64c3814
==== TABLE OF CONTENTS ====
Triple-ergodic-averages-with-distinct-integer-slopes//main.tex
Triple-ergodic-averages-with-distinct-integer-slopes//references.tex
Triple-ergodic-averages-with-distinct-integer-slopes//sections/analytic.tex
Triple-ergodic-averages-with-distinct-integer-slopes//sections/arithmetic.tex
Triple-ergodic-averages-with-distinct-integer-slopes//sections/completion.tex
Triple-ergodic-averages-with-distinct-integer-slopes//sections/introduction.tex
Triple-ergodic-averages-with-distinct-integer-slopes//sections/local.tex
Triple-ergodic-averages-with-distinct-integer-slopes//sections/transfer.tex
===========================
==== BEGIN Triple-ergodic-averages-with-distinct-integer-slopes//main.tex ====
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\hypersetup{pdftitle={Triple ergodic averages with distinct integer slopes},pdfauthor={OpenAI}}
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\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
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\title[Triple ergodic averages with distinct slopes]{Triple ergodic averages with distinct integer slopes}
\author{OpenAI}
\date{October 4, 2026}
\begin{document}
\begin{abstract}
For every invertible mixing probability-preserving transformation, triple
ergodic averages of bounded measurable functions along any three pairwise
distinct nonzero integer slopes converge almost everywhere to the product of
their integrals. The slopes may be positive or negative.
\end{abstract}
\maketitle
\input{sections/introduction}
\input{sections/local}
\input{sections/transfer}
\input{sections/analytic}
\input{sections/arithmetic}
\input{sections/completion}
\input{references}
\end{document}
==== END Triple-ergodic-averages-with-distinct-integer-slopes//main.tex ====
==== BEGIN Triple-ergodic-averages-with-distinct-integer-slopes//references.tex ====
\begin{thebibliography}{JKRW}
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I. Assani, \emph{Multiple recurrence and almost sure convergence for
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(1998), 111--124.
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V. Bergelson, \emph{Weakly mixing PET}, Ergodic Theory Dynam. Systems
\textbf{7} (1987), no.~3, 337--349.
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G.~D. Birkhoff, \emph{Proof of the ergodic theorem}, Proc. Natl. Acad.
Sci. USA \textbf{17} (1931), no.~12, 656--660.
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J. Bourgain, \emph{Double recurrence and almost sure convergence},
J. Reine Angew. Math. \textbf{404} (1990), 140--161.
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A.~P. Calder\'on, \emph{Ergodic theory and translation-invariant
operators}, Proc. Natl. Acad. Sci. USA \textbf{59} (1968), no.~2,
349--353.
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J.-M. Derrien and E. Lesigne, \emph{Un th\'eor\`eme ergodique
polyn\^omial ponctuel pour les endomorphismes exacts et les
$K$-syst\`emes}, Ann. Inst. H. Poincar\'e Probab. Statist.
\textbf{32} (1996), no.~6, 765--778.
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theorem of Szemer\'edi on arithmetic progressions}, J. Analyse Math.
\textbf{31} (1977), 204--256.
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Y. Gutman, W. Huang, S. Shao, and X. Ye,
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certain weakly mixing systems}, Acta Math. Sin. (Engl. Ser.)
\textbf{34} (2018), no.~1, 79--90.
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B. Host and B. Kra, \emph{Nonconventional ergodic averages and
nilmanifolds}, Ann. of Math. (2) \textbf{161} (2005), no.~1, 397--488.
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\emph{Oscillation in ergodic theory}, Ergodic Theory Dynam. Systems
\textbf{18} (1998), no.~4, 889--935.
\bibitem[LSS]{LSS}
J. Leng, A. Sah, and M. Sawhney, \emph{Quasipolynomial bounds on the
inverse theorem for the Gowers $U^{s+1}[N]$-norm}, preprint,
\href{https://arxiv.org/abs/2402.17994v3}{arXiv:2402.17994v3},
August 10, 2024.
\bibitem[H]{H}
OpenAI, \emph{An $L^3$ bound for the trilinear Hilbert transform},
OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/An-L3-bound-for-the-trilinear-Hilbert-transform-October-5-2026/paper.pdf}{OAI:An-L3-bound-for-the-trilinear-Hilbert-transform-October-5-2026},
2026.
\bibitem[V]{V}
OpenAI, \emph{Pointwise convergence of triple ergodic averages for
mixing transformations}, OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/pointwise-triple-ergodic-averages-mixing-transformations.pdf}{OAI:Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026},
2026.
\bibitem[Tao]{Tao}
T. Tao, \emph{Cancellation for the multilinear Hilbert transform},
Collect. Math. \textbf{67} (2016), 191--206.
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T. Ziegler, \emph{Universal characteristic factors and Furstenberg
averages}, J. Amer. Math. Soc. \textbf{20} (2007), no.~1, 53--97.
\end{thebibliography}
==== END Triple-ergodic-averages-with-distinct-integer-slopes//references.tex ====
==== BEGIN Triple-ergodic-averages-with-distinct-integer-slopes//sections/analytic.tex ====
\section{Quadratic structure and localization for the new slopes}
\label{sec:analytic}
We begin the proof of Theorem~\ref{thm:local} by establishing the
quadratic detection and localization estimates for the configuration
\(y_j=x+b_jt\). These estimates supply the input to the short-block
and frequency arguments of Section~\ref{sec:arithmetic}. The relevant
constructions are those of \cite[Sections~3--6]{H} and
\cite[Sections~4--6]{V}. We retain their interval projections and
stopping constructions, and prove below the coefficient-dependent
identities and estimates on which those constructions rely.
Throughout this section, the slopes are fixed. Constants may depend on
them and on the admissibility parameters. A rational number has height
at most \(H\) if its numerator and positive denominator in reduced form
are at most \(H\) in absolute value; matrix heights are entrywise.
We use the quantitative convention of \cite[Section~3.1]{H}: a
quasipolynomial bound in parameters \(H_1,\ldots,H_m\ge2\) is a bound
of the form
\[
\exp\!\left(C_{q,b}\left(1+\sum_{a=1}^m\log H_a\right)^C\right).
\]
The degree \(C\) is chosen independently of \(q\), the number of
depths, and the input schedules. A fixed number of such constructions
may be composed. Constructions repeated a variable number of times
retain the explicit count estimates in H and V.
\subsection{Local bounds and the chart class}
For a scalar or Hilbert-valued function, write
\(\norm z_{p,I}=(|I|^{-1}\int_I\norm{z(y)}^p\,dy)^{1/p}\),
with the essential-supremum convention for \(p=\infty\).
Norms without an interval subscript use ordinary Lebesgue measure.
The elementary bound used throughout the argument is unchanged except
for its constant.
\begin{lemma}[Local absolute bound]\label{lem:local-absolute}
If \(1\le p_j\le\infty\) and \(\sum_jp_j^{-1}\le2\), then
\[
|I|^{-2}\int_{\R^2}|w_I(x,t)|
\prod_{j=0}^3\norm{z_j(x+b_jt)}\,dx\,dt
\le C_b\prod_{j=0}^3\norm{z_j}_{p_j,I}.
\]
The assertion also holds for Hilbert-valued inputs, using their norms
in the integrand.
\end{lemma}
\begin{proof}
For two distinct roles \(i,j\), the map
\((x,t)\mapsto(y_i,y_j)\) has determinant \(b_j-b_i\ne0\).
This proves the endpoint with \(p_i=p_j=1\) and the other exponents
infinite. The support and kernel bounds give the all-infinite
endpoint, and normalized norm monotonicity gives the endpoints with
only one exponent equal to one. Log-convexity of the positive integral
fills the stated polytope, as in \cite[Lemma~2.1]{H}.
\end{proof}
On the support of a local kernel, all four coordinates lie in an
interval of length \(r=|I|\). Consequently
\begin{equation}
|t|\le\frac{r}{\max_j b_j-\min_j b_j}\le\frac r3.
\label{eq:analytic-step-support}
\end{equation}
In any pair of coordinate variables, the rescaled kernel thus has
support in a fixed box and the same type of derivative bounds. Its
integral in either coordinate with the other fixed is zero, by the
corresponding one-coordinate marginal hypothesis. These facts also
give \(C_N(q)\lesssim_b N\) and preserve the normalization-removal
argument of \cite[Lemma~2.4]{H}.
We recall the concrete class in which the quadratic structure is
detected. Write \(e(u)=\exp(2\pi i u)\). A \emph{quadratic chart atom}
is a function
\begin{equation}
\phi(y)=e(\lambda y^2+\beta y)
\sum_{n\in\Z^d}\chi(\theta y-n)
e\bigl((\theta y)^TC(\theta y-n)\bigr),
\label{eq:analytic-chart}
\end{equation}
where \(\theta\in\R^d\), \(\lambda,\beta\in\R\), \(C\in\mathbb Q^{d\times d}\),
and \(\chi\) is smooth and compactly supported. An overall constant
of modulus one is allowed. We use exactly the complexity convention
of \cite[Definition~4.1]{H}: for complexity \(H\ge e^2\),
\(d\le\log H\), the height of \(C\) is at most \(H\),
\(\operatorname{supp}\chi\subset[-H,H]^d\), and
\[
\norm{\partial^\alpha\chi}_\infty
\le H^{c_{\rm ch}(|\alpha|+1)}
(|\alpha|+1)^{c_{\rm ch}(|\alpha|+1)}
\]
with a fixed chart constant \(c_{\rm ch}\). The real oscillation
parameters \(\lambda,\beta,\theta\) are unrestricted. A bounded atom
has supremum norm at most one; any normalization factor is put in its
external scalar coefficient.
The elementary operations, Weyl alternative, and chart expansion of
\cite[Lemmas~4.2--4.4]{H} concern this class itself. Their proofs
therefore apply without a coefficient change. We will need the precise
splitting supplied by the Weyl alternative. In its multivariable
version replace \(\theta y\) by \(Vz\), and the ordinary quadratic
phase by \(z^T\Lambda z\), where \(\Lambda\) is real symmetric and
\(z\) has at most three coordinates. Put
\[
C_{\rm sy}=\tfrac12(C+C^T),\qquad
C_{\rm sk}=\tfrac12(C-C^T),\qquad
\operatorname{sym}M=\tfrac12(M+M^T).
\]
Let \(Q(z)\) be this multivariable chart. Suppose \(0<\delta<1/2\)
and a Lipschitz function \(\vartheta\), supported in a fixed box
with supremum and Lipschitz norms at most \(H\), satisfies
\[
\sup_\xi\left|\int\vartheta(z)Q(z)e(-\xi^Tz)\,dz\right|\ge\delta.
\]
The Weyl alternative then supplies a rational subspace \(U\), the
orthogonal projection \(P_f\) onto \(U\), \(P_s=1-P_f\), and the bounds
\begin{equation}
\begin{gathered}
(C-C^T)|_{U\times U}=0,\qquad
V_f=P_fV,\quad V_s=P_sV,\qquad \norm{V_s}\le H',\\
\Lambda_*=
\Lambda+\tfrac12V^TC_{\rm sy}V
+\operatorname{sym}(V_f^TC_{\rm sk}V_s),
\qquad \norm{\Lambda_*}\le H'.
\label{eq:analytic-splitting}
\end{gathered}
\end{equation}
The projection heights are at most \(H'\), and \(H'\) is
quasipolynomial in \(H,\delta^{-1}\). The subsequent
chart expansion has the absolute coefficient and pooled Fourier-tail
bounds of \cite[Lemma~4.4]{H}. These are the bounds used below, including
when expansions must be summed over depths.
\subsection{Separation along the configuration and local detection}
The first coefficient-dependent step moves two structured inputs into
the other two coordinates and the step variable. This is the form of
separation needed for both detection and the later predictor estimates.
\begin{lemma}[Progression separation]\label{lem:progression-separation}
Let \(i,k,a,l\) be the four distinct roles and let \(\phi_i,\phi_k\)
be bounded chart atoms of complexity at most \(H\). On
\(y_j=x+b_jt\), their product has an absolutely convergent expansion
\[
\phi_i(y_i)\phi_k(y_k)
=\sum_\nu a_\nu\psi_{a,\nu}(y_a)
\psi_{l,\nu}(y_l)Q_\nu(t)
\]
with bounded chart atoms as factors and the same shell bounds as
\cite[Lemma~4.5]{H}, with constants depending on \(b\).
In particular, for a quasipolynomial \(D=D_b(H)\), a dyadic Fourier
shell of size \(T\ge1\) has absolute coefficient sum at most
\(DT^{-100}\), and its factors have complexity at most \((DT)^C\).
\end{lemma}
\begin{proof}
First separate \(\phi_i\), temporarily suppressing its linear phase.
Choose the fixed rationals
\[
\alpha_i=1,\quad\alpha_k=0,\quad
\alpha_a=\frac{b_i-b_l}{b_l-b_a},\quad
\alpha_l=\frac{b_a-b_i}{b_l-b_a}.
\]
They satisfy \(\sum_j\alpha_j=\sum_jb_j\alpha_j=0\). Insert the
smooth integer partitions of unity from H at
\(\theta y_a,\theta y_l,\theta t\), and write the lifts as
\(d_j=\theta y_j-m_j\) and \(d'=\theta t-m\). For \(j=i,l\), set
\[
k_j=d_j-d_a-(b_j-b_a)d',\qquad k_a=0.
\]
These carries are integer vectors. The support restrictions bound
them by the chart budget, with a fixed slope factor. Conversely,
\(k_i,m_a,m\) determine \(m_i\) uniquely. Each remaining specified
carry equality is imposed exactly by a smooth cutoff equal to one at
the specified integer vector and zero at all other integer vectors.
For \(S_\alpha=\sum_j\alpha_jb_j^2\), direct expansion gives
\begin{equation}
\begin{split}
\sum_j\alpha_j\bigl(\lambda y_j^2+(\theta y_j)^TCd_j\bigr)
={}&S_\alpha\bigl(\lambda t^2+(\theta t)^TCd'\bigr)\\
&+\sum_j\alpha_j(\theta y_j)^TCk_j.
\end{split}
\label{eq:analytic-separation-identity}
\end{equation}
Indeed, substituting \(d_j=d_a+(b_j-b_a)d'+k_j\) cancels the
common-lift and \(x\)-terms. Here
\(S_\alpha=(b_i-b_a)(b_i-b_l)\). Solving for the phase in role
\(i\) leaves chart matrices \(-\alpha_aC,-\alpha_lC,S_\alpha C\)
in the three desired variables. For fixed carries, the last sum is
linear in \(x,t\), and hence splits between \(y_a,y_l\); so does the
suppressed linear phase.
The coupled amplitude is smooth in \(d_a,d_l,d'\). Its Fourier
expansion on a larger fixed box separates those variables, exactly as
in the proof of \cite[Lemma~4.5]{H}. The new substitutions cost at
most \(C_b^n\) in derivatives of order \(n\), and their rational
heights grow by fixed factors. The possible integer choices increase
by at most \(C_b^{O(d)}\). Since \(d\le\log H\), these changes are
within the quasipolynomial chart budget. H's integration by parts,
with derivative order exceeding the Fourier dimension by at least
\(110\), gives the stated absolute shell bound. Separate the second
atom with \(\alpha_i=0\) and combine the factors by direct sums of
horizontal coordinates. This completes the expansion.
\end{proof}
For one separated term, let
\[
A_r=\sup_{\xi\in\R}
\left|\int_\R Q(ru)\zeta(u)e(-\xi u)\,du\right|,
\]
where \(\zeta\) is smooth, nonnegative, equal to one on
\([-1/3,1/3]\), and has \(\norm\zeta_1\le1\), as in
\cite[Section~5.4]{V}. The two-input form in the remaining roles
has norm at most \(C_bA_r\) on normalized \(L^2(I)\) inputs.
To see this, use
\[
t=\frac{y_l-y_a}{b_l-b_a}.
\]
Fourier-expand the smooth normalized kernel in those two coordinates.
Each term is a modulated convolution with a fixed dilation of
\(Q(t)\zeta(t/r)\), so Plancherel gives the claimed bound. If
\(A_r\ge\delta\), the Weyl alternative and chart expansion further
give the finite smooth separation in \cite[Lemma~4.6]{H}: for every
\(0<\varepsilon<1/2\), the two-coordinate kernel has a separated
approximation with uniform error at most \(\varepsilon\), whose
number of terms, atom complexities, and absolute coefficient sum
are quasipolynomial in \(H,\delta^{-1},\varepsilon^{-1}\).
The retained phases are
\(\lambda_*t^2+\gamma t\), with \(|\lambda_*|r^2\) bounded by the
stated budget, and the linear phase splits as
\[
e(\gamma t)=e\left(-\frac{\gamma y_a}{b_l-b_a}\right)
e\left(\frac{\gamma y_l}{b_l-b_a}\right).
\]
Thus the fixed rational-height description of the effective quadratic
and linear frequencies in that lemma is preserved.
\begin{lemma}[Local detection]\label{lem:local-detection}
Fix \(q>2\) and \(C_1\ge1\). Suppose that the four inputs
\(R,h,u,v\), in any slot order, satisfy
\[
\norm R_{2,I}\le C_1,\qquad \norm h_{q,I}\le1,\qquad
\norm u_{2,I},\norm v_{2,I}\le1,
\qquad |H_I(R,h,u,v)|>\delta,
\]
where \(0<\delta<1/2\). There is a bounded chart atom \(\phi\)
of complexity at most \(D\) such that
\[
\left|\frac1{|I|}\int_I R(y)\overline{\phi(y)}\,dy\right|
\ge D^{-1},\qquad
D\le\exp\!\left(C_{q,b}
(1+\log C_1+\log(1/\delta))^C\right).
\]
The degree \(C\) is independent of \(q\).
\end{lemma}
\begin{proof}
We verify the progression estimate in \cite[Lemma~4.11]{H} and then
apply its detection proof. Normalize \(I=[0,1)\). Extend the kernel
to the same fixed circle used there. By
\eqref{eq:analytic-step-support}, its real support fits in the
periodic \((x,t)\)-box, and all real \(y_j\) on that support already
belong to \(I\). Hence zero-extended inputs on the circle give the
correct original integral, even when some slopes are large or negative.
The kernel's absolute Fourier coefficient sum is uniformly bounded.
The integer left inverse in Lemma~\ref{lem:algebra} distributes a
circle character with indices \((p,q')\) into slot characters with
indices \(p\mathbf1_{j=0}+q'e_j\). To a character-weighted
progression average apply H's three Cauchy--Schwarz steps, taking
Haar differences in the directions \((-b_j,1)\). The last cube has
three independent increments multiplied by nonzero differences
\(b_i-b_j\). Each such integer multiplication pushes circle Haar
measure to itself. The last integral is therefore the eighth power
of the same circle \(U^3\) norm as in H. Character multiplication
preserves that norm, the \(L^2\) norm, and the supremum norm. This
proves H's local generalized von Neumann estimate; its accompanying
two-\(L^1\) bound follows from Lemma~\ref{lem:local-absolute}.
The continuous inverse theorem and bounded-input decomposition in
\cite[Sections~4.5--4.6]{H} concern functions on this circle and its
chart atoms, and require no change. In the proof of
\cite[Lemma~4.12]{H}, clip the three opposing inputs at height \(B\).
The local absolute bound gives total error at most
\(C_bC_1B^{-(q-2)/q}\). Taking
\(B\) of order \((C_bC_1/\delta)^{q/(q-2)}\), apply that
bounded-input decomposition twice. The error estimates use the
generalized von Neumann and two-\(L^1\) bounds just proved.
Two bounded atoms remain opposite \(R\) and an input of normalized
\(L^2\) norm at most one. Lemma~\ref{lem:progression-separation}
and the preceding single-scale separation give a finite sum of
products of one-coordinate moments, with controlled absolute
coefficient sum. One moment of \(R\) consequently has the asserted
size. These are a fixed number of quasipolynomial constructions.
The clipping exponent contributes only a \(q\)-dependent multiplier
to a logarithm, so the final logarithmic degree is independent of
\(q\), as required.
\end{proof}
\subsection{The base reduction and inputs that vary with depth}
Detection is used in H and V to construct finite approximation spaces
on interval trees. We recall their meaning before verifying the scale
estimates. A column with label \(v\), born on an interval \(J_v\),
is an augmented function
\[
\Phi_v(y)=(\phi_v(y),\sqrt{\epsilon_v}\,\mathbf u_v),
\qquad y\in J_v,
\]
where \(\phi_v\) is a scalar chart atom, \(\epsilon_v>0\), and the
\(\mathbf u_v\) are distinct orthonormal private coordinates. Only
the first, physical coordinate enters \(H_I\). A local projection
on \(I\) is the orthogonal projection onto constant-coefficient
combinations of the available columns restricted to \(I\). Lists
are inherited on descendants. Accordingly the direct-sum projections
on successively finer interval partitions have nested ranges. The
private-coordinate, compression, and height-bin estimates of
\cite[Section~3]{H} depend only on these definitions and the column
budgets, and apply unchanged.
There are two coefficient-dependent scale estimates in the base
construction. For two fixed chart atoms, the separation above reduces
the other two inputs to a kernel of the form
\(w_I(x,t)Q(t)\), with fixed bounded one-coordinate multipliers.
The proof of \cite[Lemma~5.1 and Corollary~5.2]{H} expands \(Q\),
groups nearby linear frequencies, and uses the zero marginals of
\(w_I\). In coordinates \((y_a,y_l)\), its double primitive has
compact support in an interior rectangle. Fourier expansion followed
by one derivative in each coordinate expresses it as a sum of tensor
products of mean-zero smooth bumps. In role \(p\in\{a,l\}\), the
resulting scalar rows have fixed bounded carrier functions
\[
m_{\nu,p}(y)=e(\nu y)\psi_p(y),
\qquad \nu=\pm\frac{\gamma}{b_l-b_a},
\]
where \(\psi_p\) is the fixed separated multiplier in that role.
These are the only coefficient replacements in the row estimate.
They alter frequency separations and rational heights by fixed
factors. H's height-bin argument, ordinary Bessel estimate, and
compression step therefore give its absolute scale sum and finite
predictor list with the same quantitative orders, including the
pooled-tail bound needed for summation over all depths. The finer
partition comparison in \cite[Lemma~5.3]{H} uses the resulting
one-coordinate predictor moments, so its projection-increment proof
applies with the same replacement.
For the second estimate, fix three bounded chart columns and take
the transpose in role \(l\) at \(y+h\). Its other coordinates are
\[
y+h+(b_j-b_l)t\qquad(j\ne l).
