==== TABLE OF CONTENTS ====
A-power-saving-for-square-difference-free-sets//contraction.tex
A-power-saving-for-square-difference-free-sets//functional.tex
A-power-saving-for-square-difference-free-sets//graph.tex
A-power-saving-for-square-difference-free-sets//induction.tex
A-power-saving-for-square-difference-free-sets//introduction.tex
A-power-saving-for-square-difference-free-sets//kernel.tex
A-power-saving-for-square-difference-free-sets//lifts.tex
A-power-saving-for-square-difference-free-sets//list-regrouping.tex
A-power-saving-for-square-difference-free-sets//main.tex
A-power-saving-for-square-difference-free-sets//parameter-roles.tex
A-power-saving-for-square-difference-free-sets//proof-map.tex
A-power-saving-for-square-difference-free-sets//transfer.tex
A-power-saving-for-square-difference-free-sets//tuples.tex
A-power-saving-for-square-difference-free-sets//references.bib
===========================

==== BEGIN A-power-saving-for-square-difference-free-sets//contraction.tex ====
\section{Contraction on ordered intervals}
\label{sec:ordered-contraction}

A tuple of residue classes
\(\boldsymbol s=(s_v)_{v\in V}\in(\Z/M_0\Z)^V\) is called
\emph{good} if
\[
 s_{v_{j+1}}-s_{v_j}
 \quad\text{is a strict square class modulo }M_0
 \qquad(0\le j<24),
\]
where \(v_{24}=v_0\). Thus each difference is \(1\) modulo \(8\)
and a nonzero square at every odd prime dividing \(M_0\). Let
\begin{equation}
 \rho=M_0^{-d}\bigl|\{\boldsymbol s\in(\Z/M_0\Z)^V:
                         \boldsymbol s\text{ is good}\}\bigr|.
 \label{eq:good-residue-proportion}
\end{equation}
We will bound the contribution of each good tuple when every prime uses
the nonexceptional part of its measure. First we show that these assignments
have a fixed positive proportion; the proof also explains the cycle length
$24$. We retain the parameters and lifts from the preceding section.

\begin{lemma}[Existence of good residue assignments]
\label{lem:good-residues}
The proportion $\rho$ defined in~\eqref{eq:good-residue-proportion} satisfies
\[
 \rho\ge M_0^{-23}>0.
\]
\end{lemma}

\begin{proof}
We construct a length-$24$ sequence of admissible increments with sum zero at
each factor of $M_0$.  Modulo $8$, take all the increments equal to $1$; their
sum is $24\equiv0\pmod8$.

Let $p6\beta\), this
length is at least \(10Q^6\). Lemma~\ref{lem:lift-moments} therefore
applies uniformly to every interval lift and every exact-denominator
truncation used below. Its application to \(f_s\), which is bounded by
the fixed constant \(M_0\), changes only the constants. All occurrences
of \(N^{o(1)}\) in this proof are uniform over these interval choices,
residues and sets.

First consider assignments for which all 24 cycle slots choose the same
interval. Their total probability is
\[
 \sum_{i=1}^K w_i^{24}
 \le(\max_i w_i)^{23}\ll N^{-23\tau}.
\]
For every assignment, ordinary H\"older and the uniform single-slot
marginals give
\[
 \left|\int\prod_v G_v(z_v)\,d\Lambda^\circ\right|
 \le\prod_v\|G_v\|_d=N^{o(1)},
 \qquad
 G_v=\mathcal L_{I(v),\mathcal P_N,Q}f_{s_v}.
\]
Here \(d\) is even, so the required moment is supplied directly by
Lemma~\ref{lem:lift-moments}. The contribution of these assignments is
therefore \(O(N^{-23\tau+o(1)})\); there is no restriction on the interval
choices in the other slots.

In every remaining assignment there is a directed cycle edge \(u\to v\)
whose first interval precedes its second. Indeed, if the cyclic sequence
of interval indices had no ascent, it would be nonincreasing all the way
around the cycle, and hence constant. Fix one ascending edge, and denote
its two intervals by \(I_1,I_2\) for the rest of this argument. All
integers in \(I_1\) precede all integers in \(I_2\).

\textit{Eliminating the other slots.}\quad
Retain these two lifts in full. In each of the other \(d-2\) slots,
truncate the lift to exact conductors at most \(T=N^\tau\).
Lemma~\ref{lem:conductor-decay}, applied during a telescoping replacement
of these slots, bounds the total error by
\begin{equation}
 O(T^{-\gamma}N^{o(1)})=O(N^{-\gamma\tau+o(1)}).
 \label{eq:interval-truncation-error}
\end{equation}
Every norm used here is controlled by Lemma~\ref{lem:lift-moments}:
the retained and discarded pieces are subfamilies of complete exact
denominator pieces. This error is estimated before the Fourier expansion
that follows.

Each Fourier coefficient in a truncated slot has modulus at most \(M_0\),
and there are at most \(\sum_{q\le T}\varphi(q)\ll T^2\) frequencies.
The product expansion of all other slots consequently has total
coefficient absolute sum at most
\begin{equation}
 C_{d,M_0}T^{2(d-2)}\ll N^{2d\tau}.
 \label{eq:interval-fourier-cost}
\end{equation}
For one term in this expansion, let \(B\subseteq\mathcal P_N\) be the
union of the prime supports of its characters, and put
\[
 R=\prod_{p\in B}p.
\]
The conductors are squarefree, so their least common multiple satisfies
\begin{equation}
 R\le T^{d-2}\le N^{d\tau}.
 \label{eq:interval-mode-modulus}
\end{equation}

Condition on all slot labels at the primes in \(B\). The expanded
characters become constants of modulus one. Since \(\Lambda^\circ\) is
a product over primes, the tuple laws at the remaining primes are
unchanged and independent. The two selected labels now have fixed
coordinates \(a_{1,B},a_{2,B}\) at \(B\), whose difference is a nonzero
square at each prime in \(B\). At every remaining prime their pair
marginal is still uniform on strict square pairs.

Let \(\mathcal S=\mathcal P_N\setminus B\), let \(s_1=s_u\) and
\(s_2=s_v\), and define functions on \(X_{\mathcal S}\) by
\[
 g_i(z)=\bigl(\mathcal L_{I_i,\mathcal P_N,Q}f_{s_i}\bigr)(a_{i,B},z),
 \qquad i=1,2.
\]
With Fourier coefficients normalized for uniform measure on
\(X_{\mathcal S}\), their conditional pair integral is exactly
\begin{equation}
 \sum_{\xi\text{ on }X_{\mathcal S}}
   \widehat g_1(-\xi)\widehat g_2(\xi)\mathfrak g(\xi).
 \label{eq:interval-pair-fourier}
\end{equation}
To see this, simultaneous translation kills character pairs whose
frequencies do not sum to zero. For a matching pair, the multiplier is
the character average of the difference of the labels. At every odd
prime a uniform nonzero square is the square of a uniform nonzero
unit. The Chinese remainder theorem therefore identifies this average
with the unit-square multiplier \(\mathfrak g(\xi)\) used in
Proposition~\ref{prop:square-kernel}.

\textit{Estimating the selected pair.}\quad
Specialization on a uniform coordinate block of size \(R\) costs at
most \(R^{1/2}\) in \(L^2\): for every function \(G\),
\[
 \|G(a_B,\cdot)\|_2^2\le R\|G\|_2^2.
\]
This inequality holds at every specified fiber, independently of the
law under which that fiber was selected. Hence
\(\|g_i\|_2\le R^{1/2}N^{o(1)}\). By the bound
\(|\mathfrak g(\xi)|\le q(\xi)^{-1/3}\), Parseval and
Cauchy--Schwarz, the part of~\eqref{eq:interval-pair-fourier} with
conductor exceeding \(H\) has modulus at most
\begin{equation}
 H^{-1/3}\|g_1\|_2\|g_2\|_2
 \le H^{-1/3}R N^{o(1)}.
 \label{eq:interval-pair-tail}
\end{equation}

We next identify the remaining coefficients with actual integer fibers.
Put \(D=M_0R\). The parameter inequalities
\begin{equation}
 d\tau<\frac{\beta\gamma}{1000}<\frac\beta2
 \label{eq:interval-parameter-room}
\end{equation}
imply \(RH\le Q\) and, for sufficiently large \(N\), \(D\le N^\beta\).
For a remaining frequency \(\xi\) with \(q(\xi)\le H\), all frequencies
\(\xi+\eta\), where \(\eta\) ranges over the character group on \(B\),
therefore occur in the original lift. Summing this entire character group
gives
\begin{align}
 \widehat g_i(\xi)
 &=\sum_{\eta\text{ on }B}\frac1{|I_i|}
     \sum_{n\in I_i}f_{s_i}(n)e(-n(\xi+\eta))e(a_{i,B}\eta)\notag\\
 &=\frac{D}{|I_i|}
     \sum_{\substack{n\in I_i\\n\equiv a_i\pmod D}}
           1_A(n)e(-n\xi),
 \label{eq:interval-selected-fiber}
\end{align}
where \(a_i\pmod D\) combines \(s_i\pmod {M_0}\) and the coordinates
\(a_{i,B}\) by the Chinese remainder theorem. Character orthogonality
supplies the factor \(R\), while the definition of \(f_{s_i}\) supplies
\(M_0\). Goodness of \(\boldsymbol s\) and the constraints at the
conditioned primes show that \(a_2-a_1\) is a strict square class
modulo \(D\).

The remaining frequencies of conductor at most \(H\) are precisely the
frequencies with squarefree conductor at most \(H\) coprime to \(D\).
Indeed, \(D\) contains every prime less than \(P\); every prime divisor
of such a conductor is consequently an available prime at least \(P\),
at most \(H\le Q\), and outside \(B\). This includes the zero
frequency under the convention \(q(0)=1\).

Define functions on \([N]\) by
\[
 F_i(n)=1_A(n)1_{I_i}(n)1_{n\equiv a_i\pmod D}.
\]
They are bounded by one, their supports are ordered, and there is no
positive square difference from the first support to the second.
Moreover, \(D\) is eight times an odd squarefree number and
\(D\le N^\beta\). All hypotheses of
Proposition~\ref{prop:square-kernel} hold. If the Fourier coefficients
of \(F_i\) use normalization \(N^{-1}\), then
\eqref{eq:interval-selected-fiber} gives
\(\widehat g_i(\xi)=DN\widehat F_i(\xi)/|I_i|\). Thus the part of
\eqref{eq:interval-pair-fourier} of conductor at most \(H\) is bounded
in modulus by
\begin{align}
 &\frac{D^2N^2}{|I_1||I_2|}
 \left|\sum_{\substack{q(\xi)\le H,\ q(\xi)\text{ squarefree}\\
                       (q(\xi),D)=1}}
      \mathfrak g(\xi)\widehat F_1(-\xi)\widehat F_2(\xi)\right|
 \notag\\
 &\hspace{2cm}\ll H^{-1/3}D\frac{N^2}{|I_1||I_2|}
 \ll H^{-1/3}D N^{2\tau}.
 \label{eq:interval-kernel-cost}
\end{align}
The factor \(D\) in Proposition~\ref{prop:square-kernel} has canceled
one of the two fiber-normalization factors.

Combining~\eqref{eq:interval-pair-tail}
and~\eqref{eq:interval-kernel-cost} bounds each conditional pair integral
by \(O(N^{-\beta/6+(d+2)\tau+o(1)})\). This is uniform over all
conditionings of positive probability. Averaging them, and then summing
the Fourier expansion using~\eqref{eq:interval-fourier-cost}, bounds the
truncated integral for each unequal-cycle interval assignment by
\[
 O\bigl(N^{-\beta/6+(3d+2)\tau+o(1)}\bigr).
\]
Averaging interval assignments incurs no further loss. Including the
equal-cycle contribution and~\eqref{eq:interval-truncation-error}, the
original integral has modulus at most
\begin{equation}
 C\left(N^{-23\tau+o(1)}+N^{-\gamma\tau+o(1)}
       +N^{-\beta/6+(3d+2)\tau+o(1)}\right).
 \label{eq:interval-three-errors}
\end{equation}
Finally, \(23\tau=5888\sigma_0\), \(\gamma\tau=4\sigma_0\), and
\[
 \frac\beta6-(3d+2)\tau
 >\beta\left(\frac16-\frac3{64000}\right)
 >\frac\beta7>2\sigma_0.
\]
The finitely many uniform moment losses in each term can be bounded by
\(N^{\sigma_0}\), after increasing the absolute threshold in \(N\).
Equation~\eqref{eq:interval-three-errors} is consequently
\(O(N^{-\sigma_0})\), as required. All estimates used absolute values
of signed Fourier lifts; no pointwise positivity of a lift was needed.
\end{proof}
==== END A-power-saving-for-square-difference-free-sets//contraction.tex ====

==== BEGIN A-power-saving-for-square-difference-free-sets//functional.tex ====
\section{Signed multilinear forms and conductor decay}
\label{sec:functional}

The tuple laws provide two estimates for functions on products of prime
fields. The first follows the reflection and chessboard method
of~\cite[Theorem~2.2]{FrohlichLieb1978} and
\cite[Theorem~4.1]{FILS1978ReflectionI}; the finite proof below verifies
the required inequality for this particular tuple law.  Reflection positivity gives a H\"older inequality for signed
inputs and a seminorm that dominates the uniform mean.  The conditional
mixing estimate gives a separate bound for inputs supported on large
Fourier denominators.  This second estimate also applies to the
nonexceptional laws.

