diff --git a/pointwise_multiple_ergodic_mixing_v2.txt b/pointwise_multiple_ergodic_mixing_v2.txt
index 72989fa..df94c95 100644
--- a/pointwise_multiple_ergodic_mixing_v2.txt
+++ b/pointwise_multiple_ergodic_mixing_v2.txt
@@ -44,19 +44,19 @@
 
 # Pointwise Multiple Ergodic Averages for Mixing Transformations
 
-\maketitle
+maketitle
 
 Abstract: 
 Let $T$ be an invertible mixing probability-preserving transformation
-of a probability space $(X,\mathcal F,\mu)$.  For every integer $n\ge 2$
-and every tuple $f_1,...,f_n\in L^∞(\mu)$, the consecutive
+of a probability space $(X,mathcal F,calF,mu)$.  For every integer $nge 2$
+and every tuple $f_1,...,f_nin L^∞(mu)$, the consecutive
 multiple ergodic averages
-\[
- A_N(f_1,...,f_n)(x) = \frac1N\sum_{k=1}^N\prod_{j=1}^n f_j(T^{jk}x)
-\]
-converge $\mu$-almost everywhere to $\prod_{j=1}^n\int_X f_j\dd\mu$ as
-$N\to∞$ through all positive integers.  No rate of mixing is
-assumed, and $(X,\mathcal F,\mu)$ need not be a standard probability
+[
+ A_N(f_1,...,f_n)(x) = frac1Nsum_{k=1}^Nprod_{j=1}^n f_j(T^{jk}x)
+]
+converge $mu$-almost everywhere to $prod_{j=1}^nint_X f_jddmu$ as
+$Nto∞$ through all positive integers.  No rate of mixing is
+assumed, and $(X,mathcal F,calF,mu)$ need not be a standard probability
 space.
 
 
@@ -65,28 +65,28 @@ space.
 
 ## The problem and the answer
 
-\noindentSetting.
-$(X,\mathcal F,\mu)$ is a probability space, $T:X\to X$ is invertible,
+noindentSetting.
+$(X,mathcal F,calF,mu)$ is a probability space, $T:Xto X$ is invertible,
 bimeasurable and measure preserving.  $T$ is mixing if for every
-$A,B\in\mathcal F$
+$A,Binmathcal F$calF$
 
- \mu(A\cap T^{-r}B)→\mu(A)\mu(B)\qquad(|r|\to∞).
+ mu(Acap T^{-r}B)→mu(A)mu(B)qquad(|r|to∞).
 
-Fix $n\ge 2$ and $f_1,...,f_n\in L^∞(\mu)$.  The object of study
+Fix $nge 2$ and $f_1,...,f_nin L^∞(mu)$.  The object of study
 is the sequence of consecutive multiple ergodic averages
-\[
- A_N(f_1,...,f_n)(x)=\frac1N\sum_{k=1}^N\prod_{j=1}^n f_j(T^{jk}x).
-\]
+[
+ A_N(f_1,...,f_n)(x)=frac1Nsum_{k=1}^Nprod_{j=1}^n f_j(T^{jk}x).
+]
 These average the integrand along the finite arithmetic progression
 $x,Tx,...$ at the $n$ distinct times $k,2k,...,nk$.
 
-\smallskip
-\noindentMain theorem (Theorem~thm:main below).
-If $T$ is mixing, then for $\mu$-almost every $x$
-\[
- A_N(f_1,...,f_n)(x)→\prod_{j=1}^n\int_X f_j\dd\mu
-\]
-as $N\to∞$ through all positive integers.  The space $(X,\mathcal F,\mu)$
+smallskip
+noindentMain theorem (Theorem~thm:main below).
+If $T$ is mixing, then for $mu$-almost every $x$
+[
+ A_N(f_1,...,f_n)(x)→prod_{j=1}^nint_X f_jddmu
+]
+as $Nto∞$ through all positive integers.  The space $(X,mathcal F,calF,mu)$
 need not be standard and no rate of mixing is assumed.
 
 The exceptional null set depends on the tuple $(f_1,...,f_n)$ and on
@@ -100,12 +100,12 @@ The reason the theorem is delicate is the gulf between two very different
 notions of convergence.
 
 
-\item Norm convergence asks $A_N\to\int f_1...\int f_n$ in $L^2(\mu)$.
+item Norm convergence asks $A_Ntoint f_1...int f_n$ in $L^2(mu)$.
   Host--Kra [cite] (Theorem 1.1) and Ziegler [cite]
   (Corollary 1.8) proved this for every finite length on every probability-preserving
   system.
-\item Pointwise convergence asks that the same limit holds for
-  $\mu$-almost every $x$.  It is not a corollary of norm convergence:
+item Pointwise convergence asks that the same limit holds for
+  $mu$-almost every $x$.  It is not a corollary of norm convergence:
   bounded sequences can converge in $L^2$ and still oscillate on a set of
   positive measure.  Bourgain's double recurrence theorem
   [cite] proved $n=2$; higher lengths have only been settled
@@ -118,7 +118,7 @@ notions of convergence.
   length on ergodic measure-distal systems.
 
 Theorem~thm:main extends all of these: it covers every mixing
-system, every length $n\ge 2$, and does not need ergodicity or standardness
+system, every length $nge 2$, and does not need ergodicity or standardness
 beyond the reductions we make at the start.
 
 The paper continues the fourfold mixing theorem of
@@ -131,7 +131,7 @@ pointwise convergence of three-term averages at distinct positive integer
 slopes on any system.  (C) The Furstenberg--Zimmer structure facts and
 relative disjointness between relatively weakly mixing and relatively
 distal extensions [cite].  All three are quoted
-verbatim in \Ssec:inputs.  Everything else is proved here.
+verbatim in Ssec:inputs.  Everything else is proved here.
 
 ## Proof strategy: five steps
 
@@ -139,48 +139,48 @@ The proof is by contradiction, and it has five parts.  Read them once before
 diving into details.
 
 
-\item[(1)] Assume failure: a positive witness at variable lengths.
+item[(1)] Assume failure: a positive witness at variable lengths.
   If Theorem~thm:main fails for the tuple $(f_1,...,f_n)$, there is a
-  smaller index set $J_0⊆ \{1,...,n\}$, centered real functions $f_i$
-  bounded by $1$, a set $E_*$ of positive measure and $\delta>0$, such that
-  $B_L(x)=(1/L)\sum_{l=1}^L\prod_{i\in J_0}f_i(T^{il}x)$ exceeds $2\delta$ for
-  arbitrarily large integers $L$, at each $x\in E_*$.  This is \Ssec:selected.
+  smaller index set $J_0⊆ {1,...,n}$, centered real functions $f_i$
+  bounded by $1$, a set $E_*$ of positive measure and $delta>0$, such that
+  $B_L(x)=(1/L)sum_{l=1}^Lprod_{iin J_0}f_i(T^{il}x)$ exceeds $2delta$ for
+  arbitrarily large integers $L$, at each $xin E_*$.  This is Ssec:selected.
 
-\item[(2)] Turn the failure into a probability law.
-  Move to the roof-one suspension $C=X\times[0,1)$.  Choose a measurable
+item[(2)] Turn the failure into a probability law.
+  Move to the roof-one suspension $C=Xtimes[0,1)$.  Choose a measurable
   selector $N_a$ at each starting point, say $N_a$ takes values in a finite
   dyadic grid, and choose it so the trajectory average of the witness exceeds
-  $\delta$ on $E_*$ for arbitrarily large $a$.  Along any free ultrafilter
-  on $\N$, this defines a probability law $\lambda_Y$ on $Y^I$ for every flow
-  $Y\to C$.  $\lambda_Y$ has fixed marginals, is flow invariant, comes with
+  $delta$ on $E_*$ for arbitrarily large $a$.  Along any free ultrafilter
+  on $N$, this defines a probability law $lambda_Y$ on $Y^I$ for every flow
+  $Yto C$.  $lambda_Y$ has fixed marginals, is flow invariant, comes with
   small-marginal laws for at most three slots, and is functorial in $Y$.
 
-\item[(3)] Channels: a book-keeping device for three groups of outputs.
+item[(3)] Channels: a book-keeping device for three groups of outputs.
   A quantitative dependence on the original point is captured by three
-  groups of outputs, i.e.~factor maps $D_\gamma:Y\to C$.  The
+  groups of outputs, i.e.~factor maps $D_gamma:Yto C$.  The
   two conditions are: the outputs have product spatial law given their phases,
   and the original spatial coordinate has product law jointly with any proper
   collection of groups.  Two facts are needed.  Conditioning can be strengthened
   to conditioning on $W_Y$, the maximal measure-distal factor.  And the
-  construction $Y\rightsquigarrow\mathsf M_b(Y)$
+  construction $Yrightsquigarrowmathsf M_b(Y)$sfM_b(Y)$
   (each slot gets its own copy of $Y$, scaled by $1/b$) is a channel.
   The second fact is where tensor packing enters; it is the technical heart.
 
-\item[(4)] Minimum-support correlation and saturation.
+item[(4)] Minimum-support correlation and saturation.
   Among all nonzero correlations of the form
-  $\E_{\lambda_Y}[\Phi(P)\prod_{i\in J}h_i(L_i(y_i))]$, over all channels $Y$,
+  $E_{lambda_Y}[Phi(P)prod_{iin J}h_i(L_i(y_i))]$, over all channels $Y$,
   choose one with the minimum number $k$ of slots; among those, maximize the
   number $r$ of slots whose role is output-only.  Then upgrade to a
   saturated channel: no further extension of the channel can improve the
-  conditional expectation of any test given the parameters $\sigma(P)$, or the
-  parameters together with one slot $\sigma(P,y_l)$.  A short repackaging turns
+  conditional expectation of any test given the parameters $sigma(P)$, or the
+  parameters together with one slot $sigma(P,y_l)$.  A short repackaging turns
   the support-$k$ correlation on the saturated channel into a correlation
   supported on $k$ slots of the regrouped channel
-  $Z=\mathsf M_b(Y)$, where one slot flips from input to output.
+  $Z=mathsf M_b(Y)$,sfM_b(Y)$, where one slot flips from input to output.
 
-\item[(5)] Square argument and contradiction.
-  On the selected law of $Z$ we have an $I\times I$ array.  Row $b$ and column
-  $b$ both have the old law $\lambda_Y$; they meet at a common corner.  Saturation
+item[(5)] Square argument and contradiction.
+  On the selected law of $Z$ we have an $Itimes I$ array.  Row $b$ and column
+  $b$ both have the old law $lambda_Y$; they meet at a common corner.  Saturation
   gives a row--column identity for a bilinear pairing of two conditional
   expectations.  A Hilbert-space lemma on independent random vectors in $L^2$
   of a conditional fibre forces the pairing not to vanish.  Packaging it back
@@ -189,21 +189,21 @@ diving into details.
 