\]
After squaring the normalized \(L^2\) norm, the six atoms form a
multivariable chart in \((h,t,t')\). At a fixed path point \(y\),
all parameters before multiplication by \(r\) or \(r^2\) remain
fixed as the depth changes. H's Weyl alternative, chart expansion,
and height-bin argument therefore apply. Away from the bounded
number of intermediate scales, the bounded effective quadratic
parameters are tiny and each retained \(t\)-frequency is tiny or
large. Smoothness handles the large frequencies. Removing the tiny
oscillations leaves exactly
\[
\int_\R w_I(y+h-b_lt,t)\,dt=0.
\]
This proves the active-scale count and the small-norm scale sum in
\cite[Lemma~5.7]{H}, with constants depending on \(b\).
For scheduled inputs we use \cite[Section~4]{V}. Its row estimates
concern Hilbert-space operators on refining interval partitions,
evaluated on inputs at prescribed nondecreasing read depths. Their
hypotheses are support on the input partition, representation of the
adjoint ranges on the observation partition by the inherited columns,
the stated order of successive partitions, and an ordinary Bessel
bound before evaluation on the varying inputs. None involves the
maps \(y_j\). Restriction to a new spatial root preserves the
pointwise block bound in Theorem~\ref{thm:local}. The original
\(W_j\) and original depth schedules are retained; subsequences may
have empty intersections with some blocks, as allowed in V's row
argument. Its nearest-endpoint extension is retained as well.
The smooth rows in \cite[Lemma~4.6]{V} require the ordinary Bessel
bound just verified, with the fixed carriers above. Thus the
scheduled scale-sum and predictor results
\cite[Lemma~5.3 and Corollary~5.4]{V}, and its delayed replacement
Lemma~5.5, hold with the new coefficients. The remaining steps of
its Section~5 use the same local detection, fixed-cell fits,
projection energies, and staggered pair comparisons. A finer
projection in a comparison is still applied to the input at the
prescribed read depth, and its row estimate has no factor depending
on \(W_j\).
We now describe the output of the base construction. For the
fixed-input argument, embed the scalar input as
\(F_{j,s}=(f_j,0)\), independent of \(s\), on a subtree satisfying
\(\norm{f_j}_{q,I}\le1\) on every cell, as in
\cite[Proposition~2.3]{H}. For the scheduled argument, put
\(F_{j,s}=(z_{j,s},0)\) with the inputs of Theorem~\ref{thm:local}.
The adaptive fits and two-earliest-increments expansion of
\cite[Lemmas~5.4--5.5]{H}, or \cite[Sections~5.1--5.2]{V}, express
the original form, at any finite accuracy cutoff, in tuples with
two differences of fitted projections and two original-input
residuals, together with a remainder whose absolute length-weighted
sum tends to zero on the fixed finite tree. Each tuple has an
earlier and a later accuracy index, and a choice of the four roles. Their
exact staggered ordering is retained from those results.
High-position truncation, using the predictor estimates above, is
\cite[Proposition~5.6]{H} or \cite[Sections~5.5--5.6]{V}. It assigns
each original summation cell to a segment root \(R\) with inherited
projection families; write \(\mathcal I_R\) for the cells assigned
to \(R\). Auxiliary projection cells are not additional summands.
Fix one tuple and let \(k\) be its later accuracy logarithm.
The two positions called low in this construction are denoted
\(i,j\), and the other two by \(a,l\). On a cell of depth \(s\),
the resulting arguments satisfy
\begin{equation}
\begin{gathered}
Y_{p,I}=B_{p,I}+U_{p,I},\qquad U_{i,I}=U_{j,I}=0,\\
B_{p,I}=\sum_{h=1}^{m_p}\varepsilon_{p,h}Q^{p,h}_{s}F_{p,s},
\qquad m_p\le2,\quad \varepsilon_{p,h}\in\{-1,1\},\\
U_{p,I}=(P^{\rm new}_{p,s+K_0}-P^{\rm new}_{p,s})F_{p,s}
\quad(p=a,l).
\end{gathered}
\label{eq:analytic-basic-arguments}
\end{equation}
All expressions are restricted to \(I\); a projection indexed by
a finer depth is the direct sum on that finer partition. The
\(Q^{p,h}\) and \(P^{\rm new}_p\) are the inherited basic families
just constructed, and \(K_0\) is their positive integer delay.
Their column sizes, inverse private weights, scalar complexities,
coefficient sums, and path counts have a joint budget
\(e^k\le D_{\rm base}\le\exp(C_{q,b}k^C)\).
On every segment they give
\begin{equation}
\begin{gathered}
\norm{B_{p,I}}_{2,I},\ \norm{U_{p,I}}_{2,I},\
\norm{Y_{p,I}}_{2,I}\le C_{q,b},\qquad
\Lambda_k\le C_{q,b}k^C,\\
\sum_{I\in\mathcal I_R}|I|\norm{U_{p,I}}_{2,I}^2
\le\Lambda_k^2|R|\quad(p=a,l).
\end{gathered}
\label{eq:analytic-basic-energy}
\end{equation}
One may take \(\Lambda_k\) to dominate
\(C_{q,b}(1+K_0)\log(2D_{\rm base})\). For the fixed-input
argument the smaller bound in H also suffices.
For noninitial accuracy levels, every pair of full \(Y\) arguments
has form norm at most \(\eta=\exp(-k^\theta)\), and
\(|H_I(Y_I)|\le\eta\). Here the pair norm is the supremum against
normalized \(L^2(I)\) inputs in the complementary two positions;
\(0<\theta<1\) is fixed before \(q\). Define \(\mathcal A_R\)
to consist of those cells for which some available triple of basic
scalar columns has transpose norm greater than
\(D_{\rm base}^{-C_{\rm act}}\), with the fixed sufficiently large
exponent in H or V. The active-scale estimate gives
\(\sum_{I\in\mathcal A_R}\mathbf1_I\le D_{\rm base}^C\).
There are three remaining patterns: \(z^\varnothing=B\),
\(z^a\) replaces only role \(a\) of \(B\) by \(U_a\), and
\(z^l\) replaces only role \(l\) by \(U_l\). Put
\begin{equation}
\mathfrak M_R(z)=\sum_{I\in\mathcal A_R}|I|
\min\{\eta,|H_I(z_I)|\}.
\label{eq:analytic-base-target}
\end{equation}
The output of \cite[Proposition~5.8]{H} and
\cite[Proposition~5.6]{V} is
\begin{equation}
\sum_{I\in\mathcal I_R}|I|\,|H_I(Y_I)|
\le C_{q,b}\eta\Lambda_k^2|R|+e^{-k}|R|
+\sum_{z\in\{z^\varnothing,z^a,z^l\}}\mathfrak M_R(z).
\label{eq:analytic-active-reduction}
\end{equation}
Indeed the term with both \(U\) factors uses the full complementary
pair bound and \eqref{eq:analytic-basic-energy}. The inactive
triple scale sum bounds the omitted cells. On the others, apply
\(\min(\eta,\sum x_m)\le\sum\min(\eta,x_m)\) to the full
tuple before separating its patterns. No individual pattern is
asserted to have the full-pair smallness.
For each fixed tuple the segment roots have total length bounded
by a fixed multiple of the original root length, and the sum of
the high-truncation replacement errors is at most \(C_{q,b}e^{-k}\)
times that length. These are the segment-packing assertions in
the cited base constructions. The finitely many initial accuracy
levels use the active path bound and local sizes, without the
factor \(\eta\), and hence have depth-independent bounds. All
these estimates are valid anew on a normalized subroot with its
inherited lists and original input schedule. We have therefore
reduced both input classes to estimating
\eqref{eq:analytic-base-target} uniformly over these concrete base
patterns.
\subsection{Curvature tags and separation of distinct tags}
We now localize the base arguments according to their quadratic
structure. Recall the nonzero integers \(c_j\) of
Lemma~\ref{lem:algebra}. A column in role \(i\) has
\emph{normalized curvature tag} \(v=(\theta_v,C_v,\lambda_v)\)
when its quadratic chart phase is written as
\[
c_i\bigl(\lambda_v y^2+(\theta_vy)^TC_v(\theta_vy-n)\bigr).
\]
Its linear modulation is not part of its tag. This is H's
normalization with \((-1,3,-3,1)\) replaced by \(c\). Division by
the fixed nonzero \(c_i\) preserves the chart budgets.
At length \(r\), the difference data of two tags \(v,v'\) are
\begin{equation}
V'=r\binom{\theta_v}{\theta_{v'}},\qquad
C'=\operatorname{diag}(C_v,-C_{v'}),\qquad
\Lambda'=r^2(\lambda_v-\lambda_{v'}).
\label{eq:analytic-tag-difference}
\end{equation}
The ambient graph in \cite[Section~6.1]{H} joins two tags when these
data admit \eqref{eq:analytic-splitting} with bounds at most the
chosen edge height \(H_{\rm ed}\); equal tags have distance zero.
This graph includes all hypothetical tags within the basic budget.
Its distances depend on tags and depth, and do not increase when
the interval length is halved.
\begin{lemma}[Separation of distinct tags]\label{lem:tag-separation}
For single-column complexities at most \(\exp(C_{q,b}k^C)\) and
\(\log(1/\tau)\le C_{q,b}k^C\), the edge height can be chosen with
\(\log H_{\rm ed}\le C_{q,b}k^C\) so that tags at graph distance
greater than one have both of the following properties. Their
same-slot normalized physical inner product is at most \(\tau\)
in absolute value. In distinct slots, their form norm against two
arbitrary normalized \(L^2(I)\) spectators is at most \(\tau\).
\end{lemma}
\begin{proof}
For the first assertion use the Weyl alternative on the conjugate
product, after the smoothing of the interval indicator in
\cite[Lemma~6.1]{H}. Its fixed nonzero factor \(c_i\) changes
the allowed rational heights by a fixed amount, and multiplying
back gives the splitting for the tag difference.
For distinct source slots \(i,j\), let the spectators be at
\(a,l\), and write the centered source coordinates in terms of
the normalized spectator coordinates \(z,w\). Their linear
parts are \(r(\xi_pz+\zeta_pw)\), where
\[
\xi_p=\frac{b_l-b_p}{b_l-b_a},\qquad
\zeta_p=\frac{b_p-b_a}{b_l-b_a}\qquad(p=i,j).
\]
These are nonzero fixed rationals. The cancellation of polynomials
of degree at most two gives
\begin{equation}
c_i\xi_i\zeta_i=-c_j\xi_j\zeta_j=:\kappa\ne0.
\label{eq:analytic-mixed-curvature}
\end{equation}
Use the difference data \eqref{eq:analytic-tag-difference}. For
a small return \(V'h=m+d_h\), choose an integer period \(\Pi\)
and require \(m\in\Pi\Z^{d_v+d_{v'}}\), where \(d_v,d_{v'}\)
are the horizontal dimensions of the two tags. Choose \(\Pi\)
sufficiently divisible that \(\xi_i\Pi,\xi_j\Pi\) are integers and
\(\Pi C_v,\Pi C_{v'},c_i\xi_i\Pi C_v,c_j\xi_j\Pi C_{v'}\)
have integral entries.
This clears the lattice-index and phase denominators together.
Translate \(z\) by \(h\), and the source lift indices by
\(\xi_im_v\) and \(\xi_jm_{v'}\), respectively. The phase-increment
calculation in the proof of \cite[Lemma~4.3]{H}, together with
\eqref{eq:analytic-mixed-curvature}, gives the modulation in \(w\)
\[
\kappa\Omega(h,m),\qquad
\Omega(h,m)=(2\Lambda'+(V')^TC'V')h
-(V')^T(C'-(C')^T)m.
\]
Indeed each source contribution has the multiplier
\(c_p\xi_p\zeta_p\); the two contributions therefore combine
into these difference data. The remaining linear terms in \(z\)
are absorbed in the first spectator. The amplitude and kernel
errors are bounded by a quasipolynomial times
\(|h|+\norm{d_h}\), as in H.
Call the two spectators \(f,g\), and clip them at a sufficiently large inverse polynomial
in \(\tau\). Their discarded \(L^1\) tails are small by their
\(L^2\) bounds and Lemma~\ref{lem:local-absolute}. If the form is
larger than \(\tau\), applying the return calculation to sufficiently
many multiples of \((h,m)\) gives large correlations of one fixed
bounded kernel with tests \(f_\ell(z)g_\ell(w)\), where the
\(f_\ell\) have bounded \(L^2\) norms and
\[
g_\ell(w)=g(w)e(\ell\kappa\Omega(h,m)w)
\]
up to a common bounded multiplier. The small return threshold is
chosen before these multiples so that all translation errors remain
small. No regularity of the first spectator is required: it is
translated by a change of variables. If \(|\Omega(h,m)|\) exceeds
a fixed constant, the second tests are Bessel on the fixed box,
with bound controlled by the clipping height. Cauchy--Schwarz in
\(z\) then contradicts the number of large correlations. Thus
\(\Omega\) is bounded on every sufficiently small return.
The concluding return-to-splitting argument in the proof of
\cite[Lemma~4.3]{H} uses precisely this bound: its alternating
return identity yields an isotropic lattice space, successive
minima bound the rational fast projection and slow part, and
coordinate-direction returns bound the effective quadratic form.
The enlarged denominator-clearing period changes only its
quasipolynomial constants. It supplies
\eqref{eq:analytic-splitting}, contradicting the assumed graph
distance. This is the distinct-slot argument of
\cite[Lemma~6.1]{H} for the present coefficients.
\end{proof}
\subsection{Same-tag tests and the localization output}
The complementary fact is that a column of one tag can detect a
residual in another role using a template of that same tag. This
keeps the approximation spaces finite without repeatedly increasing
their nonlinear complexity. A pure source column means one scalar
column label, rather than a linear combination of labels.
\begin{lemma}[Uniform same-tag test]\label{lem:same-tag-test}
Fix an original tag \(v\) and its original source columns within
the complexity budget \(\exp(C_{q,b}k^C)\).
There are fixed template families in
all four slots, consisting of bounded columns of tag \(v\) with
arbitrary external linear modulations. Suppose a pure source
column from the original or template family in role \(i\) has
form norm greater than \(t_0>0\) against a physical residual in \(L^2(I)\) in
role \(j\) and two normalized \(L^2(I)\) spectators. The residual
then has a normalized moment of magnitude at least \(t_0/H_0\)
against a template in role \(j\), where
\(H_0\le\exp(C_{q,b}k^C)\). This bound is independent of the
external linear modulations and of the number of earlier template
insertions.
\end{lemma}
\begin{proof}
Use the nonnegative partition function \(\rho\) and template
amplitudes \(\rho(d)e(\ell^Td/P_0)\) of
\cite[Lemma~6.6]{H}, with \(\ell\) ranging over residue vectors
modulo a sufficiently large integer period \(P_0\). For the four
lifts \(d_p=\theta_vy_p-n_p\), set
\begin{equation}
d_\diamond=\sum_pe_pd_p,\qquad
k_p=d_p-d_0-b_pd_\diamond.
\label{eq:analytic-common-carries}
\end{equation}
The B\'ezout identities show that \(d_\diamond\) is a lift of
\(\theta_vt\) modulo the integer lattice and every \(k_p\) is
integral. All are bounded on the branch boxes. The common
quadratic phases cancel by Lemma~\ref{lem:algebra}; for fixed
carries the remaining phase is
\[
\sum_pc_p(\theta_vy_p)^TC_vk_p,
\]
which is linear in the coordinates. Smooth equality cutoffs in
the four lifts impose \eqref{eq:analytic-common-carries} exactly,
because their arguments are integer valued on the configuration.
Their Fourier expansion on a sufficiently large box therefore
produces the same template amplitudes. Increasing the period
and the possible carry count costs only the allowed chart budget.
As in H, write each Fourier index as \(\ell=P_0q'+r'\).
The factor \(e((q')^Td_p)\) is an external linear modulation,
so only \(r'\) enters the template amplitude. This gives the
same bound even when the source was itself a previously inserted
template. For an original source, the remaining smooth horizontal
amplitude, after phase cancellation, is periodized and expanded
in ordinary torus characters. Its remaining dependence on the
source coordinate is a linear modulation, which is absorbed into
the spectator tests. All these absolute coefficient sums are
bounded by the same quasipolynomial budget.
After separating the smooth kernel, express the target
coordinate as \(y_j=\alpha y_a+\beta y_l\), with fixed nonzero
rational \(\alpha,\beta\). Fourier-expand the demodulated
residual on a fixed larger interval. If \(M\) is the supremum
of its normalized template moments, one separated term is bounded
by
\[
M\sum_m|A_mB_m|
\le M\left(\sum_m|A_m|^2\right)^{1/2}
\left(\sum_m|B_m|^2\right)^{1/2}
\le C_bM.
\]
The last inequality is Bessel's inequality for spectator
frequencies with nonzero spacings proportional to
\(\alpha,\beta\). Summing the absolute Fourier coefficients
gives \(H_0M\), and proves the assertion.
\end{proof}
We record how these two tests enter the remainder of the imported
construction. This also fixes the meaning of its output used in
Section~\ref{sec:arithmetic}. Put \(P=\lceil\eta^{-\alpha}\rceil\),
where \(\alpha>0\) will be chosen in the closing argument. The
vertices are occurrences of the original columns and are ordered by
admission. Added main and helper columns attach to those vertices
and have their normalized tags; they create no new graph vertices.
H's short-path, color, and historical-mask constructions
\cite[Lemmas~6.2--6.5]{H} form groups of these vertices and assign
current groups to each summation cell. An actual use is such an
occurrence of a current group at a cell. A mask is
the subset of available vertices allowed in a specified local
projection. A pad enlarges that subset by the prescribed graph
radius; a positive smeared filter is the average of \(P\) nested
pad projections. A history is the ordered product of such filters
or their complements used in the binary localization. The number
\(\mathcal L\) of colors is polynomial in \(k\), and these main
histories have \(O(\log\mathcal L)\) factors.
Each group records a distinguished tag \(z\), the primary winner
in H's color construction. Every vertex admitted to its masks
has, already at admission, a graph path to \(z\) of length
polynomial in \(P,k\). The short-path lemma therefore gives the
endpoint splitting \eqref{eq:analytic-splitting} relative to \(z\),
with logarithmic height polynomial in \(P,k\). This is the
splitting used for the first matching of a column in
Section~\ref{sec:arithmetic}.
We use the precise masks and historical-use convention of those
lemmas. Its relevant properties are as follows. Masks are
inherited on descendants; at an actual use, enlarging a pad by one
includes every neighbor of the preceding mask. An endpoint is usable when its
mask is nonempty and its cell has a downstream actual use of its
group. Distinct groups of one color have tags separated by graph
distance greater than one on overlapping usable endpoints.
The endpoint bound along short paths follows from the rational
splitting \eqref{eq:analytic-splitting}. The usable-space Bessel
bound follows from same-slot separation in
Lemma~\ref{lem:tag-separation}. These proofs involve only rational
linear algebra, inherited interval projections, and the indicated
splitting and separation bounds; thus their path, rank, and pad
counts keep the quantitative orders in H.
For completeness, the closure condition imposed on these spaces
has an exact form. At rank \(r\), let \(n_{r-1}\) bound the
number of labels through the preceding rank along a path, let
\(D=D_{\rm base}\), and let \(\tau_3=D^{-C_\tau}\). For a
basic argument \(a\), each prescribed history \(\mathcal H\),
target mask \(\mathcal M\), and pure lower-rank source whose
tag belongs to that mask, the form norm against the residual
\((1-P_{r,\mathcal M})\mathcal Ha\) and two normalized
\(L^2\) spectators is required to be at most
\begin{equation}
\frac{\tau_3}{D\sqrt{n_{r-1}}}\norm a_{2,I}.
\label{eq:analytic-closure}
\end{equation}
On failure, Lemma~\ref{lem:same-tag-test} supplies a new column
and hence a projection increment. The products-of-prefixes
estimate of H Section~3 bounds the sum of those increments,
exactly as in its Lemma~6.7. Its rank recurrence and main/helper
catalog counts are therefore unchanged. This count depends on the
names of masks and histories, including anticipated helper bases,
but not on the numerical edge height or on later mode precision.
As in \cite[Section~6.4]{H} and \cite[Section~6.1]{V}, first fix
these counts, then the mismatch tolerance, and finally the edge
height needed in Lemma~\ref{lem:tag-separation}.
The lower-rank count in
\eqref{eq:analytic-closure} is retained, so no full source count
is lost when these errors are summed.
It follows that the binary localization and outgoing-filter
identities of \cite[Proposition~6.8 and Lemma~6.9]{H} hold for
the present forms: they use \eqref{eq:analytic-closure}, distinct-tag
form separation, the local absolute bound, and the same nested
projection algebra. In the color construction of
\cite[Lemma~6.3]{H}, the first failure of its marking conditions
may remove a subtree. Increasing the fixed thresholds makes the
sum of these roots' lengths an arbitrarily small fixed fraction
of \(|R|\). Section~\ref{sec:completion} restarts the original
summands on those roots. After these marking-failure cuts, let
\(\mathcal A_R^{\rm main}\) denote the remaining active cells.
For each base pattern and \(I\in\mathcal A_R^{\rm main}\) of depth
\(s\), the localization is
\begin{equation}
\begin{gathered}
H_I(z_I)=\sum_\lambda\ \sum_{G\ {\rm current}}
H_I(Z^\lambda_{0,G,s},\ldots,Z^\lambda_{3,G,s})+e_I,\\
Z^\lambda_{p,G,s}
=\overline L^\lambda_{p,G,s}\mathcal B^\lambda_{p,s}z_{p,I},
\qquad
\sum_{I\in\mathcal A_R^{\rm main}}|I||e_I|
\le C_{q,b}e^{-k}|R|.
\end{gathered}
\label{eq:analytic-localization}
\end{equation}
Here \(\lambda\) ranges over polynomially many fixed binary search
leaves. The inner history \(\mathcal B^\lambda_{p,s}\) is independent
of \(G\), and \(\overline L^\lambda_{p,G,s}\) is the outer ball
average. The minimum in \eqref{eq:analytic-base-target} is
distributed across the fixed leaves, while the group sum remains
inside the original cell contribution. The filters act on the
whole basic arguments in this identity. We next record their
endpoint bounds, then the algebra that moves the complementary
filters onto a side operand.
For varying inputs, this localization acts at each cell on its
one actual tuple at that cell's original depth. The insertion
tests freeze that operand while the column list grows. The
products-of-prefixes estimate therefore remains a fixed-vector
estimate, and gives the same closure count; this is the argument
of \cite[Section~6.1]{V}. After the operator families have been
fixed, its Lemma~6.1 and Corollary~6.2 control their changes between
endpoint depths. More explicitly, for one slot, search leaf,
and color, write the full localized endpoint output as
\(Z_{G,s}=\overline L_{G,s}\mathcal B_s B_s\), where
\(\overline L\) is the outer positive ball average and
\(\mathcal B\) its group-independent inner history. If depth
intervals \([r,t]\) have edge-overlap multiplicity at most \(a\),
and \(E_{G,r,t}\) is a union of \(t\)-cells with downstream actual
uses certifying the indicated group, those results give
\begin{equation}
\sum_{G,[r,t]}
\norm{\mathbf1_{E_{G,r,t}}(Z_{G,t}-Z_{G,r})}_2^2
\le aG_0|R|,
\qquad G_0=\exp\!\bigl(C_{q,b}(1+\log(2k))^C\bigr).