Retain the index set \(V=\Pi^\ell\), where \(\Pi\) consists of the
permutation words on \([h]\), \(h=24\), and \(\ell=161\), and put
\(d=|V|\).  In particular, \(d\) is even.  Fix the prime threshold
\(P\) from Proposition~\ref{prop:tuple-laws} and
Lemma~\ref{lem:conditional-mixing}, and put \(\gamma=1/64\).
For a finite set \(\mathcal P\) of primes at least \(P\), define
\[
 X_{\mathcal P}=\prod_{p\in\mathcal P}\mathbb F_p,
 \qquad
 \mu_{\mathcal P}=\bigotimes_{p\in\mathcal P}\mu_p.
\]
We regard \(\mu_{\mathcal P}\) as a probability law on
\(X_{\mathcal P}^V\) by collecting, for each slot \(v\), its labels
at all primes into \(z_v\in X_{\mathcal P}\).  Every single-slot
marginal is uniform on \(X_{\mathcal P}\).  Unless another measure
is specified, norms and expectations on \(X_{\mathcal P}\) use
uniform probability measure.  Set
\begin{equation}
 \mathcal Z_{\mathcal P}((g_v)_{v\in V})
   =\int\prod_{v\in V}g_v(z_v)\,d\mu_{\mathcal P}(z),
 \qquad
 Z_{\mathcal P}(g)=\mathcal Z_{\mathcal P}((g)_{v\in V}).
 \label{eq:functional-definition}
\end{equation}
For an empty prime set, \(X_{\varnothing}\) is a singleton and
the product law has mass one; thus \(Z_{\varnothing}(g)=g^d\).

\subsection{A finite signed reflection inequality}

\begin{proposition}[Signed H\"older inequality]
\label{prop:signed-holder}
For every finite prime set \(\mathcal P\) as above and real functions
\(g,g_v:X_{\mathcal P}\to\mathbb R\), one has
\begin{equation}
 Z_{\mathcal P}(g)\ge0,
 \qquad
 \left|\mathcal Z_{\mathcal P}((g_v)_{v\in V})\right|
 \le\prod_{v\in V}Z_{\mathcal P}(g_v)^{1/d}.
 \label{eq:signed-holder}
\end{equation}
The inequality includes inputs with zero diagonal integral.
\end{proposition}

\begin{proof}
Across any reflection cut, the matrix of \(\mu_{\mathcal P}\) is
the tensor product of the corresponding matrices of the \(\mu_p\).
It is therefore symmetric positive semidefinite.  This also holds for
the empty prime set.  We first record precisely what a reflection does
to an assignment of functions.

Fix a block and two distinct symbols.  Let \(\theta\) exchange those
symbols in that block, and let \(V_+\) be the vertices in which the
first symbol occurs before the second.  Put \(V_-=\theta V_+\).
Define the two fold maps \(\phi_+,\phi_-:V\to V\) by
\[
 \phi_+(v)=
 \begin{cases}v,&v\in V_+,\\\theta v,&v\in V_-,\end{cases}
 \qquad
 \phi_-(v)=
 \begin{cases}\theta v,&v\in V_+,\\v,&v\in V_-.\end{cases}
\]
For an assignment \(a:V\to\mathbb R^{X_{\mathcal P}}\), abbreviate
\(\mathcal Z_{\mathcal P}((a(v))_{v\in V})\) by \(\Lambda(a)\).
The bilinear form of the cut matrix is evaluated on the two real
functions
\[
 F(x)=\prod_{v\in V_+}a(v)(x_v),
 \qquad
 G(x)=\prod_{v\in V_+}a(\theta v)(x_v).
\]
Its mixed value is \(\Lambda(a)\); its two quadratic values are
\(\Lambda(a\circ\phi_+)\) and \(\Lambda(a\circ\phi_-)\).
Cauchy--Schwarz for this positive semidefinite form gives
\begin{equation}
 |\Lambda(a)|^2
 \le\Lambda(a\circ\phi_+)\Lambda(a\circ\phi_-),
 \qquad
 \Lambda(a\circ\phi_\pm)\ge0.
 \label{eq:folding-cauchy-schwarz}
\end{equation}
This remains valid when the form has a nontrivial null space.
For a constant assignment, \(F=G\); hence \(Z_{\mathcal P}(g)\ge0\).

We next specify a finite sequence of folds whose composition is constant.
For a word \(w\) in block \(b\), write \(p_i=w^{-1}(i)\) for the
position of symbol \(i\).  Let \(\phi_{b,i}\) be the fold onto
\(p_i0\), choose a
maximizer.  Each of its folded assignments still takes values in the
same finite list, so each folded integral is at most \(M\) in absolute
value.  Equation~\eqref{eq:folding-cauchy-schwarz} says that those two
integrals are nonnegative and their product is at least \(M^2\).
They must therefore both equal \(M\).  In particular, either fold of
any maximizing assignment is again maximizing.

Apply the folds in reverse sorting order,
\(\phi_T,\phi_{T-1},\ldots,\phi_1\).  Pullbacks compose on the
right, so the resulting assignment is
\(a\circ\phi_T\circ\cdots\circ\phi_1\), which is constant by
\eqref{eq:constant-fold-composition}.  Every intermediate integral is
\(M\), and the last is one of the diagonal values \(Z_{\mathcal P}(f_i)\).
Conversely, every constant assignment was included in the maximum.
It follows that
\begin{equation}
 M=\max_{1\le i\le k}Z_{\mathcal P}(f_i).
 \label{eq:finite-folding-maximum}
\end{equation}
When \(M=0\), all these diagonal values are zero, so the same identity
holds.  This is a finite argument and requires no limiting sequence of
folds. The maximizing-assignment argument is analogous to the proof of
the abstract chessboard estimate in
\cite[Theorem~4.1]{FILS1978ReflectionI}.

Finally, for \(\epsilon>0\), apply
\eqref{eq:finite-folding-maximum} to the list
\[
 f_v=\frac{g_v}{(Z_{\mathcal P}(g_v)+\epsilon)^{1/d}},
 \qquad v\in V.
\]
Every denominator is positive and \(Z_{\mathcal P}(f_v)\le1\).
Multilinearity consequently gives
\[
 \left|\mathcal Z_{\mathcal P}((g_v)_{v\in V})\right|
 \le\prod_{v\in V}(Z_{\mathcal P}(g_v)+\epsilon)^{1/d}.
\]
Letting \(\epsilon\downarrow0\) proves
\eqref{eq:signed-holder}, including zero diagonal values.
\end{proof}

\subsection{The seminorm and its symmetries}

Proposition~\ref{prop:signed-holder} allows us to define, for real \(g\),
\[
 \|g\|_{\mathcal P}=Z_{\mathcal P}(g)^{1/d}.
\]
For \(B\subseteq\mathcal P\), let \(\mathbb E_Bg\) denote
uniform averaging over the coordinates in \(B\), holding the other
coordinates fixed.  We may regard this either as a function on
\(X_{\mathcal P}\) independent of \(B\), or as a function on
\(X_{\mathcal P\setminus B}\).

\begin{lemma}[Seminorm properties]
\label{lem:functional-properties}
The map \(g\mapsto\|g\|_{\mathcal P}\) is a seminorm on the real
functions on \(X_{\mathcal P}\), and
\begin{equation}
 |\mathbb E g|\le\|g\|_{\mathcal P}.
 \label{eq:functional-mean}
\end{equation}
For \(a_p\in\mathbb F_p\) and \(r_p\in Q_p^*\), let
\(A(x)_p=r_px_p+a_p\).  Then
\begin{equation}
 \|g\circ A\|_{\mathcal P}=\|g\|_{\mathcal P}.
 \label{eq:functional-affine}
\end{equation}
For every \(B\subseteq\mathcal P\),
\begin{equation}
 \|\mathbb E_Bg\|_{\mathcal P\setminus B}
   =\|\mathbb E_Bg\|_{\mathcal P}
   \le\|g\|_{\mathcal P}.
 \label{eq:functional-projection}
\end{equation}
\end{lemma}

\begin{proof}
Nonnegativity is already proved.  Since \(d\) is even, homogeneity of
the diagonal form gives \(\|cg\|_{\mathcal P}=|c|\|g\|_{\mathcal P}\)
for every real \(c\).  For real functions \(g_1,g_2\), expand
\(Z_{\mathcal P}(g_1+g_2)\) multilinearly and apply
\eqref{eq:signed-holder} to each term.  With
\(a=\|g_1\|_{\mathcal P}\) and \(b=\|g_2\|_{\mathcal P}\), this gives
\[
 0\le Z_{\mathcal P}(g_1+g_2)
 \le\sum_{S\subseteq V}a^{|S|}b^{d-|S|}=(a+b)^d.
\]
Taking the nonnegative \(d\)-th root proves the triangle inequality,
also when one of the seminorms vanishes.

The probability normalization gives \(Z_{\mathcal P}(1)=1\).
Insert \(g\) in one slot and \(1\) in all others in
\eqref{eq:signed-holder}.  The uniform single-slot marginal identifies
the left side with \(|\mathbb E g|\), proving
\eqref{eq:functional-mean}.  The simultaneous affine invariance of each
\(\mu_p\) proves \eqref{eq:functional-affine}; its parameters may be
chosen independently at different primes.

Let \(G_B\) be the additive subgroup of \(X_{\mathcal P}\) consisting
of vectors supported on \(B\).  Then
\[
 (\mathbb E_Bg)(x)=\frac1{|G_B|}\sum_{t\in G_B}g(x+t).
\]
The triangle inequality and translation invariance show that this average
has seminorm at most \(\|g\|_{\mathcal P}\).  Since the averaged
function is independent of \(B\), integration over the prime laws in
\(B\), each of mass one, gives
\[
 Z_{\mathcal P}(\mathbb E_Bg)
   =Z_{\mathcal P\setminus B}(\mathbb E_Bg).
\]
This proves \eqref{eq:functional-projection}.  All statements include
the empty prime set, for which the seminorm is absolute value.
\end{proof}

\subsection{Conductor decay}

The characters of \(X_{\mathcal P}\) are indexed by
\(\xi\in\mathbb Q/\mathbb Z\) whose reduced denominator \(q(\xi)\)
divides \(\prod_{p\in\mathcal P}p\), with \(q(0)=1\).  We write
such a character as \(e(z\xi)\): it means \(e(n\xi)\) for any integer
\(n\) satisfying \(n\equiv z_p\pmod p\) for all \(p\in\mathcal P\).
This is well defined by the Chinese remainder theorem.  The Fourier
coefficient of \(g\) is
\[
 \widehat g(\xi)=\mathbb E_{z\in X_{\mathcal P}}g(z)e(-z\xi).
\]
For \(U\subseteq\mathcal P\), put \(q_U=\prod_{p\in U}p\),
including \(q_{\varnothing}=1\).  The exact prime support of a
character is the set of primes at which it is nontrivial; its denominator
is exactly the product of these primes.

Recall the exceptional mass and the normalized nonexceptional law,
\[
 \eta_p=\varepsilon_p(\mathbb F_p^V),
 \qquad
 \mu_p^\circ=\frac{\mu_p-\varepsilon_p}{1-\eta_p}.
\]
Proposition~\ref{prop:tuple-laws} gives \(\eta_p\le p^{-4}<1\).

\begin{lemma}[Conductor decay]
\label{lem:conductor-decay}
Let \(\mathcal P\) be a finite set of primes at least \(P\), and
at each prime choose \(\rho_p\) to be either \(\mu_p\) or
\(\mu_p^\circ\).  Let \(R\ge1\), and let
\(g_w:X_{\mathcal P}\to\mathbb C\), \(w\in V\), be functions.
Suppose that for one slot \(v\),
\(\widehat g_v(\xi)=0\) whenever \(q(\xi)\le R\).  Then
\begin{equation}
 \left|\int\prod_{w\in V}g_w(z_w)\,
       d\!\bigotimes_{p\in\mathcal P}\rho_p(z)\right|
 \le R^{-\gamma}\|g_v\|_2
        \prod_{w\ne v}\|g_w\|_{2(d-1)}.
 \label{eq:conductor-decay}
\end{equation}
All norms on the right use uniform probability measure on
\(X_{\mathcal P}\).  The constant in the inequality is one.
\end{lemma}

\begin{proof}
Fix the distinguished slot \(v\).  For each \(p\), let
\(Y_p=\mathbb F_p^{V\setminus\{v\}}\), let \(\lambda_p\) be
the marginal law of the other labels under \(\rho_p\), and define
\[
 T_pf(y)=\mathbb E_{\rho_p}[f(z_v)\mid (z_w)_{w\ne v}=y].
\]
This is an operator from \(L^2(\mathbb F_p,\mathrm{unif})\) to
\(L^2(Y_p,\lambda_p)\); its values on marginal-null points are
irrelevant.  It preserves constants, and the tower property gives
\begin{equation}
 T_p1=1,
 \qquad
 \int T_pf\,d\lambda_p=\mathbb E_{x\in\mathbb F_p}f(x).
 \label{eq:conditional-centering}
\end{equation}
For uniform-mean-zero \(f\), Lemma~\ref{lem:conditional-mixing}
gives
\begin{equation}
 \|T_pf\|_{L^2(\lambda_p)}
   \le p^{-\gamma}\|f\|_{L^2(\mathrm{unif})}.
 \label{eq:local-conductor-contraction}
\end{equation}
Let \(H_p^0\) be the mean-zero subspace of
\(L^2(\mathbb F_p,\mathrm{unif})\), and let \(K_p^0\) be the
mean-zero subspace of \(L^2(Y_p,\lambda_p)\).  Thus
\[
 L^2(\mathbb F_p,\mathrm{unif})=\mathbb C1\mathbin{\oplus}H_p^0,
 \qquad
 L^2(Y_p,\lambda_p)=\mathbb C1\mathbin{\oplus}K_p^0,
\]
with orthogonal sums.  Equation~\eqref{eq:conditional-centering}
says that \(T_pH_p^0\subseteq K_p^0\).  Notice that centering in
the codomain refers to \(\lambda_p\), which need not be uniform.

