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\title{Pointwise convergence of fourfold ergodic averages\\ for mixing transformations}
\author{OpenAI}
\date{October 4, 2026}

\begin{document}
\maketitle

\begin{abstract}
Let $T$ be an invertible, bimeasurable, measure-preserving, mixing
transformation of a probability space. For every fixed bounded
$f_1,f_2,f_3,f_4$ the fourfold ergodic averages along the times
$n,2n,3n,4n$ converge almost everywhere to the product of the four
integrals, along all positive integer lengths. No rate of mixing is
assumed, and the space need not be standard.
\end{abstract}

\section{Introduction and main theorem}\label{sec:intro}

Let $(X,\mathcal F,\mu,T)$ be an invertible probability-preserving
system and let $f_1,\dots,f_d\in L^\infty(\mu)$. The multiple ergodic
averages are
\[
 A_N(f_1,\dots,f_d)(x)=\frac1N\sum_{n=1}^N\prod_{j=1}^d f_j(T^{jn}x).
\]
The pointwise problem asks whether $A_N$ converges almost everywhere
for every finite $d$ on every system. Here we settle the case $d=4$
under a mixing hypothesis. Call $T$ \emph{mixing} if
\begin{equation}\label{eq:mixing}
 \mu(A\cap T^{-n}B)\longrightarrow\mu(A)\mu(B)\qquad(|n|\to\infty)
\end{equation}
for all measurable $A,B$. The functions are fixed in each convergence
statement, and we assume no rate in \eqref{eq:mixing}.

\begin{theorem}\label{thm:main}
Let $(X,\mathcal F,\mu)$ be a probability space and let $T:X\to X$ be
invertible, bimeasurable, measure-preserving and mixing in the sense of
\eqref{eq:mixing}. For all $f_1,f_2,f_3,f_4\in L^\infty(\mu)$,
\[
 \frac1N\sum_{n=1}^N
 f_1(T^nx)f_2(T^{2n}x)f_3(T^{3n}x)f_4(T^{4n}x)
 \longrightarrow\prod_{j=1}^4\int_X f_j\dd\mu
\]
for $\mu$-almost every $x$, along every positive integer $N$. The
space need not be standard.
\end{theorem}

\subsection*{Where this sits}

Birkhoff's theorem \cite{Birkhoff} is the case $d=1$. Furstenberg's
ergodic proof of Szemer\'edi's theorem \cite{Furstenberg} made
arithmetic-progression products central. Host and Kra
\cite[Theorem~1.1]{HostKra}, and independently Ziegler \cite{Ziegler},
proved \emph{norm} convergence of every finite consecutive length using
nilpotent characteristic factors; norm convergence alone says nothing
about a single point. Bourgain \cite{Bourgain} proved pointwise
convergence for two powers. Further positive pointwise results: $K$
systems \cite{DerrienLesigne}, weakly mixing systems with spectral
restrictions \cite{Assani}, weakly mixing systems all of whose
pairwise-independent self-joinings are products
\cite{GutmanHuangShaoYe}, and every finite consecutive length on
ergodic measure-distal systems \cite[Theorem~C]{HuangShaoYe}, extending
\cite{Lesigne}. Recent polynomial results \cite{KrauseMirekTao},
\cite{KoszMirekPeluseWanWright} impose degree hypotheses that exclude
the four linear iterates treated here. Kuca's survey \cite{Kuca}
collects the norm and pointwise problems.

Our proof rests on three companion inputs. The multiple-mixing theorem
of \cite[Theorem~1.1]{MixingCompanion} upgrades \eqref{eq:mixing} to all
finite orders. The analytic construction of \cite{TripleCompanion},
built on the local estimates of \cite{HilbertCompanion}, gives an
oscillation estimate for threefold averages on \emph{arbitrary}
invertible systems; mixing only later identifies the limit.
Appendix~\ref{sec:lower-length} adapts that estimate to each
three-element subset of $\{1,2,3,4\}$ and derives pointwise convergence
for every proper subaverage. The local analytic theorems are cited with
their hypotheses; the new content here is the fourfold deduction.

The structural engine is the Furstenberg--Zimmer theory of relatively
compact and relatively weakly mixing extensions, recorded in the
nonergodic and power-stable form in Section~\ref{sec:structure}. The
construction also stabilizes certain conditional expectations under
extension, in the spirit of sated extensions \cite{Austin}.

\subsection*{Why selected lengths are hard}

Suppose the theorem fails. After removing means we get four centred
real functions and a positive-measure set on which the averages stay
bounded away from zero at arbitrarily late lengths. Pick one such
length $N_a(x)$ past scale $2^a$. Averaging the configurations
$(T^nx,T^{2n}x,T^{3n}x,T^{4n}x)$, $1\le n\le N_a(x)$, and passing to a
limit preserves a nonzero correlation; a constant-roof suspension turns
these limit laws into flow-invariant measures and allows later changes
of speed.

Selection is the whole difficulty: because the length is chosen from
the initial state, conditioning on it can distort the joint law of
auxiliary variables, and norm convergence without a rate cannot bound
all errors. We introduce three groups of auxiliary spatial coordinates
whose joint law is a product and any two of which are jointly
independent of the original coordinate, conditionally on the phase.
A length event may change the full three-group law but leaves every
two-group marginal a product. Toy model: independent symmetric signs
$U,V,W$ have a product independent of each pair, whereas conditioning
on $UVW=1$ constrains the triple.

The tool that survives selection is a finite-dimensional packing
estimate (Section~\ref{sec:packing}): two bounded almost-orthogonal
families, with the large inner products confined to a bounded-degree
graph, have few tensor products landing substantially in any
slowly-growing subspace. A Gaussian comparison and a trace-moment bound
make it quantitative, uniform over all Hilbert-space dimensions and
free of any mixing rate. Applied to the leading covariance subspace it
controls the projected part of each selected average; the leftover has
a square-summable error over the length classes.

\subsection*{From a correlation to a contradiction}

A \emph{channel} (Definition~\ref{def:channel}) carries an original
spatial coordinate plus the three auxiliary groups, with independence
as above. In its selected four-coordinate law we look at correlations
where each slot uses a centred function of the original coordinate or
of the auxiliaries, weighted by a bounded function of the distal
parameters. Failure supplies one with no auxiliary slots. Choose a
channel and correlation maximizing the number of auxiliary slots.
Independence forbids four. An iterated array in which products along
distinct axes through a fixed corner are orthogonal but share the same
nonzero correlation with the corner function, killed by Bessel, forbids
three. So at most two.

The final step adds one. Enlarge the channel until the required
conditional expectations are stable under any further extension.
Then treat a selected four-coordinate law as a new system and select
again; the result is a square of states whose distinguished row and
column meet at one corner. Saturation identifies their conditional
laws. A reanchoring identity on the distal parameters shows the row--
column correlation cannot vanish, so regrouping one coordinate turns
an original-coordinate slot into an auxiliary slot, contradicting
maximality.

Sections~\ref{sec:structure} and~\ref{sec:selected} supply the
structural facts and selected laws; Section~\ref{sec:packing}
the packing estimate; Section~\ref{sec:channels} channel preservation;
Section~\ref{sec:activation} the maximality argument and the theorem.
The coefficient work is in Appendix~\ref{sec:lower-length}.

\section{Standard reduction and relative structure}\label{sec:structure}

The auxiliary systems built later are extensions and joinings of a
suspension of the original $T$, and they need not be ergodic. We record
the structural facts in the nonergodic setting and separate the
pointwise input for three functions from the norm input for any number.

\subsection{Reduction to a standard system}

Put $I=\{1,2,3,4\}$. It suffices to prove Theorem~\ref{thm:main} on a
standard space. For a fixed tuple of bounded functions, the map that
records all their integer translates takes values in a countable product
of compact discs. The pushforward measure with the shift is a standard
factor, the functions descend to its coordinates, and mixing passes
down; almost-sure convergence there pulls back. We therefore write
$(X,\mu,T)$ for a standard mixing system, all factors and conditional
expectations taken modulo null sets on completed spaces.

\subsection{The two companion inputs}

The first is the multiple-mixing theorem \cite[Theorem~1.1]{MixingCompanion}:
for every fixed finite $g_1,\dots,g_s\in L^\infty(\mu)$,
\begin{equation}\label{eq:all-orders}
 \int_X\prod_{j=1}^s g_j(T^{n_j}x)\dd\mu(x)\longrightarrow\prod_{j=1}^s\int_X g_j\dd\mu
 \quad\text{whenever}\quad\min_{j\ne l}|n_j-n_l|\to\infty.
\end{equation}
Simple-function approximation upgrades the set version of \eqref{eq:mixing}
to \eqref{eq:all-orders}; all uses are for fixed functions, no rate assumed.

The second is a pointwise theorem on systems that are neither mixing nor
ergodic; we isolate its exact scope and prove it in
Appendix~\ref{sec:lower-length}.

\begin{lemma}[Lower-length convergence]\label{lem:lower-length}
Let $(Y,m,R)$ be an invertible probability-preserving transformation of
a standard space. If $\varnothing\ne J\subsetneq I$ and $g_i\in L^\infty(m)$
for $i\in J$, then $\frac1N\sum_{n=1}^N\prod_{i\in J}g_i(R^{in}y)$
converges for $m$-almost every $y$.
\end{lemma}

\subsection{Distal factors and relative weak mixing}

Let $(Y,m,R)$ be a standard system and $B$ a factor. We use the
conditional-correlation form of \emph{relative weak mixing}: for
bounded $f,g$ with $\E[f\mid B]=0$,
\begin{equation}\label{eq:rwm}
 \frac1H\sum_{h=1}^H\bigl\|\E[\overline f\,(g\circ R^h)\mid B]\bigr\|_2^2\longrightarrow0.
\end{equation}
For $F_1,\dots,F_r\in L^2(m)$ the finitely generated
$L^\infty(B)$-module is $\{\sum_j a_jF_j:a_j\in L^\infty(B)\}$, called
\emph{invariant} if composition with $R$ and $R^{-1}$ preserves it.
An extension is \emph{relatively compact} if its $L^2$ is generated by
invariant modules (closure after spanning the union, not each module
separately): the corrected compact-module characterization
\cite[Theorem~4.1]{Jamneshan}\cite[Theorem~2.5]{JamneshanCorrigendum}.
A \emph{relatively distal} extension is a tower of relatively compact
extensions with inverse limits; distality with no base named means over
the trivial factor. The identity action is compact, so nonergodic
systems are included.

\begin{proposition}[Relative structure]\label{prop:structure}
For standard probability-preserving systems:
\begin{enumerate}
\item There is a maximal measure-distal factor $W_Y$ containing the
invariant factor, with $Y\to W_Y$ relatively weakly mixing. A distal
factor pulls back into the maximal distal factor of any extension, and
every measure-preserving automorphism commuting with $R$ preserves $W_Y$.
\item Factors and joinings of distal systems are distal; a distal system
is relatively distal over each factor; $R$ is distal iff $R^q$ is, for
each nonzero integer $q$.
\item Relative weak mixing passes to nonzero integer powers and to
conditionally independent base change, and the product of relatively
weakly mixing fibre laws over any invariant joining of their bases is
relatively weakly mixing over that joining.
\item A relatively weakly mixing and a relatively distal extension of
one base are relatively disjoint: the only invariant joining fixing the
base is the relatively independent one.
\item If $R$ is distal then for every $d\ge1$ and bounded $g_1,\dots,g_d$
the averages $\frac1N\sum_{n=1}^N\prod_{i=1}^dg_i(R^{in}y)$ converge
almost surely.
\end{enumerate}
For a flow $(S_Y^t)_{t\in\R}$, the maximal distal factor $W_Y$ of
$S_Y^1$ is a flow factor, and its action is distal at every nonzero
rational time.
\end{proposition}

\begin{proof}
Existence and the relatively weakly mixing extension are
Furstenberg--Zimmer \cite{Furstenberg,Zimmer}; the nonergodic
probability-algebra form is \cite[Theorems~1.1,1.2]{Jamneshan}. We
verify the consequences used later, tracking base and time.

\emph{Compact modules under base change and joining.} At a compact
stage an invariant finitely generated module stays finitely generated
after enlarging the base: truncate generators under their invariant
pointwise length when they transform by unitary matrices over the base.
Products of bounded generators lie in finitely generated tensor
modules, so joins of compact stages are compact over joined bases, and
the same for distal towers follows by closing at limit stages. Hence
distal factors of a system lie in its maximal distal factor (apply to
an automorphism commuting with $R$ and inverse to get preservation of
$W_Y$), and the invariant factor sits inside $W_Y$ since it is compact.

\emph{Time.} Compact $R$-modules are compact $R^q$-modules, so distality
passes to powers. Conversely if $R^q$ is distal its compact tower is
preserved by $R$; at each successor stage replace an $R^q$-invariant
finitely generated module by the sum of its first $q$ translates, a
finitely generated $R$-module over the same base, and the resulting
tower proves distality of $R$. Inversion handles negative powers.