\label{eq:analytic-endpoint-variation}
\end{equation}
The shared histories without an outer ball satisfy the corresponding
estimate counted once across groups. Endpoints before the attempt
root are zero, and all other endpoints use inherited continuations
and local refits. The proof first establishes the ordinary Bessel
bounds for operator changes, using usable endpoint ranges and
compression, and then evaluates the rows at their prescribed
depth inputs. Input changes use only the original block variation
bound. Thus the constants in
\eqref{eq:analytic-endpoint-variation} are independent of the
partitions and \(W_j\).
\subsection{Transferring the complementary filters}
The side estimate will encounter two distinguished operands while
the other two retain their whole localized histories. We state
the projection identity for this situation explicitly. Fix an
actual group use at a cell \(I\), one search leaf, and four distinct
roles \(a,a',c,d\). Let \(S_a,S_{a'}\) be fixed Hilbert-valued
\(L^2(I)\) operands in the first two roles. They are called side
operands here because they are retained while the complementary
histories are simplified; their particular construction and
cross-depth bounds are given in Section~\ref{sec:arithmetic}.
The other two roles have the whole localized arguments with basic
inputs \(B_c,B_d\) from \eqref{eq:analytic-localization}.
Choose the side operands before closing the helper catalog.
The helper version of \eqref{eq:analytic-closure} is imposed with
these fixed states as bases and lower-rank main columns as sources,
as in \cite[Lemma~6.7]{H}.
Expand each complementary factor \(1-\overline Q\) as the
difference of the identity and a positive average. There are only
\(2^{O(\log\mathcal L)}\) resulting products. In one such product,
remove the largest-rank outermost positive filter in roles \(c,d\).
The positive main ranks are distinct in the source construction,
so the remaining source in the other complementary role has
strictly lower rank. Suppose the filter being removed is on
role \(c\). Let \(Q_{c,p}\), \(1\le p\le P\), be its individual
pad projections, and \(Q_{a,p}\) the helper-rank projections in
role \(a\) with the same masks and pads. For \(j=a,c\), set
\begin{equation}
\begin{gathered}
Q_{j,0}=0,\qquad Q_{j,P+1}=1,\qquad
A_j^p=Q_{j,p}-Q_{j,p-1}\quad(1\le p\le P+1),\\
\overline Q_j=\frac1P\sum_{u=1}^P Q_{j,u}
=\sum_{p=1}^{P+1}\frac{P-p+1}{P}A_j^p .
\end{gathered}
\label{eq:analytic-bands}
\end{equation}
The \(A_j^p\) are mutually orthogonal projections and sum to the
identity. Their final band is included, with weight zero in
the positive average.
Let \(x_c\) be the inner history on \(B_c\) after deleting the
outgoing filter, and let \(x_d\) be the entire remaining history
on \(B_d\). Write \(x_a=\mathcal D_aS_a\), where \(\mathcal D_a\)
is the product of helper averages already transferred, and
\(x_{a'}=S_{a'}\). In the next displays arguments are written in
role order \(a,a',c,d\); each still occupies its original slot in
\(H_I\). Multilinearity and \eqref{eq:analytic-bands} give
\begin{equation}
\begin{split}
&H_I(x_a,x_{a'},\overline Q_cx_c,x_d)
-H_I(\overline Q_ax_a,x_{a'},x_c,x_d)\\
&\qquad=\sum_{p,q=1}^{P+1}\frac{p-q}{P}
H_I(A_a^px_a,x_{a'},A_c^qx_c,x_d).
\end{split}
\label{eq:analytic-band-transfer}
\end{equation}
Indeed the coefficient of the pair of bands is the weight of
band \(q\) in role \(c\) minus the weight of band \(p\) in
role \(a\). This proves H's Equation~(15), including its
terminal bands, for the present form without a coefficient change.
The estimate accompanying \cite[Lemma~6.9]{H} permits only
\(|p-q|\le2\) to be retained: the other terms at this cell and
group have total absolute value at most
\[
D_{\rm base}^{-C_3}
\norm{S_a}_{2,I}\norm{S_{a'}}_{2,I}
\]
for any prescribed fixed \(C_3\), after the numerical accuracies
are chosen. Its hypotheses are precisely the closure and
separation estimates proved above. If \(p>q+2\), expand the
lower \(c\)-band into pure main sources and use the helper
closure test on the higher \(a\)-band, written as a difference
of residual projections. If \(q>p+2\), expand the lower-rank
source in role \(d\). A source tag within one edge of the
lower helper mask belongs, by neighbor closure, to the next
outgoing mask; the main closure test then controls the higher
\(c\)-band. For a tag farther away, expand the helper primal
and use Lemma~\ref{lem:tag-separation}. The coefficient counts
are canceled by the lower-rank denominator in
\eqref{eq:analytic-closure}, except for fixed count powers
already included when the mismatch tolerance was chosen.
The \(O(P^2)\) band pairs and polynomially many transfer
positions fit those accuracy choices because \(\log P=O(k)\).
For each fixed offset \(h=q-p\in\{-2,-1,0,1,2\}\), retain the
coupled sum
\begin{equation}
\mathcal B_{I,h}
=-\frac hP
\sum_{\substack{1\le p\le P+1\\1\le p+h\le P+1}}
H_I(A_a^px_a,x_{a'},A_c^{p+h}x_c,x_d).
\label{eq:analytic-retained-bands}
\end{equation}
The sum over \(p\) stays inside the cell contribution. When the
current group is also summed, its sum stays there as well.
This is the expression to which the short-block estimate is
applied; taking absolute values of its individual bands would
lose the matched-band summation used in that estimate.
Forward only the transferred main term in
\eqref{eq:analytic-band-transfer} to the next step. Removing
the remaining positive filters in decreasing rank ends at
\begin{equation}
H_I(\mathcal D_aS_a,S_{a'},B_c,B_d),
\qquad \mathcal D_a
\text{ a product of }O(\log\mathcal L)\text{ helper averages}.
\label{eq:analytic-transfer-main}
\end{equation}
Restoring \(B_c,B_d\) to the full \(Y_c,Y_d\) uses their pair
bound from the base reduction. Each correction has an additional
basic \(U\) factor and is therefore a three-side term.
For the retained defects, the endpoint convention matters.
If the outgoing filter is a ball filter, split
\(A_c^{P+1}=1-Q_{c,P}\) into the shared identity and its positive
ball endpoint. The other complementary band endpoints have
usable ball ranges. If the outgoing filter is cumulative,
its outer ball has already been removed and the entire
complementary band expression is shared across groups.
Role \(d\) likewise either has one positive outer ball filter
or is shared. In the endpoint energy estimates, a shared
identity is evaluated on the union of required cells and counted
once; the original form still retains every actual use.
These are exactly the endpoint
classes controlled by \eqref{eq:analytic-endpoint-variation}
and \cite[Corollary~6.2]{V}; they give the side estimate in
Section~\ref{sec:arithmetic} its coupled band operands.
We have obtained the base minimum, the localized group expressions,
their outgoing-filter identity, and the endpoint estimates with the
same main-column budgets as H and V. The remaining work is to
estimate the coupled near bands and the remaining localized quadratic
contributions. Section~\ref{sec:arithmetic}
establishes the coefficient-dependent comparisons used for those
two estimates.
==== END Triple-ergodic-averages-with-distinct-integer-slopes//sections/analytic.tex ====
==== BEGIN Triple-ergodic-averages-with-distinct-integer-slopes//sections/arithmetic.tex ====
\section{Short blocks, rational shifts, and the core estimate}
\label{sec:arithmetic}
Starting from the localization proved in the preceding section, we now
split the arguments into side terms and core terms. Side terms are controlled by
square estimates for projection increments, together with a comparison on
a bounded block of depths. Core terms are controlled by separation of
the frequencies left after removing a common quadratic phase. We prove
the coefficient-dependent parts of both arguments here. The projection,
graph, and amplitude summations that we use from \cite{H,V} are identified
at their points of application.
Throughout this section, the primitive integer vector
\(b=(0,b_1,b_2,b_3)\), the integers \(e_j\), and the nonzero integers
\(c_j\) are those of Lemma~\ref{lem:algebra}. Constants may depend on
these fixed integers and on the admissible kernel bounds. We write
\(e(t)=\exp(2\pi i t)\). For an interval \(I\), the notation
\(\|f\|_{2,I}\) denotes \((|I|^{-1}\int_I|f|^2)^{1/2}\), also for
Hilbert-space-valued functions.
\subsection{A comparison on a short block}
We first establish the version of \cite[Lemma~7.2]{H} needed by the
side argument. This estimate compares randomized inputs with the
fixed-input constant \(C_N(q)\); in particular, the exponent of the
block length must tend to zero as \(q\) tends to two.
\begin{proposition}[Short-block comparison]\label{prop:short-block-b}
Let \(2<q<3\). Let a finite dyadic tree piece have root \(R\), with
internal summation intervals in at most \(K\) consecutive depths and
within the original range of at most \(N\) depths. Its terminal
intervals, called leaves, partition \(R\) and occur in this depth block
or one depth below it. In role \(j\), let \(\mathbf z_j\) be a
deterministic Hilbert-space-valued function supported on \(R\), with
\(\|\mathbf z_j\|_{2,L}\le s_j\) on every leaf \(L\).
Suppose scalar randomizations \(z_j^\sigma\) satisfy
\[
\bigl(\mathbb E_\sigma|z_j^\sigma(y)|^a\bigr)^{1/a}
\le C_a\|\mathbf z_j(y)\|
\]
for \(a=2,4,8\), with constants independent of the tree and \(K\).
For measurable signs
\(|\epsilon_I(\sigma)|\le1\) and any subcollection of internal intervals,
\[
\mathbb E_\sigma\left|
\sum_I |I|\epsilon_I(\sigma)H_I(z_0^\sigma,z_1^\sigma,
z_2^\sigma,z_3^\sigma)\right|
\le (1+C_N(q))C_{q,b}(1+\log(K+2))^C
(K+2)^{11(q-2)/(q-1)}|R|\prod_j s_j .
\]
The constant \(C_{q,b}\) may also depend on the fixed moment
bounds \(C_a\) and the kernel bounds.
\end{proposition}
\begin{proof}
Homogeneity reduces to \(s_j=1\); a zero size makes the form zero.
Multiplying all signs by one random scalar of modulus one reduces the
expected absolute value to the absolute value of an expected signed sum.
We use the partition of the four leaf depths from the proof of
\cite[Lemma~7.2]{H}. Here is the information from that partition that
will be needed. If the least depth occurs more than once, designate two
roles attaining it. If it occurs once, designate that role and
separate the least and second least depths by the smallest dyadic
interval of depth indices containing them in opposite halves.
Specify the nonempty set of roles whose depths belong to its upper
half, and designate one of those roles by a fixed ordering.
Every role that is neither the minimum role nor in that specified set
has depth greater than the upper endpoint of that dyadic interval.
This partitions the depth tuples into rectangles of masks
\(E_j\), each a union of leaves. There are
\(O(1+\log(K+2))\) length classes and a bounded number of role assignments.
Within each class, there are \(O(K+1)\) positions; the masks in each of
the two designated roles, say \(a,a'\), are disjoint over those positions.
For a fixed rectangle there is an index \(t\) such that every role has
leaf depth at least \(t\), with the possible exception of role \(a\).
Only summation depths \(s<t\) contribute. Indeed a leaf meeting an
internal interval must be its strict descendant. The same assertion
holds after the small coordinate enlargements below, by the kernel's
fixed interior support margin.
Write \(r_s\) for the length at depth \(s\), and set
\(\delta=\beta r_t\), where \(\beta>0\) is a sufficiently small fixed
dyadic number depending on \(b\) and the support margin. Decompose the
physical coordinates as
\[
x=\delta(n+u),\qquad \tau=\delta(m+v),\qquad
n_j=n+b_jm,\qquad w_j=u+b_jv.
\]
The slow coordinates \(n,m\) are integers. Choose the fast
representatives from the parallelogram
\begin{equation}\label{eq:fast-tile}
\mathcal P_b=
\{(u,v):0\le u<1,\quad 0\le u+b_1v<|b_1|\}.
\end{equation}
This is a fundamental domain for translation by \(\mathbb Z^2\), of
area one. To decompose a point, first choose \(n\) so that its first
coordinate minus \(n\) belongs to \([0,1)\), and then choose \(m\) so
that \(u+b_1v\) belongs to \([0,|b_1|)\). The latter choices differ
by \(b_1\), so existence and uniqueness hold for either sign of \(b_1\).
Let \(\mathbb T_0=\mathbb R/(T_0\mathbb Z)\), with Haar probability
measure, where \(T_0\) will be a sufficiently large fixed number.
Define \(F_j^\sigma(p,w)\) by sampling
\(z_j^\sigma(\delta(p+w))\mathbf1_{E_j}(\delta(p+w))\)
on a bounded interval of real lifts and setting it to zero elsewhere
on \(\mathbb T_0\). Use lift intervals \([0,1)\) and
\([0,|b_1|)\) in roles zero and one, and bounded intervals containing
the ranges of \(w_2,w_3\) on \(\mathcal P_b\) in the other roles.
The fast Haar integral reproduces integration over \(\mathcal P_b\),
up to its fixed normalization. To check the potentially noninjective
pair coordinates, suppose a torus progression has all four bounded
lifts \(\widetilde w_j\) in these windows. Put
\[
\widetilde u=\widetilde w_0,\qquad
\widetilde v=\sum_j e_j\widetilde w_j.
\]
The integer inverse of Lemma~\ref{lem:algebra} gives
\(\widetilde w_j=\widetilde u+b_j\widetilde v\pmod {T_0}\).
All the discrepancies are bounded in terms of the fixed windows and
\(b,e\). Choose \(T_0\) larger than these bounds. Each discrepancy
then vanishes, so these are equalities of real numbers and
\((\widetilde u,\widetilde v)\in\mathcal P_b\). Conversely, every
point of the tile gives exactly these lifts. This verifies the sampling
identity without asserting that any coordinate pair is unimodular.
Replace the kernel by its value at \((\delta n,\delta m)\).
At depth \(s\), the error is \(O_b(\delta/r_s)\), with a fixed support
enlargement. The normalized sampled form has factor
\((\delta/r_s)^2\). On the product of local counting measure and
the fast torus, any two distinct coordinate maps are injective on the
lattice and Haar-surjective on the torus. Consequently the two-
\(L^1\), two-\(L^\infty\) bounds hold, and interpolation gives the
absolute form estimate whenever the four reciprocal exponents sum to
at most two.
For clarity, let \(M_{j,I}\) be the squared mass of \(\mathbf z_j\)
in all masked sampling windows relevant to \(I\), including their
bounded overlap multiplicity. Choosing \(\beta\) small enough puts
these windows inside \(I\). The leaf normalization therefore gives
\[
M_{j,I}\le C_b|I|,\qquad
\sum_{\operatorname{depth}(I)=s}M_{j,I}\le C_b|E_j|.
\]
The absolute \(L^2\) estimate and the moment hypothesis bound the
sampling errors at depth \(s\) by
\[
C_b\frac{\delta}{r_s}|E_a|^{1/2}|E_{a'}|^{1/2}.
\]
The sum over \(s<t\) is bounded, and summing rectangle positions uses
the disjointness of both designated masks.
We next improve the integrability of the fast inputs. On their fast
Fourier coefficients define
\[
\widehat{U_{j,\gamma}F}(p,k)
=e(\gamma k^2/c_j)\widehat F(p,k),
\qquad \gamma\in\mathbb R/(C_c\mathbb Z),
\quad C_c=\operatorname{lcm}_j|c_j|.
\]
The fast progression integral is unchanged under simultaneous
application of these unitaries. Its Fourier support obeys
\(\sum k_j=\sum b_jk_j=0\), so Lemma~\ref{lem:algebra} gives
\(k_j=c_j(h+b_jh')\) for real \(h,h'\). Thus
\[
\sum_j k_j^2/c_j=\sum_jc_j(h+b_jh')^2=0.
\]
For a single role, integrating the fourth power first in the fast
variable and then in \(\gamma\) equates the sums and the sums of
squares of two pairs of Fourier indices. Those pairs agree as
multisets, whence
\begin{equation}\label{eq:fast-fourth}
\mathbb E_\gamma\int_{\mathbb T_0}|U_{j,\gamma}F(p,w)|^4\,dw
\le2\|F(p,\cdot)\|_2^4.
\end{equation}
Fourier polynomial approximation is justified by the pairwise
\(L^2\) form estimate and this same fourth-moment estimate for differences.
Let \(\mathfrak e_j(p)\) now denote the fast \(L^2\) size of the deterministic
stack in the sampling window. Window overlap gives
\(\delta\sum_p \mathfrak e_j(p)^2\le C_b|E_j|\).
If all leaves of a role have depth at least \(t\), then
\(\mathfrak e_j(p)\le C_b\): disjoint leaves meeting a window of length
\(O_b(\delta)\) have total length at most \(O_b(\delta)+2r_t\).
In every role, including the possible remaining role \(a\),
\[
\frac{\delta}{|I|}\sum_{p\ \text{relevant to}\ I}\mathfrak e_j(p)^2\le C_b.
\]
Minkowski's inequality transfers the assumed sign moments to these
window sizes. Intermediate moment orders below eight are bounded
by the eighth moment using H\"older's inequality.
Write \(G_j=U_{j,\gamma}F_j\). If role \(a\) has leaves above
depth \(t\), clip it radially at \(A=(K+2)^{10}\). The clipped tail
has local \(L^1\) norm at most \(A^{-1}\|F_a\|_2^2\), whose
\(L^4\) norm in the random parameters is \(O_b(A^{-1})\).
In the other roles the local
\(L^3\) norms have bounded fourth moments, by
\eqref{eq:fast-fourth} and the preceding window bounds. The absolute
form estimate with exponents \((1,3,3,3)\) therefore gives expected
tail error \(O_b(A^{-1})\) per normalized cell. Even summing
\(O((K+1)^2)\) positions and depths is harmless.
For the rest of the proof, replace \(G_a\) by this clipped function
when that role is present.
To return to the original continuous form, fix the random parameters
and \((u,v)\in\mathbb T_0^2\), and plant bursts
\[
g_j(y)=\sum_pG_j(p,u+b_jv)
\mathbf1_{\{|y-\delta p|<c'\delta\}}.
\]
Choose \(c'>0\) so small that
\(c'(2+\max_j|b_j|\sum_l|e_l|)<1\).
If a real configuration lies in bursts centered at \(p_j\), set
\(n=p_0\), \(m=\sum_j e_jp_j\). Then
\[
|x/\delta-n|<c',\qquad
|\tau/\delta-m|<c'\sum_j|e_j|,
\qquad |p_j-n-b_jm|<1.
\]
The last quantity is an integer and hence is zero. For this slow
pair the allowed offsets form the polygon
\(\{|u'+b_jv'|<c'\ (0\le j\le3)\}\), of fixed positive finite area.
The center-valued kernel integral on the polygon therefore gives
exactly the sampled product, times that area and \(\delta^2\).
The second kernel replacement has the same error estimate as the
first: unitarity and clipping preserve fast \(L^2\) bounds and do
not enlarge the set of slow indices. We can now apply \(C_N(q)\)
to these fixed compactly supported burst inputs, using the original
admissible kernels.
For completeness, the exponent calculation is unchanged. Put
\[
p_a=p_{a'}=p_*=2+2(q-2),\qquad
p_j=\frac{2p_*}{p_*-2}\quad(j\notin\{a,a'\}).
\]
Use fast exponent \(p_*\) in the clipped role and exponent four
in the other roles. Fast pairwise H\"older, random H\"older of
exponent four, spatial H\"older with the \(p_j\), and the mixed-norm
maximal theorem \cite[Lemma~7.1]{H} apply: all exponents in that
maximal theorem exceed \(q\). The resulting regular-role factors
are \(C_b|E_j|^{1/p_j}\); the clipped factor is
\(C_{q,b}A^{1-2/p_*}|E_a|^{1/p_*}\).
Summing \(O(K+1)\) positions in the two disjoint designated masks
costs \((K+2)^{1-2/p_*}\). Hence the total power is
\[
A^{1-2/p_*}(K+2)^{1-2/p_*}
=(K+2)^{11(q-2)/(q-1)}.
\]
The logarithmic number of length classes finishes the proof.
\end{proof}
This proposition supplies exactly the input used by the side estimates
\cite[Lemmas~7.3--7.4 and Proposition~7.5]{H} and
\cite[Section~6.5]{V}. Their remaining arguments concern nested
projections, frozen stacks, and variation in depth. In addition to the
local absolute form bound, they assume the square estimate H(17) for
the linear sides. We identify its source below; no coordinate
substitution is needed in those projection arguments.
\subsection{A common gauge and a lattice of rational shifts}
We describe the data from the mode construction of
\cite[Section~8]{H} before checking its arithmetic. Fix a localized
group \(G\), with central tag \(z=(\theta_z,C_z,\lambda_z)\), and
a role \(l\). A \emph{primary} is an available main column of this
role's outer projection. Its first matching cell is the cell where
the finite expansion described below is chosen; that choice and its
labels are then inherited on descendants. Thus later freezing does
not resample a primary's expansion.
Choose an integer \(Q_0\) clearing the denominators of \(C_z\), and
put \(\vartheta=\theta_z/Q_0\). For an offset \(\rho\) in the unit
cube of horizontal coordinates, define
\[
d^\rho(y)=\rho+\{\theta_z y-\rho\},\qquad
p^\rho(y)=\lambda_z y^2+(\theta_z y)^TC_zd^\rho(y).
\]
Choose \(\delta_c=\exp(-k^{C_{\rm cut}})\) sufficiently small after
the main budgets, with fixed exponent \(C_{\rm cut}\).
Let \(\zeta^\rho\) be the product Gevrey cutoff of
\cite[Section~8]{H}, vanishing within distance \(\delta_c\) of each
branch cut and equal to one outside distance \(2\delta_c\).
It is bounded between zero and one. For each role and group,
one offset is kept fixed throughout its history. Its expectation
is at least \(1/2\), and
\(\mathbb E_\rho|1-(\zeta^\rho)^2|^2\le C d_{\max}\delta_c\),
where \(d_{\max}\) denotes an upper bound for the dimensions.