 
 Only input (B) is used at length 3 on an arbitrary system.  Everything for
-length $n\ge 4$ is either norm-convergence ($L^2$) or pointwise convergence on
+length $nge 4$ is either norm-convergence ($L^2$) or pointwise convergence on
 distal systems.
 
 ## Map of the paper
 
 {ll}
-\Ssec:inputs     & standard models, and inputs (A)--(C) stated precisely.\\
-\Ssec:selected   & the failure witness, the suspension, the selectors, the laws $\lambda_Y$.\\
-\Ssec:channels   & channels, channel conditioning, channel closure by tensor packing.\\
-\Ssec:correlations& minimum support, saturation, the regrouped channel $Z=\mathsf M_b(Y)$.\\
-\Ssec:square     & the square argument and the contradiction.\\
-\Ssec:packing    & appendix: the tensor packing theorem.
+Ssec:inputs     & standard models, and inputs (A)--(C) stated precisely.
+Ssec:selected   & the failure witness, the suspension, the selectors, the laws $lambda_Y$.
+Ssec:channels   & channels, channel conditioning, channel closure by tensor packing.
+Ssec:correlations& minimum support, saturation, the regrouped channel $Z=mathsf M_b(Y)$.sfM_b(Y)$.
+Ssec:square     & the square argument and the contradiction.
+Ssec:packing    & appendix: the tensor packing theorem.
 
 
-\smallskip
+smallskip
 A remark on what is and is not elementary.  The proof uses genuine results
 of ergodic theory (Furstenberg--Zimmer structure, Khintchine-style matrix
 estimates, sated-extension constructions).  We do not pretend otherwise.
@@ -216,52 +216,52 @@ check every quantitative estimate.
 # Dynamical inputs and standard models
 
 
-Fix $n\ge 2$ and put $I=\{1,...,n\}$.  We first record three facts
+Fix $nge 2$ and put $I={1,...,n}$.  We first record three facts
 from other papers, and set up the standard-model reduction.  The pointwise
 input on an arbitrary system is available only for three slots;
 for any finite number of slots we have only $L^2$ convergence.
 
 ## Reduction to a standard probability space
 
-Let $(X,\mathcal F,\mu,T)$ be the fixed system, and $f_1,...,f_n$ bounded
+Let $(X,mathcal F,calF,mu,T)$ be the fixed system, and $f_1,...,f_n$ bounded
 real functions.  Record all integer translates of the $f_j$ as coordinates
-in a single map $\Phi:X\to K^{\Z\times\{1,...,n\}}$, where $K=[-M,M]$ is
+in a single map $Phi:Xto K^{Ztimes{1,...,n}}$, where $K=[-M,M]$ is
 a compact real interval containing the ranges of all $f_j$.  The pushforward
-$\nu=\Phi_*\mu$ is a Borel probability on a compact metrizable space, and $T$
-pushes forward to the invertible shift $\sigma$ on $K^{\Z\times I}$, preserving
-$\nu$.  Since $T$ is mixing, so is $\sigma$.
+$nu=Phi_*mu$ is a Borel probability on a compact metrizable space, and $T$
+pushes forward to the invertible shift $sigma$ on $K^{Ztimes I}$, preserving
+$nu$.  Since $T$ is mixing, so is $sigma$.
 
 The original functions become coordinate functions on this factor.  A
 countable union of translated null sets runs through the change of
-representatives, so $\mu$-almost-everywhere convergence of the pushed-forward
+representatives, so $mu$-almost-everywhere convergence of the pushed-forward
 averages pulls back to the same statement on $X$.  From now on the
-system is standard: $(X,\mathcal F,\mu,T)$ is a standard Borel probability
+system is standard: $(X,mathcal F,calF,mu,T)$ is a standard Borel probability
 space with an invertible measure-preserving transformation.
 
 Throughout the proof, factors and conditional laws are understood modulo null
 sets; every function is taken to have a Borel representative.  For a pair of
 standard probability spaces, conditional expectations of bounded Borel
-functions with respect to sub-$\sigma$-algebras admit bounded Borel versions.
+functions with respect to sub-$sigma$-algebras admit bounded Borel versions.
 
 ## Input (A): mixing of every finite order
 
 This is the content of Rokhlin's multiple-mixing problem for one
 transformation.
 
-[INPUT A: higher-order mixing; \cite[Theorem 1.1]{MixingCompanion}]
+[INPUT A: higher-order mixing; cite[Theorem 1.1]{MixingCompanion}]
 
 Let $T$ be invertible and mixing on a probability space.  For every integer
-$s\ge 2$, every $g_1,...,g_s\in L^∞(\mu)$ and every $s$-tuple of
+$sge 2$, every $g_1,...,g_sin L^∞(mu)$ and every $s$-tuple of
 integers $t_1,...,t_s$,
-\[
- \int_X\prod_{j=1}^s g_j(T^{t_j}x)\dd\mu(x)→\prod_{j=1}^s\int_X g_j\dd\mu
-\]
-as $\min_{i\ne j}|t_i-t_j|\to∞$.  For $s=1$ the identity holds at
-every single time $t\in\Z$.
+[
+ int_Xprod_{j=1}^s g_j(T^{t_j}x)ddmu(x)→prod_{j=1}^sint_X g_jddmu
+]
+as $min_{ine j}|t_i-t_j|to∞$.  For $s=1$ the identity holds at
+every single time $tinZ$.
 
 
 The cited theorem is stated for sets and for ordered times.  Simple-function
-approximation, invariance of $\mu$ under $T$ and the finitely many orderings
+approximation, invariance of $mu$ under $T$ and the finitely many orderings
 of the times give the stated formulation.  No rate of mixing is asserted or
 used.
 
@@ -273,33 +273,33 @@ system, not just a mixing one.
 
 [Averaging profile]
 An averaging profile is a real compactly supported smooth function
-$\varphi$ on $\R$ whose mass is one and whose moments of order $1,2,3$ vanish:
-\[
- \int_\R\varphi(s)\dd s=1,\qquad
- \int_\R s^r\varphi(s)\dd s=0\quad(r=1,2,3).
-\]
-It is standard if, for some constant $G\ge 1$ depending on $\varphi$,
-\[
- \norm{\varphi^{(r)}}_∞\le (G(r+2))^{G(r+2)}\qquad(r\ge0).
-\]
+$varphi$ on $R$ whose mass is one and whose moments of order $1,2,3$ vanish:
+[
+ int_Rvarphi(s)dd s=1,qquad
+ int_R s^rvarphi(s)dd s=0quad(r=1,2,3).
+]
+It is standard if, for some constant $Gge 1$ depending on $varphi$,
+[
+ norm{varphi^{(r)}}_∞le (G(r+2))^{G(r+2)}qquad(rge0).
+]
 
 
-[INPUT B: smooth triple convergence; \cite[Prop 3.5, Lem 3.6, Lem 3.9]{SlopesCompanion}]
+[INPUT B: smooth triple convergence; cite[Prop 3.5, Lem 3.6, Lem 3.9]{SlopesCompanion}]
 
 Let $R$ be any invertible probability-preserving transformation of a
 standard probability space.  Let $b_1,b_2,b_3$ be distinct nonzero integers
 with greatest common divisor one, and let $g_1,g_2,g_3$ be bounded real
-functions.  Then for every standard averaging profile $\varphi$,
-\[
- V_k^\varphi(y)=\sum_{l\in\Z}2^{-k}\varphi(l/2^k)\prod_{j=1}^3 g_j(R^{b_jl}y)
-\]
-converges almost everywhere as $k\to∞$.
+functions.  Then for every standard averaging profile $varphi$,
+[
+ V_k^varphi(y)=sum_{linZ}2^{-k}varphi(l/2^k)prod_{j=1}^3 g_j(R^{b_jl}y)
+]
+converges almost everywhere as $kto∞$.
 
-Moreover, for each $c\in[1,2]$ and $ε>0$ there exists a standard
-averaging profile $\varphi$ with
-\[
- \norm{\varphi-c^{-1}\mathbf 1_{(0,c]}}_{L^1(\R)}<ε.
-\]
+Moreover, for each $cin[1,2]$ and $ε>0$ there exists a standard
+averaging profile $varphi$ with
+[
+ norm{varphi-c^{-1}mathbf 1_bf1_{(0,c]}}_{L^1(R)}<ε.
+]
 
 
 The second assertion approximates the uniform density on $(0,c]$ in $L^1$
@@ -316,61 +316,61 @@ the only point where any pointwise input for arbitrary systems is used.
 
 Let $R$ be any invertible probability-preserving transformation of a
 standard probability space.  For bounded real functions $g_1,...,g_d$
-with $d\le 3$, and distinct positive integer slopes $a_1,...,a_d$,
+with $dle 3$, and distinct positive integer slopes $a_1,...,a_d$,
 the averages
-\[
- \frac1N\sum_{l=1}^N\prod_{j=1}^dg_j(R^{a_jl}y)
-\]
+[
+ frac1Nsum_{l=1}^Nprod_{j=1}^dg_j(R^{a_jl}y)
+]
 converge almost everywhere along all positive integer lengths $N$.
 
 
 Proof. 
 Add constant-one functions and repeat slopes if $d<3$; both additions
 multiply the average by a constant and change nothing about its
-convergence.  Write the resulting slopes as $g\,b_1,g\,b_2,g\,b_3$ where
-$g\ge 1$ and $b_1,b_2,b_3$ are distinct positive integers with
-$\gcd(b_1,b_2,b_3)=1$.
+convergence.  Write the resulting slopes as $g,b_1,g,b_2,g,b_3$ where
+$gge 1$ and $b_1,b_2,b_3$ are distinct positive integers with
+$gcd(b_1,b_2,b_3)=1$.
 
-By scaling we may assume $|g_j|\le1$.  Put $S=R^g$ and
-\[
- F_l=\prod_{j=1}^3 g_j(S^{b_j l}y).
-\]
-Then $|F_l|\le 1$.  Let $N_k=\lfloor c2^k\rfloor$ for $c\in[1,2]$.
+By scaling we may assume $|g_j|le1$.  Put $S=R^g$ and
+[
+ F_l=prod_{j=1}^3 g_j(S^{b_j l}y).
+]
+Then $|F_l|le 1$.  Let $N_k=lfloor c2^krfloor$ for $cin[1,2]$.
 