For \(U\subseteq\mathcal P\), let \(H_U\) be the tensor subspace
with factor \(H_p^0\) when \(p\in U\) and factor \(\mathbb C1\)
otherwise; define \(K_U\) in the same way using \(K_p^0\).
The \(H_U\) form an orthogonal direct sum of
\(L^2(X_{\mathcal P},\mathrm{unif})\).  Distinct \(K_U,K_W\)
are also orthogonal: at a prime in the symmetric difference of \(U,W\),
one tensor factor is constant and the other has mean zero.  Inner products
of pure tensors therefore vanish, and linearity gives the assertion for
the whole subspaces.

Under the product law \(\bigotimes_p\rho_p\), the marginal of all
the other labels is \(\lambda=\bigotimes_p\lambda_p\).  Conditional
independence across primes shows that the operator from the distinguished
slot to all other labels is
\[
 T=\bigotimes_{p\in\mathcal P}T_p.
\]
For completeness, on an input \(\prod_pf_p(z_{v,p})\), the conditional
expectation is \(\prod_pT_pf_p\), since the conditional law is a
product over primes.  Such pure tensors span the finite-dimensional
domain, proving the operator identity.  Consequently
\begin{equation}
 T(H_U)\subseteq K_U,
 \qquad
 \|T|_{H_U}\|\le\prod_{p\in U}p^{-\gamma}=q_U^{-\gamma}.
 \label{eq:exact-support-contraction}
\end{equation}
The norm bound follows by applying
\eqref{eq:local-conductor-contraction} one tensor factor at a time;
on the constant factors the norm is one.

Nontrivial characters at \(p\) form an orthonormal basis for \(H_p^0\).
Thus \(H_U\) consists precisely of Fourier sums with exact denominator
\(q_U\).  Decompose the distinguished input as
\(g_v=\sum_{q_U>R}g_{v,U}\), with \(g_{v,U}\in H_U\).  The
images of distinct summands are orthogonal by
\eqref{eq:exact-support-contraction} and the orthogonality of the
\(K_U\).  Hence
\begin{align*}
 \|Tg_v\|_{L^2(\lambda)}^2
 &=\sum_{q_U>R}\|Tg_{v,U}\|_{L^2(\lambda)}^2\\
 &\le\sum_{q_U>R}q_U^{-2\gamma}\|g_{v,U}\|_2^2
 \le R^{-2\gamma}\|g_v\|_2^2.
\end{align*}
Only distinct exact supports need have orthogonal images; no such
assertion is made for different characters within a single support.

Finally condition the multilinear integral on all labels except those
in slot \(v\), and apply Cauchy--Schwarz.  Its absolute value is at most
\[
 R^{-\gamma}\|g_v\|_2
 \left\|\prod_{w\ne v}g_w(z_w)\right\|_{L^2(\lambda)}.
\]
H\"older's inequality with \(d-1\) factors and the uniform marginal
in each remaining slot give
\[
 \left\|\prod_{w\ne v}g_w(z_w)\right\|_{L^2(\lambda)}
 \le\prod_{w\ne v}
       \left(\mathbb E_{\lambda}|g_w(z_w)|^{2(d-1)}\right)^{1/(2(d-1))}
 =\prod_{w\ne v}\|g_w\|_{2(d-1)}.
\]
This proves \eqref{eq:conductor-decay}.  If \(\mathcal P\) is empty,
the support hypothesis forces \(g_v=0\), so the assertion holds there
too.  No reflection positivity of the laws \(\rho_p\) was used.
\end{proof}
==== END A-power-saving-for-square-difference-free-sets//functional.tex ====

==== BEGIN A-power-saving-for-square-difference-free-sets//graph.tex ====
\section{Finite-field square constraints}
\label{sec:finite-field-graphs}

This section controls systems of square constraints on independent field
variables.  A single constraint is close to a constant kernel in operator
norm.  Replacing constraints one at a time then counts any fixed directed
graph without loops or opposite edges.  Applying this count to a four-cycle
of two lists gives the operator estimate needed for the tuple measures.

Throughout, finite sets carry uniform probability measure, and
\(e(t)=\exp(2\pi i t)\).  For an odd prime \(p\), write
\[
 Q_p^*=\{u^2:u\in\mathbb F_p^\times\},
 \qquad
 A_p(t)=1_{Q_p^*}(t),
 \qquad
 \delta_p=\frac{\sqrt p+1}{2p}\le p^{-1/2}.
\]
For finite sets \(X,Y\), the operator associated with a kernel
\(K:X\times Y\to\mathbb R\) is
\[
 T_K f(x)=\mathbb E_{y\in Y}K(x,y)f(y).
\]
We use \(\|K\|_{\mathrm{op}}\) to mean the norm of \(T_K\) from
\(L^2(Y)\) to \(L^2(X)\).  A constant kernel \(c\) therefore denotes
the operator \(f\mapsto c\,\mathbb E f\).

\begin{lemma}[Counting directed square constraints]
\label{lem:graph-count}
Let \(p\) be an odd prime and let \(G=(W,E)\) be a finite directed graph
with no loops and at most one edge on each unordered pair of distinct
vertices.  Put \(m=|E|\), and let \((z_w)_{w\in W}\) be independent
uniform elements of \(\mathbb F_p\).  Then
\[
 \left|
 \mathbb E\prod_{(u,v)\in E}A_p(z_v-z_u)-2^{-m}
 \right|
 \le m\delta_p.
\]
The assertion includes the empty graph, whose product is \(1\).
\end{lemma}

\begin{proof}
We first show that the kernel
\(b_p(x,y)=A_p(y-x)-1/2\) has operator norm at most \(\delta_p\).
The characters \(x\mapsto e(kx/p)\), \(k\in\mathbb F_p\), are an
orthonormal eigenbasis.  The constant eigenvalue is
\[
 \frac{|Q_p^*|}{p}-\frac12=-\frac1{2p}.
\]
For \(k\ne0\), put \(G_p(k)=\sum_{u\in\mathbb F_p}e(ku^2/p)\).
The corresponding eigenvalue is
\[
 \frac1p\sum_{t\in Q_p^*}e(kt/p)
 =\frac{G_p(k)-1}{2p}.
\]
Here the subtraction of \(1\) removes the zero square, and division
by \(2\) removes the multiplicity of every nonzero square.  Since
\((u,v)\mapsto(u-v,u+v)\) is a bijection on \(\mathbb F_p^2\),
\[
 |G_p(k)|^2
 =\sum_{u,v\in\mathbb F_p}e\bigl(k(u^2-v^2)/p\bigr)
 =\sum_{a,b\in\mathbb F_p}e(kab/p)
 =p.
\]
Thus all eigenvalues have absolute value at most \(\delta_p\).
In particular,
\begin{equation}
 \left|
 \mathbb E_{x,y} f(x)g(y)\bigl(A_p(y-x)-\tfrac12\bigr)
 \right|
 \le\delta_p\|f\|_2\|g\|_2.
 \label{eq:single-square-mixing}
\end{equation}
The same bound holds with \(x-y\) in place of \(y-x\), since the
transposed operator has the same norm.

Now replace the \(m\) edge indicators one at a time by \(1/2\).
In a telescoping error for an edge \((u,v)\), fix every variable
except \(z_u,z_v\).  No other edge has both these endpoints.
Consequently all the other factors separate as
\(C f(z_u)g(z_v)\), where \(|C|,|f|,|g|\le1\).
Equation~\eqref{eq:single-square-mixing} bounds the conditional error
by \(\delta_p\).  Averaging the fixed variables and summing over
the edges proves the assertion.
\end{proof}

We next divide the variables into two independent lists.  Only
constraints between the lists enter the following kernel; constraints
within either list must be treated separately.

\begin{lemma}[Mixing between two lists]
\label{lem:list-mixing}
Let \(I,J\) be finite disjoint sets, and let \(G\) be a directed graph
on \(I\sqcup J\) whose \(m\) edges all join \(I\) to \(J\), with at
most one edge on each unordered pair.  The directions may vary from
edge to edge.  For
\(x\in\mathbb F_p^I\), \(y\in\mathbb F_p^J\), write \(z_i=x_i\) for
\(i\in I\), \(z_j=y_j\) for \(j\in J\), and set
\[
 K(x,y)=\prod_{(u,v)\in E(G)}A_p(z_v-z_u).
\]
Then
\begin{equation}
 \|K-2^{-m}\|_{\mathrm{op}}
 \le (32m)^{1/4}p^{-1/8}.
 \label{eq:list-mixing}
\end{equation}
For \(m=0\), both sides are zero.  In particular, the implied
constant for any fixed pair of lists is independent of \(p\).
\end{lemma}

\begin{proof}
Put \(c=2^{-m}\) and \(B=K-c\).  With respect to the normalized
point-mass bases, \(T_B\) has matrix entries
\(B(x,y)/\sqrt{|\mathbb F_p^I|\,|\mathbb F_p^J|}\).
If \(s_r\) are its singular values, then
\begin{align}
 \|B\|_{\mathrm{op}}^4
 &\le \sum_r s_r^4
 =\operatorname{tr}\bigl((T_BT_B^*)^2\bigr) \notag\\
 &=\mathbb E_{x^0,x^1,y^0,y^1}
 B(x^0,y^0)B(x^0,y^1)B(x^1,y^0)B(x^1,y^1),
 \label{eq:list-four-cycle}
\end{align}
where the four lists are independent and uniform.  The kernel is
real, so no conjugates appear in the last expression.

Let \(\Omega=\{0,1\}^2\).  Expanding the last line gives
\[
 \sum_{F\subseteq\Omega}(-c)^{4-|F|}
 \mathbb E\prod_{(a,b)\in F}K(x^a,y^b).
\]
For each \(F\), its product is the constraint indicator of a graph
on
\[
 (I\times\{0,1\})\sqcup(J\times\{0,1\}).
\]
Every base edge between \(i\in I\) and \(j\in J\) is copied, in its
original direction, between \((i,a)\) and \((j,b)\) for every
\((a,b)\in F\).  The two endpoints determine the base unordered
pair and the corner \((a,b)\).  Hence this graph has no loops or
repeated unordered edges, and has exactly \(m|F|\) edges.
Lemma~\ref{lem:graph-count} therefore gives
\[
 \mathbb E\prod_{(a,b)\in F}K(x^a,y^b)
 =c^{|F|}+E_F,
 \qquad |E_F|\le m|F|\delta_p.
\]
The main terms sum to \((c-c)^4=0\).  The remaining terms have
absolute value at most
\[
 m\delta_p\sum_{t=0}^4\binom4t t c^{4-t}
 =4m(1+c)^3\delta_p
 \le32m p^{-1/2}.
\]
Combining this with \eqref{eq:list-four-cycle} and taking fourth
roots proves \eqref{eq:list-mixing}.
\end{proof}
==== END A-power-saving-for-square-difference-free-sets//graph.tex ====

==== BEGIN A-power-saving-for-square-difference-free-sets//induction.tex ====
\section{Exceptional terms and the power-saving induction}
\label{sec:induction}

We now combine the scale comparison, which applies to every small-residue
assignment, with the cancellation obtained for good assignments. For good
assignments the nonexceptional contribution is already negligible; we control
the exceptional parts by expanding the prime measures. The other assignments
are bounded directly by the scale comparison. Expanding measures rather than
comparing integrands pointwise keeps the argument valid for signed Fourier
lifts throughout.

The finite set $V$ and the constants $P,M_0,\beta,\gamma,\tau,\sigma_0$
have now been fixed. Lemma~\ref{lem:good-residues} has also established the
fixed positive proportion $\rho$ of good small-residue assignments.
The prime threshold $P$ was chosen so that
\begin{equation}
 \prod_{p\ge P}(1+p^{-3})-1\le\frac14.
 \label{eq:exceptional-euler-budget}
\end{equation}
For every positive integer $n$, define
\begin{equation}
 \mathcal Y(n)=
 \max_{\substack{A\subseteq[n]\\ A\text{ square-difference-free}}}Y(n,A).
 \label{eq:maximal-functional}
\end{equation}
This is a finite nonnegative number, by Proposition~\ref{prop:signed-holder}.
Write $\eta_p=\varepsilon_p(\F_p^V)$ for the mass of the exceptional measure,
so $0\le \eta_p\le p^{-4}$.  For
$B\subseteq\mathcal P_N$, put
\begin{equation}
 q_B=\prod_{p\in B}p,\qquad
 \eta_B=\prod_{p\in B}\eta_p,\qquad
 n_B=\left\lceil\frac{N}{(M_0q_B)^2}\right\rceil.
 \label{eq:exceptional-subset-notation}
\end{equation}
Empty products have value $1$, so $q_\varnothing=\eta_\varnothing=1$ and
$n_\varnothing=\lceil N/M_0^2\rceil$.

\begin{proposition}\label{prop:induction-recurrence}
There are absolute constants $K<\infty$ and $N_1\ge2$ such that, for every
integer $N\ge N_1$,
\begin{align}
 \mathcal Y(N)
 &\le K N^{-\sigma_0}
 +(1+u_N)\left((1-\rho)\mathcal Y(n_\varnothing)
       +\rho\!
       \sum_{\substack{\varnothing\ne B\subseteq\mathcal P_N\\q_B\le N^\tau}}
            \eta_B\mathcal Y(n_B)\right),
 \label{eq:functional-recurrence}\\
 u_N&=(1+M_0^2N^{2\tau-1})^d-1.
 \label{eq:uniform-ceiling-error}
\end{align}
In particular, $u_N\to0$ as $N\to\infty$.  The constants $K,N_1$ do not
depend on any proposed induction bound for $\mathcal Y$.
\end{proposition}

\begin{proof}
Fix a square-difference-free set $A\subseteq[N]$.  Write
\[
 G_s=\mathcal L_{[N],\mathcal P_N,Q}f_s,
 \qquad
 \Phi_{\mathbf s}((z_v)_{v\in V})=\prod_{v\in V}G_{s_v}(z_v),
\]
where $\mathbf s=(s_v)_{v\in V}$ is a small-residue assignment.  By
\eqref{eq:small-residue-input}, $1_A=\E_s f_s$.  The lift is linear, and the
product-space integral is multilinear, so
\begin{equation}
 Y(N,A)=\E_{\mathbf s}
       \int\Phi_{\mathbf s}\,d\bigotimes_{p\in\mathcal P_N}\mu_p.
 \label{eq:residue-average-functional}
\end{equation}

For $B\subseteq\mathcal P_N$, define the nonnegative finite measure
\[
 \kappa_B=
   \bigotimes_{p\in B}\varepsilon_p
   \otimes\bigotimes_{p\in\mathcal P_N\setminus B}\mu_p.
\]
Its mass is $\eta_B$.  The finite product expansion gives the following
identity of signed measures:
\begin{equation}
 \bigotimes_{p\in\mathcal P_N}\mu_p
 -\bigotimes_{p\in\mathcal P_N}(\mu_p-\varepsilon_p)
 =\sum_{\varnothing\ne B\subseteq\mathcal P_N}
       (-1)^{|B|+1}\kappa_B.
 \label{eq:signed-measure-inclusion-exclusion}
\end{equation}
For any real integrand $\Phi$, integration followed by the triangle inequality
therefore yields
\begin{equation}
 \int\Phi\,d\bigotimes_p\mu_p
 \le
 \left|\int\Phi\,d\bigotimes_p(\mu_p-\varepsilon_p)\right|
 +\sum_{B\ne\varnothing}\left|\int\Phi\,d\kappa_B\right|.
 \label{eq:exceptional-signed-bound}
\end{equation}
This step makes no sign assumption on $\Phi$.  Notice also that the measure
outside $B$ in each summand is the full measure $\mu_p$.

