prod_{i\in J}G_i(y_i)
=\E_{\lambda_Y}\prod_{i\in J}\bigl(\E_{m_Y}[G_i\mid W_Y](y_i)\bigr).
\end{equation}
\emph{Small marginals given $P$.}
Multiplying $G_i$ by arbitrary bounded test functions of $P_i$ gives
\begin{equation}\label{eq:small-own-parameters}
\law_{\lambda_Y}\bigl((y_i)_{i\in J}\mid(P_i)_{i\in J}\bigr)
=\bigotimes_{i\in J}m_{Y,P_i}.
\end{equation}
Now upgrade from $(P_i)_{i\in J}$ to the full array $P$. Under
$S_Y^{it}$ acting on slot $i$, the $J$-marginal system of
\eqref{eq:small-own-parameters} is relatively weakly mixing over its own
parameter law, by Assertions 2 and 3 of
Proposition~\ref{prop:input-structure}. The full $P$-system is a joining of
the distal systems $W_Y$ at integer slopes $i$ and is thereby distal. Its
joint law under $\lambda_Y$ is an invariant joining identifying that common
subarray; by relative disjointness (Assertion 4) the joining splits, which is
exactly the assertion. This completes the proof.
\end{proof}
The full parameter law is unaffected by the selectors, even though the full
state law need not be. We turn the state law into a new flow while preserving
one chosen slot as the original system.
\begin{lemma}[Normalized joining flow]
\label{lem:normalized-joining}
For $b\in I$, define
\begin{equation}\label{eq:normalized-joining}
\mathsf M_b(Y)=\left(Y^I,\lambda_Y,
S_{\mathsf M_b(Y)}^t(y_i)_{i\in I}=(S_Y^{it/b}y_i)_{i\in I}\right).
\end{equation}
This is a standard probability-preserving flow over $C$ via
$(y_i)_{i\in I}\mapsto\pi_Y(y_b)$; the projection onto slot $b$ is an
extension onto $Y$. The array $P$ belongs to the maximal measure-distal
factor of $\mathsf M_b(Y)$ for time one.
\end{lemma}
\begin{proof}
Invariance follows from Proposition~\ref{prop:selected-law} after rescaling
the time parameter by $1/b$. The factor map is $\pi_Y\circ p_b$. Slot $b$
has speed $1$ and marginal $m_Y$, giving $(y_i)_i\mapsto y_b$ as a factor
onto $Y$. In slot $i$, the $W_Y$-factor is distal for time $i/b$, by
Assertion 2 (equivalence for rational powers); the law of $P$ is an
invariant joining of these rational-time distal systems, hence distal.
Maximality of the distal factor gives the last assertion.
\end{proof}
\begin{lemma}[The retained positive correlation]
\label{lem:positive-witness}
With the functions and phase test of Lemma~\ref{lem:selected-witness},
\begin{equation}\label{eq:positive-witness}
\int_{C^I}\Psi\bigl((\upsilon(z_i))_i\bigr)
\prod_{i\in J_0}f_i(\xi(z_i))\dd\lambda_C>0.
\end{equation}
The phase array is a function of the distal parameter array of $C^I$.
\end{lemma}
\begin{proof}
Continuous functions on the compact product of circles are uniformly
approximable by finite sums of products of continuous one-coordinate
functions. Apply this to $\Psi$, use \eqref{eq:selected-law} for each
resulting product, then apply the strictly positive lower bound in
\eqref{eq:selected-positive-limit}. The phase at time zero on $C$ is
distal, hence lies in $W_C$.
\end{proof}
\section{Channels: three groups of outputs}
\label{sec:channels}
The selected length is determined by the original point in $C$. It
therefore can change the joint law of other coordinates, even when their
unconditioned law is a product. To control the dependence we pack the
outputs into three groups. Independence of the original spatial coordinate
from any two groups will turn out to hold automatically, and this is enough
to cancel the effect of selecting the length. The construction adapts
\cite[Section 5]{FourfoldCompanion}; we spell out every step.