 The smooth-to-ordinary step.
-For $\varphi$ a profile, we have
-\[
- V_k^\varphi(y)=\sum_{l\in\Z}2^{-k}\varphi(l/2^k)F_l.
-\]
-For $h_c=c^{-1}\mathbf 1_{(0,c]}$ set
-\[
- W_k(y)=\sum_{l\in\Z}2^{-k}h_c(l/2^k)F_l.
-\]
-Then $W_k=(1/N_k)\sum_{l=1}^{N_k}F_l+O(1/(c2^k))$ because $h_c$ is the
-normalized indicator of $(0,c]$, using $\lfloor c2^k\rfloor=N_k$ and the
+For $varphi$ a profile, we have
+[
+ V_k^varphi(y)=sum_{linZ}2^{-k}varphi(l/2^k)F_l.
+]
+For $h_c=c^{-1}mathbf 1_bf1_{(0,c]}$ set
+[
+ W_k(y)=sum_{linZ}2^{-k}h_c(l/2^k)F_l.
+]
+Then $W_k=(1/N_k)sum_{l=1}^{N_k}F_l+O(1/(c2^k))$ because $h_c$ is the
+normalized indicator of $(0,c]$, using $lfloor c2^krfloor=N_k$ and the
 error on the last $c2^k-N_k$ terms being at most $1/(c2^k)$.  Also
-\[
- |V_k^\varphi(y)-W_k(y)|\le 2^{-k}\sum_{l\in\Z}|h_c(l/2^k)-\varphi(l/2^k)|.
-\]
-By Riemann sums the right side tends to $\norm{h_c-\varphi}_1$ uniformly in
-$y$.  Now pick a sequence of profiles $\varphi^{(m)}$ with
-$\norm{h_c-\varphi^{(m)}}_1\to0$.  On a common conull set (intersection
+[
+ |V_k^varphi(y)-W_k(y)|le 2^{-k}sum_{linZ}|h_c(l/2^k)-varphi(l/2^k)|.
+]
+By Riemann sums the right side tends to $norm{h_c-varphi}_1$ uniformly in
+$y$.  Now pick a sequence of profiles $varphi^{(m)}$ with
+$norm{h_c-varphi^{(m)}}_1to0$.  On a common conull set (intersection
 of the conull sets supplied by Proposition~prop:input-smooth at each
-$m$) all $V_k^{\varphi^{(m)}}(y)$ converge.  For $m\to∞$ the above
+$m$) all $V_k^{varphi^{(m)}}(y)$ converge.  For $mto∞$ the above
 display and the triangle inequality show that $W_k(y)$ is Cauchy.  Since
 $W_k$ and the ordinary average differ by $O(2^{-k})$, the ordinary averages
-along $N_k=\lfloor c2^k\rfloor$ form a Cauchy sequence for $\mu$-almost
+along $N_k=lfloor c2^krfloor$ form a Cauchy sequence for $mu$-almost
 every $y$.  This holds for every $c$ in a countable dense subset of
 $[1,2]$.
 
 All lengths, not just dyadic.
-For any bounded sequence $|H_l|\le1$ and positive integers $M\le N$,
-\[
- |\frac1M\sum_{l=1}^M H_l-\frac1N\sum_{l=1}^N H_l|\le(2|N-M|)/(\max(N,M)).
-\]
-For $2^k\le N<2^{k+1}$ pick a finite rational grid $c\in[1,2]$ with mesh
-less than some $\eta$, and compare $N$ to the sequence values at
-$\lfloor c2^k\rfloor$.  The error is at most $O(\eta)$.  Letting $\eta\to0$
+For any bounded sequence $|H_l|le1$ and positive integers $Mle N$,
+[
+ |frac1Msum_{l=1}^M H_l-frac1Nsum_{l=1}^N H_l|le(2|N-M|)/(max(N,M)).
+]
+For $2^kle N<2^{k+1}$ pick a finite rational grid $cin[1,2]$ with mesh
+less than some $eta$, and compare $N$ to the sequence values at
+$lfloor c2^krfloor$.  The error is at most $O(eta)$.  Letting $etato0$
 on a countable union of scales preserves almost everywhere convergence.
 Therefore the average along every complex of lengths converges almost
 everywhere.
@@ -387,53 +387,53 @@ existence of the a.e. limit is.
 ## Input (C): Furstenberg--Zimmer structure facts
 
 The third input packages the standard relative ergodic theory.  Recall an
-extension $Y\to B$ of an invertible system $R$ is relatively weakly
-mixing if for bounded $f,g$ with $\E[f\mid B]=0$,
+extension $Yto B$ of an invertible system $R$ is relatively weakly
+mixing if for bounded $f,g$ with $E[fmid B]=0$,
 
- \frac1H\sum_{h=1}^H\norm{\E[\bar f\,(g\circ R^h)\mid B]}_2^2→0.
+ frac1Hsum_{h=1}^Hnorm{E[barbar ff,(gcirc R^h)mid B]}_2^2→0.
 
 It is relatively compact if the $L^∞(B)$-modules spanned by
-finite sums $\sum_j a_jF_j$ ($a_j\in L^∞(B)$, $F_j\in L^2$ fixed) are
+finite sums $sum_j a_jF_j$ ($a_jin L^∞(B)$, $F_jin L^2$ fixed) are
 dense.  A relatively distal extension is a tower of relatively
 compact extensions and inverse limits.  A system is distal if it is
 distal over the trivial factor.  A joining of two systems is an
 invariant probability on the product with the prescribed marginals.
 
-[INPUT C: relative structure; \cite[Prop 2.2]{FourfoldCompanion}]
+[INPUT C: relative structure; cite[Prop 2.2]{FourfoldCompanion}]
 
 The following hold for standard probability-preserving systems, without
 an ergodicity hypothesis.
 
-\item Every system $Y$ has a maximal measure-distal factor $W_Y$, and the
-  extension $Y\to W_Y$ is relatively weakly mixing.  Distal factors of any
+item Every system $Y$ has a maximal measure-distal factor $W_Y$, and the
+  extension $Yto W_Y$ is relatively weakly mixing.  Distal factors of any
   extension pull into $W_Y$.  Automorphisms commuting with $R$ preserve $W_Y$.
-\item Factors and joinings of distal systems are distal.  A distal system is
+item Factors and joinings of distal systems are distal.  A distal system is
   relatively distal over each factor.  Distality is equivalent for $R$ and any
   nonzero integer power.
-\item Relative weak mixing passes to nonzero integer powers.  The
+item Relative weak mixing passes to nonzero integer powers.  The
   conditional product of relatively weakly mixing fibre laws over an invariant
   joining of the base is relatively weakly mixing over that joining.
-\item A relatively weakly mixing extension and a relatively distal extension
+item A relatively weakly mixing extension and a relatively distal extension
   of the same base are relatively disjoint: every invariant joining identifying
   their bases equals the conditional product over that base.
-\item For a distal transformation $R$, every $d\ge 1$ and every tuple of
+item For a distal transformation $R$, every $dge 1$ and every tuple of
   bounded $g_1,...,g_d$, the averages
-  \[
-   \frac1N\sum_{l=1}^N\prod_{j=1}^dg_j(R^{jl}y)
-  \]
+  [
+   frac1Nsum_{l=1}^Nprod_{j=1}^dg_j(R^{jl}y)
+  ]
   converge almost everywhere.
 
-For a flow $Y$ (a jointly Borel action of $\R$), the maximal distal factor of
+For a flow $Y$ (a jointly Borel action of $R$), the maximal distal factor of
 time one is preserved by the flow, and its action at every nonzero rational
 time is distal.
 
 
 The cited results include the nonergodic form of the Furstenberg--Zimmer
 structure theorem [cite] and the pointwise theorem
-for distal systems \cite[Theorem C]{HuangShaoYe2019}; see
-\cite[Prop 2.2]{FourfoldCompanion} for the proof in the nonergodic setup.  In
+for distal systems cite[Theorem C]{HuangShaoYe2019}; see
+cite[Prop 2.2]{FourfoldCompanion} for the proof in the nonergodic setup.  In
 the flow assertion, invariance of $W_Y$ under every flow time is immediate
-because each time commutes with time one.  If $r=p/q\ne 0$ is rational, the
+because each time commutes with time one.  If $r=p/qne 0$ is rational, the
 $q$-th power of $S_Y^r$ equals $S_Y^p$, which is distal; hence $S_Y^r$ is
 distal on $W_Y$.
 
@@ -444,42 +444,42 @@ throughout.
 
 [Relative $L^2$ convergence]
 
-Suppose $Y\to B$ is a relatively weakly mixing extension of $R$.  For
+Suppose $Yto B$ is a relatively weakly mixing extension of $R$.  For
 distinct nonzero integers $a_1,...,a_d$ and bounded real $G_1,...,G_d$,
-\[
- \left\|\frac1N\sum_{l=1}^N(\prod_{i=1}^dG_i\circ R^{a_il}-\prod_{i=1}^d\E[G_i\mid B]\circ R^{a_il})\right\|_2→0.
-\]
+[
+ left|frac1Nsum_{l=1}^N(prod_{i=1}^dG_icirc R^{a_il}-prod_{i=1}^dE[G_imid B]circ R^{a_il})right|_2→0.
+]
 
 
 Proof. 
 By telescoping and scaling, it suffices to show the average of
-$v_l=\prod_i G_i\circ R^{a_il}$ tends to zero in $L^2$ when $|G_i|\le1$ for
+$v_l=prod_i G_icirc R^{a_il}$ tends to zero in $L^2$ when $|G_i|le1$ for
 all $i$ and one of them, say $G_s$, is centered over $B$.
 
-Induction on $d$.  For fixed $h\ge1$,
-\[
- F_{i,h}=G_ibar\,(G_i\circ R^{a_ih}),
-\]
+Induction on $d$.  For fixed $hge1$,
+[
+ F_{i,h}=G_ibar,(G_icirc R^{a_ih}),
+]
 so by invariance
-\[
- \ip{v_l}{v_{l+h}}=\int_Y F_{s,h}\prod_{i\ne s}F_{i,h}\circ R^{(a_i-a_s)l}\dd m_Y.
-\]
+[
+ ip{v_l}{v_{l+h}}=int_Y F_{s,h}prod_{ine s}F_{i,h}circ R^{(a_i-a_s)l}dd m_Y.
+]
 The induction hypothesis replaces the average over $l$ of the product over
-$i\ne s$ by its conditional expectation over $B$.  The remaining product is
+$ine s$ by its conditional expectation over $B$.  The remaining product is
 $B$-measurable and bounded by one; hence
-\[
- limsup_{N\to∞}|\frac1N\sum_{l=1}^N\ip{v_l}{v_{l+h}}|\le\norm{\E[F_{s,h}\mid B]}_1.
-\]
+[
+ limsup_{Nto∞}|frac1Nsum_{l=1}^Nip{v_l}{v_{l+h}}|lenorm{E[F_{s,h}mid B]}_1.
+]
 For $d=1$ this bound holds directly; for $d>1$ it is the induction output.
 The Hilbert-space van der Corput inequality bounds the size of the limit in
 $L^2$ by a constant times
-\[
- \frac1H+\frac1H\sum_{h=1}^H\norm{\E[F_{s,h}\mid B]}_1.
-\]
+[
+ frac1H+frac1Hsum_{h=1}^Hnorm{E[F_{s,h}mid B]}_1.
+]
 Relative weak mixing of $B⊆ Y$ over $R^{a_s}$ (Assertion~3 of
-Proposition~prop:input-structure) makes each $\norm{\E[F_{s,h}\mid B]}_1$
+Proposition~prop:input-structure) makes each $norm{E[F_{s,h}mid B]}_1$
 small in average, and Cauchy--Schwarz (since $F_{s,h}$ is bounded) makes the
-whole expression tend to zero as $H\to∞$.  This completes the induction.
+whole expression tend to zero as $Hto∞$.  This completes the induction.
 
