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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.
+%
+% Reading keys, so nothing is hidden:
+%
+%   (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).
+%
+% The single result proved below is Theorem 0.1: on every mixing system,
+% A_N converges almost everywhere to the product of the integrals.
+% ============================================================================
+
+\documentclass[11pt]{article}
+\usepackage{amsmath,amssymb,amsthm}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newcommand{\E}{\mathbb E}
+\newcommand{\N}{\mathbb N}
+\newcommand{\Z}{\mathbb Z}
+\newcommand{\R}{\mathbb R}
+\newcommand{\C}{\mathbb C}
+\newcommand{\T}{\mathbb T}
+\newcommand{\norm}[1]{\lVert #1\rVert}
+\newcommand{\abs}[1]{\lvert #1\rvert}
+\newcommand{\ip}[2]{\langle #1,#2\rangle}
+\newcommand{\dd}{\,d}
+\newcommand{\law}{\mathcal L}
+\DeclareMathOperator{\tr}{tr}
+
+\begin{document}
+\title{Pointwise Multiple Ergodic Averages for Mixing Transformations}
+\author{OpenAI}
+\date{October 4, 2026}
+\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}