\emph{Relative weak mixing under powers and base change.} The nonnegative
summands in \eqref{eq:rwm} show restriction to multiples of a positive
integer keeps Ces\`aro limit zero; invariance and conjugation handle
negative multiples. For joined bases $Y_j\to B_j$ with a base joining
$\sigma$, give the fibres the law $\int\bigotimes_jm_j(\dd y_j\mid b_j)\dd\sigma$.
For bounded tensor tests the conditional correlation over the full base
is $a(b)\,c(S_B^hb)\prod_j\E[\overline f_j(g_j\circ R_j^h)\mid B_j](b_j)$.
If some $f_j$ is conditionally centred the squared $L^2(\sigma)$ norm is
bounded by a constant times $\|\E[\overline f_j(g_j\circ R_j^h)\mid B_j]\|^2_{L^2(B_j)}$,
using only the $j$th marginal of $\sigma$, whose Ces\`aro mean vanishes.
Centred tensors span a dense subspace of the orthocomplement of the base
after subtracting the conditional expectation and telescoping, so
bounded approximation proves the claim; an extra base coordinate proves
stability under conditionally independent base change.

\emph{Relative disjointness.} Take a compact extension of the base. The
absolute weak-mixing disjointness principle is
Furstenberg's \cite[Part~I, Sec.~7]{FurstenbergDisjointness}. A joining
gives a base-linear, time-intertwining conditional expectation
from its $L^2$ to the relatively weakly mixing side, carrying finitely
generated invariant modules to such modules; relative weak mixing puts
the image in the base, so the joining is relatively independent. Induct
along a distal tower: after independence at a stage, the far extension
is a conditionally independent base change and the compact-stage
argument repeats; conditional expectations pass to limits. No ergodicity
was used. This also gives closure of distal systems under factors: if $D$
is distal with factor $B$, then $D$ is distal over the pullback of $W_B$,
and relative disjointness of $D\to W_B$ and $B\to W_B$ forces the latter
to be trivial.

\emph{Pointwise assertion.} The ergodic case is \cite[Theorem~C]{HuangShaoYe}
(measure-theoretic distality). For nonergodic $R$, start the compact
tower at the invariant factor and disintegrate. On a standard space the
tower is countable, since increasing successor stages give nonzero
orthogonal differences in separable $L^2$. At each stage pick a
countable generating algebra and approximate its indicators by functions
in the compact modules (successor stages) or preceding factors (limit
stages) with summable squared $L^2$ errors. Fubini makes these summable
in the conditional $L^2$ spaces on a common conull set of components, so
the restricted algebras are dense and the invariance identities
disintegrate; successor stages stay compact and limit stages stay
generated by predecessors on that set. Every intermediate factor shares
the invariant algebra, so each component tower is ergodic with isometric
compact extensions, hence an ergodic distal system. The cited theorem
applies on almost every component.

\emph{Flows.} Every flow time commutes with time one, hence preserves
$W_Y$. If $r=p/q\ne0$ is rational, the $q$th power of its action on
$W_Y$ is the distal time-$p$ action, so distality at time $r$ follows
from the power assertion.
\end{proof}

The following norm estimate explains why proper subaverages look simple
over the distal factor; it uses only relative weak mixing.

\begin{lemma}[Relatively weakly mixing characteristic factor]
\label{lem:relative-norm}
If $Y\to B$ is relatively weakly mixing for invertible $R$ and
$a_1,\dots,a_d$ are distinct nonzero integers with bounded
$g_1,\dots,g_d$, then
\begin{equation}\label{eq:relnorm}
 \Bigl\|\frac1N\sum_{n=1}^N\Bigl(\prod_{i=1}^d g_i\circ R^{a_in}
 -\prod_{i=1}^d\E[g_i\mid B]\circ R^{a_in}\Bigr)\Bigr\|_2\longrightarrow0.
\end{equation}
\end{lemma}

\begin{proof}
By telescoping and scaling, take the $g_i$ bounded by one with one, say
$g_s$, conditionally centred. Induct on $d$, empty product first. Set
$v_n=\prod_ig_i\circ R^{a_in}$ and, for fixed $h\ge1$,
$F_{i,h}=\overline{g_i}(g_i\circ R^{a_ih})$. Invariance gives
\[
 \langle v_n,v_{n+h}\rangle
 =\int F_{s,h}\prod_{i\ne s}F_{i,h}\circ R^{(a_i-a_s)n}\dd m .
\]
The other coefficients are distinct and nonzero; the induction
hypothesis replaces their factors, in the average over $n$, by their
conditional expectations over $B$ up to an error tending to zero. Those
are $B$-measurable and bounded by one, so
$\limsup_N|\frac1N\sum_n\langle v_n,v_{n+h}\rangle|\le\|\E[F_{s,h}\mid B]\|_1$
(and directly for $d=1$). The Hilbert-space van der Corput inequality
bounds the squared norm of the averages of $v_n$ by a constant times
$\frac1H+\frac1H\sum_{h\le H}\|\E[F_{s,h}\mid B]\|_1$ in the limit
superior as $N\to\infty$; the second term vanishes by \eqref{eq:rwm} for
$R^{a_s}$ and Cauchy--Schwarz. Let $H\to\infty$.
\end{proof}

Weak mixing of $T$ thus gives $L^2$ convergence to the product of the
means for all finite distinct nonzero linear powers. In a flow whose
base is preserved by every time, writing $t=n+s$, $0\le s<1$, applying
the lemma to $g_i\circ S_Y^{a_is}$ for each fixed $s$ and integrating in
$s$ (conditional expectation commuting with time shifts) gives the
continuous-time analogue.

\section{Selectors and selected laws}\label{sec:selected}

Failure of pointwise convergence lets the averaging length depend on the
initial state. We encode that in a law on four outputs. The selection may
distort their joint law, but Section~\ref{sec:structure} determines every
proper marginal, even after all distal parameters are fixed.

\subsection{From failure to a selector sequence}

Let $C=X\times[0,1)$ with $m_C=\mu\otimes\dd u$ and the unit-roof
suspension flow
\begin{equation}\label{eq:suspension}
 S_C^t(x,u)=\bigl(T^{\lfloor u+t\rfloor}x,\{u+t\}\bigr).
\end{equation}
Write $\xi:C\to X$ for the spatial coordinate and
$\upsilon:C\to\R/\Z$ for the phase. Working in a flow lets us normalize
the action at any one of the four outputs.

\begin{proposition}[Selectors witnessing failure]\label{prop:selectors}
If Theorem~\ref{thm:main} fails there are real $f_i\in L^\infty(\mu)$
with $\|f_i\|_\infty\le1$, $\int f_i\dd\mu=0$, a finite $F\subset[1,2]$,
measurable $N_a:C\to\{c2^k:c\in F,\ k\ge a\}$ ($a\ge1$), a continuous
real $\Psi$ on $(\R/\Z)^I$ and $\epsilon_0>0$ such that for all large $a$,
\begin{equation}\label{eq:selected-positive}
 \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 I}f_i\bigl(\xi(S_C^{it}z)\bigr)\dd t\dd m_C(z)\ge\epsilon_0.
\end{equation}
The selectors may be chosen independent of the initial phase.
\end{proposition}

\begin{proof}
Split each function into mean plus centred part. Lemma~\ref{lem:lower-length}
and the $L^2$ conclusion of Lemma~\ref{lem:relative-norm} show every proper
subaverage converges almost surely to the product of its means, so a
fourfold average of centred functions fails to converge to zero. Real and
imaginary parts and scaling give a real tuple bounded by one with the same
property; its averages $A_N(x)=\frac1N\sum_{n=1}^N\prod_{i\in I}f_i(T^{in}x)$
tend to zero in $L^2$. After flipping one sign there are $\delta>0$ and a
positive-measure set $E$ with $A_N(x)>2\delta$ for arbitrarily large $N$,
all $x\in E$.

Changing the length from $N$ to $M$ moves a bounded-summand average by at
most $2|N-M|/\max(N,M)$. Take $F\subset[1,2]$ fine enough that arbitrarily
late inequalities $A_N(x)>2\delta$ yield late lengths $L=c2^k$ with
$A_{\lfloor L\rfloor}(x)>\delta$; rounding error tends to zero uniformly
as $k\to\infty$. For $x\in E$ choose measurably one such length with
$k\ge a$, else set $N_a(x,u)=2^a$ (with $1\in F$). Countably many lengths
make the selectors measurable.

Take nonnegative continuous $\alpha,\beta$ on $\R/\Z$, not identically
zero, with supports in $(1/4,1/2)$ and $(0,1/16)$, and set
\[
 \Psi(v_1,v_2,v_3,v_4)=\alpha(2v_1-v_2)\,\beta(v_2-v_1).
\]
For output phases $v_i=u+it\pmod1$ this is $\alpha(u)\beta(\{t\})$, and on
its support $t=n+s$, $0\le s<1$ forces $0<u+is<1$ for every $i\in I$, so
the spatial outputs in \eqref{eq:selected-positive} are exactly $T^{in}x$.
With $b_\beta=\int_0^1\beta(s)\dd s$, integration over complete unit
intervals gives, uniformly in $x,u$, $L\ge2$,
\begin{equation}\label{eq:phase-selector-comparison}
 \frac1L\int_0^L\Psi\bigl((u+it)_{i\in I}\bigr)\prod_if_i\bigl(\xi(S_C^{it}(x,u))\bigr)\dd t
 =\alpha(u)b_\beta A_{\lfloor L\rfloor}(x)+O(L^{-1}),
\end{equation}
the shift of index range from $0..\lfloor L\rfloor-1$ to
$1..\lfloor L\rfloor$ costing another $O(L^{-1})$. Integrating the right
side over $E\times[0,1)$ is at least
$\delta\mu(E)b_\beta\int\alpha+o(1)$; off $E$ the selected length is
deterministic and $A_{2^a}\to0$ in $L^2$, so that part vanishes. Choose
$\epsilon_0$ below the positive product with $\delta\mu(E)$.
\end{proof}

For the rest of the proof assume failure, fix the functions, selectors and
phase test, and fix a free ultrafilter $\mathcal U$ on the indices $a$.

\subsection{A functorial law on four outputs}

Take a standard probability-preserving flow $(Y,m_Y,S_Y^t)$ with an
equivariant factor $\pi_Y:Y\to C$; the selector on $Y$ is $N_a\circ\pi_Y$.
For bounded $G_i$ on $Y$ define the selected law
\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}
Let $\omega_Y:Y\to W_Y$ be the maximal distal factor of time one, with
$\nu_Y=(\omega_Y)_*m_Y$ and $m_Y=\int m_{Y,w}\dd\nu_Y(w)$. For a state
$(y_i)_{i\in I}$ of the selected law put $P_i=\omega_Y(y_i)$ and $P=(P_i)_i$.

\begin{proposition}[Selected laws]\label{prop:selected}
Formula~\eqref{eq:selected-law} defines a probability measure $\lambda_Y$
on $Y^I$ with:
\begin{enumerate}
\item Every marginal is $m_Y$, and $\lambda_Y$ is invariant under
$(y_i)_i\mapsto(S_Y^{it}y_i)_i$ for all $t\in\R$. For an extension
$\theta:Y'\to Y$ over $C$, $(\theta^I)_*\lambda_{Y'}=\lambda_Y$, and the
same for countable inverse limits of extensions.
\item The law $\Lambda_Y$ of $P$ is the deterministic-length law: for
bounded $H_i$ on $W_Y$,
\begin{equation}\label{eq:ordinary-parameter-law}
 \int\prod_iH_i(w_i)\dd\Lambda_Y
 =\lim_{L\to\infty}\int_Y\frac1L\int_0^L\prod_iH_i(\omega_Y(S_Y^{it}y))\dd t\dd m_Y(y).
\end{equation}
\item For every proper $J\subsetneq I$, the conditional law of
$(y_i)_{i\in J}$ given the \emph{entire} parameter array $P$ is
$\bigotimes_{i\in J}m_{Y,P_i}$.
\end{enumerate}
In a countable inverse limit the stage laws are dense in $L^2$ of the limit.
\end{proposition}

\begin{proof}
Call the finite-$a$ law in \eqref{eq:selected-law} $\lambda_{Y,a}$. For a
bounded measurable $G$, the pointwise ergodic theorem for the flow (since
$N_a\ge2^a$) gives
$\frac1{N_a(\pi_Yy)}\int_0^{N_a(\pi_Yy)}G(S_Y^{it}y)\dd t\to\E[G\mid\mathcal I_Y](y)$
almost surely, $\mathcal I_Y$ the flow's invariant factor; integrating
gives convergence of each marginal to $m_Y$, no ergodicity needed.

Realize $Y$ as a Borel subset of a compact metrizable $K$. The ultralimit
of integrals of continuous functions on $K^I$ is a positive mass-one
functional, hence a probability measure; its marginals are $m_Y$, so it
lives on $Y^I$. Approximate bounded slot functions in $L^1(m_Y)$ by
continuous ones with the same bound; the product-test error is at most the
sum of the marginal-test errors, valid in the ultralimit and under the
limiting law. This gives \eqref{eq:selected-law} for all bounded tests.

The product action at time $s$ shifts $[0,N_a]$ to $[s,N_a+s]$, an error
$\le2|s|/2^a$ for unit-bounded tests, so $\lambda_Y$ is invariant (the
selector is still evaluated at the same initial state; no invariance of it
is needed). Pulling product tests back under an extension over $C$ leaves
the selector unchanged, giving functoriality; countable inverse limits are
standard flows with the stated projections, and finite products of stage
functions are dense, proving the last assertion. Strong continuity of the
product flow follows on tensors then by density.

Parameter and proper-slot laws. The pointwise theorem for the distal
$W_Y$ passes to continuous time: write $t=n+s$, $0\le s<1$; for fixed $s$
apply Proposition~\ref{prop:structure}(5) to $H_i\circ S_{W_Y}^{is}$.
Fubini plus bounded convergence give pointwise convergence after
integrating $s$, the last incomplete unit interval costing $O(L^{-1})$;
hence continuous averages converge almost surely, and their value at a
selected length $N_a(\pi_Yy)$ has the same limit. Integration gives
\eqref{eq:ordinary-parameter-law}.