 
 Inserting constant-one functions in omitted positions, and using the
@@ -493,143 +493,143 @@ original mixing system, the trivial factor already satisfies
 ## A first corollary for the original system
 
 Before we start the proof, we note the easy consequence we will use to
-produce the failure witness in \Ssec:selected.
+produce the failure witness in Ssec:selected.
 
 
-Suppose $T$ is mixing.  For any $f_1,...,f_n\in L^∞(\mu)$ with
-$\int f_i\dd\mu=0$ for some $i$, the averages $A_N(f_1,...,f_n)$ tend to
-zero in $L^2(\mu)$ as $N\to∞$.
+Suppose $T$ is mixing.  For any $f_1,...,f_nin L^∞(mu)$ with
+$int f_iddmu=0$ for some $i$, the averages $A_N(f_1,...,f_n)$ tend to
+zero in $L^2(mu)$ as $Nto∞$.
 
 
 Proof. 
 Apply Lemma~lem:relative-norm with $B$ the trivial factor, the
 identity transformation $R=T$, and $a_i=i$.  Because $T$ is mixing, the
 trivial factor is a relatively weakly mixing extension.  Then the product
-$\prod_i\E[f_i\mid B]=\prod_i\int f_i=0$, and the lemma follows.
+$prod_iE[f_imid B]=prod_iint f_i=0$, and the lemma follows.
 
 
 # The failure witness and selected laws
 
 
 Assume, toward a contradiction, that Theorem~thm:main fails for the
-fixed slopes $I=\{1,...,n\}$ and some $f_1,...,f_n\in L^∞(\mu)$.
+fixed slopes $I={1,...,n}$ and some $f_1,...,f_nin L^∞(mu)$.
 We extract a positive witness, move to the roof-one suspension $C$, define
 measurable averaging lengths $N_a$, and along a free ultrafilter obtain a
-probability law $\lambda_Y$ on $Y^I$ for every flow $Y$ over $C$.  The
-construction follows \cite[Sections 3--6]{FourfoldCompanion}, with the
+probability law $lambda_Y$ on $Y^I$ for every flow $Y$ over $C$.  The
+construction follows cite[Sections 3--6]{FourfoldCompanion}, with the
 extension and conditioning facts spelled out for every $n$.
 
 ## A positive witness at selected lengths
 
 Expanding each $f_j$ into its mean and centered part, and each centered part
 into real and imaginary parts, we find a nonempty $J_0⊆ I$ and real
-functions $f_i$, $i\in J_0$, with
-\[
- \int_Xf_i\,\dd\mu=0,\qquad \norm{f_i}_∞\le 1,
- \qquad
- B_L(x)=\frac1L\sum_{l=1}^L\prod_{i\in J_0}f_i(T^{il}x)
-\]
+functions $f_i$, $iin J_0$, with
+[
+ int_Xf_i,ddmu=0,qquad norm{f_i}_∞le 1,
+ qquad
+ B_L(x)=frac1Lsum_{l=1}^Lprod_{iin J_0}f_i(T^{il}x)
+]
 such that $B_L$ does not tend to zero on a set of positive measure.  Here and
 below $B_L$ has integer subscript.  Lemma~lem:norm-vanishes gives
-$B_L\to0$ in $L^2(\mu)$, so the failure is genuinely pointwise: $B_L$
+$B_Lto0$ in $L^2(mu)$, so the failure is genuinely pointwise: $B_L$
 oscillates on a set of positive measure, and its limit is $0$ in $L^2$.
-After changing the sign of one $f_i$, there exist $\delta>0$ and a
+After changing the sign of one $f_i$, there exist $delta>0$ and a
 measurable set $E_*⊆ X$ of positive measure with
 
- B_L(x)>2\delta\quadfor arbitrarily large L,\qquad
- x\in E_*.
+ B_L(x)>2deltaquadfor arbitrarily large L,qquad
+ xin E_*.
 
 
 ## The suspension and the selectors
 
 Set
 
- C=X\times[0,1),\qquad m_C=\mu ⊗ \dd u,
- \qquad
- S_C^t(x,u)=(T^{\lfloor u+t\rfloor}x,\{u+t\}),
+ C=Xtimes[0,1),qquad m_C=mu ⊗ dd u,
+ qquad
+ S_C^t(x,u)=(T^{lfloor u+trfloor}x,{u+t}),
 
-and write $\xi(x,u)=x$, $υ(x,u)=u$, viewing $υ$ also as a
-coordinate on $\R/\Z$.  Then $(C,m_C,S_C)$ is a measure-preserving flow.
+and write $xi(x,u)=x$, $υ(x,u)=u$, viewing $υ$ also as a
+coordinate on $R/Z$.  Then $(C,m_C,S_C)$ is a measure-preserving flow.
 
 [Selected positive witness]
 
-There are a finite set $F⊆[1,2]$ with $1\in F$, and measurable
-functions $N_a:C\to[2^a,∞)$, $a\in\N$, each independent of the
+There are a finite set $F⊆[1,2]$ with $1in F$, and measurable
+functions $N_a:Cto[2^a,∞)$, $ainN$, each independent of the
 phase coordinate, with
 
- N_a(x,u)\in\{c2^k:c\in F,\ k\ge a\},\qquad
- N_a(x,u)=2^a\ (x\notin E_*),
+ N_a(x,u)in{c2^k:cin F, kge a},qquad
+ N_a(x,u)=2^a (xnotin E_*),
 
-and such that $B_{\lfloor N_a(x,u)\rfloor}(x)>\delta$ on $E_*$.  Moreover
-there is a continuous nonnegative function $\Psi$ on $(\R/\Z)^I$ with
+and such that $B_{lfloor N_a(x,u)rfloor}(x)>delta$ on $E_*$.  Moreover
+there is a continuous nonnegative function $Psi$ on $(R/Z)^I$ with
 
- liminf_{a\to∞}\int_C\frac1{N_a(z)}
- \int_0^{N_a(z)}
- \Psi((υ(S_C^{it}z))_{i\in I})
- \prod_{i\in J_0}f_i(\xi(S_C^{it}z))\dd t\dd m_C(z)>0.
+ liminf_{ato∞}int_Cfrac1{N_a(z)}
+ int_0^{N_a(z)}
+ Psi((υ(S_C^{it}z))_{iin I})
+ prod_{iin J_0}f_i(xi(S_C^{it}z))dd tdd m_C(z)>0.
 
 
 
 Proof. 
 Length-change.
-For a bounded sequence $|H_l|\le1$ and positive integers $L\le M$,
-\[
- |\frac1L\sum_{l=1}^LH_l-\frac1M\sum_{l=1}^MH_l|
- \le(2|M-L|)/(\max(M,L)).
-\]
+For a bounded sequence $|H_l|le1$ and positive integers $Lle M$,
+[
+ |frac1Lsum_{l=1}^LH_l-frac1Msum_{l=1}^MH_l|
+ le(2|M-L|)/(max(M,L)).
+]
 Pick a finite grid $F⊆[1,2]$ containing $1$, so fine that for every
-integer $L$ there exist $c\in F$ and $k=\lfloor\log_2L\rfloor$ with
-$|L-c2^k|<\eta L$; choose $\eta$ so small that the length-change bound
-stays below $\delta$.  Then for every $x\in E_*$ and every $L$ at which
-$B_L(x)>2\delta$, the corresponding dyadic average
-$B_{\lfloor c2^k\rfloor}(x)$ exceeds $\delta$.
+integer $L$ there exist $cin F$ and $k=lfloorlog_2Lrfloor$ with
+$|L-c2^k|<eta L$; choose $eta$ so small that the length-change bound
+stays below $delta$.  Then for every $xin E_*$ and every $L$ at which
+$B_L(x)>2delta$, the corresponding dyadic average
+$B_{lfloor c2^krfloor}(x)$ exceeds $delta$.
 
 The measurable choice.
 Fix an enumeration of the countable set
-$\{(c,k):c\in F,\,k\in\N\}$.  For $a\in\N$ and $x\in E_*$, let $(c,k)$ be
-the first pair in this enumeration with $k\ge a$ and
-$B_{\lfloor c2^k\rfloor}(x)>\delta$.  On $E_*$ we set $N_a(x,u)=c2^k$;
-on $X∖ E_*$ we set $N_a=2^a$.  As a map $C\to[2^a,∞)$ this is
+${(c,k):cin F,,kinN}$.  For $ainN$ and $xin E_*$, let $(c,k)$ be
+the first pair in this enumeration with $kge a$ and
+$B_{lfloor c2^krfloor}(x)>delta$.  On $E_*$ we set $N_a(x,u)=c2^k$;
+on $X∖ E_*$ we set $N_a=2^a$.  As a map $Cto[2^a,∞)$ this is
 measurable and independent of $u$.  By construction
-$B_{\lfloor N_a(x,u)\rfloor}(x)>\delta$ on $E_*$.
+$B_{lfloor N_a(x,u)rfloor}(x)>delta$ on $E_*$.
 
 The phase witness.
-Choose continuous nonnegative circle functions $\alpha,\beta$, not
+Choose continuous nonnegative circle functions $alpha,beta$, not
 identically zero, supported inside $(1/4,1/2)$ and $(0,1/(4n))$
 respectively.  Put
 
- \Psi(v)=\alpha(2v_1-v_2)\beta(v_2-v_1),
- \qquad v=(v_i)_{i\in I}\in(\R/\Z)^I.
+ Psi(v)=alpha(2v_1-v_2)beta(v_2-v_1),
+ qquad v=(v_i)_{iin I}in(R/Z)^I.
 