We first bound the summands with $q_B>N^\tau$.  A summand of zero mass is
zero, so assume $\eta_B>0$.  Each normalized exceptional measure
$\varepsilon_p/\eta_p$ has uniform single-label marginals: its invariance under
simultaneous translation implies that any one label is uniformly distributed
on $\F_p$.  The same uniformity holds for $\mu_p$.  Consequently every
single-slot marginal of $\kappa_B/\eta_B$ is uniform on
$X_{\mathcal P_N}$.  No independence between different slots is needed.
Ordinary H\"older's inequality gives
\begin{equation}
 \left|\int\Phi_{\mathbf s}\,d\kappa_B\right|
 \le\eta_B\prod_{v\in V}\|G_{s_v}\|_{L^d(X_{\mathcal P_N})}
 \le C_\epsilon\eta_BN^\epsilon
 \qquad(\epsilon>0).
 \label{eq:exceptional-holder-moment}
\end{equation}
The last inequality is Lemma~\ref{lem:lift-moments}, applied at the
fixed even order $d$ and with the loss in each factor chosen sufficiently
small.  The bound $|f_s|\le M_0$ only introduces a fixed factor.  The constant
$C_\epsilon$ is independent of $B$, $A$, $N$, and $\mathbf s$.

The exceptional masses are summable with the extra factor $q_B$:
\begin{equation}
 \sum_{\varnothing\ne B\subseteq\mathcal P_N}q_B\eta_B
 =\prod_{p\in\mathcal P_N}(1+p\eta_p)-1
 \le\prod_{p\ge P}(1+p^{-3})-1\le\frac14.
 \label{eq:weighted-exceptional-mass}
\end{equation}
It follows from \eqref{eq:exceptional-holder-moment} that
\[
 \sum_{\substack{\varnothing\ne B\subseteq\mathcal P_N\\q_B>N^\tau}}
   \left|\int\Phi_{\mathbf s}\,d\kappa_B\right|
 \le\frac{C_\epsilon}{4}N^{-\tau+\epsilon}.
\]
Choose $\epsilon=\tau/2$.  Since $\sigma_0=\gamma\tau/4<\tau/2$, we obtain
\begin{equation}
 \sum_{q_B>N^\tau}
   \left|\int\Phi_{\mathbf s}\,d\kappa_B\right|
 \le K_1N^{-\sigma_0},
 \label{eq:large-exceptional-tail}
\end{equation}
uniformly in the residue assignment.  Thus the number of exceptional subsets
has already been accounted for by the convergent product
\eqref{eq:weighted-exceptional-mass}.

Now suppose $q_B\le N^\tau$ and $\eta_B>0$.  Condition on all the labels at
primes in $B$ under $\kappa_B/\eta_B$.  The remaining prime law is exactly
$\bigotimes_{p\in\mathcal P_N\setminus B}\mu_p$.  For each realization, the
function in slot $v$ is $G_{s_v}$ specialized at its prescribed labels in $B$.
The mixed assertion of Proposition~\ref{prop:scale-transfer} applies to these
arbitrary slotwise residues and labels.  With
\[
 D_B=M_0q_B,\qquad \lambda_B=\frac{D_B^2n_B}{N},
\]
it gives
\begin{equation}
 \left|\int\Phi_{\mathbf s}\,d\kappa_B\right|
 \le\eta_B\left(\lambda_B^d\mathcal Y(n_B)+K_2N^{-\sigma_0}\right).
 \label{eq:small-exceptional-transfer}
\end{equation}
The same inequality is trivially true when $\eta_B=0$.  Crucially, the error
and its lower threshold for $N$ are uniform in $B$ and in every conditioned
tuple.  No induction hypothesis has been used.

The ceiling in the smaller scale produces a uniform multiplicative error.
For $q_B\le N^\tau$,
\[
 1\le\lambda_B\le1+\frac{D_B^2}{N}
       \le1+M_0^2N^{2\tau-1},
\]
and hence $\lambda_B^d\le1+u_N$.  Moreover,
$\sum_{B\ne\varnothing}\eta_B\le1/4$ by
\eqref{eq:weighted-exceptional-mass}.  Summing the errors in
\eqref{eq:small-exceptional-transfer} is therefore harmless.

If $\mathbf s$ is good, Proposition~\ref{prop:nonexceptional-contraction}
bounds the first term on the right of \eqref{eq:exceptional-signed-bound} by
$K_3N^{-\sigma_0}$.  Combining this with
\eqref{eq:large-exceptional-tail} and
\eqref{eq:small-exceptional-transfer}, we obtain
\[
 \int\Phi_{\mathbf s}\,d\bigotimes_p\mu_p
 \le K_4N^{-\sigma_0}
 +(1+u_N)
   \sum_{\substack{\varnothing\ne B\subseteq\mathcal P_N\\q_B\le N^\tau}}
      \eta_B\mathcal Y(n_B).
\]
For a nongood $\mathbf s$, apply the mixed scale comparison directly with
$B=\varnothing$:
\[
 \left|\int\Phi_{\mathbf s}\,d\bigotimes_p\mu_p\right|
 \le(1+u_N)\mathcal Y(n_\varnothing)+K_2N^{-\sigma_0}.
\]
Averaging these two estimates in
\eqref{eq:residue-average-functional} gives
\eqref{eq:functional-recurrence}, since the good assignments have probability
$\rho$.  The estimate is uniform in $A$, so we may take its maximum.
Finally, $2\tau<1$ by the fixed parameter choice, and $M_0,d$ are fixed, so
$u_N\to0$.  All constants in the proof came from the preceding estimates and
fixed parameters; none depended on an induction constant.
\end{proof}

\begin{theorem}\label{thm:functional-decay}
There are absolute constants $a_0>0$ and $C_0<\infty$ such that, for every
positive integer $N$ and every square-difference-free set $A\subseteq[N]$,
\[
 Y(N,A)\le C_0N^{-a_0}.
\]
\end{theorem}

\begin{proof}
We make all remaining parameter choices before starting the induction.
The finite construction fixes $h,t_0,\ell,d$; the conditional and kernel estimates
fix $\gamma$ and $\beta$.  The threshold $P$ was then chosen, and with it
$M_0$, $\tau$, and $\sigma_0$.  In particular the probability $\rho$ of
Lemma~\ref{lem:good-residues} is already a fixed positive number.  We shall
not enlarge $P$ after choosing the exponent below.

Put $r=1-3\rho/4$, so $1/4\le r<1$, and choose
\begin{equation}
 a_0=\min\left\{\frac14,\frac{\sigma_0}{2},
                  \frac{\log(1/r)}{4\log M_0}\right\}.
 \label{eq:absolute-exponent-choice}
\end{equation}
Table~\ref{tab:parameter-roles} summarizes the roles of the parameters;
among the listed quantities, only those in the last row vary with
the interval length.
\input{figures/parameter-roles}

Every quantity in~\eqref{eq:absolute-exponent-choice} is fixed and positive. Thus
$00$.  Choose an integer $N_*\ge N_1$ so large that,
for every $N\ge N_*$,
\begin{equation}
 \theta u_N\le\kappa,\qquad
 K N^{a_0-\sigma_0}\le\kappa,
 \label{eq:induction-threshold-choice}
\end{equation}
where $K,N_1$ are supplied by
Proposition~\ref{prop:induction-recurrence}.  This is possible because
$u_N\to0$ and $a_0<\sigma_0$.  The choice of $N_*$ is independent of $C_0$.
Choose $C_0\ge1$ large enough that
\begin{equation}
 \mathcal Y(n)\le C_0 n^{-a_0}\qquad(1\le n0$ are absolute constants for which
$|A|\le CN^{1-c}$ for every such $N$ and $A$.
\end{proof}
==== END A-power-saving-for-square-difference-free-sets//induction.tex ====

==== BEGIN A-power-saving-for-square-difference-free-sets//introduction.tex ====
\section{Introduction}
\label{sec:introduction}

For a positive integer $N$, write $[N]=\{1,\ldots,N\}$. A set
$A\subseteq[N]$ is \emph{square-difference-free} if
\[
 (A-A)\cap\{m^2:m\in\N,\ m\ge1\}=\varnothing.
\]
Thus the restriction concerns differences in the integers. Define
\[
 s(N)=\max\{|A|:A\subseteq[N]\text{ is square-difference-free}\}.
\]
Our main result is a fixed power saving in this extremal problem.

\begin{theorem}
\label{thm:main}
There are absolute constants $c>0$ and $C<\infty$ such that, for every
integer $N\ge1$ and every square-difference-free set $A\subseteq[N]$,
\[
 |A|\le C N^{1-c}.
\]
Equivalently, every subset of $[N]$ larger than $C N^{1-c}$ contains
distinct elements whose absolute difference is the square of a positive
integer.
\end{theorem}

The exponent obtained here is positive and absolute, but extremely small.
The argument does not yield a useful numerical value or an optimal exponent.
All finite constructions in the proof are fixed independently of $N$ and
$A$; their size affects the constants and the exponent.

\begin{remark}[Square-difference colorings]
Theorem~\ref{thm:main} gives a polynomial bound for the classical
square-difference coloring problem~\cite[Section~4]{BeigelGasarch2008}.
Fix its constants $c,C$, reducing $c$ so that $0(Cr)^{1/c}$, every $r$-coloring
of $[N]$ has same-colored elements $a,a+d^2$ with $d\ge1$.
Indeed, a largest color class has size at least $N/r>C N^{1-c}$.
Only the two endpoints must share a color; no color restriction is
imposed on $d$. Equivalently, if $G_N$ is the graph on $[N]$ joining
distinct integers whose absolute difference is a square, then
\[
 \chi(G_N)\ge \frac{N}{s(N)}\ge C^{-1}N^c,
\]
since its independent sets are exactly the square-difference-free sets.
\end{remark}

\subsection{Previous work}

The problem originates in a question of Lov\'asz.
Furstenberg~\cite[Theorem~1.2]{Furstenberg1977} and S\'ark\"ozy~\cite{Sarkozy1978I}
proved independently that $s(N)=o(N)$, using ergodic theory and the
circle method, respectively. S\'ark\"ozy's quantitative estimate has the form
$s(N)\le N/(\log N)^{1/3+o(1)}$; see also the historical account
in~\cite{BloomMaynard2022}. Pintz, Steiger and
Szemer\'edi~\cite[Theorem~1]{PSS1988} strengthened this to
\[
 s(N)\ll \frac{N}{(\log N)^{c_{\rm PSS}\log\log\log\log N}}
\]
for an absolute $c_{\rm PSS}>0$. Their argument combines concentration on
arithmetic progressions with an iteration that increases the collection
of large Fourier coefficients. Bloom and Maynard~\cite{BloomMaynard2022}
subsequently obtained
\[
 s(N)\ll \frac{N}{(\log N)^{c_{\rm BM}\log\log\log N}}
\]
for an absolute $c_{\rm BM}>0$, using bounds for the additive energy of
rational numbers with small denominators to improve the density-increment
argument.


Green and Sawhney~\cite[Theorem~1.1]{GreenSawhney2025} prove
\[
 s(N)\ll N\exp\bigl(-c_1\sqrt{\log N}\bigr)
\]
for an absolute $c_1>0$. Their arithmetic level-$d$ inequality transfers
product-space hypercontractive estimates to rational Fourier coefficients.
Theorem~\ref{thm:main} answers affirmatively the fixed-power question
posed in~\cite[Section~1.1]{GreenSawhney2025}.
Adajar et al.~\cite[Theorem~1.1]{AdajarEtAl2026} extend the arithmetic
level-$d$ approach to fixed integer polynomials having a root modulo
every positive integer. For degree at least two they obtain a density
saving $\exp(-c(\log N)^\mu)$ for every fixed $0<\mu<1/2$, with
constants depending on the polynomial and $\mu$.