For proper $J$, apply the same time decomposition to
Lemma~\ref{lem:lower-length} on $Y$ itself; the selected $J$-slot marginal
is the deterministic-length law. The continuous-time version of
Lemma~\ref{lem:relative-norm} over $W_Y$ replaces each slot test $G_i$ by
$\E[G_i\mid W_Y]$, and we may multiply each by an arbitrary bounded
function of its parameter, so given $(P_i)_{i\in J}$ the marginal has the
conditional product law $\bigotimes_{i\in J}m_{Y,P_i}$. To condition on
the parameters outside $J$: under the time-one action with speeds $i$ this
law is a relatively weakly mixing extension of its parameter subarray
(each $Y\to W_Y$ is relatively weakly mixing for that power, and
Proposition~\ref{prop:structure}(3) applies over the joining of base laws).
The full parameter array is a joining of the distal systems
$(W_Y,S^i_{W_Y})$, hence distal and relatively distal over the subarray.
The joint law of the $J$ states and all parameters is an invariant joining
of these two extensions over the common base; relative disjointness makes
it relatively independent, which is exactly the product fibre law.
\end{proof}

For $b\in I$ define the normalized flow
\begin{equation}\label{eq:normalized-selected-flow}
 \mathsf M_b(Y)=\bigl(Y^I,\ \lambda_Y,\ (y_i)_i\mapsto(S_Y^{it/b}y_i)_i\bigr).
\end{equation}
Its projection to slot $b$ is an equivariant extension of $Y$, hence of
$C$, so the selected-law construction can be iterated with the same
selectors and ultrafilter.

\begin{lemma}[Distal parameters after normalization]\label{lem:parameter-distal}
The full parameter array $(\omega_Y(y_i))_{i\in I}$ is a distal factor of
the time-one map of $\mathsf M_b(Y)$, hence lies in its maximal distal
factor. The same holds for the arrays of original parameters at every
finite iterate of normalized selected flows.
\end{lemma}

\begin{proof}
In slot $i$ the parameter action is $S_{W_Y}^{i/b}$, distal by
Proposition~\ref{prop:structure}; a joining of distal systems is distal.
At the next iteration the old distal factors pull into the new maximal
distal factors; induction.
\end{proof}

The phase test in Proposition~\ref{prop:selectors} is a uniform limit of
finite sums of products of one-slot phase functions, so
\eqref{eq:selected-law} applies to its product with the four spatial
tests. In particular
\begin{equation}\label{eq:initial-selected-witness}
 \int_{C^I}\Psi\bigl((\upsilon(z_i))_i\bigr)\prod_if_i(\xi(z_i))\dd\lambda_C\ge\epsilon_0.
\end{equation}
The phase array is part of the distal parameter array (phase rotation is
compact); \eqref{eq:initial-selected-witness} is the nonzero centred
correlation from which Section~\ref{sec:activation} starts.

\section{Tensor packing into a small subspace}\label{sec:packing}

Selecting an averaging length changes the joint law of the averaged
variables. The deterministic Hilbert-space estimate here controls that
change. In the application two independent groups supply the two $L^2$
spaces whose tensor product we test. The estimate is uniform over the
projection: a subspace of slowly growing rank captures a substantial part
of only a small fraction of an almost-orthogonal tensor family.
Hilbert spaces below are real, tensor products are Hilbert tensor
products, and $\log$ is natural.

\begin{theorem}[Tensor packing]\label{thm:packing}
There is a universal $C$ with the following property. Let
$0<\varepsilon\le1$ and integers $d\ge1$, $L\ge0$. Let $x_1,\dots,x_m$,
$y_1,\dots,y_m$ lie in the unit balls of real Hilbert spaces $H_1,H_2$.
For each family, a graph on $\{1,\dots,m\}$ of maximum degree $\le L$
contains every off-diagonal pair whose absolute inner product exceeds
$\eta$, where $0\le\eta\le\varepsilon^2/10^4$; the two graphs may
differ. Then for every orthogonal projection $P$ on $H_1\otimes H_2$ of
rank $\le d$,
\begin{equation}\label{eq:packing-quantitative}
 \#\{n:\norm{P(x_n\otimes y_n)}\ge\varepsilon\}
 \le\frac{C(1+L)}{\varepsilon^2}
 \exp\!\bigl(C\varepsilon^{-2}\sqrt d\log(2+d)\bigr).
\end{equation}
In particular, with $d_m=\lceil(1+\log m)^{3/2}\rceil$ the left side is
$o_{\varepsilon,L}(m)$, uniformly over Hilbert spaces, vectors,
projections and allowed $\eta$.
\end{theorem}

The proof separates the directions where the test space has large
variance, then applies a Gaussian comparison to the rest (the finite
Sudakov--Fernique inequality \cite[Sec.~1,~(1.2)]{Fernique},
\cite{Sudakov}). We first record the matrix estimate, via the
Gaussian integration-by-parts moment argument behind matrix Khintchine
bounds \cite[Lemma~6.1, Prop.~7.3]{Tropp}, tracking the covariance trace
rather than ambient dimensions.

\begin{lemma}[Gaussian matrix estimate]\label{lem:packing-gaussian}
Let $B_1,\dots,B_r$ be matrices between finite-dimensional real Hilbert
spaces and
\[
 H_j=\begin{pmatrix}0&B_j\\ B_j^*&0\end{pmatrix},\qquad
 S=\sum_{j=1}^r H_j^2 .
\]
If $\norm{S}_{\rm op}\le\sqrt d$ and $\tr S\le2d$ for some $d\ge1$, then
for independent standard normals $g_j$,
\begin{equation}\label{eq:packing-gaussian-bound}
 \E\Bigl\|\sum_{j=1}^r g_jB_j\Bigr\|_{\rm op}\le Cd^{1/4}\sqrt{\log(2+d)},
\end{equation}
with $C$ independent of the dimensions.
\end{lemma}

\begin{proof}
Write $H=\sum_jg_jH_j$. Gaussian integration by parts and differentiation
of the matrix power give, for $q\ge2$,
\[
 \E\tr H^{2q}=\sum_{j=1}^r\sum_{a=0}^{2q-2}\E\tr\bigl(H_jH^aH_jH^{2q-2-a}\bigr).
\]
For self-adjoint $H$ with eigenvalues $\lambda_u$ and self-adjoint $K$,
each summand is at most
$\sum_{u,v}|K_{uv}|^2|\lambda_v|^a|\lambda_u|^{2q-2-a}\le\tr(K^2|H|^{2q-2})$
(weighted arithmetic--geometric mean plus symmetry $|K_{uv}|=|K_{vu}|$).
Summing in $j$:
\begin{equation}\label{eq:packing-moment-recursion}
 \E\tr H^{2q}\le(2q-1)\norm S_{\rm op}\,\E\tr|H|^{2q-2}.
\end{equation}
Since $\E\tr H^2=\tr S$, iteration gives
$\E\tr H^{2q}\le(2q-1)!!\,\norm S_{\rm op}^{q-1}\tr S$. The operator norm
of $H$ equals that of $\sum_jg_jB_j$; a $2q$-th root and the hypotheses
bound its expectation by $C\sqrt q\,d^{1/4}(2\sqrt d)^{1/(2q)}$. Take an
integer $q\ge2$ comparable to $\log(2+d)$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:packing}]
We use the elementary fact that if unit-ball vectors $v_n$ have inner
products bounded in absolute value by $\eta$ outside a graph of degree
$L$, then for any nonempty $A$,
\begin{equation}\label{eq:packing-cluster}
 \Bigl\|\frac1{|A|}\sum_{n\in A}v_n\Bigr\|^2\le\frac{1+L}{|A|}+\eta,
\end{equation}
as there are at most $(1+L)|A|$ ordered diagonal or exceptional pairs in
the squared-norm expansion. So a lower bound on the norm of such an
average bounds $|A|$.

Work in the finite spans of the $x_n$, $y_n$: project $P$'s range by $Q$
onto their tensor product; for $z$ in that product the projection norm
onto the new subspace is at least $\norm{Pz}$ (replace a unit test vector
by its $Q$-image), and the rank does not increase.

Identify tensors with matrices $H_2\to H_1$ and let $A_1,\dots,A_r$
($r\le d$) be a Hilbert--Schmidt orthonormal basis of the test space. Then
$R_1=\sum_jA_jA_j^*$, $R_2=\sum_jA_j^*A_j$ have trace $r$; let $E_1,E_2$
be their spectral subspaces for eigenvalues $>\sqrt d$, each of dimension
$\le\sqrt d$. Discard indices where the projection of $x_n$ onto $E_1$
exceeds $\varepsilon/10$, likewise $y_n$ onto $E_2$. Cover either unit
ball by at most $(1+80/\varepsilon)^{\sqrt d}$ balls of radius
$\varepsilon/40$; if a ball contains $s$ discarded projections their
average has norm $\ge\varepsilon/20$, and \eqref{eq:packing-cluster}
(with $\eta\le\varepsilon^2/10^4$) gives $s\le C(1+L)\varepsilon^{-2}$. So
discarded indices number at most
\begin{equation}\label{eq:packing-discarded}
 C(1+L)\varepsilon^{-2}\exp\bigl(\sqrt d\log(1+80/\varepsilon)\bigr).
\end{equation}

Let $Q_1,Q_2$ project onto $E_1^\perp,E_2^\perp$ and $B_j=Q_1A_jQ_2$. The
coefficient map $z\mapsto(\ip{A_j}{z}_{\rm HS})_j$ is contractive, and for
a surviving index
$\norm{x_n\otimes y_n-Q_1x_n\otimes Q_2y_n}\le\norm{(1-Q_1)x_n}+\norm{(1-Q_2)y_n}\le\varepsilon/5$.
Hence every remaining counted index has coefficient vector
$v_n=(\ip{x_n}{B_jy_n})_j$ with $\norm{v_n}\ge\varepsilon/2$. Outside the
union of the two exceptional graphs the tensor inner products have
absolute value $\le\eta^2$, and that union has degree $\le2L$. A set of
$v_n$ in a ball of radius $\varepsilon/8$ has average norm $\ge3\varepsilon/8$,
so \eqref{eq:packing-cluster} on their preimage tensors bounds it by
$C(1+L)\varepsilon^{-2}$; a maximal $\varepsilon/8$-separated subset of
cardinality $l$ leaves at most $C(1+L)\varepsilon^{-2}l$ uncounted.

Bound $l$: for $G=\sum_jg_jB_j$ the centred Gaussians $Z_n=\ip{x_n}{Gy_n}$
have increment variances $\norm{v_n-v_{n'}}^2$, and Gaussian comparison
gives for $l\ge2$
\begin{equation}\label{eq:packing-comparison}
 c\varepsilon\sqrt{\log l}\le\E\max_nZ_n\le\E\norm G_{\rm op}.
\end{equation}
(Compare two finite centred Gaussian families by applying integration by
parts to $\beta^{-1}\log\sum_ne^{\beta z_n}$ along an independent
interpolation; the derivative
$\frac\beta4\E\sum_{n,n'}p_np_{n'}(\E|Z_n-Z_{n'}|^2-\E|Z_n'-Z_{n'}'|^2)$,
$p_n=e^{\beta z_n}/\sum_je^{\beta z_j}$, is nonnegative when the first
increment variances dominate. Let $\beta\to\infty$.) In
\eqref{eq:packing-comparison} take $Z_n'$ independent normals of variance
$\varepsilon^2/128$ and use $\E\max_{n\le l}g_n\ge c\sqrt{\log l}$.

Compression gives $\sum_jB_jB_j^*\le Q_1R_1Q_1$ and
$\sum_jB_j^*B_j\le Q_2R_2Q_2$, with norms at most $\sqrt d$ and traces at
most $d$, so Lemma~\ref{lem:packing-gaussian} applied to
\eqref{eq:packing-comparison} yields
$\log l\le C\varepsilon^{-2}\sqrt d\log(2+d)$; the case $l\le1$ obeys it
too. Combining with \eqref{eq:packing-discarded} and
$\log(1+80/\varepsilon)\le C\varepsilon^{-2}$ proves
\eqref{eq:packing-quantitative}. For $d=d_m$ the exponent is
$O_\varepsilon((\log m)^{3/4}\log\log m)=o(\log m)$.
\end{proof}

We also use the averaged form. Since projection norms are at most one, for
any fixed $\varepsilon$, vectors obeying the small-pair hypothesis, and
rank-$\le d_m$ projection $P$,
\begin{equation}\label{eq:packing-average}
 \frac1m\sum_{n=1}^m\norm{P(x_n\otimes y_n)}
 \le\varepsilon+\frac{C(1+L)}{m\varepsilon^2}
 \exp\!\bigl(C\varepsilon^{-2}\sqrt{d_m}\log(2+d_m)\bigr).
\end{equation}
Here $\varepsilon$ and then $\eta$ are fixed before $m$ grows; the
exceptional degree may depend on the chosen $\eta$. In
Section~\ref{sec:channels}, mixing supplies exactly this order of
quantifiers; no uniform mixing rate is needed.

\section{Channels and preservation of independence}\label{sec:channels}

Selected laws can retain correlations that ordinary averaging destroys.
We specify extensions carrying independent outputs and show the class
survives passage to a selected law. There are three groups of outputs.
Every proper collection of groups is independent of the original input,
whereas the full triple may be correlated with it. That gap is what lets
us select lengths and still keep the correlations the final argument uses.