-For $z=(x,u)$ and $t=l+s$ with $l\in\Z$, $0\le s<1$, the phase of slot $i$
-is $υ(S_C^{it}z)=\{u+is\}$.  The support of $\Psi$ then forces
-$0<u+is<1$ for every $i\in I$.  In particular, on the support of $\Psi$,
-$\xi(S_C^{it}z)=T^{il}x$ for every $i$.
+For $z=(x,u)$ and $t=l+s$ with $linZ$, $0le s<1$, the phase of slot $i$
+is $υ(S_C^{it}z)={u+is}$.  The support of $Psi$ then forces
+$0<u+is<1$ for every $iin I$.  In particular, on the support of $Psi$,
+$xi(S_C^{it}z)=T^{il}x$ for every $i$.
 
 Summing complete unit intervals of length $1$ in $t$ and using the support
-condition, we obtain, uniformly in $z$ and $L\ge2$,
-
- &\frac1L\int_0^L
- \Psi((υ(S_C^{it}z))_i)
- \prod_{i\in J_0}f_i(\xi(S_C^{it}z))\dd t
- \notag\\
- &\hspace{18mm}
- =\alpha(u)(\int_0^1\beta(s)\dd s)B_{\lfloor L\rfloor}(x)
+condition, we obtain, uniformly in $z$ and $Lge2$,
+
+ &frac1Lint_0^L
+ Psi((υ(S_C^{it}z))_i)
+ prod_{iin J_0}f_i(xi(S_C^{it}z))dd t
+ notag
+ &hspace{18mm}
+ =alpha(u)(int_0^1beta(s)dd s)B_{lfloor Lrfloor}(x)
  +O(L^{-1}).
  
 
 The error covers the incomplete last unit interval, the change of index
-summation from $0,...,\lfloor L\rfloor-1$ to $1,...,\lfloor L\rfloor$,
+summation from $0,...,lfloor Lrfloor-1$ to $1,...,lfloor Lrfloor$,
 and the normalisation.
 
 Integrating.
 On $E_*$, with $L=N_a(z)$, the integral of the main term in
 (eq:suspension-witness) is at least
-\[
- \delta\mu(E_*)(\int_0^1\alpha(u)\dd u)(\int_0^1\beta(s)\dd s)>0,
-\]
-since $B_{\lfloor N_a(z)\rfloor}>\delta$ on $E_*$.  On $X∖ E_*$, the
-function $N_a\equiv 2^a$ is independent of $u$, and $B_{2^a}\to0$ in $L^2(\mu)$
+[
+ deltamu(E_*)(int_0^1alpha(u)dd u)(int_0^1beta(s)dd s)>0,
+]
+since $B_{lfloor N_a(z)rfloor}>delta$ on $E_*$.  On $X∖ E_*$, the
+function $N_aequiv 2^a$ is independent of $u$, and $B_{2^a}to0$ in $L^2(mu)$
 by Lemma~lem:norm-vanishes; hence the contribution of this region
 tends to zero.  The uniform error is $O(2^{-a})$.  Taking $liminf$ over
 $a$ completes the proof.
@@ -637,79 +637,79 @@ $a$ completes the proof.
 
 ## Selected laws on every flow over $C$
 
-Fix the selectors and a free ultrafilter $\mathcal U$ on $\N$.
+Fix the selectors and a free ultrafilter $mathcal U$calU$ on $N$.
 
 [Flow over $C$; extensions]
 A flow over $C$ is a standard probability space $(Y,m_Y)$ with a
-jointly Borel probability-preserving action $S_Y^t$, $t\in\R$, and a
-probability-preserving factor map $\pi_Y:Y\to C$ with $\pi_YS_Y^t=S_C^t\pi_Y$
-mod $m_Y$ for each fixed $t$.  An extension $\rho:Y'\to Y$ of such
+jointly Borel probability-preserving action $S_Y^t$, $tinR$, and a
+probability-preserving factor map $pi_Y:Yto C$ with $pi_YS_Y^t=S_C^tpi_Y$
+mod $m_Y$ for each fixed $t$.  An extension $rho:Y'to Y$ of such
 flows is a probability-preserving factor map, equivariant modulo null sets,
-with $\pi_{Y'}=\pi_Y\circ\rho$ modulo null sets.  We do not require a common
+with $pi_{Y'}=pi_Ycircrho$ modulo null sets.  We do not require a common
 conull set on which every real-time factor identity holds; joint
 measurability and Fubini justify their uses in time integrals.
 
 
 [Selected joining laws]
 
-Every flow $Y$ over $C$ has a unique probability $\lambda_Y$ on $Y^I$
-satisfying, for all bounded measurable $G_i:Y\to\R$,
+Every flow $Y$ over $C$ has a unique probability $lambda_Y$ on $Y^I$
+satisfying, for all bounded measurable $G_i:YtoR$,
 
- \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).
+ int_{Y^I}prod_{iin I}G_i(y_i)ddlambda_Y
+ =lim_{atomathcal U}calUint_Yfrac1{N_a(pi_Yy)}
+ int_0^{N_a(pi_Yy)}prod_{iin I}G_i(S_Y^{it}y)dd tdd m_Y(y).
 
-Its marginals are $m_Y$, and $\lambda_Y$ is invariant under
-$(y_i)_{i\in I}\mapsto(S_Y^{it}y_i)_{i\in I}$ for every $t\in\R$.  If
-$\rho:Y'\to Y$ is an extension over $C$, then $(\rho^I)_*\lambda_{Y'}=\lambda_Y$.
+Its marginals are $m_Y$, and $lambda_Y$ is invariant under
+$(y_i)_{iin I}mapsto(S_Y^{it}y_i)_{iin I}$ for every $tinR$.  If
+$rho:Y'to Y$ is an extension over $C$, then $(rho^I)_*lambda_{Y'}=lambda_Y$.
 
 Countable inverse limits exist in this category.  The selected law at the
 limit projects to the selected law at each stage.  Pullbacks of the stage
-$\sigma$-algebras have dense union in $L^2(\lambda_Y)$ at the limit.
+$sigma$-algebras have dense union in $L^2(lambda_Y)$ at the limit.
 
 
 Proof. 
 Construction at each $a$.
 For each $a$, the expression before the ultralimit in (eq:selected-law)
-defines a probability measure $\lambda_{Y,a}$ on $Y^I$, since $N_a\circ\pi_Y$
-is measurable, $y\mapsto S_Y^{it}y$ is jointly Borel, and the integrand is
+defines a probability measure $lambda_{Y,a}$ on $Y^I$, since $N_acircpi_Y$
+is measurable, $ymapsto S_Y^{it}y$ is jointly Borel, and the integrand is
 bounded.
 
 Marginals.
-We show $\lambda_{Y,a}\to m_Y$ on marginals as $a\to∞$ along every
-bounded $G:Y\to\R$.  By Birkhoff applied to the auxiliary function
-$y\mapsto\int_0^1G(S_Y^{is}y)\dd s$ and to $S_Y^i$, the continuous-time
-averages $(1/L)\int_0^LG(S_Y^{it}y)\dd t$ converge almost everywhere as
-$L\to∞$, and their a.e. limit $\bar G$ satisfies
-$\int_Y\bar G\dd m_Y=\int_YG\dd m_Y$ since every deterministic average has
-this integral.  As $N_a\ge 2^a\to∞$, bounded convergence gives
-$\int_Y(1/N_a)\int_0^{N_a}G(S_Y^{it}y)\dd t\dd m_Y(y)\to\int_YG\dd m_Y$.
+We show $lambda_{Y,a}to m_Y$ on marginals as $ato∞$ along every
+bounded $G:YtoR$.  By Birkhoff applied to the auxiliary function
+$ymapstoint_0^1G(S_Y^{is}y)dd s$ and to $S_Y^i$, the continuous-time
+averages $(1/L)int_0^LG(S_Y^{it}y)dd t$ converge almost everywhere as
+$Lto∞$, and their a.e. limit $barbar G$G$ satisfies
+$int_Ybarbar GGdd m_Y=int_YGdd m_Y$ since every deterministic average has
+this integral.  As $N_age 2^ato∞$, bounded convergence gives
+$int_Y(1/N_a)int_0^{N_a}G(S_Y^{it}y)dd tdd m_Y(y)toint_YGdd m_Y$.
 
 Ultralimit.
 Embed $Y$ into a compact metrizable $K$ (as a Borel subset with $m_Y$
-extended by zero).  Along the ultrafilter $\mathcal U$,
-$\lambda_{Y,a}$ against continuous functions on $K^I$ converges to a
+extended by zero).  Along the ultrafilter $mathcal U$,calU$,
+$lambda_{Y,a}$ against continuous functions on $K^I$ converges to a
 positive normalised functional, hence a probability $K^I$.  Its marginals
-are $m_Y$, so it is concentrated on $Y^I$.  Denote it $\lambda_Y$.
+are $m_Y$, so it is concentrated on $Y^I$.  Denote it $lambda_Y$.
 
 Formula for all bounded tests.
-For bounded $G_i:Y\to\R$ with $|G_i|\le1$, approximate each by continuous
+For bounded $G_i:YtoR$ with $|G_i|le1$, approximate each by continuous
 functions on $K$ with the same bound in $L^1(m_Y)$ to within $ε$.
 The difference of the two product integrals (eq:selected-law) at stage
-$a$ is bounded by $\sum_iε$.  Under $\lambda_Y$ these errors are
-$L^1$-errors, and under $\lambda_{Y,a}$ they converge to those errors by
-marginal convergence.  Letting $ε\to0$ gives the formula.
+$a$ is bounded by $sum_iε$.  Under $lambda_Y$ these errors are
+$L^1$-errors, and under $lambda_{Y,a}$ they converge to those errors by
+marginal convergence.  Letting $εto0$ gives the formula.
 Product tests determine a unique probability on $Y^I$, giving uniqueness.
 All identities depend only on $G_i$ up to a.e. equivalence.
 
 Invariance.
-Fix $t_0\in\R$ and shift the time interval in (eq:selected-law) by
-$t_0$.  The selector $N_a\circ\pi_Y$ does not change under
-$S_Y^{t_0}$.  For $|G_i|\le1$ the error is at most $2|t_0|2^{-a}$,
+Fix $t_0inR$ and shift the time interval in (eq:selected-law) by
+$t_0$.  The selector $N_acircpi_Y$ does not change under
+$S_Y^{t_0}$.  For $|G_i|le1$ the error is at most $2|t_0|2^{-a}$,
 proving invariance under $S_Y^{t_0}$ in each slot.
 
 Functoriality.
-Substitute $G_i\circ\rho$ for $G_i$ in the formula and use $\pi_{Y'}=\pi_Y\rho$
+Substitute $G_icircrho$ for $G_i$ in the formula and use $pi_{Y'}=pi_Yrho$
 plus Fubini and equivariance.  The right-hand side of (eq:selected-law)
 is identical for $Y$ and $Y'$.
 