\subsection{Channels}
\begin{definition}[Channel]
\label{def:channel}
A \emph{channel} is a flow $Y$ over $C$ with factor $\pi_Y:Y\to C$, together
with a finite list of probability-preserving maps $D_\gamma:Y\to C$ called
\emph{outputs}, and positive rational numbers $q_\gamma$, satisfying
\begin{equation}\label{eq:output-equivariance}
D_\gamma(S_Y^ty)=S_C^{q_\gamma t}D_\gamma(y)
\end{equation}
for each fixed $t$, mod $m_Y$. Partition the output indices into three
groups, numbered $1,2,3$; empty groups are allowed. Set
\[
x_0=\xi\circ\pi_Y,\qquad x_\gamma=\xi\circ D_\gamma,\qquad
d_Y=(x_\gamma)_\gamma,
\]
and for $H\subseteq\{1,2,3\}$ let $d_{Y,H}$ be the sublist of $d_Y$ from
groups in $H$. Let $V$ be the joint of the phase coordinates of $\pi_Y$
and all $D_\gamma$ (i.e. $V=(\upsilon\circ\pi_Y,(\upsilon\circ D_\gamma)_\gamma)$).
Require, under $m_Y$,
\begin{equation}\label{eq:channel-laws}
\begin{split}
\law(d_Y\mid V)&=\bigotimes_\gamma\mu,\\
\law(x_0,d_{Y,H}\mid V)&=
\mu\otimes\bigotimes_{\gamma\text{ in groups }H}\mu,
\qquad H\subsetneq\{1,2,3\}.
\end{split}
\end{equation}
An empty spatial list has the one-point law.
\end{definition}
Thus the spatial outputs have a full product law independent of the phases.
The original spatial coordinate $x_0$ has product law jointly with the
outputs in any \emph{proper} collection of groups; independence from all
three groups is \emph{not} required. The suspension $C$, with no outputs,
is a channel.
\subsection{Conditioning and extensions}
The first observation is that the phase variable $V$ in Definition~\ref{def:channel}
can be replaced by the maximal measure-distal factor $W_Y$.
\begin{lemma}[Conditioning and extensions]
\label{lem:channel-conditioning}
For a channel $Y$, both identities in \eqref{eq:channel-laws} hold with
conditioning on $W_Y$ in place of $V$. Every flow extension of $Y$ over
$C$ is a channel with the pulled-back output data, and the same
strengthening holds there.
\end{lemma}
\begin{proof}
Choose a positive integer $h$ such that $hq_\gamma\in\Z$ for every output.
The phase vector $V$ is a factor fixed by time $h$, and its time-one action
is distal, so $V$ belongs to $W_Y$. For any spatial tuple appearing in
\eqref{eq:channel-laws}, its joint factor with $V$ has product law over
$V$; the joint factor with $W_Y$ is a joining of this over $V$.
At time $h$, the spatial coordinates evolve by a product of positive
integer powers of $T$ (the powers are $hq_\gamma\in\N$ and $h$ for $x_0$).
This product is mixing by INPUT A. The assertion follows first for
product tests, by mixing of each power, and then for all bounded tests by
approximation. Consequently the spatial factor is relatively weakly
mixing over $V$ for time $h$.
On the other hand, $W_Y$ is distal for time $h$ and relatively distal over
$V$. Relative disjointness (Assertion 4 of
Proposition~\ref{prop:input-structure}) makes the spatial factor and
$W_Y$ relatively independent over $V$, so its conditional spatial law given
$W_Y$ is the product law in \eqref{eq:channel-laws}.
Every flow extension preserves the joint laws of the original data, so the
channel identities pull back. The same argument, applied with a common
supersystem, gives the stronger conditioning on $W_{Y'}$ for the extension
$Y'$.
\end{proof}
\subsection{The multiplicative construction $\mathsf M_b$}
Recall from Lemma~\ref{lem:normalized-joining} that for $b\in I$,
$\mathsf M_b(Y)$ has state space $Y^I$, measure $\lambda_Y$, flow
$(y_i)_{i\in I}\mapsto(S_Y^{it/b}y_i)_{i\in I}$, and factor $y_b\to Y$.
\begin{proposition}[Channel closure]
\label{prop:channel-closure}
Let $Y$ be a channel and $b\in I$. Define data on $Z=\mathsf M_b(Y)$ by:
\begin{itemize}
\item the original factor is $x_0(y_b)=\xi\circ\pi_Y(y_b)$;
\item the outputs are all $D_\gamma(y_i)$, $i\in I$, each in the old group
of $\gamma$;
\item the speed of $D_\gamma(y_i)$ is $iq_\gamma/b$.