Write a point of the suspension $C$ as $(x,u)$, $x\in X$, $u\in[0,1)$.
A factor of a flow $Y$ with values in $C$ and speed $q>0$ is a
measure-preserving $D:Y\to C$ with $D(S_Y^ty)=S_C^{qt}D(y)$ for all $t$,
modulo null sets.

\begin{definition}\label{def:channel}
A \emph{channel} is a flow extension $\pi_Y:(Y,m_Y,S_Y^t)\to C$ together
with finitely many output factors $D_\alpha:Y\to C$ of positive rational
speeds $q_\alpha$, partitioned into groups $\mathcal D_1,\mathcal D_2,
\mathcal D_3$ (empty allowed). Let $x_0$ be the spatial coordinate of
$\pi_Y$, $x_\alpha$ that of $D_\alpha$, and $V$ the phases of $\pi_Y$ and
all $D_\alpha$. Require
\begin{align}
 \mathcal L((x_\alpha)_\alpha\mid V)&=\bigotimes_\alpha\mu,
 \label{eq:channel-all}\\
 \mathcal L\bigl(x_0,(x_\alpha)_{\alpha\in\cup_{j\in J}\mathcal D_j}\mid V\bigr)
 &=\mu\otimes\!\!\bigotimes_{\alpha\in\cup_{j\in J}\mathcal D_j}\!\mu
 \quad(J\subsetneq\{1,2,3\}).
 \label{eq:channel-proper}
\end{align}
The right sides do not depend on $V$.
\end{definition}

So $x_0$ is independent of every proper collection of groups even after
the phases are given; independence from all three jointly is not required.
The flow $C$ with no outputs is a channel.

\begin{lemma}\label{lem:channel-distal}
For a channel $Y$, both identities \eqref{eq:channel-all}--\eqref{eq:channel-proper}
also hold conditioning on the maximal measure-distal factor $W_Y$. Every
flow extension of $Y$, with pulled-back original factor and outputs, is a
channel obeying the same over its own maximal distal factor.
\end{lemma}

\begin{proof}
The phase factor $V$ is a factor of a finite product of circle rotations,
hence sits in $W_Y$. Let $h$ clear the output speed denominators; at
time $h$ the phases are fixed, and each spatial tuple on the right of
\eqref{eq:channel-all} or \eqref{eq:channel-proper} has product conditional
law and evolves under a product of positive integer powers of $T$, which
is mixing. So the tuple plus $V$ is a relatively weakly mixing extension
of the fixed phase factor. The factor $W_Y$ is distal for time $h$ and
relatively distal over $V$, so relative disjointness
(Proposition~\ref{prop:structure}) makes the joining of that tuple with
$W_Y$ over $V$ the conditionally independent one, i.e. the claimed product
law. Extensions do not change the joint distribution defining a channel;
apply the same to its (possibly larger) maximal distal factor.
\end{proof}

Recall $\lambda_Y$ on $Y^I$, and $\mathsf M_b(Y)$ equips it with the flow
of speed $i/b$ in slot $i$; projection to slot $b$ is an extension of $Y$.

\begin{proposition}[Closure under selected laws]\label{prop:channel-closure}
Let $Y$ be a channel and $b\in I$. On $\mathsf M_b(Y)$ use $\pi_Y(y_b)$ as
original factor. For $j\in\{1,2,3\}$ let new group $j$ contain the outputs
$D_\alpha(y_i)$, $\alpha\in\mathcal D_j$, $i\in I$. Then $\mathsf M_b(Y)$
is a channel.
\end{proposition}

The new output speeds $iq_\alpha/b$ are positive rationals; the content is
independence. A proper collection of groups preserves independence from the
input after conditioning on the selected length; for all three groups the
selected law may change, and tensor packing controls the effect.

\begin{proof}
Test on products of bounded spatial functions and bounded continuous phase
functions; products of separate phase tests determine the phase law and fit
the slot-test definition of $\lambda_Y$. After real/imaginary parts,
expanding each spatial test into mean plus centred part, and scaling, every
spatial factor is $1$ or a real centred function bounded by one, and it
suffices to show the tested integral vanishes whenever a centred factor
occurs.

Work first under $m_Y$, conditionally on the phase vector $V$. Choose $h$
with $hq_\alpha$ integral for every output and split time as
\begin{equation}\label{eq:channel-time-split}
 t=hn+s,\qquad 0\le s<h.
\end{equation}
For fixed $V,s$ each phase test is independent of $n$; a spatial factor
from output $\alpha$ in slot $i$ is evaluated at
\begin{equation}\label{eq:channel-affine-times}
 T^{e_\alpha in+a_{\alpha,i}}x_\alpha,\qquad e_\alpha=hq_\alpha\in\N,
\end{equation}
the offsets $a_{\alpha,i}$ ranging over fixed finite sets as $V,s$ vary;
the original coordinate in slot $b$ is at $T^{hbn+a_0}x_0$.

Replace a selected length $N=c2^k$ by the union of its
$m=\lfloor N/h\rfloor$ complete intervals of length $h$, normalized by $m$;
the error for a unit-bounded test is $\le2h/N\to0$ uniformly in the
selected length as $a\to\infty$. So integrate $s$ uniformly over $[0,h)$
and study averages of $m$ terms; discarding finitely many initial terms
costs the same.

\emph{Proper collections of groups.} Under \eqref{eq:channel-proper} the
spatial variables in these groups are independent of $x_0$ and each other
conditionally on $V$, and $N_a$ is a function of $(x_0,u_0)$ with $u_0$
fixed by $V$. If some output factor is centred, the output-product
expectation tends to zero as $n\to\infty$: factor across the independent
outputs, then use mixing of all orders (\cite[Thm.~1.1]{MixingCompanion})
for the distinct slot times within each output; the gaps in
\eqref{eq:channel-affine-times} diverge within each output and there are
finitely many offset patterns, so convergence is uniform in $V,s$. With
$R_n(V,s)$ that expectation and $n_0$ so that $|R_n|\le\delta$ for
$n\ge n_0$, conditioning on $x_0,V,s$ bounds the output-averaged tested
sum by $\frac1{m(x_0,V)}\sum_{n<m(x_0,V)}|R_n(V,s)|\le\delta+n_0/\lfloor2^a/h\rfloor$. Integrating and then $a\to\infty$,
$\delta\downarrow0$ give zero. If all outputs equal $1$ and the
original-coordinate test is centred, Birkhoff for $T^{hb}$ gives zero
almost-sure limit simultaneously for finitely many offsets, and the
selected $m\to\infty$, so bounded convergence finishes. This is the
original-coordinate channel rule.

\emph{All three groups.} No original-coordinate test. Terms where one group
has no centred factor were just handled. Suppose every group has a centred
factor. For fixed $V,s$ let $\Omega_j$ be the product of copies of
$(X,\mu)$ for group $j$'s input outputs and write the three group products
$x_n\in L^2(\Omega_1)$, $y_n\in L^2(\Omega_2)$, $w_n\in L^2(\Omega_3)$,
all of norm $\le1$; before a length event their conditional law given $V$
is a product. Phase weights stay outside these vectors (the spatial
functions are fixed; $V,s$ enter only through finitely many integer
offsets in \eqref{eq:channel-affine-times}).

For fixed $\eta>0$ the families $x_n,y_n$ satisfy the small-pair hypothesis
of Theorem~\ref{thm:packing} after removing finitely many initial indices:
expand a pair inner product over the independent outputs, one output has a
centred factor, and mixing of all orders makes its expectation small when
the distinct times from the two indices are far apart (true for large
$n,n'$ within either index). The only obstruction forces
$|e_\alpha in-e_\alpha i'n'+a|\le R$ for finitely many $i,i',a$ and fixed
$R$; for each $n$ there are boundedly many such $n'$ since $e_\alpha i'>0$,
so the obstructors form a bounded-degree graph. Constants and cutoff depend
on $\eta$ and the fixed tests, uniform in $V,s$ by finite offsets.

Condition on $E_{k,c}=\{N_a=c2^k\}$, conditional probability
$p=p_{k,c}>0$ (repeated values assigned one index so the events partition
the space); $E_{k,c}$ is determined by $x_0$ once $V$ is fixed. By
\eqref{eq:channel-proper} conditioning on it preserves the product marginal
on any two groups and the marginal on each single group; the triple law may
not stay a product. Its density $\rho$ against the original triple product
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}
since for any set of group coordinates its probability jointly with
$E_{k,c}$ is at most its original product probability.

For the $m$ terms of this length put $z_n=x_n\otimes y_n$. Choose the
dimension so the packing bound is $o(m)$ and its reciprocal sums. The
choice $d_m=\lceil(1+\log m)^{3/2}\rceil$ has both (for $m\asymp2^k$ its
reciprocal is $O((1+k)^{-3/2})$). Let $P_m$ project onto $d_m$ eigenvectors
of $\sum_nz_nz_n^*$ of largest eigenvalue (multiplicity counted; whole
range if smaller; fixed choice within ties). Fix $0<\varepsilon\le1$ and
choose $0<\eta\le\varepsilon^2/10^4$. Equation~\eqref{eq:packing-average}
gives
\begin{equation}\label{eq:channel-leading-average}
 \frac1m\sum_n\norm{P_mz_n}\le\varepsilon+o_{\varepsilon,L}(1).
\end{equation}
If a cutoff $n_0$ was needed, apply \eqref{eq:packing-quantitative} to the
remaining $m-n_0$ vectors with rank $d_m$: the numerator is still
$m^{o(1)}$, the discarded indices cost at most $n_0/m$; norms are for the
original product measures.

Under the event law the first two groups keep their product marginal and
the third its marginal, so Cauchy--Schwarz bounds a term
$(P_mz_n)w_n$ (including event probability) by $p\norm{P_mz_n}$; averaged
this is $p$ times the right side of \eqref{eq:channel-leading-average}.

Write $z_n'=(1-P_m)z_n$. The covariance trace is $\le m$, so its
$(d_m+1)$-st eigenvalue is $\le m/(d_m+1)$; the residual Gram matrix
$A=(\ip{z_n'}{z_{n'}'})_{n,n'}$ has norm $\le m/d_m$, while
$B=(\ip{w_n}{w_{n'}})$ is positive with trace $\le m$. Hence
\begin{equation}\label{eq:channel-residual}
 \Bigl\|\frac1m\sum_nz_n'\otimes w_n\Bigr\|_2^2=\frac{\tr(AB)}{m^2}\le\frac1{d_m}.
\end{equation}
Combining \eqref{eq:channel-density} and \eqref{eq:channel-residual} bounds
this event's residual contribution by $\sqrt{p/d_m}$.

Sum over all lengths. For fixed $h$, $m=\lfloor c2^k/h\rfloor\asymp2^k$ at
large $k$, uniformly in $c\in F$, so $d_m\asymp(1+k)^{3/2}$ and
\begin{equation}\label{eq:channel-tail}
 \sum_{k\ge a,\,c\in F}\sqrt{p_{k,c}/d_{m(k,c)}}
 \le\Bigl(\sum_{k,c}p_{k,c}\Bigr)^{1/2}\Bigl(\sum_{k\ge a,\,c\in F}d_{m(k,c)}^{-1}\Bigr)^{1/2}\to0,
\end{equation}
a Hilbert--Schwarz split with a convergent series tail. The leading-space
contributions sum to at most $\varepsilon+o_{\varepsilon,L}(1)$ since the
$p_{k,c}$ sum to one. Let $a\to\infty$, then $\varepsilon\downarrow0$. This
proves the all-group product rule.

All estimates were uniform in the finitely many offset patterns; for each
pattern and integer $m$ the finite-dimensional projections are chosen once,
so no measurability issue arises. Integrate in $V,s$ and restore the phase
test; the uniformly vanishing endpieces do not affect the selected law.
\end{proof}

\begin{corollary}\label{cor:channel-array-distal}
In Proposition~\ref{prop:channel-closure} let $P=(\omega_Y(y_i))_{i\in I}$
be the old distal-parameter array. The spatial coordinates of all copied
outputs are independent with product law conditionally on $P$. The same
holds after any finite iterate, conditionally on the complete array of old
$W_Y$-parameters.
\end{corollary}

\begin{proof}
Lemma~\ref{lem:parameter-distal} puts the old parameter array in the maximal
distal factor of the new flow, at every finite iterate. Repeated
Proposition~\ref{prop:channel-closure} supplies the channel structure;
Lemma~\ref{lem:channel-distal} gives the conditional product law over that
maximal distal factor, hence over the old parameter array.
\end{proof}

The channel and its tests are fixed before the selected limit is taken; the
packing estimate and the length summation need no uniform mixing over
different channels or iteration depths.

\section{Activating an additional output role}\label{sec:activation}

Channels supply independent spatial outputs even after the length is
selected. We use that to kill the nonzero correlation of
Proposition~\ref{prop:selectors}. The argument counts how many of the four
factors can be read from channel outputs; it bounds that count, then passes
to an extension stabilizing certain conditional expectations, and finally
manufactures one more output role to contradict maximality.

Fix the selectors and the assignment $Y\mapsto\lambda_Y$. For a channel
$Y$ write $d_Y(y)$ for its full spatial output list and $x_Y(y)$ for the
spatial coordinate of $\pi_Y(y)\in C$; a length-$m$ list has product law
$\mu^{\otimes m}$, with the one-point space for $m=0$. Functions are real
and bounded.