In the opposite direction, Ruzsa~\cite[Theorems~1 and~2]{Ruzsa1984}
constructed square-difference-free sets of size
$\gg N^{\gamma_0}$, where
\[
 \gamma_0=\frac12\left(1+\frac{\log 7}{\log 65}\right)
          =0.733077\ldots.
\]
His construction uses a set of residues with no square differences
modulo a squarefree integer to restrict selected digits of an integer
expansion. Beigel and Gasarch~\cite[Lemma~3.6 and the proof of Theorem~3.7]{BeigelGasarch2008} and
Lewko~\cite[Theorem~3]{Lewko2015} obtained the improved exponent
\[
 \gamma_1=\frac12\left(1+\frac{\log 12}{\log 205}\right)
          =0.733411797\ldots
\]
by exhibiting suitable sets of twelve residues modulo $205$.
Krachun's construction~\cite[Theorem~1]{Krachun2026} passes the
three-quarter threshold and gives
\[
 \liminf_{N\to\infty}\frac{\log s(N)}{\log N}
 \ge 0.752796455874514\ldots.
\]
A substantial gap remains between this lower exponent and the upper
exponent $1-c$ in Theorem~\ref{thm:main}.

Reflection positivity yields multilinear inequalities through the chessboard
method of Fr\"ohlich and Lieb~\cite[Theorem~2.2]{FrohlichLieb1978} and its
abstract formulation by Fr\"ohlich, Israel, Lieb and
Simon~\cite[Theorem~4.1]{FILS1978ReflectionI}.
Hatami~\cite[Theorem~2.8 of the arXiv version]{Hatami2010} characterizes
graph seminorms through mixed H\"older inequalities. Conlon and
Lee~\cite[Theorems~1.2 and~1.3]{ConlonLee2017Reflection} use finite
reflection groups to obtain such inequalities for absolute-value inputs,
and also for real-valued inputs when the two defining sets of simple
reflections are disjoint.
The signed mixed inequality is a classical feature of the chessboard
method; its role here is to permit Fourier lifts that can change sign.
We construct a finite-field probability law with the particular cycle
support, reflection positivity and conditional mixing needed for the
integer problem, and prove the signed inequality for that law directly.

The rational-frequency energy estimates of Bloom and
Maynard~\cite{BloomMaynard2022} and the product-space methods of Green and
Sawhney~\cite{GreenSawhney2025} are methodological precedents for the
arithmetic part. The argument retains rational Fourier analysis and
rescaling along square-step progressions. Here the quantity compared
between scales is a multilinear functional of a signed Fourier lift.

\subsection{Outline of the proof}

The proof constructs a nonnegative quantity that dominates a fixed power
of the density, and then proves that this quantity decreases by a power
of the interval length. The comparison with shorter intervals is available
for every residue specification. A separate estimate makes a fixed positive
proportion of those specifications negligible; these two estimates produce
the contraction.

Here is the quantity to be controlled. Choose a fixed even integer $d$
and a fixed set $V$ of $d$ labels. For a finite set $\mathcal P$ of sufficiently
large primes, put $X_{\mathcal P}=\prod_{p\in\mathcal P}\F_p$.
We will construct a probability law for a tuple $(z_v)_{v\in V}$ in
$X_{\mathcal P}$, and for a real function $g$ on this space define
\[
 Z_{\mathcal P}(g)=\E\prod_{v\in V}g(z_v).
\]
Its $d$-th root will be a seminorm satisfying
$|\E g|\le Z_{\mathcal P}(g)^{1/d}$. To apply it to $A\subseteq[N]$,
take $\mathcal P$ to be the primes between a fixed threshold $P$ and
$N^\beta$, where $\beta>0$ is fixed and small. Let $G$ retain the
rational Fourier coefficients of $1_A$ at frequencies of squarefree
reduced denominator at most $N^\beta$ whose prime factors all belong
to $\mathcal P$, including the zero frequency.
Section~\ref{sec:lift-moments} defines this \emph{Fourier lift} precisely.
It is real and its mean is exactly $|A|/N$, although its values may have
either sign. With these choices, write $Y(N,A)=Z_{\mathcal P}(G)$.
The seminorm bound gives
\[
 (|A|/N)^d\le Y(N,A).
\]
Thus a fixed power decay for $Y$ proves the theorem.

The tuple law is constructed separately over each prime field, then
multiplied over the primes, collecting the field labels into vectors
$z_v\in X_{\mathcal P}$. Every individual field label is uniform.
The law has two further properties with distinct purposes.
First, it is reflection positive: specified involutions of $V$ exchange
two halves, and the matrix coupling the half-tuples is symmetric positive
semidefinite. Repeated Cauchy--Schwarz across these cuts proves the signed
mixed inequality and seminorm bound
(Proposition~\ref{prop:signed-holder}). Second, outside a part of mass
at most $p^{-4}$, a designated directed cycle of $24$ labels has nonzero
square differences in $\F_p$, and each of its edge pairs is uniform
among pairs with such a difference. This exact pair law supplies the
arithmetic input below. The full reflection-positive law and its normalized
nonexceptional part also satisfy conditional mixing, which makes inputs
with large Fourier denominators negligible
(Lemma~\ref{lem:conductor-decay}). Section~\ref{sec:tuple-laws} constructs
these laws from the graph estimates of Section~\ref{sec:finite-field-graphs}.

The approximation errors require control of signed lifts beyond their
means. Every fixed moment of $G$ grows more slowly than any fixed positive
power of $N$, uniformly in the prime set and in every retained family of
complete denominator classes. Lemma~\ref{lem:lift-moments} proves this
by a conditional square-function estimate and symmetrization. The same
bounds, with an explicit fiber-size cost, remain available after specifying
some prime coordinates. Combined with conductor decay, they justify the
Fourier truncations used in both estimates that follow.

For the change of scale, fixing prime coordinates whose product is at
most a sufficiently small fixed power of $N$ makes the low Fourier
coefficients exact averages on an integer residue class modulo an integer
$D$. Splitting that class into progressions of common difference $D^2$
and rescaling preserves square-difference avoidance: a square difference
in a rescaled set would give a square difference in $A$.
The seminorm's square-affine symmetries therefore compare the specialized
functional with the same functional at length $\lceil N/D^2\rceil$,
up to a factor tending to one and a power-decaying error
(Proposition~\ref{prop:scale-transfer}). This comparison applies to all
specified residues, without using the exceptional splitting.

The arithmetic cancellation uses the normalized nonexceptional prime laws.
Set aside the finitely many primes below the tuple-law threshold, together
with modulus $8$, in one fixed modulus $M_0$. Independently choose one
residue modulo $M_0$ for each slot. A fixed positive proportion of these
assignments has strict square differences along the designated cycle:
they are $1$ modulo $8$ and nonzero squares modulo every odd prime
dividing $M_0$.
For such an assignment, divide $[N]$ into ordered intervals. Unless all
cycle slots choose the same interval, one directed edge goes from an
earlier interval to a later one. Truncating and expanding the other slots
reduces its contribution to the selected pair law. A signed kernel
supported on actual shifts $m^2$ then bounds that contribution using the
absence of integer square differences
(Propositions~\ref{prop:square-kernel} and~\ref{prop:nonexceptional-contraction}).
The event that all cycle slots choose one interval is itself small.

Finally, Section~\ref{sec:induction} expands the exceptional submeasures
by inclusion--exclusion. Exceptional prime sets of large product are
bounded directly by moment estimates. A smaller exceptional prime set
with product $q_B$ leads to the shorter scale approximately $N/(M_0q_B)^2$.
Its mass remains summable even after multiplication by $q_B$; this absorbs
the factor $q_B^{2a_0}$ incurred by a sufficiently small proposed decay
exponent $a_0$. Combining this bound with the universal scale comparison
and the cancellation on a positive proportion of small-residue assignments
gives a strict weighted recurrence. Induction proves
$Y(N,A)\ll N^{-a_0}$, and the density inequality above gives
Theorem~\ref{thm:main} with $c=a_0/d$.
Figure~\ref{fig:proof-map} records the principal inputs to this assembly.
\input{figures/proof-map}

\subsection{Conventions}

All averages on finite sets are with respect to uniform probability
measure unless another measure is displayed. All $L^r$ norms of functions
on $X_{\mathcal P}$ use its uniform product measure. Empty products are
$1$; the product over an empty prime set is a one-point space. We write
$e(t)=\exp(2\pi i t)$ and identify rational frequencies modulo $1$.
The \emph{conductor} of a rational frequency is its reduced positive
denominator; the conductor of $0$ is $1$.

For nonnegative quantities, $U\ll V$ means $U\le C V$ for a constant
with the dependencies stated in context. A uniform bound $U=N^{o(1)}$
means that for every $\varepsilon>0$ there is a constant
$C_\varepsilon$ such that $U\le C_\varepsilon N^\varepsilon$ for all
allowed data. Fixed tuple sizes and fixed moment orders may affect
$C_\varepsilon$; sets, intervals, chosen residues and retained frequency
families may not. All final choices of parameters are made independently
of $N$ and $A$.
==== END A-power-saving-for-square-difference-free-sets//introduction.tex ====

==== BEGIN A-power-saving-for-square-difference-free-sets//kernel.tex ====
\section{A signed kernel for actual square differences}
\label{sec:square-kernel}

The preceding tuple measures impose square-difference constraints in residue fields.
This section supplies the passage to actual positive integer squares.
We construct a signed measure supported on such squares and approximate
its Fourier transform uniformly by a sum over small rational frequencies.
Because every actual square shift vanishes between the supports under
consideration, the signs of the measure cause no difficulty. The weighted
square sums and major/minor-arc analysis have a direct antecedent
in~\cite[Section~5]{GreenSawhney2025}. Here the signed divisor weight
is chosen to match the strict-square pair law of our finite-field tuples.

For the remainder of the paper, fix
\begin{equation}
 \beta=10^{-5}.
 \label{eq:kernel-beta}
\end{equation}
Write $e(t)=\exp(2\pi i t)$, and let $q(\xi)$ be the reduced denominator
of $\xi\in\mathbb Q/\mathbb Z$, with $q(0)=1$.
For every rational frequency define the normalized unit quadratic sum
\begin{equation}
 \mathfrak g(\xi)
 =\frac{1}{\varphi(q(\xi))}
   \sum_{m\in(\mathbb Z/q(\xi)\mathbb Z)^\times}e(\xi m^2),
 \qquad \mathfrak g(0)=1.
 \label{eq:kernel-unit-gauss}
\end{equation}
Here $\varphi$ is Euler's totient function.
If $D=8d_0$ with $d_0$ odd and squarefree, call a residue modulo $D$
a \emph{strict square class} if it is $1$ modulo $8$ and a nonzero
square modulo every prime dividing $d_0$.

\begin{proposition}[Ordered square-kernel estimate]
\label{prop:square-kernel}
Let $N\ge1$ be an integer, put $H=N^{\beta/2}$, and let
$D=8d_0\le N^\beta$, where $d_0$ is odd and squarefree.
Suppose $F,J:[N]\to\mathbb C$ satisfy $|F|,|J|\le1$ and are supported
in residue classes $a,b$ modulo $D$, respectively. Assume that $b-a$
is a strict square class, that $x3$, a primitive character averages to zero
on that class.

Apply these local facts to the decomposition
$\alpha_0=\alpha_D+\alpha_1$ already used in
\eqref{eq:kernel-model-coefficient}. Chinese remaindering shows that
the factor at primes dividing $D$ is
\begin{equation}
 \mathbb E_m\rho_D(m)e(\alpha_Dm^2)
 =\begin{cases}
   e(\alpha_Dr),&q(\alpha_D)\mid D,\\
   0,&q(\alpha_D)\nmid D.
  \end{cases}
 \label{eq:kernel-root-law}
\end{equation}
The expectation here is the uniform average over any common period.
The normalized odd-root restrictions, and the normalized oddness
restriction at two, are precisely responsible for the factor $D$
in the model kernel. In particular, choosing only one odd root
introduces no additional factor in \eqref{eq:kernel-root-law}.

\paragraph{The coprime divisor sieve.}
Every prime of an admissible $u$ is coprime to $D$.
If such a prime does not divide $q_1$, its mean-zero factor $w_p$
is independent of the exponential and all other factors by the
Chinese remainder theorem. Thus $A_u=0$ unless
$u\mid\operatorname{rad}(q_1)$.

Suppose first that $q_1\le H$. All divisors of
$\operatorname{rad}(q_1)$ then occur in the sum. By
\eqref{eq:kernel-complete-sieve}, the divisor weight is the density of
the uniform unit law modulo $q_1$ relative to the uniform law.
If $p^j$ exactly divides $q_1$ with
$j\ge2$, its local primitive unit quadratic average vanishes.
Indeed, conditional on a unit $x$ modulo $p^{j-1}$, the $p$ lifts
$x+tp^{j-1}$ have phase factors differing by $e(2atx/p)$,
where $p\nmid 2ax$. Their sum is zero.
If $q_1$ is squarefree, the unit average is exactly
$\mathfrak g(\alpha_1)$.
Together with \eqref{eq:kernel-root-law}, this proves
\begin{equation}
 c(\alpha_0)=c_0(\alpha_0)\qquad(q_1\le H).
 \label{eq:kernel-small-coefficient}
\end{equation}

For $q_1>H$ the model coefficient is zero, and we need a bound for
the truncated divisor sum. We give the prime-power calculation
explicitly. For an odd prime $p$, an integer $j\ge1$, and $p\nmid a$,
write
\[
 G_j(a)=\frac1{p^j}\sum_{m\bmod p^j}e(am^2/p^j),
 \qquad G_0(a)=1.
\]
Because two is invertible modulo $p^j$, the substitution
$(m,n)\mapsto(m-n,m+n)$ is a bijection. Consequently
\begin{equation}
 |G_j(a)|^2
 =\frac1{p^{2j}}\sum_{u,v\bmod p^j}e(auv/p^j)
 =p^{-j}.
 \label{eq:kernel-prime-power-gauss}
\end{equation}
Let
$T_j=\mathbb E_{m\bmod p^j}\mathbf1_{p\mid m}e(am^2/p^j)$.
For $j=1$ one has $T_1=1/p$; for $j\ge2$, substituting $m=pt$
gives $T_j=p^{-1}G_{j-2}(a)$.
It follows that
\begin{equation}
 B_j:=\mathbb E_{m\bmod p^j}w_p(m)e(am^2/p^j)
      =\frac{pT_j-G_j(a)}{p-1},
 \qquad |B_j|\le2p^{-j/2}.
 \label{eq:kernel-weighted-prime-power-gauss}
\end{equation}
For $j\ge2$, the last bound follows from
$(p+1)/(p-1)\le2$ and \eqref{eq:kernel-prime-power-gauss}.
For $j=1$, the bound obtained from the formula is
$(1+p^{-1/2})/(p-1)$; its ratio to $p^{-1/2}$ is
$1/(\sqrt p-1)\le2$. Thus the exponent-one case is included.