\end{itemize}
Then these data make $Z$ a channel over $C$.
\end{proposition}
The proof is the technical heart of the paper, and is given in
\S\ref{subsec:closure-proof}. It uses INPUT A (mixing of every order) at
length $n$ and the tensor packing theorem stated in \S\ref{sec:packing}.
\begin{corollary}[Copied output law]
\label{cor:channel-copied-law}
For every channel $Y$, the copied spatial list
$(d_Y(y_i))_{i\in I}$ under $\lambda_Y$ has its full product law
conditional on $P=(\omega_Y(y_i))_{i\in I}$.
\end{corollary}
\begin{proof}
Apply Proposition~\ref{prop:channel-closure} and
Lemma~\ref{lem:channel-conditioning} to $\mathsf M_b(Y)$. By
Lemma~\ref{lem:normalized-joining}, the old parameter array $P$ belongs to
the maximal distal factor of $\mathsf M_b(Y)$. Conditioning the
constant product law \eqref{eq:channel-laws} down to $P$ proves the
assertion.
\end{proof}
\subsection{Proof of channel closure}
\label{subsec:closure-proof}
This is \S\ref{subsec:closure-proof}, the proof of
Proposition~\ref{prop:channel-closure}.
\begin{proof}[Proof of Proposition~\ref{prop:channel-closure}]
Each new output $D_\gamma(y_i)$ is probability-preserving and has positive
rational speed $iq_\gamma/b$, so the equivariance \eqref{eq:output-equivariance}
holds with $q=iq_\gamma/b$. It remains to prove the two conditional laws in
\eqref{eq:channel-laws}.
\emph{Reductions.}
Test the laws with products of bounded spatial functions and products of
bounded phase-test functions. Such tests determine the laws, and complexity
is unchanged by approximation. Expanding each spatial function into its
mean and centered part, and using centered functions of bounded absolute
value, reduces the proof to real spatial factors that are either constants or
centered and bounded by $1$. We may bound every phase test by $1$; the
assertion below is linear in the phase test and so is preserved.
A test containing a spatial factor at $x_0(y_b)$ uses outputs from only a
proper subset of the groups; we treat such tests inside the "proper set of
groups" paragraph. Consequently we may assume that in the "full output
list" case, no spatial test involves $x_0(y_b)$.
\emph{Time intervals and offsets.}
Use the selected-law formula \eqref{eq:selected-law} with base flow being
$Y$, and condition on the old phase vector $V$ under $m_Y$. Fix an integer
$h>0$ clearing all output speed denominators, and write $t=hl+s$ with
$l\ge0$ an integer and $0\le s<h$. For fixed $V,s$, all phase weights in
the test are independent of $l$. The spatial function for output $\gamma$
at slot $i$ is evaluated at
\begin{equation}\label{eq:channel-times}
T^{e_\gamma i l+a_{\gamma,i}}x_\gamma,\qquad e_\gamma=hq_\gamma\in\N.
\end{equation}
The offsets $a_{\gamma,i}$ depend on $V,s$ and range over a fixed finite
set, by \eqref{eq:suspension}. If the original-factor test is present, its
argument is $T^{hbl+a_0}x_0$, with the same finite-offset property.
Equivariance and Fubini justify these formulas inside the time integrals.
On an event where the selected length is $N=c2^k$ ($c\in F$, $k\ge a$), put
$m=\lfloor c2^k/h\rfloor$. Replacing the normalized time integral by the
average over $m$ complete intervals, followed by integration against
$ds/h$, has error at most $2h2^{-a}$. For all sufficiently large $a$,
$m\ge1$ on each admissible event.
\emph{Case 1: a proper set of groups (or all outputs $1$).}
Suppose some proper subset $H\subsetneq\{1,2,3\}$ is used together with
$x_0$. Given $V$, the relevant old output coordinates have product law
and are independent of $x_0$, by \eqref{eq:channel-laws}. The selector
depends on $V,x_0$ only. Suppose at least one output occurrence is centered.