For a primary \(\Phi=(\varphi_\Phi,\epsilon_\Phi^{1/2}\mathbf e_\Phi)\)
with normalized tag \(v\),
the endpoint splitting from the localization section and the chart
expansion of \cite[Lemma~4.4]{H} give, at its first match,
\[
\zeta^\rho\varphi_\Phi e(-c_l p^\rho)
=\sum_m a_{\Phi,m}e(\lambda_my^2+\gamma_my)+O(\delta_c).
\]
The chart expansion uses this same error tolerance \(\delta_c\),
and all modes of one primary have the same quadratic coefficient. As in
\cite[Lemma~8.1]{H}, use mode atoms
\[
\psi_m(y)=\left(
\zeta^\rho(y)e(c_lp^\rho(y))e(\lambda_my^2+\gamma_my),
\epsilon_{\rm m}^{1/2}\mathbf e_m\right).
\]
The coefficient sum, mode count, normalized curvature, and rational
heights have logarithms polynomial in the accuracy and localization
parameters \(k,P\). In particular, for modes of one primary,
\begin{equation}\label{eq:mode-differences-b}
\gamma_m-\gamma_{m'}=a\cdot(\theta_v,\vartheta),
\qquad a\in\mathbb Q^{\dim(\theta_v,\vartheta)}
\text{ of controlled height}.
\end{equation}
Multiplication of the splitting by the fixed integer \(c_l\) changes
only fixed constants in these bounds.
The resulting lift \(\mathcal R_{l,G}\) is one fixed pointwise bounded
operator on augmented inputs. Indeed, set
\(q_\Phi=\sum_m a_{\Phi,m}(\psi_m)_{\rm phys}
-(\zeta^\rho)^2\varphi_\Phi\), so that
\(|q_\Phi|\le\delta_c\). For an augmented input \(F\), its lifted
physical component is
\[
(\zeta^\rho)^2F_{\rm phys}
+\sum_{\Phi\ \text{matched}}
\epsilon_\Phi^{-1/2}F_\Phi q_\Phi,
\]
and its mode coordinate belonging to \((\Phi,m)\) is
\(\epsilon_{\rm m}^{1/2}a_{\Phi,m}
\epsilon_\Phi^{-1/2}F_\Phi\).
Choose \(\delta_c\) small and \(\epsilon_{\rm m}\) small after the
main column budgets. Cauchy--Schwarz then gives a uniform operator
norm. The private coordinate of an outer projection output records
its column coefficient, so this operator gives exactly its mode
synthesis. It is cell-local and independent of the freezing block.
There is also a common periodic multiplier for the four gauges.
For four offsets, put \(d_j=d^{\rho_j}(y_j)\) and
\[
d_\diamond=\sum_j e_jd_j,\qquad
k_j=d_j-d_0-b_jd_\diamond.
\]
The \(k_j\) are bounded integer vectors. Between branch cuts the
lifts are affine and their affine parts cancel in this expression;
hence the carries are locally constant. The three identities
\(\sum c_jb_j^a=0\), \(0\le a\le2\), give
\begin{equation}\label{eq:gauge-multiplier-b}
\prod_j e(c_jp^{\rho_j}(y_j))
=e\left(\sum_jc_j(\theta_zy_j)^TC_zk_j\right).
\end{equation}
For fixed carries the right side is an ordinary character on the
torus with coordinates \((\vartheta y_{l'},\vartheta t)\), for any
anchor \(l'\): \(Q_0\) clears \(C_z\), and changing the anchor uses
integer coefficients \(b_j-b_{l'}\). Smooth cutoffs slightly larger
than the original ones extend these character pieces by zero at the
cuts. They give the smooth periodic multiplier, bounded by one,
required in \cite[Section~8.3]{H}.
The number of carries and the derivative bounds acquire only fixed
factors per coordinate or derivative. Their logarithmic costs remain
polynomial in \(P,k\). Choose a mode budget \(D_*\ge e^k\),
\(\log D_*\le\operatorname{poly}_b(P,k)\), covering the mode lists,
coefficient sums, heights, inverse private weights, and absolute
Fourier sums, and large enough to dominate the dimensions and all
fixed slope factors. Only then choose a Fourier cutoff \(B_*\) with
tails at most \(D_*^{-300}\). This preserves the order of choices
in H; a coefficient-sum budget is not defined through its own
truncation radius.
To control rational divisions at later frequency tests, fix an integer
\begin{equation}\label{eq:Ds-choice}
D_{\rm s}\ge6,\qquad
6\mid D_{\rm s},\qquad |b_i-b_j|\mid D_{\rm s}\quad(i\ne j).
\end{equation}
The mode graphs will use shifts in the lattice generated by
\(\vartheta/L_h\), where
\begin{equation}\label{eq:height-parameters-b}
L_h=D_{\rm s}^{h},\qquad E_h=A_0^{h+1}.
\end{equation}
Here \(E_h\) bounds integer shift indices. The choice of
\(D_{\rm s}\) is fixed independently of all depth and accuracy
parameters. From this point, constants indexed by \(b\) may also
depend on this fixed choice of \(D_{\rm s}\).
We recall the scale geometry to make the graph conditions explicit.
In a depth block \([s_0,s_0+K)\), freeze coefficients at
\(u_{s_0}=s_0-4K\), compare on a cell \(J\) at depth
\(v_{s_0}=s_0-2K\), and put \(t_{s_0}=s_0+K\). A mode has instantaneous frequency
\(\omega_m(x)=2\lambda_mx+\gamma_m\).
On the list frozen at \(u_{s_0}\), join two modes in the PRE graph if
\[
\sup_{x\in J}|\omega_m(x)-\omega_{m'}(x)
+n\cdot\vartheta/L_h|
\le\Xi/r_{v_{s_0}}
\quad\text{for some }n\in\mathbb Z^{\dim\theta_z},\quad
|n|_\infty\le E_h.
\]
The CURRENT graph has the same definition with threshold
\(\Xi/r_{t_{s_0}}\). Write \(\mathcal E_h(m)\),
\(\mathcal F_h(m)\) for their components. Both graphs increase
with height, because a witness \(n\) at height \(h\) becomes
\(D_{\rm s}n\) at height \(h+1\), provided \(A_0\ge D_{\rm s}\).
They also increase along successive depth blocks on inherited lists.
Here are parameter choices sufficient for the arithmetic. Let
\(d_{\max}\ge2\) bound all relevant dimensions, set \(\Delta=10\), and put
\[
C_*=(10d_{\max}D_*)^{100d_{\max}^4},\qquad
R'=\lceil\log_2C_*\rceil+2\Delta+10.
\]
Choose the largest height \(J_*\), with \(0\le h\le J_*\), large compared with
\(\delta_*^{-2}d_{\max}R'\log(2D_{\rm m})\), where \(D_{\rm m}\)
is the main-column budget and \(\delta_*>0\) is a sufficiently small
fixed number. Take \(A_0\) to be a sufficiently large fixed power
of \(C_*D_{\rm s}^{R'}B_*D_*\), and \(\Xi=D_*^{1000}\).
Choose a rational height reserve and a frequency gap with
\[
\mathcal H>
(d_{\max}C_*E_{J_*}B_*L_{J_*}Q_0\Xi)^{10},\qquad
\mathcal T>(d_{\max}\mathcal H)^{Cd_{\max}^4}\Xi.
\]
Finally choose \(K\), polynomial in \(P,k\), after these logarithmic
budgets, so the height-bin packets of \cite[Lemma~3.5]{H} and the
gap \(\mathcal T\) fit inside a block. Increase it so the pre-lag
Taylor errors are as small as any prescribed fixed inverse power of
\(D_*\). All degrees can be fixed independently of \(q\).
For a primary with horizontal parameters
\(\Theta=(\theta_v,\vartheta)\), call a block nonexceptional when
every rational vector \(a\) of height at most \(\mathcal H\) satisfies
\begin{equation}\label{eq:tiny-huge-b}
|a\cdot\Theta|\le\mathcal T^{-1}/r_{v_{s_0}}
\quad\hbox{or}\quad
|a\cdot\Theta|\ge\mathcal T/r_{t_{s_0}}.
\end{equation}
The height-bin proof of
\cite[Lemma~8.2]{H} gives \(O(d_{\max})\) exceptional blocks per
primary. At a nonexceptional block let \(K_{\rm rat}\) be the
rational span of the tiny vectors, and set
\(K_B=\{w:(0,w)\in K_{\rm rat}\}\).
Cramer's rule and the choice of \(\mathcal T\) show that every
height-\(\mathcal H\) vector in \(K_{\rm rat}\) is itself tiny.
\begin{lemma}[Height stabilization]\label{lem:height-b}
For a nonexceptional primary, all but \(O(d_{\max}R')\) heights
\(h\in\{0,\ldots,J_*\}\)
have the following property: two of its modes in one CURRENT
component at height \(h\) are directly adjacent in PRE at height
\(h-\Delta\).
\end{lemma}
\begin{proof}
This is \cite[Lemma~8.3]{H} with the shift lattice
\eqref{eq:height-parameters-b}; we give its divisibility and size
argument. A simple CURRENT path has at most \(D_*\) edges.
The equal quadratic coefficients of its endpoint modes and
\eqref{eq:mode-differences-b} yield
\[
a+(0,m_h/L_h)\in K_{\rm rat},\qquad
m_h\in\mathbb Z^{\dim\theta_z},\quad |m_h|_\infty\le D_*E_h.
\]
Indeed the path bounds its evaluation by \(D_*\Xi/r_{t_{s_0}}\), and
the gap forces the tiny alternative.
Let \(V_i\) be the rational span of the height-\(D_*\) vectors
\(q\) for which \(q+(0,m/L_i)\in K_{\rm rat}\) with
\(|m|_\infty\le D_*E_i\). Let \(U_i\) be the span of the
integer vectors in \(K_B\) of size at most \(C_*^3D_*E_i\).
These spaces increase. Discard heights below \(R'\) and those
where either rank changes between \(h-R'\) and \(h\); there are
\(O(d_{\max}R')\) such heights.
At a remaining height, express \(a\in V_{h-R'}\) in a defining
basis \(a=\sum_j\beta_ja_j\), with witnesses \(m_j\).
Cramer's rule bounds the coefficients and their common denominator
by \(C_*\). Subtracting the relations gives
\[
m_h-D_{\rm s}^{R'}\sum_j\beta_jm_j\in K_B.
\]
After clearing denominators its size is at most \(C_*^3D_*E_h\),
so it lies in \(U_h=U_{h-R'}=:U\).
The quotient \(\mathbb Z^{\dim\theta_z}/(U\cap\mathbb Z^{\dim\theta_z})\)
is free, since the intersection with the rational space \(U\)
is saturated. In an integer basis of this quotient,
\[
[m_h]=D_{\rm s}^{R'}\sum_j\beta_j[m_j].
\]
For each prime \(p\mid D_{\rm s}\), every integral coordinate on
the right has valuation at least
\(R'v_p(D_{\rm s})-\log_2C_*\ge\Delta v_p(D_{\rm s})\).
Thus \([m_h]\) is divisible by \(D_{\rm s}^\Delta\).
Choose an integral lift \(m'\) of this quotient class divided by
\(D_{\rm s}^\Delta\). Then
\[
m'-D_{\rm s}^{R'-\Delta}\sum_j\beta_jm_j\in U.
\]
Round its coefficients in an integer spanning set for \(U\)
whose vectors have size at most \(C_*^3D_*E_{h-R'}\). The adjusted
lift remains integral and satisfies
\[
|m'|_\infty\le
D_{\rm s}^{R'-\Delta}\sum_j|\beta_j|\,|m_j|_\infty
+d_{\max}C_*^3D_*E_{h-R'}\le E_{h-\Delta}.
\]
The last inequality follows from the stated fixed-power choice of
\(A_0\). No lattice basis or covolume estimate is involved.
Since \(m_h-D_{\rm s}^\Delta m'\in K_B\), the vector
\(a+(0,m'/L_{h-\Delta})\) belongs to \(K_{\rm rat}\) and remains
within the height reserve. It is tiny by
\eqref{eq:tiny-huge-b}, which gives the required PRE edge.
\end{proof}
We now specify the remaining output imported from
\cite[Section~8]{H}. Write \(\pi(\mathcal D)\) for the
primary labels represented in a set \(\mathcal D\) of modes.
A mode is good at height \(h\ge\Delta\) when
\[
\left\lfloor\log_{1.1}|\pi(\mathcal E_{h-\Delta}(m))|\right\rfloor
=
\left\lfloor\log_{1.1}|\pi(\mathcal F_h(m))|\right\rfloor
\]
and at least nine tenths of the primaries in \(\mathcal F_h(m)\)
are nonexceptional and satisfy Lemma~\ref{lem:height-b} at \(h\).
Heights below \(\Delta\) are bad.
Call a mode core if it is bad at at most \(\delta_*(J_*+1)\) heights;
call every other mode a linear-side mode. The increasing graphs and
height stabilization give the coherence of these predicates,
the bad-density count, and hereditary representatives by exactly
\cite[Lemmas~8.4--8.8, Proposition~8.9, and Lemma~8.10]{H}. Those arguments use component inclusion,
cardinality, projections, and the rational gap; their coordinate
inputs have all been established above. A minimal PRE or CURRENT
component means a component at height zero. Good modes in a CURRENT
component have one PRE predecessor at height \(h-\Delta\). Moreover,
core modes in one minimal CURRENT component have identical good-height
predicates throughout the upper range
\[
\mathcal H_{\rm up}=
\{h\in\mathbb Z:\lceil4\delta_*J_*\rceil\le h\le J_*\}.
\]
Each core mode has bad fraction at most \(2\delta_*\) in this range.
We also record the decomposition of the original localized outputs.
Let \(Z_{l,G,s}\) be the current augmented output in role \(l\).
For a basic-\(B\) role in the block beginning at \(s_0\), put
\[
\widetilde Z_{l,G,s_0}
=\mathcal R_{l,G} Z_{l,G,u_{s_0}}.
\]
Let \(A_{l,G,s_0}\) and \(V_{l,G,s_0}\) be its exact core and
linear-side mode subtotals, with all private coordinates retained.
Define the full side by
\[
S_{l,G,s}=Z_{l,G,s}-Z_{l,G,u_{s_0}}+V_{l,G,s_0}
\qquad(s_0\le s<s_0+K).
\]
In a basic-\(U\) role set \(S_{l,G,s}=Z_{l,G,s}\) and
\(A_{l,G,s_0}=\mathcal E_{l,G,s_0}=0\).
On physical components these definitions give
\[
Z_{l,G,s}=A_{l,G,s_0}+S_{l,G,s}+\mathcal E_{l,G,s_0}.
\]
For a basic-\(B\) role, the physical error is
\[
\mathcal E_{l,G,s_0}
=(Z_{l,G,u_{s_0}})_{\rm phys}
-(\widetilde Z_{l,G,s_0})_{\rm phys}.
\]
The offsets can be fixed so that every term containing this physical
error is negligible. The explicit lift, applied to
\(Z_{l,G,u_{s_0}}=\sum_\Phi\beta_\Phi\Phi\), gives
\[
\mathcal E_{l,G,s_0}
=(1-(\zeta^\rho)^2)(Z_{l,G,u_{s_0}})_{\rm phys}
-\sum_\Phi\beta_\Phi q_\Phi.
\]
The main coefficients and original actual-use cells were fixed before
the offsets. Their coefficient, size, and path-count bounds, together
with the cutoff expectation, imply the error estimate from
\cite[Section~8, subsection on fixing offsets]{H}:
\[
\mathbb E_\rho
\sum_{l,G,I\ {\rm used}}|I|\,
\|\mathcal E_{l,G,s_0(I)}\|_{2,I}^2
\le D_{\rm m}^C d_{\max}\delta_c|R|.
\]
With \(\delta_c\) chosen this small in the main budget, fix offsets
satisfying this bound. All later selections only remove nonnegative
terms; they need not be independent of the offsets.
The linear-side square estimate is uniform in the offsets.
The local absolute \(L^2\) form bound and Cauchy--Schwarz therefore
make the terms containing \(\mathcal E\) negligible, after the same
size stops as in H. This explains why the subsequent estimates may
use the exact core and full-side arguments.
Here are the quantitative conclusions used below. A successful
CURRENT component contains a good core mode, whose PRE component
at \(h-\Delta\) is its uniquely determined good predecessor.
For fixed height, role, group, and the pair of floor logarithms of
the predecessor's primary and mode counts, choose a representative
mode inherited under predecessor inclusion. On comparable actual
use cells \(I,I'\), distinct representatives satisfy H(25), namely
\begin{equation}\label{eq:representative-separation-b}
|\omega_a(x_*)-\omega_{a'}(x_*)+n\cdot\vartheta/L_h|
>\frac\Xi2\max(|I|^{-1},|I'|^{-1}),
\qquad |n|_\infty\le E_h,
\end{equation}
for \(x_*\in I\cap I'\). On an actual cell, a successful component
\(\mathcal F\) has amplitude
\[
\sigma_{\mathcal F}^2
=\sum_{\mathcal D\subseteq\mathcal F}
\left(\|A_{\mathcal D}\|_{2,I}
+D_*^{-30}\sum_{\substack{m\in\mathcal D\\m\ {\rm core}}}
|\alpha_m|\right)^2 .
\]
Here \(\mathcal D\) runs over minimal CURRENT components,
\(A_{\mathcal D}\) is the full core subtotal in the augmented mode
space, and \(\alpha_m\) is its frozen mode coefficient.
This is H(26); the added coefficient terms absorb uniform Taylor
and Fourier errors.
The Bessel estimates and bad-density count give H(17):
\[
\sum_{G,s_0}\|V_{l,G,s_0}\|_2^2\le G_0|R|,
\qquad G_0=\exp\bigl(C_{q,b}(1+\log(2k))^C\bigr).
\]
This is the square hypothesis needed for the side estimates following
Proposition~\ref{prop:short-block-b}. After the stopping construction
of \cite[Proposition~8.12]{H}, the representatives occurring in a
dyadic amplitude bin \(\sigma_j\) along a retained path satisfy H(28):
\begin{equation}\label{eq:core-path-budgets-b}
\sum_G|\mathcal A_j^G|
\le\min(D_*,P^C\sigma_j^{-2}),\qquad
\sum_{s,G}\|S_{l,G,s}\|_{2,I_s}^2\le P^C.
\end{equation}
Here \(\mathcal A_j^G\) is the set of distinct representative labels
in that bin. All local
side and component sizes are at most \(P^C\).
For scheduled inputs, these statements are used in the form proved
in \cite[Section~6.3--6.4, especially Lemma~6.3]{V}. The rows are
cell-local projections composed with the fixed bounded lift
\(\mathcal R_{l,G}\). Their ordinary Bessel estimates come from
the separated mode construction, and their adjoint main ranges lie
in the usable endpoint spaces established by localization. V evaluates
the depth input after the final main basic projection. Its row
composition result therefore applies to the same size, variation,
reading-depth, and endpoint hypotheses. In particular, the possibly
large mode budget \(D_*\) is not inserted into the strong square
constant \(G_0\). This is the distinction required by V's side
estimate and by the final choice of \(q\).
\subsection{The side estimate and the near-band gain}
We now estimate the contribution of the full sides to the cell minimum
in \eqref{eq:analytic-base-target}. The input is the localized expression
\eqref{eq:analytic-localization}, its outgoing-filter identity, the
endpoint variation estimate \eqref{eq:analytic-endpoint-variation},
and the linear-side square bound H(17). These are precisely the
hypotheses of \cite[Proposition~7.5]{H}; for scheduled inputs their
counterparts are assembled in \cite[Section~6.5]{V}.
Expand each localized argument into its core, full side, and negligible
physical error. Terms with at least three side factors have a
length-weighted \(p\)-power bound \(P^C|R|\), for any fixed
\(2/3<p<1\). Indeed Cauchy--Schwarz first sums the group index in
two side roles; writing the three resulting square sizes as
\(a_s,b_s,c_s\), their contribution is bounded using
\[
\sum_s(a_sb_sc_s)^p
\le
\left(\sum_s a_s^2\right)^{p/2}
\left(\sum_s b_s^2\right)^{p/2}
\left(\sum_s c_s^2\right)^{p/2}.
\]
This uses \(3p>2\) and the retained path caps; the fourth argument
uses its \(P^C\) size bound. The group sum is formed before the
power is taken.
In a term with exactly two sides, restore its two core complements
to the whole current arguments \(Z_{c,G,s},Z_{d,G,s}\).
The corrections have a third side. The outgoing-filter identity
then removes the complementary projections one at a time, transferring
helper averages to one side, say role \(a\). Its main term is
\[
H_I(\mathcal D_{a,s}S_{a,G,s},S_{a',G,s},B_c,B_d),
\]
where \(\mathcal D_{a,s}\) is the resulting product of helper
averages. Restoring the full base pair \(Y_c,Y_d\) introduces
another third-side error. The pair-smallness from the base reduction
bounds this main term by \(\eta\) times the two side sizes.
Summing costs at most \(\eta P^C|R|\).
At one transfer step, use the outgoing and helper bands defined in
\eqref{eq:analytic-bands}, with these two full sides as its fixed
side operands. The exact outgoing-minus-helper defect is
\eqref{eq:analytic-band-transfer}. Its far bands have negligible
error. For each remaining offset \(h\), \(|h|\le2\), the retained
expression \eqref{eq:analytic-retained-bands} has coefficient
\(-h/P=O(P^{-1})\). Its paired sum over the band index must remain
inside the cell contribution.
Freeze the complementary band stack and the other complementary
argument at the block's first depth. Their differences from current
values have square budgets by
\eqref{eq:analytic-endpoint-variation}. Freeze errors therefore
supply a third strong factor and have the preceding \(p\)-power
bound. The same argument handles cells where a frozen complement
has large size: its difference from its bounded current value is
then large. On the remaining ancestor hulls inside a block, both
frozen complements have bounded local \(L^2\) size.
The binary-prefix decomposition in \cite[Lemma~7.4]{H} now writes
the two varying sides, at their uses, as selections from fixed vector
stacks. The terminal complementary band \(1-Q_{c,P}\) is first
split into the identity and a positive ball term. Positive outer-ball
ranges have joint group budgets, whereas group-independent identity
and cumulative histories are counted once rather than once per group.
This convention is needed for the following stack energies.
One stack includes the band index. There are only \(G_0\)
overhead factors. Individual pad choices stay inside the actual
positive averaged operators: a choice is coupled across depths,
and Jensen's inequality restores the average after estimating it.
They are not additional stack configurations.
The stack energies satisfy H(18):
\[
\sum_\nu\|\mathbf A_\nu\|_2^2\le PG_0|R|,
\qquad
\sum_\nu\|\mathbf B_\nu\|_2^2\le G_0|R|.
\]
Here \(\nu\) records the group, block, and fixed stack configuration.