For a fixed divisor $u\mid\operatorname{rad}(q_1)$, the coprime
local mean is a product of factors from
\eqref{eq:kernel-prime-power-gauss} and
\eqref{eq:kernel-weighted-prime-power-gauss}. Its modulus is at most
$2^{\omega(u)}q_1^{-1/2}$.
The factor at primes of $D$ has modulus at most one by
\eqref{eq:kernel-root-law}. Dropping the restriction $u\le H$
after taking absolute values therefore gives
\begin{equation}
 |c(\alpha_0)|
 \le q_1^{-1/2}\sum_{u\mid\operatorname{rad}(q_1)}2^{\omega(u)}
 =3^{\omega(q_1)}q_1^{-1/2}
 \ll q_1^{-1/3}
 \le H^{-1/3}.
 \label{eq:kernel-large-coefficient}
\end{equation}
The penultimate implied constant is absolute. For example, one may use
\[
 C_*=
 \prod_{\substack{p<729\\p\text{ odd prime}}}\max(1,3p^{-1/6}),
\]
since $3\le p^{1/6}$ for every prime $p\ge729$.
Repeated prime powers only increase $q_1^{1/6}$.
This argument covers all prime powers in $q_1$, and if
$q(\alpha_D)\nmid D$, every coefficient under discussion is already
zero by \eqref{eq:kernel-root-law}.

\paragraph{Completion of the uniform comparison.}
On the minor arcs, \eqref{eq:kernel-minor-actual} and
\eqref{eq:kernel-minor-model} apply. On a major arc,
\eqref{eq:kernel-major-model}, \eqref{eq:kernel-major-actual},
\eqref{eq:kernel-small-coefficient} and
\eqref{eq:kernel-large-coefficient} apply, with $|V_N(y)|\le1$.
For clarity, every approximation error is bounded here using the fixed
value \eqref{eq:kernel-beta}:
\begin{align*}
 DHN^{-1/20}\sqrt{\log N}
   &\le N^{-0.049985}\sqrt{\log N},\\
 DH^2N^{-1/10}
   &\le N^{-0.09998},\\
 qD^2H^3/N
   &\le N^{-0.899965},\\
 \frac{qD^2H^2(1+|y|N)}{\sqrt N}
   &\le3N^{-0.29997}.
\end{align*}
In the last two lines $q\le N^{1/10}$ and
$|y|\le2N^{-9/10}$.
Each bound is $O(N^{-\beta/6})=O(H^{-1/3})$ with an absolute constant.
Together with the coefficient bound this proves
\eqref{eq:kernel-uniform-approximation} for all sufficiently large $N$.
All size and separation thresholds used above are absolute.
For the bounded remaining range, the elementary bounds
$\|S\|_\infty\ll DH$ and $\|S_0\|_\infty\le DH^2$
absorb that range by increasing the absolute constant.
Thus \eqref{eq:kernel-uniform-approximation} is uniform in all the
data of the proposition.

\paragraph{The directional bilinear identity.}
It remains to pass from the Fourier comparison to the supports in the
statement. Extend $F$ and $J$ by zero to $\mathbb Z$.
For any finitely supported kernel $h$ define
\[
 \mathcal B_h(F,J)=\frac1N\sum_{x\in\mathbb Z}
                              \sum_{n\in\mathbb Z}h(n)F(x)J(x+n).
\]
Both sums use counting measure; the displayed $1/N$ normalizes only
the base variable. Let
\[
 \widetilde F(\alpha)=\sum_{x\in\mathbb Z}F(x)e(-x\alpha),
 \qquad
 \widetilde J(\alpha)=\sum_{y\in\mathbb Z}J(y)e(-y\alpha),
 \qquad
 S_h(\alpha)=\sum_{n\in\mathbb Z}h(n)e(\alpha n).
\]
Expanding and integrating the characters imposes $y=x+n$, giving
the bilinear identity
\[
 \mathcal B_h(F,J)=\frac1N\int_0^1
 S_h(\alpha)\widetilde F(-\alpha)\widetilde J(\alpha)\,d\alpha.
\]
There are no conjugates in this identity.
Cauchy--Schwarz and counting-measure Plancherel now give
\begin{align}
 |\mathcal B_{k-k_0}(F,J)|
 &\le\frac{\|S-S_0\|_\infty}{N}
          \|F\|_{\ell^2(\mathbb Z)}\|J\|_{\ell^2(\mathbb Z)}
 \ll H^{-1/3},
 \label{eq:kernel-plancherel-error}
\end{align}
because each of the last two norms is at most $\sqrt N$.

Every term in $\mathcal B_k(F,J)$ contains $F(x)J(x+m^2)$ with
$m\ge1$, and this product is zero by hypothesis.
Thus $\mathcal B_k(F,J)=0$ term by term, regardless of the signed
weights in \eqref{eq:kernel-actual}.
For a pair $x,y$ with $F(x)J(y)\ne0$, the ordering assumption gives
$1\le y-x\le N-1$, and the residue conditions give
$y-x\equiv r\pmod D$.
Consequently both indicators in \eqref{eq:kernel-model} are automatic
for every contributing pair. Its expansion is exactly
\begin{align*}
 \mathcal B_{k_0}(F,J)
 &=\frac{D}{N^2}\sum_{x,y\in[N]}F(x)J(y)
                    \sum_{\xi\in\Xi}\mathfrak g(\xi)e(-\xi(y-x))\\
 &=D\sum_{\xi\in\Xi}\mathfrak g(\xi)
                    \widehat F(-\xi)\widehat J(\xi).
\end{align*}
Combining this with \eqref{eq:kernel-plancherel-error} proves
\eqref{eq:kernel-conclusion}.
\end{proof}

\begin{remark}
\label{rem:kernel-interval-normalization}
The normalization for later interval restrictions follows directly
from the proposition. Suppose $I_1,I_2\subseteq[N]$ are nonempty
intervals with $\max I_1<\min I_2$, and let $\Xi$ be the frequency
set in \eqref{eq:kernel-conclusion}. Suppose $F_i$ is the indicator of the restriction of a
square-difference-free set to $I_i$ and a class $a_i$ modulo $D$,
where $a_2-a_1$ is a strict square class. The coefficients
\[
 c_i(\xi)=\frac D{|I_i|}\sum_{n\in I_i}F_i(n)e(-n\xi)
          =\frac{DN}{|I_i|}\widehat F_i(\xi)
\]
satisfy
\[
 \left|\sum_{\xi\in\Xi}\mathfrak g(\xi)c_1(-\xi)c_2(\xi)\right|
 \ll H^{-1/3}\frac{DN^2}{|I_1||I_2|}.
\]
This conclusion uses the integer supports of the restrictions;
it requires no pointwise positivity of a Fourier lift.
\end{remark}
==== END A-power-saving-for-square-difference-free-sets//kernel.tex ====

==== BEGIN A-power-saving-for-square-difference-free-sets//lifts.tex ====
\section{Uniform moment bounds for Fourier lifts}
\label{sec:lift-moments}

We now transfer a bounded function on an integer interval to the product
of its prime residue spaces.  Keeping rational frequencies of bounded
denominator need not preserve pointwise bounds or positivity.  The
replacement we require is a uniform bound for every fixed moment, valid
also after retaining arbitrary complete denominator classes.  We first
control a square function of the exact prime supports, then recover the
sum by symmetrization. For related arithmetic moment estimates and Fourier
lifting, see \cite[Theorem~2 of the arXiv version]{BloomMaynard2022} and
\cite[Section~3]{GreenSawhney2025}; the uniform estimate used here is
proved below.

Let \(\mathcal P\) be a finite set of primes, and give
\[
 X_{\mathcal P}=\prod_{p\in\mathcal P}\mathbb F_p
\]
uniform probability measure.  A character is indexed by
\(\xi=\sum_{p\in\mathcal P}a_p/p\pmod1\), where
\(a_p\in\{0,\ldots,p-1\}\), and we write
\[
 e(z\xi)=\prod_{p\in\mathcal P}e(a_pz_p/p),
 \qquad e(t)=\exp(2\pi i t).
\]
Its exact prime support is \(\{p:a_p\ne0\}\), and its reduced
denominator is the product of these primes.  In particular \(q(0)=1\).
Summing a geometric progression in each coordinate shows that these
characters form an orthonormal basis.
For \(U\subseteq\mathcal P\), put \(q_U=\prod_{p\in U}p\), with
\(q_\varnothing=1\).

Let \(I\) be a nonempty consecutive interval of integers, of length
\(L\), and let \(f:I\to\mathbb C\).  Define
\[
 \widehat f_I(\xi)=\frac1L\sum_{n\in I}f(n)e(-n\xi),
 \qquad
 f_U(z_U)=\sum_{q(\xi)=q_U}\widehat f_I(\xi)e(z_U\xi).
\]
For \(Q\ge1\), the Fourier lift is
\[
 \mathcal L_{I,\mathcal P,Q}f
   =\sum_{\substack{U\subseteq\mathcal P\\q_U\le Q}}f_U.
\]
If \(\mathcal F\) is a subfamily of these supports, write
\(\mathcal L^{\mathcal F}_{I,\mathcal P,Q}f=\sum_{U\in\mathcal F}f_U\).
Thus retaining a support means retaining all frequencies of its exact
denominator.  Each \(f_U\) depends only on \(z_U\) and has mean zero
in each coordinate belonging to \(U\).  Real \(f\) gives real
\(f_U\), because the summation set is closed under negation.

\begin{remark}[A lift need not be nonnegative]
Take distinct primes $p,q$ with $p,q\le QH,B}
  =\{U\in\mathcal F:q_{U\setminus B}>H\}.
\]
After specialization their Fourier supports lie, respectively, at
remaining denominators at most \(H\) and greater than \(H\).
\end{lemma}

\begin{proof}
The specified fiber has uniform probability \(1/q_B\).  Its
contribution to \(\mathbb E|G|^p\) is
\[
 q_B^{-1}\mathbb E_{z_{\mathcal P\setminus B}}
 |G(z_B,z_{\mathcal P\setminus B})|^p.
\]
Discarding all other nonnegative contributions proves
\eqref{eq:lift-specialization}.
Apply Lemma~\ref{lem:lift-moments} to each displayed family.
Every character of \(f_U\), when specialized on \(B\), retains
exactly the coordinate support \(U\setminus B\).  Combining such
characters can cancel coefficients but cannot create frequencies
outside the asserted remaining-denominator ranges.  This proves the
last assertion.
\end{proof}

In particular, for \(d\ge2\), a multilinear estimate using one \(L^2\) norm and
\(d-1\) norms in \(L^{2(d-1)}\), with the same coordinates fixed
in every slot, incurs the total specialization factor
\[
 q_B^{1/2}\left(q_B^{1/(2(d-1))}\right)^{d-1}=q_B.
\]
Two specialized \(L^2\) factors likewise incur \(q_B\).
Multiplying the original bounded function by a fixed constant simply
multiplies every norm bound by the absolute value of that constant.

For later uniform estimates one can make the subpower dependence
entirely explicit.  Put
\[
 k_*(Q)=\max\{j\in\mathbb Z_{\ge0}:(j+1)!\le Q\},
 \qquad
 \mathfrak M_r(Q)=2\bigl(k_*(Q)+1\bigr)A_r^{k_*(Q)}.
\]
The moment bound is at most \(M\mathfrak M_r(Q)\), and for every
fixed \(r\) and \(\epsilon>0\), one has
\(\mathfrak M_r(Q)\le Q^\epsilon\) for sufficiently large \(Q\).
The threshold may depend on \(r,\epsilon\), but on none of the
intervals, functions, fibers, or retained families.  Thus any fixed
finite product of the envelopes \(\mathfrak M_r(Q)\) remains subpower,
even when their fixed moment orders are very large.  The displayed
input bounds and specialization powers remain explicit.
==== END A-power-saving-for-square-difference-free-sets//lifts.tex ====

==== BEGIN A-power-saving-for-square-difference-free-sets//list-regrouping.tex ====
\begin{figure}[htbp]
\centering
\[
 \underbrace{\left((Y_{(u,w),i})_{i\in[m_s]}\right)_{w\in\Pi_+}}
             _{r\text{ lists, each of length }m_s}
 \quad\longleftrightarrow\quad
 \underbrace{(Y'_{u,(w,i)})_{(w,i)\in\Pi_+\times[m_s]}}
             _{\text{one list of length }m_{s+1}=r m_s}
\]
\[
 \begin{gathered}
 j'_v=(w,i)\ \text{ uniform in }\Pi_+\times[m_s]
 \quad\xrightarrow{\ \text{restrict to }w=v_b\ }\quad
 Y'_{u,(v_b,i)}=Y_{(u,v_b),i},\\[2pt]
 \mathbb P(w=v_b)=r^{-1},\qquad
 \mathbb P(\text{all }v\in V_+\text{ select their own sublist})=r^{-n}.
 \end{gathered}
\]
\caption{The two steps when marking a block $b$. For a new address
$u\in U_{S\cup\{b\}}$, the old plus addresses above it are $(u,w)$,
where $w\in\Pi_+$ is the word in block $b$; the prime on $Y'$ marks
only a relabeling of the same entries. Regrouping the independent lists
preserves their distribution without a loss. For each $v\in V_+$,
with $u=u_{S\cup\{b\}}(v)$, restricting its independent new index to
$w=v_b$ recovers its old emitted label. This second step has the total
probability $r^{-n}$.}
\label{fig:list-regrouping}
\end{figure}
==== END A-power-saving-for-square-difference-free-sets//list-regrouping.tex ====