@@ -717,27 +717,27 @@ Inverse limits.
 For $Y_0← Y_1←...$, let $Y_∞$ be the full
 countable product of the underlying standard spaces with the consistent
 inverse-limit measure and coordinatewise flow.  Joint measurability is
-automatic for coordinatewise actions of $\R$ on a countable product.  The
+automatic for coordinatewise actions of $R$ on a countable product.  The
 projections are measurable, $S_{Y_∞}^t$ preserves $m_{Y_∞}$
 because every cylinder has the same measure as its preimage.  Stages are
 factor maps mod null sets; projection through $Y_0$ gives the factor
-$Y_∞\to C$.  Functoriality gives the claimed projections of stage
-selected laws.  The stage $\sigma$-algebras generate the product
-$\sigma$-algebra; pullbacks are increasing and generate $L^2(\lambda_Y)$,
-by the usual $L^2$ approximation by increasing $\sigma$-algebras.
+$Y_∞to C$.  Functoriality gives the claimed projections of stage
+selected laws.  The stage $sigma$-algebras generate the product
+$sigma$-algebra; pullbacks are increasing and generate $L^2(lambda_Y)$,
+by the usual $L^2$ approximation by increasing $sigma$-algebras.
 
 
 ## Distal parameters and small marginals
 
-For a flow $Y$ over $C$, let $\omega_Y:Y\to W_Y$ be its maximal measure-distal
-factor for time one, and $\nu_Y=(\omega_Y)_*m_Y$.  Write $m_{Y,w}$ for
-its conditional measures, so $m_Y=\int_{W_Y}m_{Y,w}\dd\nu_Y(w)$.  Under
-$\lambda_Y$ put
+For a flow $Y$ over $C$, let $omega_Y:Yto W_Y$ be its maximal measure-distal
+factor for time one, and $nu_Y=(omega_Y)_*m_Y$.  Write $m_{Y,w}$ for
+its conditional measures, so $m_Y=int_{W_Y}m_{Y,w}ddnu_Y(w)$.  Under
+$lambda_Y$ put
 
- P_i=\omega_Y(y_i),\qquad P=(P_i)_{i\in I}.
+ P_i=omega_Y(y_i),qquad P=(P_i)_{iin I}.
 
 By Assertion 1 of Proposition~prop:input-structure, $W_Y$ is preserved
-by every flow time.  Under an extension $\rho:Y'\to Y$, the pullback of $W_Y$
+by every flow time.  Under an extension $rho:Y'to Y$, the pullback of $W_Y$
 is a distal factor of $Y'$ for time one, hence is contained in $W_{Y'}$.
 Thus $P$ on $Y^I$ is a measurable function of the corresponding array on
 $(Y')^I$.
@@ -746,61 +746,61 @@ $(Y')^I$.
 
 For every flow $Y$ over $C$ and every bounded measurable $H_i$ on $W_Y$,
 
- \E_{\lambda_Y}\prod_{i\in I}H_i(P_i)
- =\lim_{L\to∞}\int_Y\frac1L\int_0^L
- \prod_{i\in I}H_i(\omega_Y(S_Y^{it}y))\dd t\dd m_Y(y).
+ E_{lambda_Y}prod_{iin I}H_i(P_i)
+ =lim_{Lto∞}int_Yfrac1Lint_0^L
+ prod_{iin I}H_i(omega_Y(S_Y^{it}y))dd tdd m_Y(y).
 
-For every nonempty $J⊆ I$ with $|J|\le 3$,
+For every nonempty $J⊆ I$ with $|J|le 3$,
 
- \law_{\lambda_Y}((y_i)_{i\in J}\mid P)
- =⊗_{i\in J}m_{Y,P_i}.
+ law_{lambda_Y}((y_i)_{iin J}mid P)
+ =⊗_{iin J}m_{Y,P_i}.
 
 
 
 Proof. 
 From discrete to continuous time.
-Write $t=l+s$ with $l\ge0$ integer and $0\le s<1$.  For each fixed $s$, a
+Write $t=l+s$ with $lge0$ integer and $0le s<1$.  For each fixed $s$, a
 continuous-time product becomes the discrete-time product at slopes $i$ with
-slot functions $G_i\circ S_Y^{is}$, evaluated at times $l\ge0$.  If the
+slot functions $G_icirc S_Y^{is}$, evaluated at times $lge0$.  If the
 discrete averages converge almost everywhere, Fubini and bounded convergence
 give that the $s$-average of them does.  Starting the discrete average at
 $l=0$ rather than $l=1$, and dropping an incomplete final unit interval,
 contributes an error that tends to zero.  The same argument applied to real
-$L\ge2$ in place of dyadic $N$ shows convergence along all real lengths.
+$Lge2$ in place of dyadic $N$ shows convergence along all real lengths.
 
 Deterministic parameters.
-Apply this reduction to the functions $H_i\circ\omega_Y$, which are fixed by
+Apply this reduction to the functions $H_icircomega_Y$, which are fixed by
 all flow times (Assertion 1 of Proposition~prop:input-structure);
 pointwise convergence on distal systems follows from Assertion 5.  Since
-$N_a\to∞$, bounded convergence with (eq:selected-law) gives
+$N_ato∞$, bounded convergence with (eq:selected-law) gives
 (eq:deterministic-parameters).
 
 Small marginals off the parameter.
-The extension $Y\to W_Y$ is relatively weakly mixing.  For $|J|\le3$,
-Lemma~lem:triple applied to the discrete averages at slopes $i\in J$
+The extension $Yto W_Y$ is relatively weakly mixing.  For $|J|le3$,
+Lemma~lem:triple applied to the discrete averages at slopes $iin J$
 shows that the $J$-marginal attains the integrated deterministic-length law.
 By Lemma~lem:relative-norm, the difference of the deterministic
-average $\prod_{i\in J}G_i(y_i)$ and the same product with each $G_i$
-replaced by $\E[G_i\mid W_Y]$ tends to zero in $L^2$.  Integrating in $s$
+average $prod_{iin J}G_i(y_i)$ and the same product with each $G_i$
+replaced by $E[G_imid W_Y]$ tends to zero in $L^2$.  Integrating in $s$
 gives
 
- \E_{\lambda_Y}\prod_{i\in J}G_i(y_i)
- =\E_{\lambda_Y}\prod_{i\in J}(\E_{m_Y}[G_i\mid W_Y](y_i)).
+ E_{lambda_Y}prod_{iin J}G_i(y_i)
+ =E_{lambda_Y}prod_{iin J}(E_{m_Y}[G_imid W_Y](y_i)).
 
 
 Small marginals given $P$.
 Multiplying $G_i$ by arbitrary bounded test functions of $P_i$ gives
 
- \law_{\lambda_Y}((y_i)_{i\in J}\mid(P_i)_{i\in J})
- =⊗_{i\in J}m_{Y,P_i}.
+ law_{lambda_Y}((y_i)_{iin J}mid(P_i)_{iin J})
+ =⊗_{iin J}m_{Y,P_i}.
 
-Now upgrade from $(P_i)_{i\in J}$ to the full array $P$.  Under
+Now upgrade from $(P_i)_{iin J}$ to the full array $P$.  Under
 $S_Y^{it}$ acting on slot $i$, the $J$-marginal system of
 (eq:small-own-parameters) is relatively weakly mixing over its own
 parameter law, by Assertions 2 and 3 of
 Proposition~prop:input-structure.  The full $P$-system is a joining of
 the distal systems $W_Y$ at integer slopes $i$ and is thereby distal.  Its
-joint law under $\lambda_Y$ is an invariant joining identifying that common
+joint law under $lambda_Y$ is an invariant joining identifying that common
 subarray; by relative disjointness (Assertion 4) the joining splits, which is
 exactly the assertion.  This completes the proof.
 
@@ -811,21 +811,21 @@ one chosen slot as the original system.
 
 [Normalized joining flow]
 
-For $b\in I$, define
+For $bin I$, define
 
- \mathsf M_b(Y)=(Y^I,\lambda_Y,
-   S_{\mathsf M_b(Y)}^t(y_i)_{i\in I}=(S_Y^{it/b}y_i)_{i\in I}).
+ mathsf M_b(Y)=(Y^I,sfM_b(Y)=(Y^I,lambda_Y,
+   S_{mathsf M_b(Y)}sfM_b(Y)^t(y_i)_{iin I}=(S_Y^{it/b}y_i)_{iin I}).
 
 This is a standard probability-preserving flow over $C$ via
-$(y_i)_{i\in I}\mapsto\pi_Y(y_b)$; the projection onto slot $b$ is an
+$(y_i)_{iin I}mapstopi_Y(y_b)$; the projection onto slot $b$ is an
 extension onto $Y$.  The array $P$ belongs to the maximal measure-distal
-factor of $\mathsf M_b(Y)$ for time one.
+factor of $mathsf M_b(Y)$sfM_b(Y)$ for time one.
 
 
 Proof. 
 Invariance follows from Proposition~prop:selected-law after rescaling
-the time parameter by $1/b$.  The factor map is $\pi_Y\circ p_b$.  Slot $b$
-has speed $1$ and marginal $m_Y$, giving $(y_i)_i\mapsto y_b$ as a factor
+the time parameter by $1/b$.  The factor map is $pi_Ycirc p_b$.  Slot $b$
+has speed $1$ and marginal $m_Y$, giving $(y_i)_imapsto y_b$ as a factor
 onto $Y$.  In slot $i$, the $W_Y$-factor is distal for time $i/b$, by
 Assertion 2 (equivalence for rational powers); the law of $P$ is an
 invariant joining of these rational-time distal systems, hence distal.
@@ -836,8 +836,8 @@ Maximality of the distal factor gives the last assertion.
 
 With the functions and phase test of Lemma~lem:selected-witness,
 
- \int_{C^I}\Psi((υ(z_i))_i)
- \prod_{i\in J_0}f_i(\xi(z_i))\dd\lambda_C>0.
+ int_{C^I}Psi((υ(z_i))_i)
+ prod_{iin J_0}f_i(xi(z_i))ddlambda_C>0.
 
 The phase array is a function of the distal parameter array of $C^I$.
 
@@ -845,7 +845,7 @@ The phase array is a function of the distal parameter array of $C^I$.
 Proof. 
 Continuous functions on the compact product of circles are uniformly
 approximable by finite sums of products of continuous one-coordinate
-functions.  Apply this to $\Psi$, use (eq:selected-law) for each
+functions.  Apply this to $Psi$, use (eq:selected-law) for each
 resulting product, then apply the strictly positive lower bound in
 (eq:selected-positive-limit).  The phase at time zero on $C$ is
 distal, hence lies in $W_C$.
@@ -860,34 +860,34 @@ unconditioned law is a product.  To control the dependence we pack the
 outputs into three groups.  Independence of the original spatial coordinate
 from any two groups will turn out to hold automatically, and this is enough
 to cancel the effect of selecting the length.  The construction adapts
-\cite[Section 5]{FourfoldCompanion}; we spell out every step.
+cite[Section 5]{FourfoldCompanion}; we spell out every step.
 