The functions remain fixed after subsequent cuts. Independent
Rademacher signs select the two position indices, and a third family
matches the band in \(\mathbf A_\nu\) with the frozen complementary
band stack. Iterated Khintchine gives the moment hypothesis of
Proposition~\ref{prop:short-block-b}.
For precision, the passage from these energies to a sum of local
stack sizes uses the common hull cuts of \cite[Section~7]{H}.
A hull consists of all ancestors between a use and its maximal use
cell, including their full intervals. Write \(\Omega_\nu\) for
the union of the original use cells. The main path and catalog
counts give
\(\sum_\nu|\Omega_\nu|\le M_0|R|\), with
\(\log M_0\le\operatorname{poly}_{q,b}(k)\).
For a family of total stack energy at most \(E|R|\), write
\(\widetilde m_\nu\) for its full hull maxima. They obey
\[
\sum_\nu|\{x:\widetilde m_\nu(x)>T\}|
\le\min(M_0|R|,CE|R|/T).
\]
Stopping at level \(C_1E\) and integrating up to that level costs
at most \(CE(1+\log(2+C_1M_0))|R|\). Thus only a main-budget
logarithm enters \(G_0\).
For the retained hull define
\[
m_{A,\nu}(x)=
\sup_{\substack{I\ {\rm in\ the\ retained\ hull}\\x\in I}}
\frac1{|I|}\int_I\|\mathbf A_\nu(y)\|^2\,dy,
\]
with value zero for an empty supremum; define \(m_{B,\nu}\) in
the same way. The weak \(L^1\) estimate for the dyadic maximal
function, followed by truncation and integration, gives common
individual cuts; Markov's inequality for the truncated sum gives
the joint cuts for which these squared hull maxima satisfy
\[
\sum_\nu m_{A,\nu}\le PG_0,\qquad
\sum_\nu m_{B,\nu}\le G_0.
\]
The removed roots have arbitrarily small fixed total relative length.
They are first crossings of partial ancestor quantities, so retained
caps hold on whole intervals even in branches containing a later cut.
A sparse decomposition of each retained hull into tree pieces gives
disjoint major subsets of the piece roots. If \(a_Q,b_Q\) are the
two normalized stack sizes at such a root, then
\[
\sum_{\nu,Q}|Q|a_Qb_Q
\le C\int_R\sum_\nu\sqrt{m_{A,\nu}m_{B,\nu}}
\le P^{1/2}G_0|R|.
\]
Complete each piece by its boundary children. The two side stacks
then have the required leaf-size bounds, and the frozen complements
have bounded leaf sizes. Proposition~\ref{prop:short-block-b}
applies to at most \(K\) depths of each piece. Multiplying by the
transfer coefficient \(O(P^{-1})\) yields the near-band bound
\begin{equation}\label{eq:near-band-b}
(1+C_N(q))C_{q,b}P^{-1/2}G_0
(K+2)^{C_{19}(q-2)}|R|.
\end{equation}
The logarithmic factor of the short-block estimate is included in
\(G_0\). Arbitrary signs at the original cells are allowed, so this
is an absolute estimate for their coupled band contributions.
No absolute sum over individual paired band indices has been used.
Thus \cite[Proposition~7.5]{H}, with the coefficient-dependent
comparison now proved, gives a complete conditional side bound.
Third-side and freeze errors have \(p\)-power cost \(P^C|R|\);
transferred main terms have absolute cost \(\eta P^C|R|\);
the remaining near bands satisfy \eqref{eq:near-band-b}.
Applying \(\min(\eta,x)\le\eta^{1-p}x^p\) only to the powered
terms gives their contribution \(C_{q,b}\eta^\varepsilon P^C|R|\),
with \(\varepsilon=1-p>0\). All cuts remove original summands at
first crossings and the data are reconstructed on each normalized
subroot. This is the root-uniform side estimate used in the closing
argument. The scheduled version follows from the same fixed stacks
and the row estimates already stated, as in \cite[Section~6.5]{V}.
\subsection{Frequency tests for three or four core arguments}
Fix a retained path point \(x_*\), an actual cell \(I=I_s\) with
\(r=|I|\), a group, and a common successful height
\(h\in\mathcal H_{\rm up}\) for the
core arguments. Put \(R_s=r^{-1}\) and
\[
\xi=\vartheta/L_{h-\Delta},\qquad
\gamma_j^*=\omega_{a_j}(x_*),\qquad B'_h=D_*^5E_{h-\Delta}.
\]
The representative \(a_j\) belongs to the good predecessor of the
component in role \(j\). Summing a PRE path and using the pre-lag
Taylor bound expresses its selected physical argument, up to the
controlled uniform error, as
\[
e(c_jp^{\rho_j}(y_j))e(\gamma_j^*(y_j-x_*))
g_j(\xi(y_j-x_*)).
\]
The function \(g_j\) is a cutoff times a trigonometric polynomial.
After collecting the common multiplier
\eqref{eq:gauge-multiplier-b}, all Fourier indices in any anchor
coordinates lie in the box of radius \(B'_h\), enlarged by fixed
factors depending on \(b\). This is H(29) with its complete
coefficient-sum and error bounds.
To compare physical and torus amplitudes, take the horizontal tiny
space \(K_B\) of a nonexceptional primary in a successful component,
and let
\[
\mathbb H=(K_B\cap\mathbb Z^{\dim\theta_z})^\perp
\subseteq\mathbb T^{\dim\theta_z}.
\]
This is a connected torus because the integer lattice being
annihilated is saturated. The tiny/huge alternative makes physical
interval averages of the relevant characters agree with Haar
averages on \(\mathbb H\), to the prescribed small error.
The component floors above absorb those errors, giving H(30):
\[
a'_j:=\|g_j\|_{L^2(\mathbb H)}
\le C_b\left[
\sum_{\mathcal D\ {\rm selected}}
\left(\|A_{\mathcal D}\|_{2,I}
+D_*^{-30}\sum_{\substack{m\in\mathcal D\\m\ {\rm core}}}
|\alpha_m|\right)^2
\right]^{1/2}
\le C_b\sigma_{\mathcal F_j}.
\]
The selected sets are whole minimal CURRENT subtotals, by coherence
of the height predicates on \(\mathcal H_{\rm up}\). For distinct
roles, \(u+(b_j-b_{l'})v\) and \(u+(b_i-b_{l'})v\) have independent
Haar distributions on \(\mathbb H^2\). Interpolating the six
two-\(L^1\) bounds gives
\begin{equation}\label{eq:core-haar-b}
\int_{\mathbb H^2}\prod_j|g_j(u+(b_j-b_{l'})v)|\,du\,dv
\le\prod_j a'_j.
\end{equation}
A bounded progression multiplier changes only the constant.
For four core roles put
\[
Q=\sum_j\gamma_j^*,\qquad T=\sum_jb_j\gamma_j^*.
\]
Choose, by a fixed rule independent of the mode masks, a reference
\(q'=Q+n\cdot\xi\), \(t'=T+m\cdot\xi\) in the full Fourier
box with \(\max(|q'|,|t'|)\le U_0R_s\), where
\(U_0=D_*^{250}\). Introduce dyadic parameters by
\begin{equation}\label{eq:four-core-test-b}
\mu\asymp\min\left(1,\min_j\frac{|t'-b_jq'|}{R_s}\right),
\qquad
\nu\asymp\max\left(1,\frac{|q'|}{R_s},\frac{|t'|}{R_s}\right).
\end{equation}
Discard absent references and inner minima below \(U_0^{-1}\).
The four marginals of the normalized kernel make its Fourier
transform vanish on \(t'=b_jq'\). The mean value theorem near
these lines and rapid Fourier decay give
\(C_{L,b}\mu\nu^{-L}\) for every fixed \(L\). The tiny/huge
dichotomy puts all moderate frequencies in one pair of character
cosets modulo \(K_B\); summing that pair of cosets is a Haar
integral bounded by \eqref{eq:core-haar-b}. Thus its contribution
is at most \(C_{L,b}\mu\nu^{-L}\prod_j a'_j\).
For three core roles with side role \(l\), anchor at \(y_l\) and put
\[
Q=\sum_{j\ne l}\gamma_j^*,\qquad
T_l=\sum_{j\ne l}(b_j-b_l)\gamma_j^*.
\]
Choose a moderate reference \(T_l+m\cdot\xi\) and write
\(\beta=r(T_l+m\cdot\xi)\). Set
\(\mu\asymp\min(1,|\beta|)\),
\(\nu\asymp\max(1,|\beta|)\), with the same small and large
discards. Partial Fourier integration of the kernel in the step
variable gives a function \(W_\beta\) whose fixed-order derivatives
are bounded by \(C_{L,b}\mu\nu^{-L}\). At zero this uses the
\(l\)-marginal; at large frequency it uses integration by parts.
Duality in \eqref{eq:core-haar-b} bounds the remaining horizontal
coefficient sequence in \(\ell^2\).
More explicitly, average a fresh offset cutoff
\(\zeta^{\rho_l}/c_\zeta\) in the side, where
\(c_\zeta=\mathbb E_{\rho_l}\zeta^{\rho_l}\ge1/2\) is constant.
This insertion is an exact identity in expectation.
Keep the bounded factor \(c_\zeta^{-1}\) outside the estimate and set
\(S'=\zeta^{\rho_l}e(-c_lp^{\rho_l})S_{\rm phys}\) on \(I\).
Its normalized \(L^2\) norm is at most that of the side. Write
\(\widehat S'_I(k')\) for its normalized Fourier coefficients.
The horizontal pairings decay in
\(|k'+r(Q+n\cdot\xi)|\). Schur's estimate on a dyadic shell
of radius \(\Lambda\), followed by averaging the fresh offset,
gives H(32) with
\begin{equation}\label{eq:side-windows-b}
b_\Lambda^2=
\mathbb E_{\rho_l}
\sum_{\substack{k'\in\mathbb Z:\\
|k'+r(Q+n\cdot\xi)|\le2\Lambda\ \text{for some}\ n,\\
|n|_\infty\le B'_h}}
|\widehat S'_I(k')|^2.
\end{equation}
The contribution is bounded by
\(C_{L,b}\mu\nu^{-L}\prod_{j\ne l}a'_j
\sum_{1\le\Lambda\le U_0}\Lambda^{-L}b_\Lambda\),
where the sum is dyadic. Notice that the windows use the whole
Fourier box and are independent of the mode masks.
The error estimates in \cite[Section~9.1--9.2]{H} now apply with
these explicit tests. They use the absolute local form estimate,
Fourier tails, the pre-lag, and the rational gap, all with reserves
already chosen above. Per tuple the error is bounded by a fixed
multiple of \(D_*^{-100}\) times the absolute core coefficient
sums, and by the side size if a side is present. The common-height
argument samples heights independently and uniformly from
\(\mathcal H_{\rm up}\) until all roles are good; its first-success
weights sum to one on each pure tuple. A core mode has bad fraction
at most \(2\delta_*\), so the sum over failure assignments converges.
Consequently these errors sum absolutely to \(O(e^{-k}|R|)\), and
the principal cell terms have the form H(33):
\begin{equation}\label{eq:tested-cell-b}
C_{L,b}\mu\nu^{-L}\Lambda^{-L}
\left(\prod_{j=0}^3\sigma_j\right)N_s^G.
\end{equation}
Here the \(\sigma_j\) are dyadic component-amplitude bins, with
the side bin belonging to \(b_\Lambda\); \(N_s^G\) counts the
representative tuples passing the tests. For four cores,
\(\Lambda=1\). We next bound these counts across scales.
\subsection{Sparse mass for separated tests}
The finite counting statement is useful independently of the mode
construction. Its integer divisibility is the reason for the
choice \eqref{eq:Ds-choice}.
\begin{lemma}[Sparse mass]\label{lem:sparse-b}
Let \(J\subseteq\{0,1,2,3\}\) consist of three or four core roles.
In each role let a finite label set \(\mathcal A_i\) be identified
injectively with real numbers, with \(|\mathcal A_i|\le M\),
\(M\ge1\). Let \(d\ge0\) be an integer. Fix
\(\zeta\in\mathbb R^d\), \(B,L\ge1\), \(Q,\Sigma,R_0>0\), and
\(0<\mu\le1\le\nu\). Every shift vector below belongs to
\(\mathbb Z^d\). For finitely many distinct integers \(s\), at
dyadic scales \(R_s=R_02^s\), let
\(\mathcal E_s\subseteq\prod_{i\in J}\mathcal A_i\)
and \(N_s=|\mathcal E_s|\). Assume the following five conditions.
\begin{enumerate}
\item Two labels in different core roles determine at most one full
tuple, and that pair occurs at at most \(L\) scales.
\item If there is a side role, there are \(w_s\ge0\),
\(\sum_s w_s\le M\), such that every fixed core label is incident
to at most \(w_s\) tuples at scale \(s\).
\item Distinct same-role labels occurring at scales \(s,t\) satisfy
\[
|a-a'+n\cdot\zeta|>\Sigma\max(R_s,R_t)
\quad(|n|_\infty\le Q).
\]
\item For every triple \(T\subseteq J\), with excluded role
\(l'\in\{0,1,2,3\}\setminus T\) and
\(\alpha_i=b_i-b_{l'}\), each tuple has a witness
\(|n|_\infty\le B\) such that
\[
\mu R_s\le\left|\sum_{i\in T}\alpha_i a_i
+D_{\rm s}^2 n\cdot\zeta\right|\le\nu R_s.
\]
\item At every occupied scale,
\( |n\cdot\zeta|\notin[\mu R_s/1000,1000\nu R_s]\)
for all \(|n|_\infty\le Q\).
\end{enumerate}
There are constants \(C_b,C_{{\rm res},b}\) such that, if
\[
Q\ge C_{{\rm res},b}(d+1)^2B,\qquad
\Sigma\ge C_{{\rm res},b}(d+1)^2\nu,
\]
then, with \(\ell=1+\log_2(2+(d+1)^2\nu/\mu)\),
\[
\sum_{s:N_s\le\delta M^2}N_s
\le C_b\ell L^{29}\delta^{1/2080}M^2
\qquad(0<\delta\le1).
\]
The exponents and constants are independent of the dimension and
the number of labels and scales, apart from the indicated factors.
\end{lemma}
\begin{proof}
We follow the finite argument of \cite[Lemma~9.1]{H}, spelling out
the arithmetic and the dimension dependence. The proof first covers
most label differences by neighborhoods associated with later scales.
Three such future covers then place sums of triples in a thin slab.
Separated anchor labels reduce that slab's cover from three factors
of \(M\) to two, while the pure-shift gap makes many translated
triples distinct inside the cover. This comparison forces small
mass at sparsely occupied scales.
Approximate all finite
real data on a sufficiently fine common grid, taking approximants
on a fixed divisible multiple of its step so divisions by the
\(\alpha_i\) remain on the grid. For example, a multiple of
\(D_{\rm s}^2\) suffices. Choosing the grid after the data preserves
separation and annular inequalities with fixed relaxed margins and
leaves a narrower pure-shift gap. Round scalar widths upward by at
most one grid step. The estimates below are uniform in that step.
Restrict to \(N_s\le\delta M^2\). Pair determination gives total
mass at most \(LM^2\). For parameters \(\delta\le\alpha/2\)
and \(0<\kappa<1\), greedily partition the scales in decreasing
order of \(R_s\) into counting blocks of mass at most
\(\alpha M^2\). There are at most \(J_0=3L/\alpha\) blocks.
For a block \(\mathfrak b\), let \(R_{\mathfrak b}^+\) be its
largest scale, take \(N_0=\lceil C_bB\rceil\) with
\(C_b>32D_{\rm s}\), and choose a grid width
\(\epsilon_{\mathfrak b}\asymp\nu R_{\mathfrak b}^+\).
Use the finite neighborhoods
\[
U_{\mathfrak b}(t)=
\{D_{\rm s}n\cdot\zeta+u:
|n|_\infty\le tN_0,\ u\text{ on the grid},\
|u|\le t\epsilon_{\mathfrak b}\}.
\]
They are symmetric and satisfy
\(U(t)+U(t')\subseteq U(t+t')\).
Fix a role and a counting block with uncovered mass exceeding
\(\kappa M^2\). Project onto this role and another core role,
retaining one scale witness per pair, and color edges by a third
core label. The graph has at least \((\kappa/L)M^2\) edges and
a proper coloring with at most \(M\) colors. The elementary
four-step-walk observation in the proof of \cite[Lemma~9.1]{H}
gives a subset \(B_0\) of at least \(c\eta M\) vertices,
\(\eta=\kappa/L\), such that any two have at least
\(c\eta^5M^3\) four-step walks with distinct ordered color tuples.
Repeated vertices are allowed.
That observation is purely a graph statement: choose a random
neighborhood, remove vertices with many low-codegree partners, and
count the two choices of a common neighbor on either side of a
retained midpoint.
Alternating the four annular relations along a walk cancels the
intermediate labels. Dividing by the nonzero integer \(\alpha_i\)
gives
\[
a-a'\in q(c_1,c_2,c_3,c_4)+U_{\mathfrak b}(1).
\]
Here \(q\) depends only on the ordered four edge colors; these
\(c_i\) are graph colors, unrelated to the fixed quadratic weights.
The scalar error after division is at most
\(4\nu R_{\mathfrak b}^+\), and
\(D_{\rm s}^2/\alpha_i\) is an integer multiple of
\(D_{\rm s}\), with its index covered by \(N_0\).
Packed differences outside \(U_{\mathfrak b}(2)\) cannot use the
same color tuple. The \(M^4\) possible color tuples therefore
bound their number by \(C\eta^{-5}M\). Repeating after removing
\(B_0\) covers all but \(\kappa M^2\) occurrences in each role
and block by at most
\(K_1=C(L/\kappa)^6\) subsets, each with its difference set
covered by \(K_1M\) translates of \(U_{\mathfrak b}(2)\).
Discard uncovered occurrences. Call a core label future when it
belongs to a cover subset in a later counting block. Nonfuture
label sets are disjoint over blocks. Let \(m_i^{\mathfrak b}\)
be their cardinalities in core role \(i\). Treat the side as
nonfuture, with \(m_l^{\mathfrak b}=\sum_{s\in\mathfrak b}w_s\).
Occurrences with fewer than three future roles have two nonfuture
roles. Their total mass is at most \(C\sqrt{L\alpha}M^2\), by
summing
\(\min(\alpha M^2,Lm_i^{\mathfrak b}m_j^{\mathfrak b})\)
and Cauchy--Schwarz. The discarded uncovered mass is at most
\(4J_0\kappa M^2\).
Assign each remaining occurrence three future roles and one future
cover subset in each. A class fixes its current counting block
and its three future block/subset choices. There are at most
\(4J_0^4K_1^3\) classes. Fix one of mass \(\rho M^2\), with
future subsets \(B_1,B_2,B_3\).
The subscripts now enumerate those roles; their coefficients remain
the corresponding \(\alpha_i\). Set
\(\Gamma=C_{{\rm gap},b}(d+1)^2\nu/\mu\).
If \(\rho\le C_b\delta\ell\), retain this bound. Otherwise split
the mass into four consecutive ranges by quartiles, omitting the
at most three scale bins crossing a cut. Each range retains at
least \(\rho M^2/8\) mass. The first provides anchor sets
\(B_{i0}\subset B_i\) of size at least \(c\rho M/L\).
Let \(R^\#\) be the largest scale of the second range; these
anchors are separated modulo reserved shifts at scale \(R^\#\).
The fourth range gives a set \(F\) of at least \(c\rho M^2/L\)
distinct triples projected onto these three roles.
The third range supplies the scale margin, and a greedy thinning
of the second gives \(q\ge c_b\rho/(\delta\ell)\) triples
\(t_v\), also projected onto these roles, at scales separated by
factors at least \(\Gamma\).
All future-block scales and fourth-range scales are smaller than
these by at least the required factor \(\Gamma\).
Write \(U_i\) for the neighborhoods of the three future subsets.
Subdividing index and scalar intervals covers \(U_i(t)\) by at
most \((2t+1)^{d+1}\) translates of \(U_i(1)\). Hence there is
an odd \(H_i\le100(d+1)^2\) with
\(|U_i(H_i+2)|\le2|U_i(H_i)|\). The preliminary exponential
cover is used only to find this small-growth index, and is never
paid in the cardinality estimate. The translates
\(a+U_i(H_i)\), \(a\in B_{i0}\), are disjoint: an intersection
would contradict the anchor separation with a shift index
\(O_b((d+1)^2B)\) and scalar error
\(O_b((d+1)^2\nu R^\#)\).
Set \(Z_i=B_{i0}+U_i(H_i)\). The difference covers give
\( |Z_i-B_i|\le C(K_1L/\rho)|Z_i|\).
Choose a nonempty \(X_i\subset Z_i\) minimizing
\(r_i=|X_i-B_i|/|X_i|\). Petridis's minimal-growth subset
argument, in the form reproduced in \cite[Lemma~9.1]{H}, gives
\( |X_i-mB_i|\le r_i^m|X_i|\).
One anchor translate contains a subset \(Y_i\subseteq X_i\)
of at least \(|X_i|/M\) points. If \(\mathcal P\subseteq mB_i\)
has all distinct differences outside \(U_i(2H_i)\), the sets
\(Y_i-p\), \(p\in\mathcal P\), are disjoint and lie in
\(X_i-mB_i\). Thus \(|\mathcal P|\le r_i^mM\).
A maximal packing covers \(mB_i\) by that
many translates of \(U_i(2H_i)\). Only \(m=2,3\) is used.
The outputs \(t_v+F\) lie in the slab
\[
(x_1,x_2,x_3)\in2B_1\times2B_2\times2B_3,\qquad
\left|\sum_i\alpha_ix_i+D_{\rm s}^2n\cdot\zeta\right|
\le2\nu R^\#,\quad |n|_\infty\le2B.
\]
Put \(r_0=CK_1L/\rho\). Cover the first two sumsets by
\(r_0^2M\) boxes each. Above a fixed pair, dividing the slab
equation by \(\alpha_3\) places the third coordinate in
\[
c+\{n\cdot\zeta:|n|_\infty\le C_b(d+1)^2B\}
+[-C_b(d+1)^2\nu R^\#,C_b(d+1)^2\nu R^\#].
\]
Every shift remains integral: both \(D_{\rm s}^2/\alpha_3\)
and \(D_{\rm s}\alpha_i/\alpha_3\) are integers.
If \(\mathcal P\) is a packing of the local third-coordinate
set outside \(U_3(2H_3)\), then
\(B_{30}+\mathcal P\subset3B_3\) is still a packing.
Collisions with different anchors are excluded by the same shift
and scalar bounds. Thus
\( |B_{30}|\,|\mathcal P|\le r_0^3M\).
The entire slab is covered by at most
\(CK_1^7(L/\rho)^8M^2\) product boxes.