\subsection{Correlations and an upper bound}

For $A\subseteq I$ consider a correlation
\begin{equation}\label{eq:role-correlation}
 \int\Phi(P)\prod_{i\in A}h_i(d_Y(y_i))\prod_{i\notin A}h_i(x_Y(y_i))\dd\lambda_Y,
 \qquad P=(\omega_Y(y_i))_{i\in I},
\end{equation}
where $\Phi$ is a bounded measurable function of the full distal-parameter
array and each $h_i$ is centred for its spatial law ($\mu^{\otimes m}$ if
$i\in A$, else $\mu$). Call $A$ the \emph{output roles}; a role function
may depend jointly on the whole list. Lemma~\ref{lem:channel-distal} makes
every factor also centred conditionally on its own parameter $P_i$.

The output-free channel is allowed; the phase test of
Proposition~\ref{prop:selectors} is measurable in the suspension's distal
parameters, so that proposition gives a nonzero correlation with
$A=\varnothing$. Hence, under failure, a largest number $r\in\{0,\dots,4\}$
of output roles occurs in a nonzero correlation over all channels and tests.

\begin{proposition}\label{prop:role-bound}
Every nonzero correlation \eqref{eq:role-correlation} has at most two
output roles; in particular $r\le2$.
\end{proposition}

\begin{proof}
Apply Proposition~\ref{prop:channel-closure} with any slot $b$: in
$\mathsf M_b(Y)$ all copied outputs from the four slots have product law,
even conditionally on the old parameter array $P$
(Corollary~\ref{cor:channel-array-distal}), so \eqref{eq:role-correlation}
vanishes when $|A|=4$.

Suppose $|A|=3$ and let $b$ be the missing slot. Start $Y_0=Y$,
$Y_{n+1}=\mathsf M_b(Y_n)$. The state of $Y_n$ is an array of $Y$-states
indexed by $I^n$; the slot-$b$ projection at each new axis respects the
flows since that slot has speed $b/b=1$. The $n$ coordinate lines through
the corner $(b,\dots,b)$ each have law $\lambda_Y$: fixing all but one
coordinate at $b$, projection along a fixed-at-$b$ coordinate is an
equivariant factor map, and functoriality gives the newly added coordinate
law $\lambda_Y$ while the earlier coordinate gives the preceding array law;
induct. All output lists are independent conditionally on the full array of
old $W_Y$-parameters (Corollary~\ref{cor:channel-array-distal}). Write
$(y_i^{(a)})_{i\in I}$ for line $a$, $P^{(a)}$ its parameter array, and
\[
 H_a=\Phi(P^{(a)})\prod_{i\ne b}h_i(d_Y(y_i^{(a)})).
\]
Lines differ in disjoint output lists (neither uses the corner), so
conditional independence and centring give $\E[H_aH_{a'}]=0$ for $a\ne a'$,
with $\norm{H_a}_2\le K$ (fixed tests). If
$H=h_b(x_Y(\text{corner}))$ and $c\ne0$ the original correlation, the line
marginals give $\ip{H}{H_a}=c$ for every $a$, so Bessel yields
$n|c|^2\le K^2\norm H_2^2$ for all $n$: impossible.
\end{proof}

\subsection{Stabilizing conditional expectations}

The next extension prevents a later system from altering the conditional
laws we use; it is an energy-increment construction of sated-extension type
\cite[Thm.~3.11, Lemma~3.12]{Austin}, adapted to the selected-law
assignment. On $(Y^I,\lambda_Y)$ set
\begin{equation}\label{eq:saturation-data}
 \mathcal D_0(Y)=\sigma(P),\qquad
 \mathcal D_l(Y)=\sigma\bigl(P,(y_i)_{i\ne l}\bigr),\quad l\in I.
\end{equation}
For an extension $\rho:Y'\to Y$ over $C$, functoriality gives
$\rho^I:(Y'^I,\lambda_{Y'})\to(Y^I,\lambda_Y)$, and the pullback of
$\mathcal D_l(Y)$ lies in $\mathcal D_l(Y')$ since the old distal factor is
contained in the new one.

\begin{proposition}\label{prop:saturation}
Every standard flow over $C$ has a standard extension $Y$ such that for
every further standard extension $\rho:Y'\to Y$ over $C$, every
$f\in L^2(\lambda_Y)$ and $l\in\{0\}\cup I$,
\begin{equation}\label{eq:saturation-identity}
 \E_{\lambda_{Y'}}[f\circ\rho^I\mid\mathcal D_l(Y')]
 =\E_{\lambda_Y}[f\mid\mathcal D_l(Y)]\circ\rho^I.
\end{equation}
If the initial flow is a channel, the extension is a channel with the same
outputs.
\end{proposition}

\begin{proof}
For $f$ and datum type $l$ call $\|\E[f\mid\mathcal D_l]\|_2^2$ its
energy; it increases under extension and is $\le\norm f_2^2$. Starting
from $Y_0$, build $Y_0\leftarrow Y_1\leftarrow\cdots$. At each stage add a
countable dense subset of the new selected joining's $L^2$ and keep
previous tests by pullback; schedule every retained test and datum type
infinitely often with tolerances tending to zero, and at a scheduled step
take an extension whose energy is within tolerance of the supremum over
all standard extensions (which exists as a supremum). Let $Y$ be the
countable inverse limit. By Proposition~\ref{prop:selected} its selected
law projects to all stage laws, and the stage $L^2$ union is dense in
$L^2(\lambda_Y)$ since the fourfold inverse limit's $\sigma$-algebra is
generated by its stage coordinate factors.

Fix a retained $f$, a datum type $l$, an extension $Y'$ of $Y$. If stage
$n+1$ optimized this pair with error $\epsilon_n$, then $Y'$ is available
over stage $n$, so
$\operatorname{energy}_{Y'}(f)\le\sup_{R\to Y_n}\operatorname{energy}_R(f)\le\operatorname{energy}_{Y_{n+1}}(f)+\epsilon_n\le\operatorname{energy}_Y(f)+\epsilon_n$;
along the scheduled subsequence $\epsilon_n\to0$, and monotonicity gives
the opposite inequality, so the energies agree. The old conditional-
expectation subspace sits inside the new one, so equal projection norms
imply equal projections: \eqref{eq:saturation-identity} for retained $f$.
Density, conditional expectation being an $L^2$ contraction, finishes. This
uses only inclusion of datum spaces under extension, not any inverse-limit
identity for maximal distal factors. Lemma~\ref{lem:channel-distal} keeps
every stage and the limit a channel.
\end{proof}

Call a system satisfying Proposition~\ref{prop:saturation} \emph{saturated}.
A nonzero correlation persists under passage to a saturated extension: pull
back its functions; the old parameter test stays measurable in the enlarged
distal parameters.

\subsection{A channel carrying the three other factors}

Take a nonzero correlation with the maximal $r$ output roles and replace
its channel by a saturated extension $Y$; keep $A,\Phi,h_i$.
Proposition~\ref{prop:role-bound} gives $r\le2$, so pick $b\in I\setminus A$
and set $Z=\mathsf M_b(Y)$ with $Z\to Y$, $(y_j)_j\mapsto y_b$; its state
law is $\lambda_Y$. Give $Z$ a new channel structure adapted to the
correlation: keep the full old output list from slot $b$, and for every
$j\ne b$ add
\begin{equation}\label{eq:added-output-lists}
 Q_j=\begin{cases}d_Y(y_j),&j\in A,\\ x_Y(y_j),&j\notin A,\end{cases}
\end{equation}
the second case adding that slot's copy of $C$ as flow factor. Index the
three new groups by $j\in I\setminus\{b\}$, put $Q_j$ in group $j$, and
assign the three old groups of slot $b$ bijectively to the new groups; all
speeds are positive rationals.

\begin{lemma}\label{lem:regrouped-channel}
With these outputs and groups, $Z$ is a channel over its slot-$b$ copy of $C$.
\end{lemma}

\begin{proof}
Condition first on the whole old parameter array $P$ in $\lambda_Y$. The
full output collection is one list per slot: $Q_j$ at $j\ne b$ and
$d_Y(y_b)$ at $b$. Every proper collection has product spatial law by
Proposition~\ref{prop:selected} and Lemma~\ref{lem:channel-distal}. For the
full law, test by a product of one bounded spatial function of each list and
expand each into mean plus centred part; every term except the all-centred
one is fixed by a proper marginal. A nonzero all-centred conditional term
would, against some bounded test of $P$, be a correlation
\eqref{eq:role-correlation} on $Y$ with roles $A\cup\{b\}$, contradicting
maximality. Now take a proper subset of the three new groups with the
original coordinate $x_Y(y_b)$: some slot $j\ne b$ is absent, and at $b$
only a proper subset of the old groups is kept; the proper-slot
independence plus the old channel rule at $b$ gives the product law given
$P$. All new phases are functions of $P$; integrating the two product laws
down to them gives Definition~\ref{def:channel}.
\end{proof}

Write
\begin{equation}\label{eq:outside-product}
 q((y_j)_j)=\prod_{j\in A}h_j(d_Y(y_j))\prod_{j\notin A\cup\{b\}}h_j(x_Y(y_j)),
\end{equation}
a function of the three added lists $Q_j$ of a single $Z$-state. With
$y=y_b$ set
\begin{equation}\label{eq:g-and-u}
 g(P,y)=\E_{\lambda_Y}[q\mid P,y_b],\qquad u(e,y)=\E_{m_Z}[q\mid\omega_Z,y_b],\ e=\omega_Z.
\end{equation}
The original correlation equals $\E[\Phi(P)h_b(x_Y(y_b))g(P,y_b)]$, so $g$
is nonzero in $L^2$. Both expectations use the same state law but different
parameter information: $e$ contains $P$ and possibly more.

\subsection{The square and its conditional laws}

$\lambda_Z$ is a law on a square of $Y$-states. Write
$\mathbf z_i=(y_{ij})_{j\in I}$ for row $i$ (a $Z$-state), and
\[
 P_{ij}=\omega_Y(y_{ij}),\quad E_i=\omega_Z(\mathbf z_i),\quad E=(E_i)_i.
\]
Each $E_i$ contains the row $(P_{ij})_j$ (Lemma~\ref{lem:parameter-distal}).
In column $b$ put $y_i=y_{ib}$, $p=(P_{ib})_i$, $w=P_{bb}$. Functoriality
for $Z\to Y$ makes this column $\lambda_Y$-distributed; row $b$ likewise,
as a single $Z$-state. We will show
\[
 \E\bigl[q((y_i)_i)q(\mathbf z_b)\mid E\bigr]\not\equiv0.
\]
The row factor $q(\mathbf z_b)$ is read from the added outputs, the candidate
for the extra role at $b$. Figure~\ref{fig:activation-square} separates the
two uses of the old selected law.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[x=1.22cm,y=0.88cm,every node/.style={font=\small},
 cell/.style={draw=black!35,minimum width=1.13cm,minimum height=0.74cm}]
 \foreach \i in {1,2,3,4}{\foreach \j in {1,2,3,4}{\node[cell] at (\j,-\i) {};}}
 \fill[blue!10] (1.54,-4.41) rectangle (2.46,-0.59);
 \fill[orange!17] (0.54,-2.41) rectangle (4.46,-1.59);
 \fill[violet!22] (1.54,-2.41) rectangle (2.46,-1.59);
 \foreach \i in {1,2,3,4}{\foreach \j in {1,2,3,4}{\draw[black!35] (\j-0.46,-\i-0.41) rectangle (\j+0.46,-\i+0.41);}}
 \node at (1,-0.15) {$j_1$};\node at (2,-0.15) {$b$};\node at (3,-0.15) {$j_2$};\node at (4,-0.15) {$j_3$};
 \node at (0.05,-1) {$i_1$};\node at (0.05,-2) {$b$};\node at (0.05,-3) {$i_2$};\node at (0.05,-4) {$i_3$};
 \node at (2,-1) {$y_{i_1}$};\node at (2,-2) {$y_b$};\node at (2,-3) {$y_{i_2}$};\node at (2,-4) {$y_{i_3}$};
 \node at (1,-2) {$y_{bj_1}$};\node at (3,-2) {$y_{bj_2}$};\node at (4,-2) {$y_{bj_3}$};
 \node[font=\scriptsize] at (1,-1) {$P_{i_1j_1}$};\node[font=\scriptsize] at (3,-1) {$P_{i_1j_2}$};\node[font=\scriptsize] at (4,-1) {$P_{i_1j_3}$};
 \node[font=\scriptsize] at (1,-3) {$P_{i_2j_1}$};\node[font=\scriptsize] at (3,-3) {$P_{i_2j_2}$};\node[font=\scriptsize] at (4,-3) {$P_{i_2j_3}$};
 \node[font=\scriptsize] at (1,-4) {$P_{i_3j_1}$};\node[font=\scriptsize] at (3,-4) {$P_{i_3j_2}$};\node[font=\scriptsize] at (4,-4) {$P_{i_3j_3}$};
 \node[anchor=west] at (4.7,-1) {$E_{i_1}\supseteq(P_{i_1j})_j$};
 \node[anchor=west] at (4.7,-2) {$E_b\supseteq(P_{bj})_j$};
 \node[anchor=west] at (4.7,-3) {$E_{i_2}\supseteq(P_{i_2j})_j$};
 \node[anchor=west] at (4.7,-4) {$E_{i_3}\supseteq(P_{i_3j})_j$};
 \node[blue!65!black,anchor=north] at (2,-4.7) {$p=(P_{ib})_i$};
 \node[orange!65!black,anchor=north] at (5.4,-4.7) {$q(\mathbf z_b)\text{ uses row }b$};
\end{tikzpicture}
\caption{The square under $\lambda_Z$. Blue: column $b$ with old parameter
array $p$. Orange: the row containing the added outputs used by $q$. Common
corner parameter $w=P_{bb}$. The $E_i$ contain the old parameter rows. The
figure records marginal and factor relations, not exchangeability.}
\label{fig:activation-square}
\end{figure}