 ## Channels
 
 [Channel]
 
-A channel is a flow $Y$ over $C$ with factor $\pi_Y:Y\to C$, together
-with a finite list of probability-preserving maps $D_\gamma:Y\to C$ called
-outputs, and positive rational numbers $q_\gamma$, satisfying
+A channel is a flow $Y$ over $C$ with factor $pi_Y:Yto C$, together
+with a finite list of probability-preserving maps $D_gamma:Yto C$ called
+outputs, and positive rational numbers $q_gamma$, satisfying
 
- D_\gamma(S_Y^ty)=S_C^{q_\gamma t}D_\gamma(y)
+ D_gamma(S_Y^ty)=S_C^{q_gamma t}D_gamma(y)
 
 for each fixed $t$, mod $m_Y$.  Partition the output indices into three
 groups, numbered $1,2,3$; empty groups are allowed.  Set
-\[
- x_0=\xi\circ\pi_Y,\qquad x_\gamma=\xi\circ D_\gamma,\qquad
- d_Y=(x_\gamma)_\gamma,
-\]
-and for $H⊆\{1,2,3\}$ let $d_{Y,H}$ be the sublist of $d_Y$ from
-groups in $H$.  Let $V$ be the joint of the phase coordinates of $\pi_Y$
-and all $D_\gamma$ (i.e. $V=(υ\circ\pi_Y,(υ\circ D_\gamma)_\gamma)$).
+[
+ x_0=xicircpi_Y,qquad x_gamma=xicirc D_gamma,qquad
+ d_Y=(x_gamma)_gamma,
+]
+and for $H⊆{1,2,3}$ let $d_{Y,H}$ be the sublist of $d_Y$ from
+groups in $H$.  Let $V$ be the joint of the phase coordinates of $pi_Y$
+and all $D_gamma$ (i.e. $V=(υcircpi_Y,(υcirc D_gamma)_gamma)$).
 Require, under $m_Y$,
 
  
-  \law(d_Y\mid V)&=⊗_\gamma\mu,\\
-  \law(x_0,d_{Y,H}\mid V)&=
-    \mu ⊗ ⊗_{\gamma in groups H}\mu,
-    \qquad H⊊\{1,2,3\}.
+  law(d_Ymid V)&=⊗_gammamu,
+  law(x_0,d_{Y,H}mid V)&=
+    mu ⊗ ⊗_{gamma in groups H}mu,
+    qquad H⊊{1,2,3}.
  
 
 An empty spatial list has the one-point law.
@@ -913,14 +913,14 @@ strengthening holds there.
 
 
 Proof. 
-Choose a positive integer $h$ such that $hq_\gamma\in\Z$ for every output.
+Choose a positive integer $h$ such that $hq_gammainZ$ for every output.
 The phase vector $V$ is a factor fixed by time $h$, and its time-one action
 is distal, so $V$ belongs to $W_Y$.  For any spatial tuple appearing in
 (eq:channel-laws), its joint factor with $V$ has product law over
 $V$; the joint factor with $W_Y$ is a joining of this over $V$.
 
 At time $h$, the spatial coordinates evolve by a product of positive
-integer powers of $T$ (the powers are $hq_\gamma\in\N$ and $h$ for $x_0$).
+integer powers of $T$ (the powers are $hq_gammainN$ and $h$ for $x_0$).
 This product is mixing by INPUT A.  The assertion follows first for
 product tests, by mixing of each power, and then for all bounded tests by
 approximation.  Consequently the spatial factor is relatively weakly
@@ -938,40 +938,40 @@ supersystem, gives the stronger conditioning on $W_{Y'}$ for the extension
 $Y'$.
 
 
-## The multiplicative construction $\mathsf M_b$
+## The multiplicative construction $mathsf M_b$sfM_b$
 
-Recall from Lemma~lem:normalized-joining that for $b\in I$,
-$\mathsf M_b(Y)$ has state space $Y^I$, measure $\lambda_Y$, flow
-$(y_i)_{i\in I}\mapsto(S_Y^{it/b}y_i)_{i\in I}$, and factor $y_b\to Y$.
+Recall from Lemma~lem:normalized-joining that for $bin I$,
+$mathsf M_b(Y)$sfM_b(Y)$ has state space $Y^I$, measure $lambda_Y$, flow
+$(y_i)_{iin I}mapsto(S_Y^{it/b}y_i)_{iin I}$, and factor $y_bto Y$.
 
 [Channel closure]
 
-Let $Y$ be a channel and $b\in I$.  Define data on $Z=\mathsf M_b(Y)$ by:
+Let $Y$ be a channel and $bin I$.  Define data on $Z=mathsf M_b(Y)$sfM_b(Y)$ by:
 
-\item the original factor is $x_0(y_b)=\xi\circ\pi_Y(y_b)$;
-\item the outputs are all $D_\gamma(y_i)$, $i\in I$, each in the old group
-  of $\gamma$;
-\item the speed of $D_\gamma(y_i)$ is $iq_\gamma/b$.
+item the original factor is $x_0(y_b)=xicircpi_Y(y_b)$;
+item the outputs are all $D_gamma(y_i)$, $iin I$, each in the old group
+  of $gamma$;
+item the speed of $D_gamma(y_i)$ is $iq_gamma/b$.
 
 Then these data make $Z$ a channel over $C$.
 
 
 The proof is the technical heart of the paper, and is given in
-\Ssubsec:closure-proof.  It uses INPUT A (mixing of every order) at
-length $n$ and the tensor packing theorem stated in \Ssec:packing.
+Ssubsec:closure-proof.  It uses INPUT A (mixing of every order) at
+length $n$ and the tensor packing theorem stated in Ssec:packing.
 
 [Copied output law]
 
 For every channel $Y$, the copied spatial list
-$(d_Y(y_i))_{i\in I}$ under $\lambda_Y$ has its full product law
-conditional on $P=(\omega_Y(y_i))_{i\in I}$.
+$(d_Y(y_i))_{iin I}$ under $lambda_Y$ has its full product law
+conditional on $P=(omega_Y(y_i))_{iin I}$.
 
 
 Proof. 
 Apply Proposition~prop:channel-closure and
-Lemma~lem:channel-conditioning to $\mathsf M_b(Y)$.  By
+Lemma~lem:channel-conditioning to $mathsf M_b(Y)$.sfM_b(Y)$.  By
 Lemma~lem:normalized-joining, the old parameter array $P$ belongs to
-the maximal distal factor of $\mathsf M_b(Y)$.  Conditioning the
+the maximal distal factor of $mathsf M_b(Y)$.sfM_b(Y)$.  Conditioning the
 constant product law (eq:channel-laws) down to $P$ proves the
 assertion.
 
@@ -979,13 +979,13 @@ assertion.
 ## Proof of channel closure
 
 
-This is \Ssubsec:closure-proof, the proof of
+This is Ssubsec:closure-proof, the proof of
 Proposition~prop:channel-closure.
 
 Proof. [Proof of Proposition~prop:channel-closure]
-Each new output $D_\gamma(y_i)$ is probability-preserving and has positive
-rational speed $iq_\gamma/b$, so the equivariance (eq:output-equivariance)
-holds with $q=iq_\gamma/b$.  It remains to prove the two conditional laws in
+Each new output $D_gamma(y_i)$ is probability-preserving and has positive
+rational speed $iq_gamma/b$, so the equivariance (eq:output-equivariance)
+holds with $q=iq_gamma/b$.  It remains to prove the two conditional laws in
 (eq:channel-laws).
 
 Reductions.
@@ -1006,47 +1006,47 @@ Time intervals and offsets.
 Use the selected-law formula (eq:selected-law) with base flow being
 $Y$, and condition on the old phase vector $V$ under $m_Y$.  Fix an integer
 $h>0$ clearing all output speed denominators, and write $t=hl+s$ with
-$l\ge0$ an integer and $0\le s<h$.  For fixed $V,s$, all phase weights in
-the test are independent of $l$.  The spatial function for output $\gamma$
+$lge0$ an integer and $0le s<h$.  For fixed $V,s$, all phase weights in
+the test are independent of $l$.  The spatial function for output $gamma$
 at slot $i$ is evaluated at
 
- T^{e_\gamma i l+a_{\gamma,i}}x_\gamma,\qquad e_\gamma=hq_\gamma\in\N.
+ T^{e_gamma i l+a_{gamma,i}}x_gamma,qquad e_gamma=hq_gammainN.
 
-The offsets $a_{\gamma,i}$ depend on $V,s$ and range over a fixed finite
+The offsets $a_{gamma,i}$ depend on $V,s$ and range over a fixed finite
 set, by (eq:suspension).  If the original-factor test is present, its
 argument is $T^{hbl+a_0}x_0$, with the same finite-offset property.
 Equivariance and Fubini justify these formulas inside the time integrals.
 
-On an event where the selected length is $N=c2^k$ ($c\in F$, $k\ge a$), put
-$m=\lfloor c2^k/h\rfloor$.  Replacing the normalized time integral by the
+On an event where the selected length is $N=c2^k$ ($cin F$, $kge a$), put
+$m=lfloor c2^k/hrfloor$.  Replacing the normalized time integral by the
 average over $m$ complete intervals, followed by integration against
 $ds/h$, has error at most $2h2^{-a}$.  For all sufficiently large $a$,
-$m\ge1$ on each admissible event.
+$mge1$ on each admissible event.
 
 Case 1: a proper set of groups (or all outputs $1$).
-Suppose some proper subset $H⊊\{1,2,3\}$ is used together with
+Suppose some proper subset $H⊊{1,2,3}$ is used together with
 $x_0$.  Given $V$, the relevant old output coordinates have product law
 and are independent of $x_0$, by (eq:channel-laws).  The selector
 depends on $V,x_0$ only.  Suppose at least one output occurrence is centered.
 For fixed $V,s,x_0$, integrate the spatial output product at index $l$.
-The expectation factors over the old output indices $\gamma$.  For any
+The expectation factors over the old output indices $gamma$.  For any
 index with a centered occurrence, the times (eq:channel-times) in
-distinct slots separate as $l\to∞$.  Mixing of all orders (INPUT A)
+distinct slots separate as $lto∞$.  Mixing of all orders (INPUT A)
 makes the coordinate integral tend to zero (each factor bounded, one
 centered so the product law mean is $0$).  Convergence is uniform over the
-finitely many offset patterns and finitely many $V$-sections.  A Ces\`aro
+finitely many offset patterns and finitely many $V$-sections.  A Ces`aro
 limit therefore gives zero for the full average.  All other factors have
-absolute value $\le1$, so bounded convergence completes this case.
+absolute value $le1$, so bounded convergence completes this case.
 