Each such box contains at most one of the indexed outputs
\(t_v+F\). Within one translation this follows from label
separation. Across translations, let \(R_{\rm big}\) be the
larger selected scale. Its annular residual has magnitude at
least \(\mu R_{\rm big}\); the other three residuals together
are at most \(\mu R_{\rm big}/10\). Membership in one box
adds scalar error at most
\(C_b(d+1)^2\nu\max_i R_{{\rm future},i}^+
\le\mu R_{\rm big}/10\), by the choice of \(C_{{\rm gap},b}\).
The resulting pure shift has magnitude between
\(4\mu R_{\rm big}/5\) and \(3\nu R_{\rm big}\), with index
\(O_b((d+1)^2B)\le Q\), contradicting the forbidden gap.
Comparing \(q|F|\) with the slab cover yields
\(\rho^{10}\le C_b\delta\ell K_1^7L^9\), and hence
\(\rho\le C_b\ell K_1L\delta^{1/10}\).
Including all choices and losses gives
\[
M^{-2}\sum_{s:N_s\le\delta M^2}N_s
\le C_b\left(\frac{L\kappa}{\alpha}
+\sqrt{L\alpha}
+\ell L^{29}\alpha^{-4}\kappa^{-24}\delta^{1/10}\right).
\]
Take \(\alpha=\delta^{1/1040}\), \(\kappa=\alpha^2\) for
sufficiently small \(\delta\), and use the crude mass bound for
the remaining values. This proves the stated exponent \(1/2080\).
\end{proof}
\subsection{Application of the count and the resulting core bound}
We verify the hypotheses rather than apply the original arithmetic
form of \cite[Proposition~9.2]{H}. Fix a path, height, role types,
frequency bands, and amplitude bins in \eqref{eq:tested-cell-b}.
Identify each representative label with its value
\(\omega_a(x_*)\), which is injective in a fixed role and type by
\eqref{eq:representative-separation-b}. Define
\[
M_i^G=|\mathcal A_i^G|\quad\text{for core roles},\qquad
M_l^G=\sigma_l^{-2}\sum_s\|S_{l,G,s}\|_{2,I_s}^2
\quad\text{for a side},\qquad M'=\max_iM_i^G.
\]
Groups with a nonzero count have \(M'\ge1\).
Two fixed core labels determine the others. For four cores,
subtract the equations for \(Q,T\) of two candidate tuples.
The two unknown label differences solve a system with determinant
\(b_j-b_i\), modulo bounded rational shifts and errors
\(O_b(\nu\max(R_s,R_t))\). For three cores the equation for
\(T_l\) determines the last label, dividing by \(b_j-b_l\).
The shift reserve and separation force these differences to vanish.
For every core triple with excluded role \(l'\), the tests also
give the annulus
\begin{equation}\label{eq:annulus-application-b}
c_b\mu R_s\le
\left|\sum_i(b_i-b_{l'})\omega_{a_i}(x_*)+n\cdot\xi\right|
\le C_b\nu R_s,\qquad |n|_\infty\le C_bB'_h.
\end{equation}
For four cores this is \(t'-b_{l'}q'\); for three it is the
\(T_l\) test. Once the labels are fixed, the height-bin lemma
\cite[Lemma~3.5]{H} bounds their scale occurrences by
\(L_0\le P^C(1+\log(\nu/\mu))^C\).
If one role is a side, windows \eqref{eq:side-windows-b} for
different triples incident to a fixed core label are disjoint.
An overlap gives both a relation for their \(Q\) values with
error \(O(\Lambda R_s)\) and their two \(T_l\) relations.
Solving the resulting two-by-two system again divides by a
\(b_j-b_i\); separation then identifies the tuples. Couple the
fresh offset across these windows. Parseval shows their incidence
count is at most
\(w_s=\sigma_l^{-2}\|S_{l,G,s}\|_{2,I_s}^2\), giving the
side-budget hypothesis.
Here are the promised reserve checks for all these divisions. In
Lemma~\ref{lem:sparse-b} take
\[
\zeta=\xi/D_{\rm s}^2
=D_{\rm s}^{\Delta-2}\vartheta/L_h,\qquad
Q_{\rm res}=\left\lfloor E_h/D_{\rm s}^{\Delta-2}\right\rfloor,
\qquad \Sigma=\Xi/2.
\]
Every divided \(\xi\)-shift is integral in this finer lattice,
because each determinant divides \(D_{\rm s}\).
The fixed-power choice of \(A_0\) ensures
\[
Q_{\rm res}\ge
C_{{\rm res},b}(d+1)^2C_bB'_h,
\qquad d=\dim\theta_z.
\]
Also \(\nu,\Lambda\le C_bD_*^{250}\), while
\(\Xi=D_*^{1000}\), so the separation reserve holds. All
rational shifts remain below \(\mathcal H\).
Every successful component contains a nonexceptional primary,
even if its representative is exceptional. Applying
\eqref{eq:tiny-huge-b} to its pure horizontal shifts excludes the
entire intermediate interval required by the pure-shift hypothesis,
since \(\mu\ge U_0^{-1}\) and the interval length \(|I_s|\) lies
between \(r_{t_{s_0}}\) and \(r_{v_{s_0}}\). This checks all five hypotheses.
Lemma~\ref{lem:sparse-b}, pair determination, and the side incidence
bound therefore give the two counts H(34): for some fixed
\(p_0<1\),
\begin{align}
\sum_sN_s^G&\le L_0\min_{i\ne j}M_i^GM_j^G,\label{eq:count-one-b}\\
\sum_s(N_s^G)^{p_0}
&\le P^C(1+\nu+\mu^{-1}+\Lambda)^C(M')^{2p_0}.
\label{eq:count-power-b}
\end{align}
Indeed the sparse-mass exponent \(c'=1/2080\) allows any
\(1-p_0<c'\). On the bins
\(2^{-j-1}(M')^2<N_s^G\le2^{-j}(M')^2\), apply the sparse
bound and sum the geometric series with ratio
\(2^{1-p_0-c'}<1\).
Finally the amplitude and group summation of
\cite[Proposition~9.3]{H} uses precisely
\eqref{eq:tested-cell-b}, \eqref{eq:core-path-budgets-b}, and
\eqref{eq:count-one-b}--\eqref{eq:count-power-b}. Interpolate the
two counts with a sufficiently small fixed weight \(\theta'>0\),
giving power \(p=(1-\theta')+\theta'p_0\). The path square
budgets sum the group factors, and the core count cap \(D_*\)
controls the common amplitude shift. The three amplitude-gap sums
are geometric, as in that proposition. The frequency losses after
interpolation are at most
\(\mu^{-C\theta'}\nu^{C\theta'}\Lambda^{C\theta'}\);
choose \(\theta'\) small and then the derivative order \(L\)
large so they are summable against
\(\mu^p\nu^{-Lp}\Lambda^{-Lp}\).
Applying the original cell minimum before subadditivity yields
\begin{equation}\label{eq:core-final-b}
C_{q,b}\eta^{\varepsilon}P^{C_4}|R|+O(e^{-k}|R|)
\end{equation}
for the terms with at least three core arguments, with fixed
\(\varepsilon>0\) and \(C_4\) independent of \(q\).
The same calculation is the core step of \cite[Section~6.6]{V}:
its frozen scheduled inputs provide exactly the amplitudes,
representatives, and same-cell side incidence verified above.
Together with Proposition~\ref{prop:short-block-b} and the side
square bound, this supplies the side and core estimates required
to close the proof of Theorem~\ref{thm:local}.
==== END Triple-ergodic-averages-with-distinct-integer-slopes//sections/arithmetic.tex ====
==== BEGIN Triple-ergodic-averages-with-distinct-integer-slopes//sections/completion.tex ====
\section{Parameter choice and completion of the local estimate}\label{sec:completion}
The preceding sections provide the coefficient-dependent inputs to
the completion arguments in \cite[Section~10]{H} and
\cite[Section~7]{V}. We now explain the common parameter choice and
the two distinct conclusions obtained from it. First we bound the
fixed-input constant $C_N(q)$ independently of $N$. Only after this
absorption do we complete the proof for depth-dependent inputs.
\subsection{The retained-cell estimate}
We use the base reductions of \cite[Proposition~5.8]{H} and
\cite[Proposition~5.6]{V}, with the modifications proved above.
At one noninitial accuracy level their target, on a normalized root
$R$, is a sum of terms
$\abs I\min\{\eta,\abs{H_I(z_I)}\}$ over a specified subset of
descendant cells. Here $z_I$ is the four-tuple supplied by the base
expansion at $I$, and $\eta=\exp(-k^\theta)$ is its quality parameter,
with $k$ the accuracy log and $0<\theta<1$ fixed. The constructions
then decompose each of these four-tuples into the localized terms
estimated in the preceding sections.
The stopping constructions in \cite[Section~10.1]{H} and
\cite[Section~7.1]{V} are used in their stated order. Main columns and
their uses are fixed first; mode expansions and the whole side
functions are fixed next; helper projections are constructed after
those side functions. Size and counting stops are then imposed on
these fixed objects. A stop removes original summands from an
attempt; it does not redefine a previously fixed stack function.
In particular, the maximal functions in the short-block estimates
remain defined on whole retained cells, including the portions below
later stopping cells.
Let $\mathcal A_R^{\mathrm{ret}}$ denote the cells not removed by
these stops in the attempt rooted at $R$. The conclusions of the
preceding sections give the same estimate as the retained-cell
calculations in the two companions:
\begin{equation}\label{eq:retained}
\begin{aligned}
\sum_{I\in\mathcal A_R^{\mathrm{ret}}}\abs I
\min\{\eta,\abs{H_I(z_I)}\}
&\le \Bigl[C_q\eta^{\varepsilon}P^{C_4}
+(1+C_N(q))C_qP^{-1/2}G_0
(K+2)^{C_{19}(q-2)}\Bigr]\abs R\\
&\quad+O(e^{-k}\abs R).
\end{aligned}
\end{equation}
Here $\varepsilon>0$ is fixed, $P$ is the transfer parameter, $K$
is the depth-block length, and $G_0$ is the row and stack overhead.
The constants may depend on $b,\delta_0,C_0$.
For clarity, the estimate has three sources. Terms with at least
three core factors use the powered core estimate, and terms with at
least three side factors use their square-energy bounds. With
$p=1-\varepsilon>2/3$, both are converted to the first term
of~\eqref{eq:retained} by
$\min\{\eta,a\}\le\eta^{1-p}a^p$.
Terms with exactly two sides use the transfer estimate for the
whole complementary arguments. Its main term is again included in
the first term. The remaining coupled near-band sum is estimated
at power one by the short-block comparison; it gives the second
term, including its factor $1+C_N(q)$, by~\eqref{eq:near-band-b}.
Physical and numerical
errors are estimated at power one as well. These alternatives
exhaust the four-position expansion. The minimum is applied to
the whole contribution of a cell before subadditivity is used;
no smallness of a separately selected group is being assumed.
\subsection{Restarts and the complexity hierarchy}
The integrated size, path-count, and square-energy bounds allow the
stopping thresholds to be enlarged by fixed factors so that the first
stopped cells occupy at most $\abs R/4$. At each such cell send the
original summands there and below to a new attempt. The local
normalizations and column bounds are available on every new root.
Linear-side energy is proved anew by the construction on that root,
not by asserting a proportional restriction of a global energy bound.
The resulting attempt roots satisfy
\begin{equation}\label{eq:attempt-packing}
\sum_{R\text{ an attempt root below }R_0}\abs R
\le\sum_{m=0}^{\infty}4^{-m}\abs{R_0}
=\frac43\abs{R_0}.
\end{equation}
All the trees are finite, so the restart argument can equivalently be
read as induction on their remaining depths.
In the scheduled-input construction, the original depths, functions,
and block partitions remain attached to the summands. Restriction to
a new root preserves the pointwise size and variation bounds with
the same $W_j$. The row estimates of \cite[Section~4]{V} therefore
apply at each restart with no factor depending on $W_j$.
The distinction between the two complexity scales is essential.
The bounds supplied by the constructions are
\begin{equation}\label{eq:budget-hierarchy}
\begin{aligned}
\log D_{\mathrm m}&\le C_q k^{C},
&\log D_*&\le C_q(1+k+P)^C,\\
K&\le C_q(1+k+P)^{C_5},
&G_0&\le\exp\bigl(C_q(1+\log(2k))^C\bigr).
\end{aligned}
\end{equation}
Here $D_{\mathrm m}$ bounds the main and helper catalogs, and
$D_*$ bounds the larger mode, Fourier, and rational-height data.
The latter data enter the counting and mode estimates, but are not
fed back into the main catalog construction. Strong stack estimates
compress main projection families only; mode projections have
already been included in the Bessel rows. This is precisely the
separation in \cite[Section~10.3]{H} and \cite[Section~7.3]{V}.
The coefficient modifications preserve this separation. Chart
substitutions multiply heights and derivative bounds by fixed
slope-dependent factors; the change of height base occurs only
in the mode construction. The sparse-count estimate retains its
dimension-polynomial reserves and its power saving. Thus all the
degrees $C,C_4,C_5,C_{19}$ can be fixed \emph{before} choosing $q$.
Their numerical values may be enlarged to cover both the fixed-input
and scheduled-input constructions. Only coefficients such as $C_q$
are allowed to depend on the later choice of $q$.
Choose $q\in(2,3)$ sufficiently close to $2$ that
\begin{equation}\label{eq:common-q}
C_{19}C_5(q-2)<\frac18
\end{equation}
for both constructions. This common choice is legitimate because
local detection is available for every $q>2$ and short-block
comparison for every $2<q<3$. Next put
$P=\lceil\eta^{-\alpha}\rceil$, choosing a fixed $\alpha>0$ so small
that $\alpha C_4<\varepsilon/2$, as well as the analogous finitely
many upper bounds already included in the powered costs.
Finally choose the starting accuracy sufficiently large. Since $P$
dominates every fixed power of $k$ and $G_0=P^{o(1)}$, we obtain
\[
P^{-1/2}G_0(K+2)^{C_{19}(q-2)}\le C_qP^{-1/4},\qquad
\eta^{\varepsilon}P^{C_4}\le C_q\eta^{\varepsilon/2}.
\]
Also $P^{-1/4}\le\eta^{\alpha/4}$, and $e^{-k}$ is smaller than
these quality powers for large $k$. Consequently, for some $c_*>0$,
\eqref{eq:retained} and~\eqref{eq:attempt-packing} give the full
base-segment bound
\begin{equation}\label{eq:quality-gain}
\sum_{I\text{ in the base segment}}\abs I
\min\{\eta,\abs{H_I(z_I)}\}
\le C_q\eta^{c_*}(1+C_N(q))\abs{R_0}.
\end{equation}
\subsection{Fixed inputs: accuracy summation and absorption}
The base accuracy levels satisfy $k_{t+1}=k_t^A$ for a fixed $A>1$.
For tuples whose later level is $t$, there are $O(t)$ choices of
the earlier level and a bounded number of slot patterns.
The segment-root packing in \cite[Section~5]{H}, followed by
\eqref{eq:attempt-packing}, gives a bounded total root length for
each tuple. The noninitial terms in~\eqref{eq:quality-gain} therefore
have total coefficient controlled by
\begin{equation}\label{eq:accuracy-series}
C_q\sum_{t\ge12}t\exp(-c_*k_t^\theta).
\end{equation}
This series is finite, and its value is arbitrarily small when the
starting accuracy is sufficiently large. The replacement and
two-increment errors have the same summable multiplicity.
The finitely many initial levels have depth-independent bounds from
the active-scale reduction, as in \cite[Section~10.4]{H}.
Thus the modified construction proves the following normalized
conclusion.
\begin{proposition}[Normalized absorption]\label{prop:absorption}
For the fixed $q$ chosen above and every sufficiently small
$\tau>0$, there is $A_\tau<\infty$, independent of $N$, such that
the following holds. Suppose a finite rooted dyadic subtree has root
$I_*$, its summation cells lie in at most $N$ consecutive depths,
and its fixed scalar inputs satisfy
\[
\left(\abs I^{-1}\int_I\abs{f_j}^q\right)^{1/q}\le1
\quad(0\le j\le3, I\text{ in the subtree}).
\]
Then, for any subset of summation cells and admissible kernels,
\[
\sum_{I\text{ a summation cell}}\abs I\,\abs{H_I(f_0,f_1,f_2,f_3)}
\le (A_\tau+\tau C_N(q))\abs{I_*}.
\]
\end{proposition}
The argument just given is the proof: take the coefficient
in~\eqref{eq:accuracy-series} below $\tau$ and include its constant
part, the initial levels, and the numerical errors in $A_\tau$.
The base expansion is first taken at a finite accuracy cutoff.
At each fixed cell the terminal residual tends to zero by the
pair-comparison estimates of the base reduction; since there are
only finitely many cells, the passage to all accuracy levels is
legitimate. This is the concluding expansion step in
\cite[Section~10.4]{H}.
We include the outer normalization argument to make the absorption
explicit. Start with disjoint roots at the largest scale in a finite
collection from~\eqref{eq:CN}. At a root $J$, write
$a_{j,J}=(\abs J^{-1}\int_J\abs{f_j}^q)^{1/q}$. If some $a_{j,J}$
vanishes, all forms below $J$ vanish. Otherwise stop at the first
strict descendants $J'$ on which an average of $\abs{f_j}^q$
exceeds $16a_{j,J}^q$. The union of first stops has measure at most
$\abs J/4$. Dividing $f_j$ by $16^{1/q}a_{j,J}$ normalizes the
intervening tree, so Proposition~\ref{prop:absorption} bounds its sum
by
\[
16^{4/q}(A_\tau+\tau C_N(q))\abs J\prod_ja_{j,J}.
\]
Repeat at the stopped roots. The sets $E_J$ left after deleting their
first stops are disjoint, have measure at least $3\abs J/4$, and
satisfy $M_qf_j\ge a_{j,J}$ on $E_J$. Hence
\[
\sum_J\abs J\prod_ja_{j,J}
\le\frac43\int_\R\prod_jM_qf_j.
\]
Taking the supremum in the definition of $C_N(q)$ gives
\[
C_N(q)\le C A_\tau+C\tau C_N(q),
\]
where $C$ is independent of $N$ and bounded for $2<q<3$.
Choose $C\tau<1/2$ and use the already established finiteness of
$C_N(q)$ to absorb the last term. We have proved
\begin{equation}\label{eq:uniform-CN}
C_{\mathrm H}(q):=\sup_{N\ge1}C_N(q)<\infty.
\end{equation}
\subsection{Depth-dependent inputs}
Now perform the scheduled-input construction with the same $q$,
using its own subsequent choices of $\alpha$ and starting accuracy.
Insert~\eqref{eq:uniform-CN} into~\eqref{eq:quality-gain}. The
factor $1+C_N(q)$ is now a fixed constant. The scheduled base
expansion of \cite[Section~5]{V} has the same accuracy multiplicity
and segment-root packing, with the depth-row estimates supplying
the uniform bounds. Its noninitial terms therefore sum
by~\eqref{eq:accuracy-series}. Its finitely many initial levels
are bounded by its active-scale reduction independently of the
number of depths and of every $W_j$.
Finally, at a fixed original cell, all accuracy updates are evaluated
on its same actual scheduled input $z_{j,s(I)}$. The per-cell pair
comparisons therefore make the terminal residual tend to zero.
There are finitely many cells, so the finite-cell sum commutes with
this limit. The triangle inequality and the uniform summed bounds
give~\eqref{eq:local-estimate}, exactly as in
\cite[Section~7.4]{V}. This proves Theorem~\ref{thm:local}, and hence,
by Section~\ref{sec:transfer}, Theorem~\ref{thm:main}.
==== END Triple-ergodic-averages-with-distinct-integer-slopes//sections/completion.tex ====
==== BEGIN Triple-ergodic-averages-with-distinct-integer-slopes//sections/introduction.tex ====
\section{Introduction}\label{sec:introduction}
Let $(X,\mathcal F,\mu)$ be a probability space and let $T:X\to X$
be an invertible measure-preserving transformation with measurable
inverse. Given bounded measurable functions $f_1,f_2,f_3$ and distinct
nonzero integers $\lambda_1,\lambda_2,\lambda_3$, consider
\begin{equation}\label{eq:averages}
A_N(x)=\frac1N\sum_{n=1}^N
\prod_{j=1}^3 f_j(T^{\lambda_jn}x).
\end{equation}
The issue is convergence at almost every point, rather than only in
norm. We assume that $T$ is mixing, in the sense that
\begin{equation}\label{eq:mixing}
\mu(A\cap T^{-n}B)\longrightarrow\mu(A)\mu(B)
\quad\text{as }\abs n\longrightarrow\infty
\quad(A,B\in\mathcal F).
\end{equation}
\begin{theorem}\label{thm:main}
For every system satisfying~\eqref{eq:mixing}, every triple of pairwise
distinct nonzero integers $\lambda_1,\lambda_2,\lambda_3$, and every
bounded measurable $f_1,f_2,f_3:X\to\C$, the averages
in~\eqref{eq:averages} satisfy
\[
A_N(x)\longrightarrow\prod_{j=1}^3\int_X f_j\,d\mu
\quad\text{for almost every }x,
\]
as $N$ tends to infinity through all positive integers. The exceptional
null set may depend on the slopes and the functions.
\end{theorem}
The probability space need not be standard or countably generated.
Both positive and negative slopes are allowed. The proof depends on
the analytic results of the companion manuscripts~\cite{H,V}; their
role and the extension required here are described below.
\subsection{Context and predecessors}
Birkhoff's pointwise ergodic theorem~\cite{Birkhoff} treats one
function. Products along several iterates arise naturally in
Furstenberg's ergodic proof of the multiple recurrence theorem and its
connection with arithmetic progressions~\cite{Furstenberg}. Multiple
recurrence, norm convergence, and pointwise convergence are different
questions. In particular, convergence in $L^2$ does not by itself
establish convergence at almost every point along the full sequence.
For weakly mixing transformations, the product-of-integrals limit in
$L^2$ is classical. Bergelson's
weakly mixing PET theorem~\cite[Theorem~1.2 and p.~340]{Bergelson}
includes distinct nonzero integer slopes of either sign.
For general systems, Host and Kra~\cite{HostKra} and
Ziegler~\cite{Ziegler} developed the norm-convergence theory using
characteristic factors with nilpotent structure. These results explain
the limiting structure without resolving the corresponding pointwise
question.