\begin{lemma}\label{lem:square-conditional}
Given $E$, column $b$ has its $\lambda_Y$ law given $p$, and
\begin{equation}\label{eq:square-row-locality}
 \E_{\lambda_Z}[q(\mathbf z_b)\mid E,(y_i)_{i\in I}]=u(E_b,y_b).
\end{equation}
In particular $y_b\mid E\sim m_Y(\dd y\mid w)$; in one $Z$-state the slot-$b$
state given $e=\omega_Z$ has the same fibre law over its slot-$b$ parameter.
\end{lemma}

\begin{proof}
Apply saturation of $Y$ to $Z\to Y$: under $Z^I\to Y^I$ the old joining is
column $b$ and $\mathcal D_0(Z)=\sigma(E)$, so \eqref{eq:saturation-identity}
with $l=0$ (on a countable determining family) gives the first assertion;
the singleton case of Proposition~\ref{prop:selected} gives
\begin{equation}\label{eq:corner-fibre-law}
 \mathcal L(y_b\mid E)=m_Y(\cdot\mid w).
\end{equation}
The row-$b$ marginal and the tower from $E$ to $E_b$ give the single-state
assertion. For \eqref{eq:square-row-locality}, fix $l\ne b$ and let
$\mathbf z_{-l}$ be all rows but $l$. Given $E$ the three rows are
$Z$-fibre-independent (Proposition~\ref{prop:selected} for $Z$), so
\begin{equation}\label{eq:omit-one-column-state}
 \E[q(\mathbf z_b)\mid E,(y_i)_{i\ne l}]=u(E_b,y_b).
\end{equation}
For every bounded test $F$ of a $Y$-state, saturation for datum type $l$
gives
\begin{equation}\label{eq:missing-state-markov}
 \E[F(y_l)\mid E,\mathbf z_{-l}]=\E_{\lambda_Y}[F(y_l)\mid p,(y_i)_{i\ne l}],
\end{equation}
measurable in $(E,(y_i)_{i\ne l})$; so given that smaller datum, $y_l$ is
conditionally independent of $\mathbf z_{-l}$ (multiply by a bounded
function of $\mathbf z_{-l}$, condition on the smaller datum, monotone
class). Since $q(\mathbf z_b)$ is measurable in $\mathbf z_{-l}$, adjoining
$y_l$ leaves \eqref{eq:omit-one-column-state}, giving
\eqref{eq:square-row-locality}.
\end{proof}

Conditioning on the column and then on its corner state gives
\begin{equation}\label{eq:square-product}
 \E\bigl[q((y_i)_i)q(\mathbf z_b)\mid E\bigr]=\int g(p,y)u(E_b,y)m_Y(\dd y\mid w).
\end{equation}
The next lemma supplies the parameter independence showing this cannot
vanish identically.

\begin{lemma}\label{lem:parameter-reanchoring}
Under $\lambda_Z$, $p$ and $E_b$ are conditionally independent given $w=P_{bb}$.
\end{lemma}

\begin{proof}
Let $\kappa:W_Z\to W_Y$ be the flow factor from $Z\to Y$; then
$P_{ib}=\kappa(E_i)$. Proposition~\ref{prop:selected} identifies the $E$ law
with the deterministic-length law on $W_Z$. For a bounded test $F$ of $E_b$
and a rectangle $\prod_iH_i(P_{ib})$, the integrated average at fixed $t$ is
$\int_{W_Z}F(S_{W_Z}^{bt}e)\prod_iH_i(\kappa(S_{W_Z}^{it}e))\dd m_{W_Z}(e)$.
Substituting $e'=S_{W_Z}^{bt}e$:
$\int_{W_Z}F(e')\prod_iH_i(S_{W_Y}^{(i-b)t}\kappa(e'))\dd m_{W_Z}(e')$; all
factors but $F$ depend only on $\kappa(e')$, so replace $F$ by
$\E[F\mid\kappa]$ at every $t$. Averaging and the deterministic limit give
$\E[F(E_b)\prod_iH_i(P_{ib})]=\E[\E[F(E_b)\mid w]\prod_iH_i(P_{ib})]$;
rectangle tests generate the $p$ $\sigma$-algebra and $w\in p$, so
monotone class finishes.
\end{proof}

\subsection{Nonvanishing and the extra role}

\begin{lemma}\label{lem:independent-hilbert-vectors}
Let $V$ be a square-integrable random vector in a separable real Hilbert
space and $V'$ an independent copy. If $\ip V{V'}=0$ a.s.\ then $V=0$ a.s.
\end{lemma}

\begin{proof}
If $\Prob(\norm V\ge\epsilon)>0$, a countable cover by radius-$\epsilon/4$
balls has one meeting that event; with positive probability $V,V'$ both lie
in it, giving norms $\ge\epsilon$, distance $\le\epsilon/2$, and
$\ip V{V'}=\frac12(\norm V^2+\norm{V'}^2-\norm{V-V'}^2)>0$: contradiction.
\end{proof}

\begin{proposition}\label{prop:activation}
For the saturated channel $Y$ and maximal correlation above, the regrouped
channel $Z$ has a nonzero correlation \eqref{eq:role-correlation} with
output roles $A\cup\{b\}$.
\end{proposition}

\begin{proof}
In one $Z$-state let $R(e)$ be its old $W_Y$-parameter array (a function of
$e=\omega_Z$) with slot-$b$ entry $w$. Given $w$, let $\alpha_w$ be the old
array law in $\lambda_Y$ and $\beta_w$ the law of $e$ in $m_Z$; then
$R_*\beta_w=\alpha_w$. Both $p$ and $R(E_b)$ have marginal $\alpha_w$, and
Lemma~\ref{lem:parameter-reanchoring} says $(p,E_b)\sim\alpha_w\otimes\beta_w$.

Work for a.e.\ fixed $w$ in the separable $H_w=L^2(m_Y(\cdot\mid w))$; the
bounded kernels $g(p,\cdot)$, $u(e,\cdot)$ are measurable $H_w$-valued
vectors (choose a countable generating algebra on the standard Borel $Y$,
whose rational simple span is dense in every $H_w$; scalar tests and Fubini
give strong measurability). The single-state assertion of
Lemma~\ref{lem:square-conditional} says $y\mid e$ has exactly the fixed law
$m_Y(\cdot\mid w)$, so conditioning on $R(e)$ adds no $y$-dependent weight
to $e$; the tower property for \eqref{eq:g-and-u} gives
\begin{equation}\label{eq:average-u-is-g}
 \E_{\beta_w}[u(e,\cdot)\mid R(e)]=g(R(e),\cdot),
\end{equation}
also checked against a countable dense set in $H_w$ (the joint law of
$(e,y)$ over $w$ is $\beta_w(\dd e)m_Y(\dd y\mid w)$).

If \eqref{eq:square-product} were zero a.s., parameter independence gives
$\ip{g(p,\cdot)}{u(e,\cdot)}_{H_w}=0$ for $\alpha_w\otimes\beta_w$-a.e.\
$(p,e)$. Averaging in $e$ given $R(e)$ and using \eqref{eq:average-u-is-g}
yields $\ip{g(p,\cdot)}{g(p',\cdot)}_{H_w}=0$ for
$\alpha_w\otimes\alpha_w$-a.e.\ $(p,p')$, so
Lemma~\ref{lem:independent-hilbert-vectors} forces $g(p,\cdot)=0$ a.s.\ for
a.e.\ $w$, contradicting nonvanishing of $g$. Hence some bounded test
$\Psi(E)$ has
\begin{equation}\label{eq:activated-correlation}
 \E_{\lambda_Z}\bigl[\Psi(E)q((y_i)_i)q(\mathbf z_b)\bigr]\ne0,
\end{equation}
e.g.\ the sign of the nonzero conditional expectation in
\eqref{eq:square-product}. Identify the four roles: in every row $i\ne b$ the
$q((y_i)_i)$ factor reads the old slot-$b$ state (an old output if $i\in A$,
$Z$'s original coordinate otherwise), with zero spatial mean; in row $b$,
$q(\mathbf z_b)$ is a function of the three added lists $Q_j$, independent
with product law under $Z$'s outputs and each factor centred, so it is
centred; $\Psi$ tests the full new distal array $E$. Thus
\eqref{eq:activated-correlation} is of the form \eqref{eq:role-correlation}
with output roles $A\cup\{b\}$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Suppose the conclusion failed. The standard reduction and
Proposition~\ref{prop:selectors} give fixed selectors and a nonzero
correlation with no output roles, so the maximal role number $r$ exists;
Proposition~\ref{prop:role-bound} gives $r\le2$. Proposition~\ref{prop:saturation}
keeps a maximal witness in a saturated channel, and Proposition~\ref{prop:activation}
produces a nonzero correlation with $r+1$ output roles, contradicting the
definition of $r$. Failure is impossible; the limit is the product of the
four integrals as in Lemma~\ref{lem:relative-norm}. Pulling back from the
standard coded factor proves Theorem~\ref{thm:main}.
\end{proof}

\appendix
\section{Pointwise convergence for the proper subsets}\label{sec:lower-length}

We prove Lemma~\ref{lem:lower-length}. Its systems are extensions of a
suspension, not mixing or ergodic, so we need a pointwise theorem on
arbitrary invertible systems. The analytic base is the oscillation estimate
\cite[Prop.~2.2]{TripleCompanion} (call it \textbf{P}; also
\textbf{H}=\cite{HilbertCompanion}). P states the oscillation estimate for
times $n,2n,3n$ before using mixing, so it already holds without mixing.
We need the three other triples inside $\{1,2,3,4\}$; we obtain them by
modifying the local four-linear construction of H and its depth-varying
extension in P, including their intermediate estimates (the bare boundedness
of the trilinear Hilbert transform would not give oscillation). We specify
the imported statements, verify the coefficient changes, then transfer to
arbitrary systems and all integer lengths.

\subsection{Local estimates imported from H and P}

A \emph{role set} is a set $S$ of four distinct integers; initially
$S=\{0\}\cup J$ with $J\subset\{1,2,3,4\}$ of size three. For a dyadic
interval $K$ of length $r=2^{\deg}$ let
\begin{equation}\label{eq:ll-local-form}
 \mathcal H_{K,S}(z)=r^{-2}\int_{\R^2}w_K(x,t)\prod_{j\in S}z_j(x+jt)\dd x\dd t .
\end{equation}
Fix $c_0>0$, $C_0\ge1$. A kernel $w_K$ is admissible if every $x+jt$ on
its support ($j\in S$) lies in $K$ at distance $\ge c_0r$ from the boundary,
if its $t$-marginals vanish,
\begin{equation}\label{eq:ll-kernels}
 \int_\R w_K(v-jt,t)\dd t=0\ (v\in\R,\ j\in S),\qquad
 \|\partial_x^a\partial_t^bw_K\|_\infty\le r^{-a-b}\bigl(C_0(a+b+2)\bigr)^{C_0(a+b+2)},
\end{equation}
and kernels may differ per interval.

H's \emph{Uniform local estimate} \cite[Thm.~2.2]{HilbertCompanion} for
$S=\{0,1,2,3\}$ and suitable $q\in(2,3)$ reads
\begin{equation}\label{eq:ll-fixed-local}
 \sum_{K\in\mathcal K}|K|\,|\mathcal H_{K,S}(z)|\le C\int_\R\prod_{j\in S}M_qz_j,\quad M_qz=(M(|z|^q))^{1/q},
\end{equation}
$M$ the Hardy--Littlewood maximal operator, for finite families of dyadic
intervals and compactly supported $L^q$ inputs. Its proof in H, Sec.~3--10,
supplies the chart, projection, separation, counting and side estimates used
by P.

The oscillation input is P's \emph{Local estimate with variation in depth},
Thm.~3.2. Let $K_*$ be a root, $s(K)=\log_2(|K_*|/|K|)$ a depth, and for an
integer block $B=\{a,\dots,b\}$ put
$\mathcal V_B(z)(y)=\max_{s\in B}|z_s(y)|+\sum_{s=a}^{b-1}|z_{s+1}(y)-z_s(y)|$.

\begin{proposition}[Local input with changed coefficients]\label{prop:ll-depth}
Let $S=\{0\}\cup J$, $J$ a three-element subset of $\{1,2,3,4\}$. Let
$\mathcal T$ be a finite rooted dyadic tree with root $K_*$, depth $\le d$,
and let $\mathcal K\subset\mathcal T$ carry kernels obeying
\eqref{eq:ll-kernels}. For each $j\in S$ let $(z_{j,s})_{0\le s\le d}$ be
measurable on $K_*$ (extended by zero). Suppose $\{0,\dots,d\}$ is
partitioned into $W_j\ge1$ consecutive blocks, one per role possibly, on
each of which $\mathcal V_B(z_j)\le W_j^{-1/2}$ a.e.\ on $K_*$. Then
\begin{equation}\label{eq:ll-depth}
 \sum_{K\in\mathcal K}|K|\,\bigl|\mathcal H_{K,S}((z_{j,s(K)})_{j\in S})\bigr|\le C(c_0,C_0)|K_*|,
\end{equation}
$C$ independent of tree, kernels, depth range and partitions, uniform over
the four $S$.
\end{proposition}

Its proof is the proof-by-modification below: H's local estimate with its
intermediate estimates, plus P's depth-local construction (Sec.~4--7). It is
the sole analytic input to the local-to-orbit transfer
\eqref{eq:ll-oscillation}; the following subsections check coordinate and
quadratic identities, integer-lift and counting arithmetic, and their use in
P's depth scheduling, retaining P's chart atoms, projection families,
curvature tags and finite progression tests, with coefficients changed as
computed. It is \emph{not} an assertion of uniform continuity of a
Hilbert-transform bound in its slopes.