For fixed $V,s,x_0$, integrate the spatial output product at index $l$.
The expectation factors over the old output indices $\gamma$. For any
index with a centered occurrence, the times \eqref{eq:channel-times} in
distinct slots separate as $l\to\infty$. Mixing of all orders (INPUT A)
makes the coordinate integral tend to zero (each factor bounded, one
centered so the product law mean is $0$). Convergence is uniform over the
finitely many offset patterns and finitely many $V$-sections. A Ces\`aro
limit therefore gives zero for the full average. All other factors have
absolute value $\le1$, so bounded convergence completes this case.
If, on the other hand, every output test is $1$, the only centered factor
is $x_0$ (bounded by $1$ and $\int x_0\dd\mu=0$). Birkhoff's theorem for
$T^{hb}$ applied to $x_0$ gives
\[
\frac1m\sum_{l<m}x_0(T^{hbl+a_0}x_0)\longrightarrow0
\]
for $\mu$-almost every $x_0$, simultaneously for the finitely many offsets
$a_0$. Since $N_a\to\infty$ pointwise, the selected lengths share this
limit. The conditional law of $x_0$ given $V$ is $\mu$, so bounded
convergence completes this case.
\emph{Case 2: full output list with a centered occurrence per group.}
It remains to treat the full output list, with no original-factor test. If
one group has no centered occurrence, every spatial test in that group is
$1$, and the preceding argument applies. Thus suppose each group contains
a centered occurrence. For fixed $V,s$, write the spatial product at index
$l$ as
\[
u_l\in L^2(\Omega_1),\qquad v_l\in L^2(\Omega_2),\qquad w_l\in L^2(\Omega_3),
\]
where $\Omega_j$ carries the product measure with one copy of $\mu$ for each
old output in group $j$. These are real unit-vectors (functions bounded by
$1$). Given $V$, the old spatial output law is the product of these three
spaces. The vectors depend on $V,s$ only through the finite offset patterns.
\emph{Almost orthogonality with a bounded-degree graph.}
For every $\eta>0$, after omitting finitely many initial indices, the
families $(u_l)$ and $(v_l)$ each have absolute pair inner products at most
$\eta$ outside a graph of bounded maximum degree $L_0$. All bounds are
uniform in the offset patterns. To see this, in each of the two groups
choose a centered occurrence at some output index $\gamma$. Its coordinate
integral in a pair inner product is small by INPUT A whenever the times in
the two copies of \eqref{eq:channel-times} are sufficiently separated. The
times within each copy separate once $l,l'$ exceed a fixed threshold.
Failure of separation between copies requires one of finitely many
inequalities
\[
|e_\gamma i l-e_\gamma i'l'+a'|\le R.
\]
For fixed $l$, each inequality permits only a bounded number of integers
$l'$, because $e_\gamma,i'>0$; and symmetrically. Their union gives a
bounded-degree graph. Coordinate integrals for other output indices are
bounded by $1$. The separation threshold and $L_0$ may depend on $\eta$;
no rate of mixing is needed.
\emph{Conditioning on a selected length.}
For fixed selector index $a$, partition into the events $N_a=c2^k$,
$c\in F$, $k\ge a$, choosing one representation when a length has more
than one. Given $V$, write their probabilities as $p_{k,c}(V)$. On one
such event of probability $p>0$, the conditional spatial law of any two
groups, and of each single group, is unchanged. Indeed the event is
determined by $V,x_0$, which is independent of any proper set of groups by
\eqref{eq:channel-laws}. The full three-group law may change. Its
density $\rho$ relative to the product law satisfies
\begin{equation}\label{eq:channel-density}
0\le p\rho\le1,\qquad\int p\rho=p,\qquad\norm{p\rho}_2\le\sqrt p.
\end{equation}
The bound $0\le p\rho\le1$ is the absolute-continuity statement; the second
identity is the normalisation; the third is Cauchy--Schwarz. These hold
for almost every $V$, and the length events are countable.