Bourgain's double recurrence theorem~\cite{Bourgain} establishes
almost-sure convergence for two bounded functions evaluated at two
powers of one transformation. Pointwise theorems for longer products
also have substantial precedents under additional assumptions on the
system. For example, Derrien and Lesigne~\cite[Theorem~4.2]{DerrienLesigne}
prove polynomial pointwise theorems for $K$-systems and exact
endomorphisms; their invertible case includes the linear configurations
considered here. Other structural hypotheses yielding pointwise
convergence for all finite progression lengths include singular spectrum
on the Pinsker factor of a weakly mixing system, by
Assani~\cite{Assani}, and the pairwise-independent-joining property for
weakly mixing systems, by Gutman, Huang, Shao, and Ye~\cite{GHSY}.
The significance of Theorem~\ref{thm:main} is that its
dynamical assumption is mixing alone, while the three slopes are
arbitrary distinct nonzero integers.
The harmonic-analytic method separates convergence from identification
of the limit. A uniform oscillation estimate gives pointwise
convergence; mixing identifies that limit by an $L^2$ argument.
Oscillation methods in ergodic theory are developed in
\cite{Bourgain,JKRW}, and the passage from translation estimates to
orbit estimates follows the transference principle of
Calder\'on~\cite{Calderon}. We give that passage directly, so it imposes
no additional structure on the probability space.
\subsection{Analytic input and the coefficient extension}
The manuscript~\cite{H} proves an $L^3$ bound for the trilinear Hilbert
transform through local four-linear forms. Its proof includes chart
detection, localization by quadratic data, comparison on short depth
blocks, and a separated-frequency counting estimate. The quantitative
inverse theorem of Leng, Sah, and Sawhney~\cite{LSS} enters its chart
detection argument. Arithmetic regularity and inverse-theorem methods
also occur in Tao's sublogarithmic cancellation theorem for truncated
multilinear Hilbert transforms~\cite{Tao}; that cancellation theorem is
distinct from the uniform local estimate used here.
The companion~\cite{V} adapts the local estimates to inputs that vary
with depth and deduces pointwise convergence for the slopes $1,2,3$.
The local forms in both companions use the four coordinates
$x,x+t,x+2t,x+3t$. A power of $T$ and a permutation of the functions
do not turn an arbitrary triple into this configuration. Thus their
stated theorems cannot simply be invoked for the slopes in
Theorem~\ref{thm:main}.
The new work is the extension of their local proofs to
$x+b_jt$, $0\le j\le3$, for fixed distinct integers with $b_0=0$.
Three features are particularly useful beyond this application.
First, an integer left inverse of the \emph{whole} primitive
configuration replaces a unimodular pair of coordinates; this removes
extra lattice or torus configurations in discretization.
Second, a Vandermonde relation provides the quadratic cancellation and
the Fourier randomization for every set of four distinct slopes.
Third, a height base divisible by all pair differences preserves the
divisibility required in the separated-frequency argument. The last
step must preserve a bound polynomial in the dimension: an exponential
loss there would not fit the proof's parameter choices.
These are extensions of the proofs in~\cite{H,V}, not new proofs of
their coefficient-independent projection and row estimates. We state
the local conclusion completely, prove the coefficient-sensitive
identities and estimates, and specify the results imported from the
companions at their points of use. In particular, the technical part
is a proof by reference to those manuscripts, whereas the
ergodic deduction is presented in full.
\subsection{Proof structure}
Section~\ref{sec:local} defines the admissible kernels and states the
local estimate for inputs with controlled variation on blocks of
depths. Section~\ref{sec:transfer} proves Theorem~\ref{thm:main} from
that estimate: smooth scale differences are realized by cancellative
kernels, then transferred to integer sequences and finite orbits.
The resulting oscillation bound gives convergence for smooth profiles;
moment-corrected approximations to interval indicators recover all
integer lengths.
The remaining sections prove the local estimate. Section~\ref{sec:analytic}
establishes the coefficient modifications in detection and
localization. Section~\ref{sec:arithmetic} proves the short-block,
height, and counting modifications. Section~\ref{sec:completion}
closes the parameter argument, first for fixed inputs and then for
depth-dependent inputs. All analytic constants may depend on the fixed
slopes; no bound is required uniformly over triples of integers.
==== END Triple-ergodic-averages-with-distinct-integer-slopes//sections/introduction.tex ====
==== BEGIN Triple-ergodic-averages-with-distinct-integer-slopes//sections/local.tex ====
\section{The local estimate and the coefficient algebra}\label{sec:local}
The analytic estimate concerns four inputs on the real line. Its
constant may depend on the slopes, but must be independent of the number
of scales and of the number of changes of the inputs. This uniformity
will give the oscillation estimate in Section~\ref{sec:transfer}.
\subsection{Normalization and local forms}
Set
\[
g=\gcd(\abs{\lambda_1},\abs{\lambda_2},\abs{\lambda_3}),
\qquad S=T^g,\qquad b_0=0,\qquad b_j=\lambda_j/g\quad(1\le j\le3).
\]
Then $S$ is invertible, probability preserving, and mixing, and
$T^{\lambda_j n}=S^{b_jn}$. Henceforth the four distinct integers
$b_0,b_1,b_2,b_3$ are fixed, and
$\gcd(\abs{b_1},\abs{b_2},\abs{b_3})=1$. We make no assertion of
uniformity as these integers vary.
Dyadic intervals are half-open. The depth of a descendant $I$ of a
dyadic interval $J$ is $s(I)=\log_2(\abs{J}/\abs{I})$. A finite rooted
tree contains every ancestor between any of its intervals and its root.
The dyadic lattice may be translated.
\begin{definition}[Local kernels]\label{def:kernels}
Fix $\delta_0>0$ and $C_0\ge1$. For an interval $I$ of length $r$, a kernel
$w_I\in C_c^\infty(\R^2;\C)$ is admissible if
\begin{enumerate}
\item $\dist(x+b_jt,\partial I)\ge\delta_0r$ and $x+b_jt\in I$ for every
$(x,t)\in\supp w_I$ and $0\le j\le3$;
\item for every $v\in\R$ and $0\le j\le3$,
\begin{equation}\label{eq:marginals}
\int_\R w_I(v-b_jt,t)\,dt=0;
\end{equation}
\item for nonnegative integers $a,d$ and $n=a+d$,
\begin{equation}\label{eq:derivatives}
\norm{\partial_x^a\partial_t^d w_I}_\infty
\le r^{-n}\bigl(C_0(n+2)\bigr)^{C_0(n+2)}.
\end{equation}
\end{enumerate}
For scalar measurable inputs define
\begin{equation}\label{eq:local-form}
H_I(z_0,z_1,z_2,z_3)
=r^{-2}\int_{\R^2} w_I(x,t)\prod_{j=0}^3z_j(x+b_jt)\,dx\,dt,
\end{equation}
whenever the integral is absolutely convergent.
\end{definition}
For a nonempty integer block $B=\{u,\ldots,v\}$ and functions $z_s$ on
a common domain, put
\[
\mathcal V_B(z)=\max_{u\le s\le v}\abs{z_s}
+\sum_{s=u}^{v-1}\abs{z_{s+1}-z_s}.
\]
This is a pointwise bound on both size and variation within the block;
no variation across block boundaries is included.
\begin{theorem}[Local estimate with variation in depth]\label{thm:local}
Fix the primitive distinct integers $b_0=0,b_1,b_2,b_3$ and the
parameters $\delta_0,C_0$. Let $\mathcal T$ be a finite dyadic tree with root
$J$ and largest depth $d$, and let $\mathcal I\subseteq\mathcal T$.
Assign an admissible kernel to every $I\in\mathcal I$.
For each $0\le j\le3$, let $z_{j,s}:J\to\C$, $0\le s\le d$, be
measurable, and partition $\{0,\ldots,d\}$ into $W_j\ge1$
deterministic nonempty consecutive blocks. Suppose that
\[
\mathcal V_B(z_j)\le W_j^{-1/2}\quad\text{almost everywhere on }J
\]
for each block in the $j$th partition. Extend every input by zero
outside $J$. Then
\begin{equation}\label{eq:local-estimate}
\sum_{I\in\mathcal I}\abs{I}\,
\abs{H_I(z_{0,s(I)},z_{1,s(I)},z_{2,s(I)},z_{3,s(I)})}
\le C_{b,\delta_0,C_0}\abs{J}.
\end{equation}
The constant is independent of the lattice, tree, depth range,
subcollection, kernels, and the four partitions and their sizes.
\end{theorem}
The schedules are prescribed at every depth, even depths with no
summation intervals. This is exactly the hypothesis of
\cite[Theorem~3.2]{V}, except for the coefficients in the local forms.
Sections~\ref{sec:analytic}--\ref{sec:completion} prove the theorem by
extending that proof and its analytic inputs from~\cite{H}. Before
turning to those technical arguments, Section~\ref{sec:transfer} proves
the implication from Theorem~\ref{thm:local} to the ergodic theorem.
\subsection{Integer reconstruction and quadratic cancellation}
Two features of the coefficients recur in both the analytic proof and
transference. Pairs of coordinates determine $(x,t)$ over $\R$, but
need not determine it over $\Z$. All four coordinates together do
determine the integral step. Quadratic cancellation requires a separate
identity, furnished by Vandermonde interpolation.
\begin{lemma}[Coefficient identities]\label{lem:algebra}
There exist integers $e_0,e_1,e_2,e_3$ and nonzero integers
$c_0,c_1,c_2,c_3$ such that
\begin{equation}\label{eq:bezout}
\sum_{j=0}^3e_j=0,\qquad \sum_{j=0}^3e_jb_j=1,
\end{equation}
and
\begin{equation}\label{eq:vandermonde}
\sum_{j=0}^3c_jb_j^a=0\qquad(a=0,1,2).
\end{equation}
The nullspace over $\R$ of the two rows $(1)_j,(b_j)_j$ is spanned
by $(c_j)_j$ and $(b_jc_j)_j$.
Consequently, for $y_j=x+b_jt$,
\[
x=y_0,\qquad t=\sum_je_jy_j,\qquad \sum_jc_jy_j^2=0.
\]
For any distinct $i,j$, the pair map
$(x,t)\mapsto(x+b_it,x+b_jt)$ is injective on $\Z^2$ and pushes
normalized Haar measure on $(\R/L\Z)^2$ to normalized Haar measure,
for every $L>0$.
\end{lemma}
\begin{proof}
Bezout's identity gives integers $e_1,e_2,e_3$ with
$\sum_{j=1}^3e_jb_j=1$; set $e_0=-\sum_{j=1}^3e_j$.
Let $P(z)=\prod_{j=0}^3(z-b_j)$. Lagrange interpolation of $z^a$,
$0\le a\le2$, followed by comparison of the coefficient of $z^3$,
gives
$\sum_jb_j^a/P'(b_j)=0$. Multiply by a common positive integer
multiple of the denominators to obtain~\eqref{eq:vandermonde} with
every $c_j\ne0$. The two displayed null vectors are independent
because the $b_j$ are distinct, and the nullspace has dimension two.
The identities for $y_j$ follow by expansion. Finally the pair matrix
has determinant $b_j-b_i\ne0$, so it is injective on the lattice and
surjective on the real torus. A surjective continuous homomorphism of
compact groups pushes normalized Haar measure to normalized Haar
measure.
\end{proof}
\begin{remark}\label{rem:nonunimodular}
Primitivity does not mean that some pair is unimodular. For example,
the slopes $(b_0,b_1,b_2,b_3)=(0,6,10,15)$ are primitive, but no pair
difference is $1$ in absolute value. This is why the integer inverse
in~\eqref{eq:bezout}, rather than division by a pair determinant,
is used in the lattice and torus arguments below.
\end{remark}
\subsection{The fixed-input constant}
The proof of Theorem~\ref{thm:local} first establishes a fixed-input
estimate. For $2<q<3$, let $M_q f=(M(\abs f^q))^{1/q}$, where $M$
is the uncentered Hardy--Littlewood maximal operator. Let $C_N(q)$
be the least constant for which
\begin{equation}\label{eq:CN}
\sum_{I\in\mathcal A}\abs I\,\abs{H_I(f_0,f_1,f_2,f_3)}
\le C_N(q)\int_\R\prod_{j=0}^3M_qf_j(x)\,dx
\end{equation}
for all finite collections $\mathcal A$ of intervals in one translated
dyadic lattice, whose lengths belong to at most $N$ consecutive scales,
admissible kernels, and compactly supported
scalar $L^q$ inputs. This is the definition in
\cite[Section~2]{H}, with our coefficients.
Pair-coordinate estimates imply $C_N(q)\lesssim_bN$, so this constant
is finite before any absorption argument. The analytic proof will show
$\sup_NC_N(q)<\infty$ for a suitable fixed $q>2$, and then use that
bound in the scheduled-input argument. This order avoids using
Theorem~\ref{thm:local} to prove its own analytic input.
==== END Triple-ergodic-averages-with-distinct-integer-slopes//sections/local.tex ====
==== BEGIN Triple-ergodic-averages-with-distinct-integer-slopes//sections/transfer.tex ====
\section{From the local estimate to pointwise convergence}
\label{sec:transfer}
We now deduce Theorem~\ref{thm:main} from Theorem~\ref{thm:local}.
This section is independent of the internal notation of the analytic
proof. It follows the transference scheme of
\cite[Sections~2--3]{V}, including its use of signed smooth profiles.
We give all the details, both to handle arbitrary integer slopes and to
separate the two roles of the hypotheses: the local estimate gives
almost everywhere convergence of smooth averages on every invertible
probability preserving system; mixing identifies their limit.
Throughout, \(b_0=0,b_1,b_2,b_3\) are the fixed primitive slopes,
\(S\) is an invertible measure-preserving transformation, and
\(e_j\) are the integers in \eqref{eq:bezout}. All constants in
this section may depend on the slopes.
\subsection{Smooth profiles and cancellation kernels}
\begin{definition}[Averaging profile]
\label{def:tr-profile}
An averaging profile is a real \(\varphi\in C_c^\infty(\R)\) with
\begin{equation}
\int_\R\varphi=1,
\qquad \int_\R t^j\varphi(t)\,dt=0\quad(1\leq j\leq3),
\label{eq:tr-moments}
\end{equation}
and, for some \(G\geq1\),
\begin{equation}
\|\varphi^{(n)}\|_\infty
\leq \bigl(G(n+2)\bigr)^{G(n+2)}\qquad(n\geq0).
\label{eq:tr-gevrey}
\end{equation}
Write \(\varphi_a(t)=a^{-1}\varphi(t/a)\) for \(a>0\).
Profiles are necessarily signed: a nonnegative smooth function of
mass one cannot have zero second moment.
\end{definition}
The derivative class in \eqref{eq:tr-gevrey} is closed under fixed
translations, dilations, differentiation, and finite linear combinations.
It contains a nonnegative bump \(\rho\) of mass one supported in
\([-1,1]\). For completeness, normalize
\[
\rho_0(t)=
\begin{cases}
\exp\bigl(-1/(1-t^2)\bigr),& |t|<1,\\
0,& |t|\geq1.
\end{cases}
\]
For \(|t|<1\), put \(d=1-t^2\). On a complex disk of radius \(cd\)
about \(t\), with sufficiently small absolute \(c>0\), the real part
of \((1-z^2)^{-1}\) is at least \(c'/d\). Cauchy's estimate gives
\[
|\rho_0^{(n)}(t)|\leq n!(cd)^{-n}e^{-c'/d}.
\]
For each \(n\) this tends to zero at the endpoints, and its maximum
over \(0<d\leq1\) is at most \(C^{n+1}n!(n+1)^n\). Thus the zero
extension is smooth and satisfies \eqref{eq:tr-gevrey} after enlarging
\(G\). Lemma~\ref{lem:tr-density} will construct averaging profiles
arbitrarily close to interval indicators.
Fix a profile \(\varphi\), and put
\begin{equation}
\psi=\varphi_2-\varphi,
\qquad
\Delta_k(t)=\varphi_{2^{k+1}}(t)-\varphi_{2^k}(t)
=2^{-k}\psi(2^{-k}t).
\label{eq:tr-difference}
\end{equation}
\begin{lemma}[Realizing a scale difference]
\label{lem:tr-kernel}
There exist a dyadic integer \(D\geq1\), a constant \(C_0\geq1\),
and a real function \(\mathcal W\in C_c^\infty(\R^2)\), depending
only on \(\varphi\) and the slopes, with the following property.
For an interval \(I\) of length \(r=D2^k\) and midpoint \(m_I\),
the kernel
\[
w_I(x,t)=\mathcal W\left(\frac{x-m_I}{r},\frac{t}{r}\right)
\]
is admissible for Theorem~\ref{thm:local}, with interior margin
\(\delta_0=1/8\) and derivative constant \(C_0\). Moreover,
\begin{equation}
\frac1r\int_\R\mathcal W(z,t/r)\,dz=\Delta_k(t)
\qquad(t\in\R).
\label{eq:tr-kernel-integral}
\end{equation}
\end{lemma}
\begin{proof}
The moments of \(\psi\) through order three vanish: its \(j\)th
moment is \((2^j-1)\int t^j\varphi(t)\,dt\). Choose \(B\geq1\)
with \(\operatorname{supp}\psi\subseteq[-B,B]\), and define
\[
g(t)=\frac16\int_{-\infty}^t(t-v)^3\psi(v)\,dv.
\]
For \(t>B\), expansion of the cubic and the four zero moments show
that \(g(t)=0\); it also vanishes for \(t<-B\). Hence
\(g\in C_c^\infty(\R)\), \(g^{(4)}=\psi\), and \(g\) satisfies
\eqref{eq:tr-gevrey} with a possibly larger constant. Indeed, its
first four derivative bounds follow from integration over \([-B,B]\),
and its subsequent derivatives are derivatives of \(\psi\).
Take \(u(z)=8\rho(8z)\), with the bump constructed above. Choose a
dyadic integer \(D>4B\max_j|b_j|\), and set
\[
F(z,\sigma)=u(z)D^{-3}g(D\sigma),\qquad
L_j=\partial_\sigma-b_j\partial_z,\qquad
\mathcal W=\prod_{j=0}^3L_jF.
\]
The constant-coefficient operators commute. On the support of
\(\mathcal W\),
\[
|z+b_j\sigma|\leq\frac18+\frac{|b_j|B}{D}<\frac38.
\]
After rescaling, every \(x+b_jt\) therefore lies in \(I\), at distance
greater than \(r/8\) from its endpoints. Put
\(G_j=\prod_{i\ne j}L_iF\). The chain rule gives
\[
w_I(v-b_jt,t)
=r\frac{d}{dt}
G_j\left(\frac{v-b_jt-m_I}{r},\frac{t}{r}\right).
\]
The function differentiated is compactly supported in \(t\), so its
integral is zero. This proves all four line cancellations. The product
rule and \eqref{eq:tr-gevrey} give the required derivative bounds,
with fixed \(D,b_j\) absorbed into \(C_0\).
Finally, integration in \(z\) kills every term of \(\prod_jL_jF\)
containing a \(z\)-derivative. Since \(\int u=1\),
\[
\int_\R\mathcal W(z,\sigma)\,dz=Dg^{(4)}(D\sigma)=D\psi(D\sigma).
\]
Division by \(r=D2^k\) yields \eqref{eq:tr-kernel-integral}.
\end{proof}
We use the block size and variation \(\mathcal V_B\) from
Section~\ref{sec:local}. Block endpoints are always deterministic;
the input values may depend on the spatial variable.
\begin{lemma}[A real-line test]
\label{lem:tr-real}
Let \(0\leq k_-\leq k_+\), and partition
\(\{k_-,\ldots,k_+\}\) into \(w\) nonempty consecutive blocks.
Suppose that bounded Borel functions \(h_1,h_2,h_3\) and
\(h_{0,k}\) are supported in a common interval of length \(L>0\),
that \(|h_j|\leq1\) for \(1\leq j\leq3\), and that
\(\mathcal V_{B_0}(h_0)\leq w^{-1/2}\) on every block \(B_0\).
Then
\begin{equation}
\left|\sum_{k=k_-}^{k_+}\int_{\R^2}
h_{0,k}(x)\prod_{j=1}^3h_j(x+b_jt)\Delta_k(t)\,dx\,dt\right|
\leq C_{\varphi,b}(L+D2^{k_+}).
\label{eq:tr-real}
\end{equation}
The constant is independent of the scale range, partition, and inputs.
\end{lemma}
\begin{proof}
Put \(r_k=D2^k\) and \(R=r_{k_+}\). For \(0\leq a<R\), use
the translated dyadic lattice
\[
\mathcal D_a
=\{[a+m2^q,a+(m+1)2^q):m,q\in\Z\}.
\]
At scale \(r_k\), use the kernels from Lemma~\ref{lem:tr-kernel}.
Only roots of length \(R\) meeting the common support interval can
contribute: the zeroth coordinate of a supported kernel belongs to
its interval. These roots have total length at most \(L+2R\).
Restrict all inputs to each such root. This changes no local form,
since all four coordinates on its kernel support belong to the
summation interval. In the root tree, depth is \(s=k_+-k\).
Reversing the scale order preserves the block size and variation.
Thus slot zero satisfies the schedule hypothesis of
Theorem~\ref{thm:local}, and the three fixed slots satisfy it with
one block each. Summing that theorem over the roots gives
\begin{equation}
\left|\sum_{k=k_-}^{k_+}
\sum_{\substack{I\in\mathcal D_a\\|I|=r_k}}
r_k H_I(h_{0,k},h_1,h_2,h_3)\right|
\leq C_{b,1/8,C_0}(L+2R).
\label{eq:tr-root-sum}
\end{equation}
For \(r=r_k\), the following sum is \(r\)-periodic in \(a\).
Since \(r\) divides \(R\), unfolding one period yields
\begin{align*}
\frac1R\int_0^R
\sum_{\substack{I\in\mathcal D_a\\|I|=r}}r^{-1}w_I(x,t)\,da
&=\frac1r\int_0^r\sum_{m\in\Z}r^{-1}
\mathcal W\left(\frac{x-a-(m+1/2)r}{r},\frac tr\right)\,da\\
&=\frac1r\int_\R\mathcal W(z,t/r)\,dz
=\Delta_k(t).
\end{align*}
As \(rH_I\) is the integral against \(r^{-1}w_I\), averaging
\eqref{eq:tr-root-sum} in \(a\) gives \eqref{eq:tr-real}.
All exchanges of integrals and sums involve finitely many scales,
bounded supports, and uniformly finitely many relevant intervals.
\end{proof}
\subsection{Integer sequences and finite orbit segments}
The next step is where primitivity is useful. We do not assume that
any pair of slopes has difference one.
\begin{lemma}[An integer-sequence test]
\label{lem:tr-integer}
Under the scale and block hypotheses of Lemma~\ref{lem:tr-real},
let \(h_1,h_2,h_3\) and \(h_{0,k}\) instead be sequences on \(\Z\),
supported in a common integer interval of cardinality \(L\geq1\).