==== BEGIN A-power-saving-for-square-difference-free-sets//main.tex ====
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\hypersetup{colorlinks=true,linkcolor=blue!50!black,citecolor=blue!50!black,urlcolor=blue!50!black,
  pdftitle={A power saving for square-difference-free sets},pdfauthor={OpenAI}}
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\title{A power saving for square-difference-free sets}
\author{OpenAI}
\date{September 24, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove that there are absolute constants $c>0$ and $C<\infty$ such that
every set $A\subseteq\{1,\ldots,N\}$ with no nonzero square difference
satisfies $|A|\le C N^{1-c}$. This answers the fixed-power question posed
by Green and Sawhney.
\end{abstract}
\tableofcontents
\input{sections/introduction}
\input{sections/graph}
\input{sections/tuples}
\input{sections/functional}
\input{sections/lifts}
\input{sections/kernel}
\input{sections/transfer}
\input{sections/contraction}
\input{sections/induction}
\bibliographystyle{plain}
\bibliography{references}
\end{document}
==== END A-power-saving-for-square-difference-free-sets//main.tex ====

==== BEGIN A-power-saving-for-square-difference-free-sets//parameter-roles.tex ====
\begin{table}[htbp]
\centering
\small
\begin{tabular}{@{}p{.20\textwidth}p{.71\textwidth}@{}}
\toprule
Parameters & Role and order of choice\\
\midrule
$h,t_0,\ell,d$ & Fixed cycle length, marking depth, block count, and tuple size
(Section~\ref{sec:tuple-laws}).\\[3pt]
$\gamma,\beta$ & Fixed conductor-decay and rational-frequency cutoff exponents
(Sections~\ref{sec:functional} and~\ref{sec:square-kernel}).\\[3pt]
$P,M_0$ & Prime-law threshold and modulus containing all smaller primes;
fixed before choosing the decay exponent
(Section~\ref{sec:scale-transfer}).\\[3pt]
$\tau,\sigma_0$ & Fixed partition exponent and common error exponent;
chosen in~\eqref{eq:scale-parameters}.\\[3pt]
$\rho,a_0$ & Fixed positive residue proportion and induction exponent;
$a_0$ is chosen only after $M_0$ and $\rho$ in~\eqref{eq:absolute-exponent-choice}.\\[3pt]
$Q,H,\mathcal P_N$ & Scale-dependent cutoffs and prime set, defined from $N$
in~\eqref{eq:scale-parameters}.\\
\bottomrule
\end{tabular}
\caption{Parameter roles. Every construction size and exponent is fixed
independently of $N$ and $A$; the Fourier cutoffs and available prime
coordinates change with $N$.}
\label{tab:parameter-roles}
\end{table}
==== END A-power-saving-for-square-difference-free-sets//parameter-roles.tex ====

==== BEGIN A-power-saving-for-square-difference-free-sets//proof-map.tex ====
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[
  box/.style={draw,rounded corners=2pt,align=center,font=\small,
              text width=3.55cm,minimum height=1.1cm,inner sep=5pt},
  arr/.style={-{Stealth[length=5pt]},thick},x=1cm,y=1cm]
\node[box] (graph) at (-4.9,0) {Graph mixing\\Section~\ref{sec:finite-field-graphs}};
\node[box] (tuple) at (0,0) {Tuple laws\\Section~\ref{sec:tuple-laws}};
\node[box] (functional) at (4.9,0) {Signed seminorm and\\conductor decay\\Section~\ref{sec:functional}};
\node[box] (moments) at (-3.5,-2) {Uniform lift moments\\Section~\ref{sec:lift-moments}};
\node[box] (kernel) at (3.5,-2) {Actual-square kernel\\Section~\ref{sec:square-kernel}};
\node[box] (transfer) at (-3.5,-4) {Square-step comparison\\Section~\ref{sec:scale-transfer}};
\node[box] (cancel) at (3.5,-4) {Ordered-interval cancellation\\Section~\ref{sec:ordered-contraction}};
\node[box,text width=6.4cm] (induct) at (0,-6) {Exceptional-measure expansion,\\weighted recurrence and induction\\Section~\ref{sec:induction}};
\draw[arr] (graph) -- (tuple);
\draw[arr] (tuple) -- (functional);
\draw[arr] (moments) -- (transfer);
\draw[arr] (moments.south east) -- (cancel.north west);
\draw[arr] (kernel) -- (cancel);
\draw[arr] (functional.east) -- (7.1,0) -- (7.1,-4) -- (cancel.east);
\draw[arr] (functional.south west) -- (1.9,-.95) -- (-6.1,-.95) -- (-6.1,-4) -- (transfer.west);
\draw[arr] (tuple.south) -- (0,-1.25) -- (0,-3.2) -- (cancel.north west);
\draw[arr] (transfer) -- (induct);
\draw[arr] (cancel) -- (induct);
\end{tikzpicture}
\caption{Principal proof inputs. The full tuple law supplies reflection
positivity for the signed seminorm; the normalized nonexceptional law
supplies the strict-square pair law for cancellation. Both have the
conditional mixing used in conductor decay. The two lower branches are
combined only in the weighted recurrence. Arrows indicate dependencies,
not equivalences.}
\label{fig:proof-map}
\end{figure}
==== END A-power-saving-for-square-difference-free-sets//proof-map.tex ====

==== BEGIN A-power-saving-for-square-difference-free-sets//transfer.tex ====
\section{Comparison along square-step progressions}
\label{sec:scale-transfer}

We next compare the product-space functional with the same functional on
shorter intervals.  Fixing a few prime coordinates makes its low Fourier
coefficients exact averages on a residue class.  Splitting that class into
progressions with square step preserves the forbidden integer differences
and permits a change of scale.  The comparison proved here uses no induction
hypothesis. Restriction to square-step progressions is already a key
feature of quantitative square-difference arguments; see the Main Lemma
of~\cite{PSS1988} and~\cite[Lemma~6.2]{GreenSawhney2025}. Here it is
used to compare the signed multilinear functional itself.

Keep the constants \(\gamma=1/64\) and \(\beta=10^{-5}\) fixed as above.
Enlarge the fixed prime threshold \(P\), if necessary, so that the tuple
measures are available for every prime \(p\ge P\) and
\begin{equation}
 \prod_{\substack{p\ge P\\p\ {\rm prime}}}(1+p^{-3})-1\le\frac14,
 \qquad
 |\mathfrak g(\xi)|\le q(\xi)^{-1/3}
 \label{eq:prime-threshold-conditions}
\end{equation}
whenever \(q(\xi)\) is squarefree and all its prime factors are at least
\(P\).  Such a choice is possible: the product converges, while the prime
Gauss estimate used in Proposition~\ref{prop:square-kernel} gives
\[
 \left|\frac1{p-1}\sum_{x\in\F_p^\times}e(ax^2/p)\right|
 \le\frac{\sqrt p+1}{p-1}
 =\frac1{\sqrt p-1}\le p^{-1/3}
\]
for all sufficiently large \(p\) and all \(a\ne0\) in \(\F_p\).
Chinese remaindering then gives the second inequality in
\eqref{eq:prime-threshold-conditions}, including \(q(\xi)=1\).
Define the fixed integer
\begin{equation}
 M_0=8\prod_{\substack{3\le p0\), this lower bound is at least \(N^{1/2}\)
for sufficiently large \(N\).  As \(D\ge8\), we have \(N'H}\) according to the denominator on the
remaining prime coordinates \(\mathcal S\).
Before specialization, these pieces retain exactly the denominator
families
\[
 \{U\subseteq\mathcal P_N:q_U\le Q,\ q_{U\setminus B}\le H\},
 \qquad
 \{U\subseteq\mathcal P_N:q_U\le Q,\ q_{U\setminus B}>H\}.
\]
Thus Lemma~\ref{lem:lift-moments} applies to both pieces.
For a function \(u\) on \(X_B\times X_{\mathcal S}\), uniform counting
gives the specialization bound
\[
 \|u(z_B,\cdot)\|_r
 \le q_B^{1/r}\|u\|_r
 \qquad(r\ge1);
\]
this is also the estimate in Lemma~\ref{lem:lift-specialization}.
Since \(|f_s|\le M_0\), it follows that, for
\(g_*\in\{g,g_{\le H},g_{>H}\}\) and every fixed even \(r\),
\begin{equation}
 \|g_*\|_r\le M_0q_B^{1/r}N^{o(1)}.
 \label{eq:specialized-lift-moments}
\end{equation}
All subpower bounds in this proof are uniform over the indicated
sets and fibers.

Telescope the \(d\) inputs in the diagonal integral from \(g\) to
\(g_{\le H}\).  Each difference has one input \(g_{>H}\).
Lemma~\ref{lem:conductor-decay} and
\eqref{eq:specialized-lift-moments} imply, for every fixed
\(\varepsilon>0\),
\begin{equation}
 |Z_{\mathcal S}(g)-Z_{\mathcal S}(g_{\le H})|
 \le C_\varepsilon H^{-\gamma}q_BN^\varepsilon.
 \label{eq:scale-transfer-first-error}
\end{equation}
Indeed the specialization powers in that lemma add to
\[
 \frac12+\frac{d-1}{2(d-1)}=1.
\]
The factors \(d\) and \(M_0^d\) are fixed and are included in
\(C_\varepsilon\).  Applying the moment lemma with sufficiently small
individual exponents makes the aggregate moment loss at most
\(N^\varepsilon\).

\smallskip
\noindent\emph{Splitting the exact fiber into square-step progressions.}
Let \(a\bmod D\) combine \(s\) and \(z_B\).  By
Lemma~\ref{lem:exact-fiber}, for every remaining \(\xi\) with
\(q(\xi)\le H\),
\begin{equation}
 \widehat g(\xi)=\frac D N
   \sum_{\substack{n\le N\\n\equiv a\pmod D}}1_A(n)e(-n\xi).
 \label{eq:scale-transfer-fiber}
\end{equation}
Let
\[
 \begin{gathered}
 J_a=\{j\in[D^2]:j\equiv a\pmod D\},\qquad
 c_j=j-D^2,\\
 A'_j=\{n'\in[N']:D^2n'+c_j\in A\}.
 \end{gathered}
\]
There are exactly \(D\) elements of \(J_a\).
The map \((n',j)\mapsto D^2(n'-1)+j\) bijects
\([N']\times[D^2]\) with \([D^2N']\).
Restricting \(j\) to \(J_a\) selects exactly the residue class
\(a\bmod D\).  The definition of \(A'_j\) assigns zero to all padded
points greater than \(N\), so this decomposition incurs no additive
error.

Each \(A'_j\) is square-difference-free: a difference \(m^2>0\)
between its elements would give the difference
\(D^2m^2=(Dm)^2\) between the corresponding elements of \(A\).
This argument concerns actual integer squares.

For \(p\in\mathcal S\), the integer \(D\) is invertible modulo \(p\).
Define
\[
 T_j(z)_p=D^{-2}(z_p-c_j)\pmod p,\qquad
 h_j^H=\mathcal L_{[N'],\mathcal S,H}1_{A'_j}.
\]
Each \(T_j\) is a translation followed by multiplication by the
nonzero square \((D^{-1})^2\).
Multiplication of a remaining frequency by \(D^2\) preserves its
exact denominator, so changing variables \(\eta=D^2\xi\) in the
Fourier expansion of \eqref{eq:scale-transfer-fiber} gives the exact
identity
\begin{equation}
 g_{\le H}(z)
 =\frac{\lambda}{D}\sum_{j\in J_a}h_j^H(T_j(z)).
 \label{eq:scale-transfer-affine-average}
\end{equation}
For clarity, the coefficient of \(e(z\xi)\) on the right is
\[
 \frac{DN'}N\sum_{j\in J_a}e(-c_j\xi)
 \left(\frac1{N'}\sum_{n'\le N'}
       1_{A'_j}(n')e(-D^2n'\xi)\right),
\]
which is exactly \eqref{eq:scale-transfer-fiber}.

The seminorm triangle inequality and affine invariance in
Lemma~\ref{lem:functional-properties} now imply
\[
 Z_{\mathcal S}(g_{\le H})^{1/d}
 \le \frac{\lambda}{D}
        \sum_{j\in J_a}Z_{\mathcal S}(h_j^H)^{1/d}
 \le \lambda\max_{j\in J_a}Z_{\mathcal S}(h_j^H)^{1/d}.
\]
Consequently,
\begin{equation}
 Z_{\mathcal S}(g_{\le H})
 \le\lambda^d\max_{j\in J_a}Z_{\mathcal S}(h_j^H).
 \label{eq:scale-transfer-progressions}
\end{equation}
The averaging by \(D\) is essential: the number of progressions
causes no further loss.

\smallskip
\noindent\emph{Recovering the smaller-scale functional.}
Set
\[
 h_j^{Q'}=\mathcal L_{[N'],\mathcal S,Q'}1_{A'_j}.
\]
The difference \(h_j^{Q'}-h_j^H\) has conductor greater than \(H\).
By \eqref{eq:scale-transfer-cutoffs}, the moment lemma applies at
scale \(N'\) to these lifts and their pieces.  A second telescoping
application of Lemma~\ref{lem:conductor-decay}, this time without
specialization, gives
\begin{equation}
 |Z_{\mathcal S}(h_j^H)-Z_{\mathcal S}(h_j^{Q'})|
 \le C_\varepsilon H^{-\gamma}(N')^\varepsilon
 \label{eq:scale-transfer-second-error}
\end{equation}
uniformly in \(j\).