\emph{The rank bound.}
The rank parameter must meet two requirements: the packing bound must be
$o(m)$, and the reciprocal bounds $1/d_m$ must be summable over dyadic
length scales. For $m=\lfloor c2^k/h\rfloor$ set
\begin{equation}\label{eq:channel-dimension}
d_m=\lceil(1+\log m)^{3/2}\rceil,\qquad z_l=u_l\otimes v_l\quad(0\le l<m).
\end{equation}
Let $\Pi_m$ be the orthogonal projection onto up to $d_m$ leading
eigenvectors of the positive covariance operator $\sum_{l<m}z_lz_l^*$,
taking its full range if its rank is smaller.
Fix $0<\varepsilon\le1$ and then fix $0<\eta\le\varepsilon^2/10^4$ in the
preceding graph bound. Theorem~\ref{thm:packing} of the Appendix gives
\begin{equation}\label{eq:channel-projected-average}
\frac1m\sum_{l<m}\norm{\Pi_mz_l}\le\varepsilon+o(1).
\end{equation}
Here $o(1)$ tends to $0$ uniformly over the offset patterns. Indeed, after
the fixed initial indices are removed, the number of projection norms at
least $\varepsilon$ is at most
\[
C(1+L_0)\varepsilon^{-2}
\exp\!\bigl(C\varepsilon^{-2}\sqrt{d_m}\log(2+d_m)\bigr)=o(m).
\]
The removed indices contribute $o(1)$. Crucially, $\varepsilon,\eta$ and
$L_0$ are fixed before $m\to\infty$.
\emph{The projected terms.}
Cauchy--Schwarz under the conditional event law uses its unchanged
two-group and one-group marginals. Then, term by term, including the event
probability $p$,
\[
\left|p\int(\Pi_mz_l)(\omega_1,\omega_2)\,w_l(\omega_3)\,\rho(\omega)\dd\omega\right|
\le p\norm{\Pi_mz_l}_2\norm{w_l}_2
\le p\norm{\Pi_mz_l}_2.
\]
Thus the projected contributions total at most $p(\varepsilon+o(1))$.
\emph{The residual vectors.}
For the residual vectors $z_l'=(1-\Pi_m)z_l$, set
$A=(\ip{z_l'}{z_{l'}'})_{l,l'<m}$ and $B=(\ip{w_l}{w_{l'}})_{l,l'<m}$.
The covariance trace is at most $m$, so the remaining eigenvalues of the
covariance operator, and hence the nonzero eigenvalues of $A$, are at most
$m/d_m$. If the rank is at most $d_m$, the residual is zero. Also, $B$ is
positive semidefinite with trace at most $m$. Therefore
\[
\left\|\frac1m\sum_{l<m}z_l'\otimes w_l\right\|^2
=\frac{\tr(AB)}{m^2}\le\frac1{d_m}.
\]
By \eqref{eq:channel-density}, the residual contribution, including the
event probability, is at most $\sqrt{p/d_m}$. Multiplication by the fixed
phase test of absolute value at most $1$ does not increase either bound.
\emph{Summing over lengths.}
Write $m(k,c)=\lfloor c2^k/h\rfloor$. For fixed $h$ this is comparable to
$2^k$ uniformly in $c\in F$. Since $\sum_{k,c}p_{k,c}(V)=1$,
Cauchy--Schwarz gives
\begin{equation}\label{eq:channel-residual-sum}
\sum_{k\ge a,\ c\in F}\sqrt{\frac{p_{k,c}(V)}{d_{m(k,c)}}}
\le
\left(\sum_{k\ge a,\ c\in F}\frac1{d_{m(k,c)}}\right)^{1/2}\longrightarrow0.
\end{equation}
The series converges because its summands are $O(k^{-3/2})$ thanks to the
rank bound \eqref{eq:channel-dimension}. The projected contributions have
total $\limsup\le\varepsilon$, by \eqref{eq:channel-projected-average} and
the same probability identity. These estimates are uniform in $V,s$ and may
be integrated. There is no measurable-choice issue for the projections:
choose one for each of the countably many sizes and finitely many offset
patterns. Letting $a\to\infty$ in \eqref{eq:selected-law} and then
$\varepsilon\to0$ proves the required vanishing.
Both laws in \eqref{eq:channel-laws} follow.
\end{proof}