Assume the same pointwise size and block variation bounds. Then
\begin{equation}
\left|\sum_{k=k_-}^{k_+}\sum_{m,n\in\Z}
h_{0,k}(m)\prod_{j=1}^3h_j(m+b_jn)\Delta_k(n)\right|
\leq C'_{\varphi,b}(L+D2^{k_+}).
\label{eq:tr-integer}
\end{equation}
\end{lemma}
\begin{proof}
Write \(E_b=\sum_j|e_j|\), and fix \(0<\rho<1/4\) so small that
\begin{equation}
\rho\bigl(2+\max_j|b_j|E_b\bigr)<1.
\label{eq:tr-radius}
\end{equation}
Plant every sequence value on the half-open interval
\([m-\rho,m+\rho)\), and set it equal to zero elsewhere. The
resulting Borel functions have the same size and variation bounds,
and their common real support lies in an interval of length at most
\(L-1+2\rho\leq L\).
Suppose a configuration lies in four planted intervals, with integer
centers \(p_j\), and write
\(\delta_j=x+b_jt-p_j\), so \(|\delta_j|<\rho\) away from null
boundaries. Set
\[
m=p_0,\qquad n=\sum_je_jp_j,\qquad
\xi=x-m,\qquad\eta=t-n.
\]
By \eqref{eq:bezout}, \(\eta=\sum_je_j\delta_j\). Consequently
\[
|p_j-(m+b_jn)|
\leq |\delta_j|+|\xi|+|b_j\eta|
<\rho(2+|b_j|E_b)<1.
\]
The expression inside the absolute value is an integer, so
\(p_j=m+b_jn\) for all four slots. Conversely, a configuration
lies in the intervals with these centers precisely when
\[
(\xi,\eta)\in P_\rho
:=\{(\xi,\eta)\in\R^2:|\xi+b_j\eta|<\rho\ (0\leq j\leq3)\},
\]
up to boundaries. This polygon has positive finite area
\(A_\rho\): it contains a neighborhood of zero and is bounded because
two distinct coordinate forms are linearly independent. Distinct
integer pairs \((m,n)\) give disjoint regions up to boundaries,
by uniqueness of the centers and \eqref{eq:bezout}. On
\(P_\rho\), \(|\eta|\leq\rho E_b\).
It follows that the real-line integral at scale \(k\) is exactly
\[
I_k=\sum_{m,n\in\Z}h_{0,k}(m)\prod_{j=1}^3h_j(m+b_jn)
\int_{P_\rho}\Delta_k(n+\eta)\,d\xi\,d\eta.
\]
Let \(D_k\) be the discrete sum at scale \(k\) in
\eqref{eq:tr-integer}. The derivative bound
\(\|\Delta_k'\|_\infty=2^{-2k}\|\psi'\|_\infty\) gives
\[
|\Delta_k(n+\eta)-\Delta_k(n)|
\leq \rho E_b\|\psi'\|_\infty2^{-2k}.
\]
If this difference is nonzero, then
\(|n|\leq B2^k+\rho E_b\). Since \(k\geq0\), there are
\(O_{\varphi,b}(2^k)\) such integers \(n\), and at most \(L\)
contributing integers \(m\). Every coefficient product has absolute
value at most one. Hence
\[
|I_k-A_\rho D_k|\leq A_\rho C_{\varphi,b}L2^{-k}.
\]
Sum over \(k\), use \(\sum_{k\geq0}2^{-k}=2\), and apply
Lemma~\ref{lem:tr-real} to the planted inputs. Division by
\(A_\rho>0\) proves \eqref{eq:tr-integer}.
\end{proof}
For measurable functions \(|f_j|\leq1\) on the probability space,
put
\begin{equation}
F_n(x)=\prod_{j=1}^3f_j(S^{b_jn}x),\qquad
V_k^\varphi(x)=\sum_{n\in\Z}\varphi_{2^k}(n)F_n(x)
\quad(k\geq0).
\label{eq:tr-orbit-averages}
\end{equation}
All sums are finite. If
\(\operatorname{supp}\varphi\subseteq[-B_\varphi,B_\varphi]\),
with \(B_\varphi\geq1\), then
\begin{equation}
|V_k^\varphi(x)|\leq(2B_\varphi+1)\|\varphi\|_\infty.
\label{eq:tr-uniform}
\end{equation}
\begin{proposition}[Finite orbit oscillation]
\label{prop:tr-oscillation}
For every deterministic list \(0\leq k_1<\cdots<k_{w+1}\),
\begin{equation}
\sum_{i=1}^w\int_X
\max_{k_i\leq\ell\leq k_{i+1}}
|V_\ell^\varphi-V_{k_i}^\varphi|\,d\mu
\leq C''_{\varphi,b}\sqrt w.
\label{eq:tr-oscillation}
\end{equation}
This holds on every invertible probability preserving system, with a
constant independent of that system, the functions, and the endpoints.
\end{proposition}
\begin{proof}
First take sequences \(h_j\), bounded by one and supported in an
integer interval \(\mathcal K\) of cardinality \(L\), and set
\[
\mathsf V_k(m)=\sum_n\varphi_{2^k}(n)
\prod_{j=1}^3h_j(m+b_jn).
\]
For \(m\) in a subset \(\mathcal E\subseteq\mathcal K\), let
\(\ell_i(m)\) be the smallest maximizing index of
\(|\mathsf V_\ell(m)-\mathsf V_{k_i}(m)|\) in the finite range
\(k_i\leq\ell\leq k_{i+1}\). Write
\(d_i(m)=\mathsf V_{\ell_i(m)}(m)-\mathsf V_{k_i}(m)\), and take
\(\alpha_i(m)=\overline{d_i(m)}/|d_i(m)|\) when \(d_i(m)\ne0\),
and zero otherwise. In the block \(k_i\leq k<k_{i+1}\), set
\[
h_{0,k}(m)=
\begin{cases}
\alpha_i(m)/(4\sqrt w),&m\in\mathcal E, k<\ell_i(m),\\
0,&\text{otherwise}.
\end{cases}
\]
On each block this sequence has size at most \((4\sqrt w)^{-1}\)
and at most one jump of that size. Its block size and variation is
therefore at most \(w^{-1/2}\). The block endpoints remain
deterministic, although the prefix and phase depend on \(m\).
The identity \(\Delta_k=\varphi_{2^{k+1}}-\varphi_{2^k}\)
telescopes, so Lemma~\ref{lem:tr-integer} gives
\begin{equation}
\sum_{m\in\mathcal E}\sum_{i=1}^w
\max_{k_i\leq\ell\leq k_{i+1}}
|\mathsf V_\ell(m)-\mathsf V_{k_i}(m)|
\leq4C'_{\varphi,b}\sqrt w\,(L+D2^{k_{w+1}}).
\label{eq:tr-sequence-oscillation}
\end{equation}
Indeed, the sum tested in that lemma is exactly \((4\sqrt w)^{-1}\)
times the nonnegative left side, since \(\alpha_i(m)d_i(m)=|d_i(m)|\).
We now use finite orbit segments, the elementary form of Calder\'on's
transference principle \cite{Calderon}. Fix the endpoints, and choose
an integer
\[
Q\geq B_\varphi\max_j|b_j|\,2^{k_{w+1}}.
\]
For each \(M\geq1\) and \(x\in X\), use the sequences
\[
h_{j,x}(r)=
\begin{cases}
f_j(S^rx),&1-Q\leq r\leq M+Q,\\
0,&\text{otherwise}.
\end{cases}
\]
Their common support has cardinality \(M+2Q\). For
\(1\leq m\leq M\) and all scales in the endpoint list, no term
of the average is cut off, so
\(\mathsf V_k(m)=V_k^\varphi(S^mx)\). Apply
\eqref{eq:tr-sequence-oscillation} with
\(\mathcal E=\{1,\ldots,M\}\), integrate over \(x\), and use
invariance under \(S^m\). This gives
\[
M\sum_{i=1}^w\int_X
\max_{k_i\leq\ell\leq k_{i+1}}
|V_\ell^\varphi-V_{k_i}^\varphi|\,d\mu
\leq4C'_{\varphi,b}\sqrt w\,(M+2Q+D2^{k_{w+1}}).
\]
The maxima are measurable finite maxima. Divide by \(M\) and let
\(M\to\infty\), keeping the endpoints and hence \(Q\) fixed.
This proves \eqref{eq:tr-oscillation}.
\end{proof}
\begin{lemma}[Oscillation implies convergence]
\label{lem:tr-cauchy}
Let \((Z_k)_{k\geq0}\) be a uniformly bounded sequence of measurable
complex functions on a probability space. Suppose that, for every
deterministic \(0\leq k_1<\cdots<k_{w+1}\),
\[
\sum_{i=1}^w\int_X
\max_{k_i\leq\ell\leq k_{i+1}}|Z_\ell-Z_{k_i}|\,d\mu
\leq C\sqrt w.
\]
Then \(Z_k\) converges on a measurable conull set.
\end{lemma}
\begin{proof}
We include the standard finite-block argument; compare the oscillation
methods in \cite{Bourgain,JKRW}.
The measurable function
\[
\omega(x)=\lim_{K\to\infty}\sup_{p,q\geq K}|Z_p(x)-Z_q(x)|
\]
vanishes exactly where the sequence is Cauchy. If its positive set
has positive measure, there are \(\delta>0\) and
\(E=\{\omega>2\delta\}\) with \(\mu(E)=d>0\). For every \(K\),
\[
\sup_{\ell\geq K}|Z_\ell-Z_K|
\geq\tfrac12\omega>\delta\quad\text{on }E.
\]
By monotone convergence of the finite maxima, a deterministic
\(L>K\) satisfies
\(\int\max_{K\leq\ell\leq L}|Z_\ell-Z_K|\,d\mu\geq\delta d/2\).
Starting with \(k_1=0\), choose successive endpoints this way.
The assumed estimate would give \(w\delta d/2\leq C\sqrt w\)
for every \(w\), a contradiction. Thus \(\omega=0\) almost
everywhere, and completeness of \(\C\) proves the claim.
\end{proof}
Proposition~\ref{prop:tr-oscillation}, \eqref{eq:tr-uniform}, and
Lemma~\ref{lem:tr-cauchy} already give almost everywhere convergence
of \(V_k^\varphi\). Mixing has not yet been used.
\subsection{Identifying the limit by mixing}
We include the norm argument to make explicit that it accommodates
arbitrary distinct nonzero slopes and both signs of time.
The set-mixing assumption on \(S\) implies, by uniform approximation
by simple functions, that for bounded measurable \(g,h\),
\begin{equation}
\int_Xg\,(h\circ S^m)\,d\mu
\longrightarrow\left(\int_Xg\,d\mu\right)
\left(\int_Xh\,d\mu\right)
\quad(|m|\to\infty).
\label{eq:tr-functional-mixing}
\end{equation}
\begin{lemma}[The Hilbert space van der Corput criterion]
\label{lem:tr-vdc}
Let \((u_n)_{n\geq1}\) be bounded vectors in a complex Hilbert space,
with inner product linear in the first variable. If the limits
\[
\gamma_h=\lim_{N\to\infty}\frac1N
\sum_{n=1}^{N-h}\langle u_{n+h},u_n\rangle
\]
exist for each \(h\geq1\) and tend to zero as \(h\to\infty\), then
\(N^{-1}\sum_{n=1}^Nu_n\to0\) in norm.
\end{lemma}
\begin{proof}
We record the finite estimate underlying the criterion; compare
\cite[Lemma~D.1]{HostKra}. Extend \(u_n\) by zero outside
\(\{1,\ldots,N\}\). The identity
\[
H\sum_{n=1}^Nu_n
=\sum_{m=1}^{N+H-1}\sum_{j=0}^{H-1}u_{m-j}
\]
and Cauchy's inequality give
\begin{align*}
\left\|\frac1N\sum_{n=1}^Nu_n\right\|^2
\leq\frac{N+H-1}{N^2H^2}\bigg(&H\sum_{n=1}^N\|u_n\|^2\\
&+2\sum_{h=1}^{H-1}(H-h)
\operatorname{Re}\sum_{n=1}^{N-h}\langle u_{n+h},u_n\rangle
\bigg).
\end{align*}
If \(\|u_n\|\leq M\), first let \(N\to\infty\) with \(H\)
fixed, obtaining
\[
\limsup_N\left\|\frac1N\sum_{n=1}^Nu_n\right\|^2
\leq\frac{M^2}{H}+\frac2H\sum_{h=1}^{H-1}|\gamma_h|.
\]
The right side tends to zero with \(H\), since the Ces\`aro means
of \((|\gamma_h|)\) tend to zero.
\end{proof}
\begin{lemma}[Norm convergence at distinct slopes]
\label{lem:tr-norm}
If \(S\) is mixing, \(a_1,\ldots,a_r\) are distinct nonzero
integers, and \(g_1,\ldots,g_r\) are bounded measurable functions,
then
\begin{equation}
\frac1N\sum_{n=1}^N\prod_{j=1}^rg_j\circ S^{a_jn}
\longrightarrow\prod_{j=1}^r\int_Xg_j\,d\mu
\quad\text{in }L^2(\mu).
\label{eq:tr-norm}
\end{equation}
Consequently, for \(F_n\) in \eqref{eq:tr-orbit-averages} and
\(p=\prod_{j=1}^3\int_Xf_j\,d\mu\), every
\(\chi\in C_c^1(\R)\) of integral one satisfies
\begin{equation}
\sum_{n\in\Z}a^{-1}\chi(n/a)F_n\longrightarrow p
\quad\text{in }L^2(\mu),\qquad a\to\infty.
\label{eq:tr-weighted-norm}
\end{equation}
\end{lemma}
\begin{proof}
Write \(U^mf=f\circ S^m\). These operators are unitary on
\(L^2(\mu)\), with inner product
\(\langle f,g\rangle=\int f\overline g\,d\mu\).
We induct on \(r\), starting with the empty product \(r=0\).
Suppose first that one of the \(g_j\) has mean zero. Put
\[
u_n=\prod_{j=1}^rU^{a_jn}g_j,
\qquad q_{j,h}=(U^{a_jh}g_j)\overline{g_j}.
\]
By invariance,
\[
\langle u_{n+h},u_n\rangle
=\int_Xq_{1,h}\prod_{j=2}^rU^{(a_j-a_1)n}q_{j,h}\,d\mu.
\]
For fixed \(h\), induction applies to the distinct nonzero slopes
\(a_j-a_1\), \(2\leq j\leq r\). The average of this correlation
therefore tends to
\[
\gamma_h=\prod_{j=1}^r\int_Xq_{j,h}\,d\mu.
\]
Here removing the final \(h\) terms and dividing by \(N\) instead
of \(N-h\) does not change the limit. By
\eqref{eq:tr-functional-mixing}, the \(j\)th factor tends to
\(|\int g_j\,d\mu|^2\) as \(h\to\infty\). At least one limit
is zero, so \(\gamma_h\to0\). Lemma~\ref{lem:tr-vdc} proves
\eqref{eq:tr-norm} in the centered case. For general inputs, write
each \(g_j\) as its mean plus a mean-zero function and expand.
The constant term is the asserted limit, and every other term
contains a centered factor, so the centered case applies. This
completes the induction. Only fixed \(h\) is used in the induction;
no estimate uniform in \(h\) is needed.
Apply \eqref{eq:tr-norm} to \(b_j\) and \(-b_j\). The partial sums
\[
R_N^\pm=\sum_{n=1}^N(F_{\pm n}-p)
\quad\text{satisfy}\quad \|R_N^\pm\|_2=o(N).
\]
Choose \(B\geq1\) containing the support of \(\chi\), and put
\(M_a=\lfloor Ba\rfloor+1\), \(w_n^\pm=a^{-1}\chi(\pm n/a)\).
Then \(w_{M_a}^\pm=0\), and summation by parts gives
\[
\sum_{n=1}^{M_a}w_n^\pm(F_{\pm n}-p)
=\sum_{n=1}^{M_a-1}(w_n^\pm-w_{n+1}^\pm)R_n^\pm.
\]
Its norm is at most
\(a^{-2}\|\chi'\|_\infty\sum_{n<M_a}\|R_n^\pm\|_2=o(1)\).
To see the last assertion, split at a fixed \(N_0\), and use
\(\|R_n^\pm\|_2\leq\epsilon n\) beyond \(N_0\); the tail is
\(O_B(\epsilon)\), and the finite initial part tends to zero.
The \(n=0\) term is \(O(a^{-1})\) in norm, while the Riemann sum
\(a^{-1}\sum_n\chi(n/a)\) tends to one. Adding the two signs and
the zero term proves \eqref{eq:tr-weighted-norm}.
\end{proof}
For each averaging profile, Lemma~\ref{lem:tr-norm} gives
\(V_k^\varphi\to p\) in \(L^2\). Its almost everywhere limit
is therefore \(p\). More explicitly, if \(E_\varphi\) is the
measurable conull convergence set, Fatou's lemma gives
\[
\int_{E_\varphi}|\lim_kV_k^\varphi-p|^2\,d\mu
\leq\liminf_k\|V_k^\varphi-p\|_2^2=0.
\]
\subsection{Removing the smooth weights}
The moment restrictions need not obstruct approximation in \(L^1\).
The correction below, from \cite[Lemma~2.8]{V}, moves a small amount
of signed mass to a large scale. Uniform control of the supports or
derivative constants is neither asserted nor needed.
\begin{lemma}[Profiles approximate interval indicators]
\label{lem:tr-density}
For every \(c\in[1,2]\) and \(\epsilon>0\), there is an averaging
profile \(\varphi\) with
\[
\|\varphi-h_c\|_{L^1(\R)}<\epsilon,
\qquad h_c=c^{-1}\mathbf1_{(0,c]}.
\]
\end{lemma}
\begin{proof}
Let \(\rho\) be the mass-one bump constructed above, and set
\(\rho_\delta(t)=\delta^{-1}\rho(t/\delta)\). For sufficiently
small \(\delta>0\), the real smooth function
\(v=h_c*\rho_\delta\) has mass one and
\(\|v-h_c\|_1<\epsilon/2\). Indeed,
\[
\|h_c(\,\cdot-s)-h_c\|_1=2\min(|s|,c)/c,
\]
and averaging this identity against \(\rho_\delta\) proves the
claim. Also
\(\|v^{(n)}\|_\infty\leq\delta^{-n-1}\|\rho^{(n)}\|_\infty\),
so \(v\) satisfies \eqref{eq:tr-gevrey}. Fix \(v\), and write
\(m_j=\int t^jv(t)\,dt\) for \(1\leq j\leq3\).
Choose four distinct real numbers \(z_0,z_1,z_2,z_3\), and define
\[
P_j(z)=\int_\R(z+u)^j\rho(u)\,du,
\qquad \rho_{i,A}(t)=A^{-1}\rho(t/A-z_i)
\quad(A\geq1).
\]
The \(j\)th moment of \(\rho_{i,A}\) is \(A^jP_j(z_i)\).
Since \(P_j\) is monic of degree \(j\), the fixed matrix
\(\mathsf M=(P_j(z_i))_{j,i=0}^3\) is an invertible triangular
change of the Vandermonde matrix \((z_i^j)_{j,i=0}^3\). Solve
\[
\mathsf M\beta=(0,-m_1/A,-m_2/A^2,-m_3/A^3)^{\mathsf T}.
\]
The inverse matrix is fixed, hence \(\sum_i|\beta_i|=O_v(A^{-1})\).
Thus
\[
\varphi=v+\sum_{i=0}^3\beta_i\rho_{i,A}
\]
has mass one, its first three moments vanish exactly, and its added
\(L^1\) error is \(O_v(A^{-1})\). Take \(A\) large enough that
this error is less than \(\epsilon/2\). Finite addition and fixed
dilations preserve \eqref{eq:tr-gevrey}, so this real compactly
supported function is the required profile.
\end{proof}
\begin{proof}[Completion of the proof of Theorem~\ref{thm:main}]
Continue with \(|f_j|\leq1\), and write
\[
A_N(x)=\frac1N\sum_{n=1}^NF_n(x),
\qquad p=\prod_{j=1}^3\int_Xf_j\,d\mu.
\]
For every rational \(c\in[1,2]\) and integer \(h\geq1\), choose
an averaging profile \(\varphi_{c,h}\) with
\(\|h_c-\varphi_{c,h}\|_1<2^{-h}\). The preceding results give
a measurable conull set on which \(V_k^{\varphi_{c,h}}\to p\).
Intersect these sets over the countable family, obtaining a conull
set \(E\).
For \(N_k=\lfloor c2^k\rfloor\), the choice of endpoints in
\(h_c\) gives the exact identity
\[
2^{-k}\sum_{n\in\Z}h_c(n/2^k)F_n(x)
=\frac{N_k}{c2^k}A_{N_k}(x).
\]
Since \(|F_n|\leq1\), for every profile \(\varphi\),
\begin{equation}
|A_{N_k}(x)-V_k^\varphi(x)|
\leq\frac1{c2^k}
+2^{-k}\sum_{n\in\Z}|h_c(n/2^k)-\varphi(n/2^k)|.
\label{eq:tr-approximation}
\end{equation}
The second term tends to \(\|h_c-\varphi\|_1\): its summand is
a bounded compactly supported Riemann-integrable function, with
possible discontinuities only at \(0,c\). Apply
\eqref{eq:tr-approximation} with \(\varphi_{c,h}\), then let
\(h\to\infty\). For every \(x\in E\),
\begin{equation}
A_{\lfloor c2^k\rfloor}(x)\longrightarrow p
\quad\text{for every rational }c\in[1,2].
\label{eq:tr-rational-scales}
\end{equation}
Fix an integer \(s\geq1\), and use the finite rational grid
\(c_i=1+i/s\), \(0\leq i\leq s\). If
\(2^k\leq N<2^{k+1}\), choose the largest \(i\) such that
\(n=\lfloor c_i2^k\rfloor\leq N\). Then
\(0\leq N-n\leq2^k/s+1\). Averages of terms bounded by one
satisfy, whenever \(1\leq n\leq N\),
\[
|A_N-A_n|\leq\frac{2(N-n)}{N}.
\]
By \eqref{eq:tr-rational-scales}, the maximum error over the finite
grid tends to zero for \(x\in E\). Consequently
\[
\limsup_{N\to\infty}|A_N(x)-p|\leq2/s\qquad(x\in E).
\]
Let \(s\to\infty\) to obtain convergence along all positive integers.
Finally undo the normalization of the functions and use
\(S=T^g\), \(b_j=\lambda_j/g\), as in the initial reduction.
If the functions are specified only as essentially bounded classes,
choose bounded representatives and remove the union of all integer
translates of their exceptional sets. This is still a measurable
null set. Every other exceptional set used above is measurable and
every intersection is countable. Thus the proof needs no standardness
or completeness assumption on the probability space.
\end{proof}
==== END Triple-ergodic-averages-with-distinct-integer-slopes//sections/transfer.tex ====