\subsection{Coordinates and the quadratic cancellation vector}

Every $S$ above contains an adjacent pair. Translate role labels so one such
pair is $0,1$: the change of variables $x'=x+mt$ in \eqref{eq:ll-local-form}
has Jacobian one, the four coordinates are unchanged, and support,
cancellation and derivative bounds keep their form after raising $C_0$.
Pairwise differences are then $\le4$ in absolute value with diameter $3$ or
$4$, so
\begin{equation}\label{eq:ll-support-t}
 |t|\le|K|/3\quad\text{whenever every }x+jt\in K.
\end{equation}
The H vector $(-1,3,-3,1)$ becomes a primitive integer $(c_j)_{j\in S}$
proportional to $\bigl(\prod_{\ell\in S\setminus\{j\}}(j-\ell)\bigr)^{-1}$
(sign immaterial). Before translation:
\begin{equation}\label{eq:ll-coefficient-table}
\begin{array}{c|rrrr}
 S & \multicolumn{4}{c}{(c_j)_{j\in S}\text{ in increasing order}}\\\hline
 \{0,1,2,3\}&-1&3&-3&1\\
 \{0,1,2,4\}&-3&8&-6&1\\
 \{0,1,3,4\}&-1&2&-2&1\\
 \{0,2,3,4\}&-1&6&-8&3
\end{array}
\end{equation}
Lagrange interpolation of $1,u,u^2$ gives
\begin{equation}\label{eq:ll-moments}
 \textstyle\sum_{j\in S}c_j=\sum_{j\in S}jc_j=\sum_{j\in S}j^2c_j=0,
\end{equation}
preserved by translation of labels; every $c_j\ne0$ and all coefficients and
denominators below are bounded absolute constants. We separate numerical
role labels ($j$, $j-i$, $c_j$ change with $S$) from enumerations of four
objects (sizes, leaves, expansion positions stay); that convention leaves
those parts of H's proofs unchanged.

\subsection{Chart separation, detection, scale sums}

H's Lemma~2.1 uses $(x,t)\mapsto(x+it,x+jt)$ with determinant $j-i$, so its
endpoint bounds and interpolation hold for new roles; the initial bound for
the finite-depth constant in \eqref{eq:ll-fixed-local} and H's normalized
stopping reduction survive. H's coefficient/projection bounds and
compression/height-bin estimates (Lemmas~3.2--3.5) concern finite families of
vectors or real frequencies and name no progression coefficient.

H, Sec.~4, turns two chart atoms in distinct roles into separated factors.
A chart phase is $\lambda y^2+(\theta y)^TCd+\beta y$, $\theta$ real
frequencies, $C$ the rational chart matrix, $d$ a bounded lift of $\theta y$
mod the integer lattice, plus a smooth lift amplitude; its one-variable Weyl
estimates, Fourier expansion and uniform tails are unchanged.

For H, Lemma~4.5 (\emph{Progression separation}), with source roles $i,k$
and spectators $a,b$, set $\alpha_i=1$, $\alpha_k=0$ and solve
$\sum_j\alpha_j=\sum_jj\alpha_j=0$ for $\alpha_a,\alpha_b$ (possible as
$a\ne b$). With $y_j=x+jt$ and lifts $d_a,d_b,d$ of $\theta y_a,\theta y_b,
\theta t$, the integer carry $k_j=d_j-d_a-(j-a)d$ still tests exactness
(if $d_j=\theta y_j-m_j$, $d=\theta t-m$, then $k_j=-m_j+m_a+(j-a)m$). With
$A_2=\sum_jj^2\alpha_j$,
\begin{equation}\label{eq:ll-chart-identity}
 \begin{aligned}
 \sum_{j\in S}\alpha_j[\lambda y_j^2+(\theta y_j)^TCd_j]
 &=A_2[\lambda t^2+(\theta t)^TCd]\\
 &\quad+\sum_{j\in S}\alpha_j(\theta y_j)^TCk_j,
 \end{aligned}
\end{equation}
the last sum linear in $x,t$ hence separating in $y_a,y_b$; the lift-box
amplitude keeps its Gevrey bounds up to constants, so H's absolutely
convergent separation into two spectator atoms and one $t$-atom retains its
coefficient-tail exponents and quasipolynomial complexity. H, Lemma~4.6
(\emph{Single-scale separation}) uses $t=(y_b-y_a)/(b-a)$; added rational
heights are absolutely bounded.

For the generalized von Neumann bound, H, Lemma~4.11, let $i$ carry the
input controlled by its $U^3$ norm. Cauchy--Schwarz along directions
$(-j,1)$ over the other three roles fixes $x+jt$ and moves $x+it$ by
$(i-j)h$; multiplication by nonzero integer $i-j$ preserves Haar measure, so
the same cube integral appears (character weights split at the marked pair
$x=y_0$, $t=y_1-y_0$; boxes enlarge to fit all translated role sets). H,
Lemma~4.12 (\emph{Local detection}) puts the possibly unbounded $L^2$
residual in a different role; after clipping the opposing inputs its inverse
theorem decomposition and uniform error bound by Lemma~4.11 are unchanged,
and the chart-moment detections just verified apply. The clipping, inverse
theorem and mixed-norm exponents do not move.

H, Lemma~5.1 (\emph{Scale sum for one separated term}) integrates the kernel
over one spectator coordinate holding the other fixed; \eqref{eq:ll-kernels}
gives a constant Jacobian factor $|b-a|$, and the double primitive keeps
compact support and smooth bounds. Its tensor expansion into mean-zero bumps,
Bessel estimates and frequency-height bounds apply to the new pair; the only
new demodulation frequencies $\pm\gamma/(b-a)$ have rational multipliers of
bounded height, verifying the finite predictor list and the delayed
predictor replacement. The band, gap and projection-energy bounds in the base
construction depend on these estimates and multilinearity, not the labels.
The multivariable H, Lemma~5.7 (\emph{Few active scales for a fixed triple})
squares the transpose at the remaining role $\ell$, giving six atoms with
arguments $h+(j-\ell)t$, $h+(j-\ell)t'$: still fixed integer linear forms in
three variables, so H, Lemmas~4.3--4.4 apply with the same complexity; its
final kernel cancellation is the anchor-$\ell$ marginal in \eqref{eq:ll-kernels}.

\subsection{Curvature tags, short blocks, integer heights}

H, Sec.~6 groups atoms by normalized quadratic phase and horizontal chart
data (curvature tags). Normalize $(\lambda,C)$ in role $j$ by dividing by
$c_j$, keeping $\theta$. H, Lemma~6.1 (\emph{Separation of different tags})
is checked directly: for source $i,j$, spectators $a,b$, with
$\xi_\ell=\frac{b-\ell}{b-a}$, $\zeta_\ell=\frac{\ell-a}{b-a}$ and
$y_\ell=\xi_\ell y_a+\zeta_\ell y_b$, the barycentric formula gives
\begin{equation}\label{eq:ll-tag-identity}
 c_i\xi_i\zeta_i=-c_j\xi_j\zeta_j\ne0,
\end{equation}
since $(i-a)(i-b)$ cancels from $c_i$'s defining product leaving a nonzero
multiple of $(i-j)^{-1}$ antisymmetric in $i,j$. The same nonzero prefactor
gives the same separation; its period enlarges to clear fixed denominators.
H, Lemma~6.6 (\emph{Uniform same-tag test}) uses the marked adjacent pair
with lifts
\begin{equation}\label{eq:ll-template-lift}
 d_\ell=(1-\ell)d_0+\ell d_1+k_\ell,\quad k_\ell\in\Z^d,
\end{equation}
and substituting this plus \eqref{eq:ll-moments} cancels common quadratic
terms, leaving linear carry phases with the same residue-template bound after
a fixed box enlargement, no new tag. Graph distances, color histories, masks
and projection bounds are then used in their original order.

H, Lemma~7.2 (\emph{Short-block comparison}) uses two sampling windows on
roles $0,1$; lattice and lifted torus coordinates are
$n_j=(1-j)n_0+jn_1$, $w_j=(1-j)u+jv$, integer-coefficient. A fixed torus of
length $64$ and boxes in $[-12,12]$ hold the role sets (shrink the source
support slightly); pair-to-two-roles maps are injective on lattices and
surjective on connected tori, the needed comparison property. The quadratic
randomization in role $j$ is $e(\gamma m^2/c_j)$, $e(v)=e^{2\pi iv}$, $\gamma$
uniform mod a common multiple of the $|c_j|$ (value $24$ suffices); on the
Fourier support $\sum_jm_j=\sum_jjm_j=0$, so $m_j=c_j(v+jv')$ and
$\sum_jm_j^2/c_j=\sum_jc_j(v+jv')^2=0$ by \eqref{eq:ll-moments}, preserving
the form. Per-role fourth-moment averaging imposes the same two pairings as
H, keeping its fourth-moment and clipping estimates including $q-2$ powers.
Return to small intervals: four real coordinates within $e$ of integers give
an integer progression from the marked pair, and every other discrepancy is
an integer of magnitude $<8e$, so $e<1/100$ kills it; offsets have the
positive area \eqref{eq:ll-polygon} below. Support/overlap losses are constants.

Height base change: in H, Sec.~8--9 replace $L_h=6^h$ by $12^h$ and the
factor $36$ in finite progression tests by $144$; anchor changes stay
integer. The common multiplier in H, Sec.~8.3 uses
\eqref{eq:ll-template-lift} and \eqref{eq:ll-moments}, so smoothness and
Fourier truncation are unchanged in type. In H, Lemma~8.3
(\emph{Height stabilization}) a free-quotient coordinate is an integral
rational multiple of $12^{R'}$ ($R'$ the height gap, $C_*$ bounds pre-
multiplied denominators); the intersection is saturated so the quotient is
free abelian, and for a required drop $\Delta$ its $2$- and $3$-adic
valuations are $\ge2R'-\log_2C_*$, $R'-\log_3C_*$, so
$R'>\log_2C_*+2\Delta$ gives divisibility by $12^\Delta$ with no assumption
on the initial denominator's primes (the coordinate is then already
integral). Reserves and the growth constant $A_0$ in $E_h=A_0^{h+1}$ use the
new base.

\subsection{Progression counting and the depth extension}

H, Sec.~9 eliminates a role $\ell'$ to get nonzero
$\alpha_i=i-\ell'$, $|\alpha_i|\le4$; horizontal tests use frequency sums,
anchored tests use weights $i-\ell'$, and kernel cancellations are
\eqref{eq:ll-kernels}, so expansions, coefficient tails and annular tests
keep their forms.

H, Lemma~9.1 (\emph{Sparse mass of separated progression tests}) uses shift
data $\zeta\in\R^d$ and integers $n$, $\|n\|_\infty\le B$. Replace annular
shifts $36n\cdot\zeta$ by $144n\cdot\zeta$ and neighborhood shifts
$6n\cdot\zeta$ by $12n\cdot\zeta$; $N_0$ is the neighborhood coefficient
reserve, and the fine grid step clears $1,2,3,4$. Alternately adding four
annular relations of a four-step path cancels a role-$i$ intermediate label
and two labels of a second role, leaving $\alpha_i(a-a')$ and a four-color
label expression: dividing by $\alpha_i$ constructs the difference cover.
Lattice compatibility
\begin{equation}\label{eq:ll-counting-divisibility}
 144/\alpha_i\in12\Z,\qquad 12\alpha_j/\alpha_i\in\Z
\end{equation}
holds since $|\alpha_i|\mid12$. The sum of four original shifts has norm
$\le4B$, so in units of the new shift $12$, dividing by $\alpha_i$ gives norm
$\le4B\cdot12/|\alpha_i|\le48B$, covered by $N_0\ge64B$; the scalar error is
$\le4$ times one relation's error. For the slab cover, writing the first two
coordinates as box centers plus $12n_i\cdot\zeta$ and interval errors and
solving the three-role relation for the third gives shift coefficients
$12\alpha_1/\alpha_3$, $12\alpha_2/\alpha_3$, $144/\alpha_3$, integers by
\eqref{eq:ll-counting-divisibility}, of norm $\le C(d+1)^2B$ as in H. The
separation reserve and excluded pure-shift constants grow by fixed factors;
covering iterations keep the same sumset orders, graph count uses the same
four roles, block/scale counts keep their covers, so the source factor
$L^{29}\delta^{1/2080}$ survives ($L$ bounds scale occurrences of a fixed
pair of labels, $\delta$ is the sparse-density threshold). In the
application take $\zeta=\xi/144$ ($\xi$ from the height construction), so
heights start at $12^{\Delta-2}$ instead of $6^{\Delta-2}$; the reserves
supply the shifts, pair determination divides only by role differences, and
the short-block windows were checked. This verifies the counting estimate's
coefficient hypotheses and application, not merely its conclusion.