The lift \(h_j^{Q'}\) depends only on coordinates in
\(\mathcal S'=\mathcal P_{N'}\setminus B\), because every prime in a
denominator at most \(Q'\) is itself at most \(Q'\).
On the other hand, the full lift
\[
 u'_j=\mathcal L_{[N'],\mathcal P_{N'},Q'}1_{A'_j}
\]
becomes exactly \(h_j^{Q'}\) after uniform averaging over
\(B\cap\mathcal P_{N'}\).  Character orthogonality removes precisely
the denominators containing one of those primes.
By Lemma~\ref{lem:functional-properties}, coordinate averaging contracts
the seminorm; unused probability factors integrate to one.  Therefore
\begin{equation}
 Z_{\mathcal S}(h_j^{Q'})
 =Z_{\mathcal S'}(h_j^{Q'})
 \le Z_{\mathcal P_{N'}}(u'_j)
 =Y(N',A'_j).
 \label{eq:scale-transfer-projection}
\end{equation}
This also covers primes of \(B\) larger than \(Q'\), which are absent
from \(\mathcal P_{N'}\) already.

Combining \eqref{eq:scale-transfer-first-error},
\eqref{eq:scale-transfer-progressions},
\eqref{eq:scale-transfer-second-error}, and
\eqref{eq:scale-transfer-projection} proves
\eqref{eq:scale-transfer-diagonal} with error
\begin{equation}
 C_\varepsilon\left(
 H^{-\gamma}q_BN^\varepsilon+
 \lambda^dH^{-\gamma}(N')^\varepsilon\right).
 \label{eq:scale-transfer-explicit-error}
\end{equation}

\smallskip
\noindent\emph{Uniform error bounds and mixed inputs.}
The definition of \(N'\) gives
\(1\le\lambda<1+D^2/N\), and
\[
 D^2/N\le M_0^2N^{-1+2\tau}=o(1).
\]
Since \(d\) is fixed, these inequalities prove
\eqref{eq:scale-transfer-factor} and show that \(\lambda^d\)
is bounded for large \(N\).
Choose \(\varepsilon=\beta\gamma/8\) in
\eqref{eq:scale-transfer-explicit-error}.  Its two terms, before
the bounded factor \(\lambda^d\), are at most
\[
 N^{-3\beta\gamma/8+\tau}
 \quad\hbox{and}\quad N^{-3\beta\gamma/8},
\]
respectively.  As
\(\tau\le\beta\gamma/1000\) and
\(\sigma_0=\gamma\tau/4\le\tau/4\), they are
\(O(N^{-\sigma_0})\).  This establishes the diagonal bound with
constants independent of any inductive estimate for \(Y\).

Every \(g_v\) in the mixed statement is real and satisfies the same
diagonal bound.  Its right-hand side is nonnegative and independent
of \(v\).  Proposition~\ref{prop:signed-holder} therefore gives
\eqref{eq:scale-transfer-mixed}.
\end{proof}

In particular, the mixed estimate holds pointwise after specifying all
labels at primes in \(B\), regardless of the law used to choose those
labels.  It can therefore be integrated against any such probability
law.  When the prime laws form a product, the remaining coordinates
still carry the full laws used in Proposition~\ref{prop:scale-transfer}.
No reflection positivity is required of the law on the specified labels.
==== END A-power-saving-for-square-difference-free-sets//transfer.tex ====

==== BEGIN A-power-saving-for-square-difference-free-sets//tuples.tex ====
\section{Finite reflection-positive probability laws}
\label{sec:tuple-laws}

We construct probability laws on a fixed set of field labels.  Their
nonexceptional parts enforce a directed cycle of square differences.
The full laws also satisfy positivity across a family of reflection cuts.
To obtain both properties, we add a finite hierarchy of measures in which
successively more blocks of the index set are marked.  A marked block
provides a positive quadratic form that dominates the small negative part
at the corresponding unmarked cut.

\subsection{The index set and the required properties}

Fix the integers
\begin{equation}
 h=24,\qquad t_0=80,\qquad \ell=2t_0+1=161.
 \label{eq:tuple-fixed-parameters}
\end{equation}
Let \(\Pi\) be the set of permutation words on \([h]=\{1,\ldots,h\}\),
viewed as bijections from positions to symbols.  Write
\[
 K=h!,\qquad r=K/2,\qquad V=\Pi^\ell,\qquad d=|V|=K^\ell,
 \qquad n=d/2.
\]
The coordinates of a vertex \(v\in V\) are called its blocks.
Fix a cycle \(c\) of length \(h\) on the positions.  Composition on
the right, \(w\mapsto wc\), cyclically shifts a word.  Define
\(v_0,\ldots,v_{h-1}\in V\) by giving every block of \(v_j\) the
word \(c^j\), and put \(v_h=v_0\).

A reflection is specified by a block \(b\in[\ell]\) and two
distinct symbols \(\alpha,\beta\).  It acts on \(V\) by swapping
these symbols on the left in block \(b\); denote this involution by
\(\theta\).  Its associated cut is
\[
 V_+=\{v:\alpha\text{ precedes }\beta\text{ in block }b\},
 \qquad V_-=\theta V_+.
\]
The reflection has no fixed vertices, and both halves have size \(n\).

For a finite alphabet \(X\), a measure \(\nu\) on \(X^V\) defines
a matrix whose row index is \((z_v)_{v\in V_+}\) and whose column
index is \((z_{\theta v})_{v\in V_+}\).  We say that \(\nu\) is
\emph{reflection positive} if this matrix is symmetric positive
semidefinite for every cut above.  Equivalently, symmetry holds and
\[
 \int F((z_v)_{v\in V_+})F((z_{\theta v})_{v\in V_+})\,d\nu(z)
 \ge0
\]
for every real function \(F:X^{V_+}\to\mathbb R\).

Recall that \(Q_p^*=\{u^2:u\in\mathbb F_p^\times\}\) for an odd
prime \(p\).  The following proposition summarizes the construction.

\begin{proposition}[Tuple laws]
\label{prop:tuple-laws}
There is an absolute integer \(P\) such that, for every prime
\(p\ge P\), there are a probability measure \(\mu_p\) on
\(\mathbb F_p^V\) and a nonnegative submeasure \(\varepsilon_p\le\mu_p\)
whose mass satisfies
\[
 \eta_p=\varepsilon_p(\mathbb F_p^V)\le p^{-4}.
\]
Writing
\[
 \mu_p^\circ=\frac{\mu_p-\varepsilon_p}{1-\eta_p},
\]
the following properties hold.
\begin{enumerate}
\item The full law \(\mu_p\) is reflection positive.  Both
\(\mu_p\) and \(\varepsilon_p\) are invariant under simultaneous
changes \(z_v\mapsto qz_v+a\), where \(a\in\mathbb F_p\) and
\(q\in Q_p^*\).  Every single-label marginal of \(\mu_p\) and
\(\mu_p^\circ\) is uniform on \(\mathbb F_p\).
\item On the support of
\(\mu_p^\circ\),
\[
 z_{v_{j+1}}-z_{v_j}\in Q_p^*\qquad(0\le j\ell\),
which is impossible.  Each unordered pair of list entries therefore
supports at most one directed edge in \(\mathcal G_S\).
Lemma~\ref{lem:graph-count} applies to this graph and all its induced
subgraphs.

Set
\[
 \delta=\frac1{10h^2}.
\]
For distinct \(v,w\in V\), define a symmetric interaction
\(H^S_{vw}\) as follows.  It is zero unless \(v,w\) differ by a
symbol transposition in exactly one marked block.  In that case,
if the positions of the swapped symbols have gap \(g\), put
\(H^S_{vw}=\delta^g\).  Independently of the lists, choose all
indices \(j_v\in[m_s]\) uniformly and independently, and give their
joint distribution the unnormalized weight
\begin{equation}
 W_S(j)=\exp\left(
 \sum_{\{v,w\}\subseteq V}H^S_{vw}1_{\{j_v=j_w\}}\right).
 \label{eq:tuple-index-weight}
\end{equation}
Every sum over \(\{v,w\}\) is over unordered pairs of distinct
vertices, so each interaction is counted once.

Define the finite measure \(\nu_{p,S}\) by the identity
\begin{equation}
 \int G(z)\,d\nu_{p,S}(z)
 =\mathbb E_Y\mathbb E_j I_S(Y)W_S(j)
 G\bigl((Y_{u_S(v),j_v})_{v\in V}\bigr)
 \label{eq:tuple-component-law}
\end{equation}
for all functions \(G\) on \(\mathbb F_p^V\).  Both expectations
on the right are uniform probability averages.  In particular,
\(\nu_{p,S}\) has not been normalized to have mass one.
We also write \(\nu(1)\) for the total mass of a finite measure \(\nu\).

Each vertex has at most \(s\binom h2\) neighbors with nonzero
interaction, and each interaction is at most \(\delta\).  Thus,
with the fixed constant
\begin{equation}
 B=\exp\left(\frac{d(t_0+1)\binom h2\delta}{2}\right),
 \label{eq:tuple-weight-bound}
\end{equation}
we have
\begin{equation}
 1\le W_S(j)\le B,\qquad \nu_{p,S}(\mathbb F_p^V)\le B.
 \label{eq:tuple-component-mass-upper}
\end{equation}

For a cut with reflection \(\theta\), define the quadratic form
\begin{equation}
 Q_{p,S}^{b}(F)=
 \int F((z_v)_{v\in V_+})
 F((z_{\theta v})_{v\in V_+})\,d\nu_{p,S}(z).
 \label{eq:tuple-cut-form}
\end{equation}
The notation suppresses the two symbols that specify the cut.
Every component has a symmetric matrix at every cut, as we now verify.

\begin{lemma}[Symmetry of the component matrices]
\label{lem:tuple-component-symmetry}
For every allowed mark set \(S\) and every reflection \(\theta\),
the measure \(\nu_{p,S}\) is invariant under the permutation
\((z_v)\mapsto(z_{\theta v})\).  Its matrix at that cut is symmetric.
\end{lemma}

\begin{proof}
The array \(H^S\) is invariant under simultaneous application of
\(\theta\) to both endpoints.  In the reflected block, conjugating
a symbol transposition changes only the names of the swapped symbols,
not their positional gap.  Interactions in other blocks retain their
gap as well.  Hence relabeling indices by \(j_v\mapsto j_{\theta v}\)
preserves their uniform law and weight.

If the reflected block is marked, \(u_S(\theta v)=u_S(v)\), so
no relabeling of lists is needed.  If it is unmarked, \(\theta\)
acts on addresses.  Left symbol permutations commute with the right
position shift \(c\); therefore it preserves the directed address
relation \eqref{eq:tuple-address-edge}, including its orientation.
Relabeling lists by this address involution preserves their uniform
law and \(I_S\).  In either case these bijections in
\eqref{eq:tuple-component-law} send the output label at \(v\) to
the label at \(\theta v\), proving invariance.  At the cut this
invariance exchanges the two matrix indices, which is symmetry.
\end{proof}

\subsection{Strict positivity at a marked cut}

The next estimate is the source of domination between successive
mark sets.  It must remain valid when different index assignments
produce identical field labels.

\begin{lemma}[Marked-cut lower bound]
\label{lem:marked-positive}
Put
\[
 \kappa=\frac9{10}\delta^{h-1},\qquad
 \lambda=(e^\kappa-1)^n,\qquad
 c_* =\lambda m_{t_0+1}^{-n}>0.
\]
For every \(S\) with \(|S|\le t_0+1\), every cut in a marked block
\(b\in S\), and every real \(F:\mathbb F_p^{V_+}\to\mathbb R\),
\begin{equation}
 Q_{p,S}^{b}(F)
 \ge \lambda m_s^{-n}\,
 \mathbb E_{Y,j_+}I_S(Y)
 F((Y_{u_S(v),j_v})_{v\in V_+})^2.
 \label{eq:tuple-marked-lower}
\end{equation}
Here the expectation uses all uniform lists and only the uniform
indices on \(V_+\).  In particular, the coefficient in
\eqref{eq:tuple-marked-lower} is at least \(c_*\), independently
of the prime and of the field-list values.
\end{lemma}

\begin{proof}
Index an \(n\times n\) matrix by \(x,y\in V_+\) and set
\[
 \mathsf A_{xy}=H^S_{x,\theta y}.
\]
The invariance of \(H^S\) used in
Lemma~\ref{lem:tuple-component-symmetry}, together with its symmetry,
gives \(\mathsf A_{xy}=\mathsf A_{yx}\).

Suppose the two cut symbols occupy positions \(ij\), or move
the second from \(j\) to a position \(k0\), Lemma~\ref{lem:list-mixing}, with its operator
transposed if necessary, gives
\begin{equation}
 \|a-c_v\|_2\le\Delta_v,\qquad
 \|b_f\|_2\le\Delta_v\|f\|_2,
 \qquad
 \Delta_v=(32m_v)^{1/4}p^{-1/8}.
 \label{eq:conditional-numerator-bounds}
\end{equation}
The incident constraints form a cross-list graph between the single
variable \(x\) and the other labels.  Incoming and outgoing edges are
both allowed by that lemma.  The first inequality applies the centered
kernel to the constant function, while the second uses
\(\mathbb E f=0\).  All norms in
\eqref{eq:conditional-numerator-bounds} use the uniform base measures.

Put \(Z_0=\nu_{p,\varnothing}(1)\), and let \(T_0\) be the
conditional operator for the normalized leading component.  Its
other-label marginal is \(Z_0^{-1}L(y)a(y)\,dy\), and therefore
\begin{equation}
 \|T_0f\|_2^2
 =Z_0^{-1}\mathbb E_y L(y)\frac{|b_f(y)|^2}{a(y)}.
 \label{eq:conditional-leading-norm}
\end{equation}
The ratio is defined as zero where \(a(y)=0\), a set of zero marginal
mass.  On \(\{a\ge c_v/2\}\), its contribution to
\eqref{eq:conditional-leading-norm} is at most
\(2c_0^{-1}c_v^{-1}\Delta_v^2\|f\|_2^2\).
On the complementary set, Chebyshev's inequality gives
\[
 \mathbb P_y(a