 If, on the other hand, every output test is $1$, the only centered factor
-is $x_0$ (bounded by $1$ and $\int x_0\dd\mu=0$).  Birkhoff's theorem for
+is $x_0$ (bounded by $1$ and $int x_0ddmu=0$).  Birkhoff's theorem for
 $T^{hb}$ applied to $x_0$ gives
-\[
- \frac1m\sum_{l<m}x_0(T^{hbl+a_0}x_0)→0
-\]
-for $\mu$-almost every $x_0$, simultaneously for the finitely many offsets
-$a_0$.  Since $N_a\to∞$ pointwise, the selected lengths share this
-limit.  The conditional law of $x_0$ given $V$ is $\mu$, so bounded
+[
+ frac1msum_{l<m}x_0(T^{hbl+a_0}x_0)→0
+]
+for $mu$-almost every $x_0$, simultaneously for the finitely many offsets
+$a_0$.  Since $N_ato∞$ pointwise, the selected lengths share this
+limit.  The conditional law of $x_0$ given $V$ is $mu$, so bounded
 convergence completes this case.
 
 Case 2: full output list with a centered occurrence per group.
@@ -1055,119 +1055,119 @@ one group has no centered occurrence, every spatial test in that group is
 $1$, and the preceding argument applies.  Thus suppose each group contains
 a centered occurrence.  For fixed $V,s$, write the spatial product at index
 $l$ as
-\[
- u_l\in L^2(Ω_1),\qquad v_l\in L^2(Ω_2),\qquad w_l\in L^2(Ω_3),
-\]
-where $Ω_j$ carries the product measure with one copy of $\mu$ for each
+[
+ u_lin L^2(Ω_1),qquad v_lin L^2(Ω_2),qquad w_lin L^2(Ω_3),
+]
+where $Ω_j$ carries the product measure with one copy of $mu$ for each
 old output in group $j$.  These are real unit-vectors (functions bounded by
 $1$).  Given $V$, the old spatial output law is the product of these three
 spaces.  The vectors depend on $V,s$ only through the finite offset patterns.
 
 Almost orthogonality with a bounded-degree graph.
-For every $\eta>0$, after omitting finitely many initial indices, the
+For every $eta>0$, after omitting finitely many initial indices, the
 families $(u_l)$ and $(v_l)$ each have absolute pair inner products at most
-$\eta$ outside a graph of bounded maximum degree $L_0$.  All bounds are
+$eta$ outside a graph of bounded maximum degree $L_0$.  All bounds are
 uniform in the offset patterns.  To see this, in each of the two groups
-choose a centered occurrence at some output index $\gamma$.  Its coordinate
+choose a centered occurrence at some output index $gamma$.  Its coordinate
 integral in a pair inner product is small by INPUT A whenever the times in
 the two copies of (eq:channel-times) are sufficiently separated.  The
 times within each copy separate once $l,l'$ exceed a fixed threshold.
 Failure of separation between copies requires one of finitely many
 inequalities
-\[
- |e_\gamma i l-e_\gamma i'l'+a'|\le R.
-\]
+[
+ |e_gamma i l-e_gamma i'l'+a'|le R.
+]
 For fixed $l$, each inequality permits only a bounded number of integers
-$l'$, because $e_\gamma,i'>0$; and symmetrically.  Their union gives a
+$l'$, because $e_gamma,i'>0$; and symmetrically.  Their union gives a
 bounded-degree graph.  Coordinate integrals for other output indices are
-bounded by $1$.  The separation threshold and $L_0$ may depend on $\eta$;
+bounded by $1$.  The separation threshold and $L_0$ may depend on $eta$;
 no rate of mixing is needed.
 
 Conditioning on a selected length.
 For fixed selector index $a$, partition into the events $N_a=c2^k$,
-$c\in F$, $k\ge a$, choosing one representation when a length has more
+$cin F$, $kge a$, choosing one representation when a length has more
 than one.  Given $V$, write their probabilities as $p_{k,c}(V)$.  On one
 such event of probability $p>0$, the conditional spatial law of any two
 groups, and of each single group, is unchanged.  Indeed the event is
 determined by $V,x_0$, which is independent of any proper set of groups by
 (eq:channel-laws).  The full three-group law may change.  Its
-density $\rho$ relative to the product law satisfies
+density $rho$ relative to the product law satisfies
 
- 0\le p\rho\le1,\qquad\int p\rho=p,\qquad\norm{p\rho}_2\le\sqrt p.
+ 0le prhole1,qquadint prho=p,qquadnorm{prho}_2lesqrt p.
 
-The bound $0\le p\rho\le1$ is the absolute-continuity statement; the second
+The bound $0le prhole1$ is the absolute-continuity statement; the second
 identity is the normalisation; the third is Cauchy--Schwarz.  These hold
 for almost every $V$, and the length events are countable.
 
 The rank bound.
 The rank parameter must meet two requirements: the packing bound must be
 $o(m)$, and the reciprocal bounds $1/d_m$ must be summable over dyadic
-length scales.  For $m=\lfloor c2^k/h\rfloor$ set
+length scales.  For $m=lfloor c2^k/hrfloor$ set
 
- d_m=\lceil(1+\log m)^{3/2}\rceil,\qquad z_l=u_l ⊗  v_l\quad(0\le l<m).
+ d_m=lceil(1+log m)^{3/2}rceil,qquad z_l=u_l ⊗  v_lquad(0le l<m).
 
-Let $\Pi_m$ be the orthogonal projection onto up to $d_m$ leading
-eigenvectors of the positive covariance operator $\sum_{l<m}z_lz_l^*$,
+Let $Pi_m$ be the orthogonal projection onto up to $d_m$ leading
+eigenvectors of the positive covariance operator $sum_{l<m}z_lz_l^*$,
 taking its full range if its rank is smaller.
 
-Fix $0<ε\le1$ and then fix $0<\eta\leε^2/10^4$ in the
+Fix $0<εle1$ and then fix $0<etaleε^2/10^4$ in the
 preceding graph bound.  Theorem~thm:packing of the Appendix gives
 
- \frac1m\sum_{l<m}\norm{\Pi_mz_l}\leε+o(1).
+ frac1msum_{l<m}norm{Pi_mz_l}leε+o(1).
 
 Here $o(1)$ tends to $0$ uniformly over the offset patterns.  Indeed, after
 the fixed initial indices are removed, the number of projection norms at
 least $ε$ is at most
-\[
+[
  C(1+L_0)ε^{-2}
- \exp\!(Cε^{-2}√(d_m)\log(2+d_m))=o(m).
-\]
-The removed indices contribute $o(1)$.  Crucially, $ε,\eta$ and
-$L_0$ are fixed before $m\to∞$.
+ exp!(Cε^{-2}√(d_m)log(2+d_m))=o(m).
+]
+The removed indices contribute $o(1)$.  Crucially, $ε,eta$ and
+$L_0$ are fixed before $mto∞$.
 
 The projected terms.
 Cauchy--Schwarz under the conditional event law uses its unchanged
 two-group and one-group marginals.  Then, term by term, including the event
 probability $p$,
-\[
- |p\int(\Pi_mz_l)(\omega_1,\omega_2)\,w_l(\omega_3)\,\rho(\omega)\dd\omega|
- \le p\norm{\Pi_mz_l}_2\norm{w_l}_2
- \le p\norm{\Pi_mz_l}_2.
-\]
+[
+ |pint(Pi_mz_l)(omega_1,omega_2),w_l(omega_3),rho(omega)ddomega|
+ le pnorm{Pi_mz_l}_2norm{w_l}_2
+ le pnorm{Pi_mz_l}_2.
+]
 Thus the projected contributions total at most $p(ε+o(1))$.
 
 The residual vectors.
-For the residual vectors $z_l'=(1-\Pi_m)z_l$, set
-$A=(\ip{z_l'}{z_{l'}'})_{l,l'<m}$ and $B=(\ip{w_l}{w_{l'}})_{l,l'<m}$.
+For the residual vectors $z_l'=(1-Pi_m)z_l$, set
+$A=(ip{z_l'}{z_{l'}'})_{l,l'<m}$ and $B=(ip{w_l}{w_{l'}})_{l,l'<m}$.
 The covariance trace is at most $m$, so the remaining eigenvalues of the
 covariance operator, and hence the nonzero eigenvalues of $A$, are at most
 $m/d_m$.  If the rank is at most $d_m$, the residual is zero.  Also, $B$ is
 positive semidefinite with trace at most $m$.  Therefore
-\[
- \left\|\frac1m\sum_{l<m}z_l' ⊗  w_l\right\|^2
- =(\tr(AB))/(m^2)\le\frac1{d_m}.
-\]
+[
+ left|frac1msum_{l<m}z_l' ⊗  w_lright|^2
+ =(tr(AB))/(m^2)lefrac1{d_m}.
+]
 By (eq:channel-density), the residual contribution, including the
 event probability, is at most $√(p/d_m)$.  Multiplication by the fixed
 phase test of absolute value at most $1$ does not increase either bound.
 
 Summing over lengths.
-Write $m(k,c)=\lfloor c2^k/h\rfloor$.  For fixed $h$ this is comparable to
-$2^k$ uniformly in $c\in F$.  Since $\sum_{k,c}p_{k,c}(V)=1$,
+Write $m(k,c)=lfloor c2^k/hrfloor$.  For fixed $h$ this is comparable to
+$2^k$ uniformly in $cin F$.  Since $sum_{k,c}p_{k,c}(V)=1$,
 Cauchy--Schwarz gives
 
- \sum_{k\ge a,\ c\in F}\sqrt{\frac{p_{k,c}(V)}{d_{m(k,c)}}}
- \le
- (\sum_{k\ge a,\ c\in F}\frac1{d_{m(k,c)}})^{1/2}→0.
+ sum_{kge a, cin F}sqrt{frac{p_{k,c}(V)}{d_{m(k,c)}}}
+ le
+ (sum_{kge a, cin F}frac1{d_{m(k,c)}})^{1/2}→0.
 
 The series converges because its summands are $O(k^{-3/2})$ thanks to the
 rank bound (eq:channel-dimension).  The projected contributions have
-total $limsup\leε$, by (eq:channel-projected-average) and
+total $limsupleε$, by (eq:channel-projected-average) and
 the same probability identity.  These estimates are uniform in $V,s$ and may
 be integrated.  There is no measurable-choice issue for the projections:
 choose one for each of the countably many sizes and finitely many offset
-patterns.  Letting $a\to∞$ in (eq:selected-law) and then
-$ε\to0$ proves the required vanishing.
+patterns.  Letting $ato∞$ in (eq:selected-law) and then
+$εto0$ proves the required vanishing.
 
 Both laws in (eq:channel-laws) follow.