The remaining operator and summation arguments of H then apply: projection
families, Fourier modes and counting tests keep their logarithmic budgets,
and the changes inserted only fixed rational heights, derivative losses and
an exponential-base change, none altering a budget exponent as a function of
the final $q$. H, Sec.~10 fixes those exponents before taking $q\in(2,3)$
close to two, so its absorption of the finite-depth constant covers the
modified estimates. This proves \eqref{eq:ll-fixed-local} for the four role
sets with the intermediate estimates P uses.

Carrying through P's depth-local proof: the maximal estimate for local
catalog rows, the depth-varying row estimate and the smooth mean-zero row
estimate (P, Sec.~4; Lemmas~4.3, 4.4, 4.6) concern Hilbert-space operators
on nested dyadic cells, catalogs of vector-valued column functions with
dyadic birth cells, and local projections onto constant-coefficient column
combinations with cellwise-constant persistent coefficients an ordinary
fixed-input Bessel bound and nondecreasing input reads, with no role
coefficients; the replacement supplies the same scalar and Bessel inputs.
P, Lemma~5.3, Cor.~5.4 and Lemma~5.5 use $(y_a,y_b)$ and demodulations
$\pm\gamma/(b-a)$, already checked; the cutoff $1$ on $[-1/3,1/3]$ covers the
support by \eqref{eq:ll-support-t}, the kernel separates into mean-zero bump
rows with fixed carriers, and Bessel bounds for clump-replacement and
predictor rows follow from the verified cancellation and height estimates;
row support/observation partitions and nondecreasing depth reads are
unchanged. P, Sec.~6 uses the normalized tags from \eqref{eq:ll-moments}; the
mask, template-transfer, short-block and counting estimates are the versions
above, and scheduling column updates adds no new arithmetic on a label.

The final dependence is visible in H, Sec.~10.3 and P, Sec.~7. With $k$ the
accuracy logarithm and $\mathfrak p$ the growth parameter, $K_0$ the depth-
block length, $G_0$ the strong-energy/projection overhead, the term needing
$q$ near two is
\[
 \mathfrak p^{-1/2}G_0(K_0+2)^{C_{19}(q-2)},\quad
 K_0\le C_q(1+k+\mathfrak p)^{C_5},\quad
 G_0=\exp\!\bigl(C_q(1+\log(2k))^C\bigr),
\]
here renaming the source $P$ to avoid conflict. The degrees $C,C_5,C_{19}$
have absolute bounds fixed before $q$, and the short-block feeding term has
exponent $11(q-2)/(q-1)$ (H, Lemma~7.2), coefficients affecting constants
not the exponent. Take $q$ so near two that
$\max\{C_{19}^{\rm H}C_5^{\rm H},C_{19}^{\rm P}C_5^{\rm P}\}(q-2)<1/8$,
permissible in both proofs; the scheme fixes $0<\theta<1$ before $q$ and
$\alpha>0$ small after $q$, as in the sources. Their $\mathfrak p=\lceil
\exp(\alpha k^\theta)\rceil$ gives $G_0=\mathfrak p^{o(1)}$ and makes the
term $\le C_q\mathfrak p^{-1/4}$; remaining accuracy exponents and summations
keep their order. Thus P's depth-local argument gives \eqref{eq:ll-depth}.
Translating labels back restores $S=\{0\}\cup J$; P allows variation in any
role and any testing slot, so no role must stay marked.

\subsection{From local forms to orbit oscillation}\label{subsec:ll-transfer}

An \emph{averaging profile} is real, compactly supported, smooth, with
$\int\varphi=1$, $\int t^a\varphi=0$ ($1\le a\le3$), and
\begin{equation}\label{eq:ll-profile}
 \|\varphi^{(n)}\|_\infty\le(G(n+2))^{G(n+2)}\ (n\ge0),
\end{equation}
$G\ge1$; profiles may have either sign, $\varphi_A(t)=A^{-1}\varphi(t/A)$.

\begin{proposition}[Oscillation for the four triples]\label{prop:ll-oscillation}
Let $(Y,\nu,R)$ be invertible probability-preserving, $J\subset\{1,2,3,4\}$
of size three, $g_j:Y\to\C$ measurable with $|g_j|\le1$, and
$V_k^\varphi(y)=\sum_{n\in\Z}\varphi_{2^k}(n)\prod_{j\in J}g_j(R^{jn}y)$, $k\ge0$.
There is $C_\varphi<\infty$ with, for all integers $0\le k_1<\dots<k_{W+1}$,
\begin{equation}\label{eq:ll-oscillation}
 \sum_{b=1}^W\int_Y\max_{k_b\le\ell\le k_{b+1}}|V_\ell^\varphi-V_{k_b}^\varphi|\dd\nu\le C_\varphi\sqrt W,
\end{equation}
$C_\varphi$ independent of system, inputs, endpoints and $W$.
\end{proposition}

\begin{proof}
Following P, Sec.~3, with changed coefficients. Put $b=\varphi_2-\varphi$;
its moments through order three vanish, so it has a compactly supported
fourth primitive $g$, $g^{(4)}=b$. Take $B\ge1$ containing its support, a
smooth Gevrey bump $u$ of integral one in $[-1/8,1/8]$, and dyadic
$D>32B$; for $S=\{0\}\cup J$ set
\[
 F(z,\sigma)=u(z)D^{-3}g(D\sigma),\quad
 \mathcal W(z,\sigma)=\prod_{j\in S}(\partial_\sigma-j\partial_z)F(z,\sigma).
\]
The four commuting operators differentiate along lines fixing one role, so
\eqref{eq:ll-kernels} cancellations hold, and on the support
$|z+j\sigma|\le1/8+4B/D<3/8$, giving the interior margin for $r=DA$; the
derivative bounds \eqref{eq:ll-kernels} hold with profile-dependent
constants. Integrating in $z$ kills $z$-derivative terms, so
\begin{equation}\label{eq:ll-kernel-normalization}
 \frac1r\int_\R\mathcal W(z,t/r)\dd z=A^{-1}b(t/A)=\varphi_{2A}(t)-\varphi_A(t).
\end{equation}

Let $\Delta_k=\varphi_{2^{k+1}}-\varphi_{2^k}$, let $f_j$ ($j\in J$) be
bounded by one supported in an interval of length $L$, and suppose the
$f_{0,k}$ share that support with block-variation bound $W^{-1/2}$ over a
partition of $k_-\le k\le k_+$ into $W$ blocks. Apply
Proposition~\ref{prop:ll-depth} to dyadic cells of length $D2^k$ and average
the translated lattice over one period of its largest scale: nonzero cells
lie below roots of total length $\le L+2D2^{k_+}$, and scale order reverses
depth order, preserving block variation. The translation average of a
scale's kernels is \eqref{eq:ll-kernel-normalization}, giving
\begin{equation}\label{eq:ll-continuous-test}
 \Bigl|\sum_{k=k_-}^{k_+}\int_{\R^2}f_{0,k}(x)\prod_{j\in J}f_j(x+jt)\Delta_k(t)\dd x\dd t\Bigr|
 \le C_\varphi(L+D2^{k_+}).
\end{equation}

Integer transfer needs no consecutive roles. Plant each integer sequence on
radius-$e$ intervals, $e<1/200$. If $x+jt$ lies in all four such intervals
use an adjacent pair to reconstruct integers $m,n$; every other center is
$m+jn$ since its discrepancy is $<8e$. Write $x=m+\xi$, $t=n+\eta$; the
offset domain $\Omega_{S,e}=\{(\xi,\eta):|\xi+j\eta|<e,\ j\in S\}$ has, by
convexity from the two extreme roles $s_-=\min S$, $s_+=\max S$,
\begin{equation}\label{eq:ll-polygon}
 |\Omega_{S,e}|=4e^2/(s_+-s_-).
\end{equation}
Replacing $\Delta_k(n+\eta)$ by $\Delta_k(n)$ costs $O_\varphi(L)$ summed
over $k\ge0$ (derivative $O_\varphi(2^{-2k})$, $O_\varphi(2^k)$ integer
arguments, $|\eta|\le2e$); dividing by the fixed area \eqref{eq:ll-polygon}
gives the discrete version of \eqref{eq:ll-continuous-test} with bound
$C_\varphi(L+D2^{k_+})$.

Fix a finite interval $\mathcal E$ of integer test points in the common
support. For each $m\in\mathcal E$ and block $b$ pick a maximizing endpoint
$\ell_b(m)$ and unimodular $\alpha_b(m)$ making the tested difference
nonnegative, and set on that scale block
$f_{0,k}(m)=\frac{\alpha_b(m)}{4\sqrt W}\mathbf1_{\{k_b\le k<\ell_b(m)\}}$
(and $0$ off $\mathcal E$). Its supremum plus variation is
$\le1/(2\sqrt W)$; the test telescopes into $\frac1{4\sqrt W}$ times the sum
of block maxima, giving oscillation bound
$C_\varphi\sqrt W(L+D2^{k_{W+1}})$.

Transfer along orbits (Calder\'on \cite{Calderon}): apply the sequence
inequality to $g_j(R^ry)$ for $1-Q\le r\le M+Q$, zero elsewhere,
$Q=\lceil4B_\varphi2^{k_{W+1}}\rceil$, $\supp\varphi\subset[-B_\varphi,B_\varphi]$,
with $\mathcal E=\{1,\dots,M\}$. At these points the averages are the
untruncated orbit averages at $R^my$; integrate in $y$, use invariance,
divide by $M$, let $M\to\infty$, giving \eqref{eq:ll-oscillation}. Only
invertibility, measurability and measure preservation are used.
\end{proof}

\subsection{Convergence on arbitrary systems}\label{subsec:ll-convergence}

Prove Lemma~\ref{lem:lower-length} from Proposition~\ref{prop:ll-oscillation};
the oscillation-to-convergence route is classical
\cite{JonesKaufmanRosenblattWierdl}. A uniformly bounded sequence of
measurable functions satisfying \eqref{eq:ll-oscillation} is Cauchy a.e.: if
its tail oscillation exceeds $4\epsilon$ on a positive-measure $E$, then for
every fixed initial endpoint $k$ the sup of $|V_\ell-V_k|$ over $\ell\ge k$
exceeds $2\epsilon$ on $E$, and by monotone convergence one chooses a
deterministic later endpoint with integral maximum $\ge\epsilon\nu(E)$;
recursing contradicts $C_\varphi\sqrt W$ for large $W$. The finite sums
$V_k^\varphi$ are uniformly bounded (bounded compactly supported profile), so
each fixed profile gives a.e.\ convergence.

P, Lemma~2.8 (\emph{Approximation with three vanishing moments}): for each
$c\in[1,2]$ and $\epsilon>0$ there is a profile with
\begin{equation}\label{eq:ll-profile-approximation}
 \|\varphi-h_c\|_1<\epsilon,\qquad h_c(t)=c^{-1}\mathbf1_{(0,c]}(t),
\end{equation}
and signed profiles are essential. Smooth $h_c$ preserving integral one,
then cancel the other three moments with four broad translated bumps at
distance and width $\sim A$: the moment matrix for orders $0..3$, after
dividing row $j$ by $A^j$, is a fixed invertible Vandermonde-type matrix; the
prescribed zeroth correction is zero, the others are $O(A^{-1})$, so the
corrections are $O(A^{-1})$ in $L^1$ cost and the bumps keep
\eqref{eq:ll-profile} with profile-dependent constant.

Fix $J$ and $a_n(y)=\prod_{j\in J}g_j(R^{jn}y)$, $|a_n|\le1$. For
$N_k=\lfloor c2^k\rfloor$ the Riemann comparison gives, uniformly in $y$,
\begin{equation}\label{eq:ll-profile-error}
 \limsup_{k\to\infty}\Bigl|\frac1{N_k}\sum_{n=1}^{N_k}a_n(y)-V_k^\varphi(y)\Bigr|\le\|h_c-\varphi\|_1:
\end{equation}
replace $N_k^{-1}$ by $(c2^k)^{-1}$ at cost $O(2^{-k})$, then bound by
$2^{-k}\sum_{n\in\Z}|h_c(n2^{-k})-\varphi(n2^{-k})|$, converging to the
integral (compact support, piecewise continuous).

Take countably many profiles with errors $\to0$ for each rational
$c\in[1,2]$. On their common conull set the smooth limits and
\eqref{eq:ll-profile-error} make the averages along $N_k$ Cauchy. Host--Kra
\cite[Thm.~1.1]{HostKra}, for arbitrary invertible systems, gives an $L^2$
limit at all integer lengths (apply to the four consecutive powers with the
constant one at the omitted role); the bounded pointwise limit along each
$N_k$ equals this $L^2$ limit a.e., so all rational-$c$ subsequences share
one limit, with no identification and no mixing needed.

Finally for $M\le N$, $|a_n|\le1$,
$|\frac1N\sum_{n\le N}a_n-\frac1M\sum_{n\le M}a_n|\le2(N-M)/N$. Any rational
mesh of $[1,2]$ catches every large integer within its relative mesh size
plus $O(N^{-1})$ of some $\lfloor c2^k\rfloor$ from the mesh; the finitely
many subsequences share the limit, and meshes shrinking to zero prove
convergence along all lengths. Homogeneity removes $|g_j|\le1$; a proper
subset with fewer than three roles embeds in a triple by the constant one.
This proves Lemma~\ref{lem:lower-length}.

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