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% Copyright 2026 Eric Naslund. SPDX-License-Identifier: Apache-2.0
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\title{Square-difference-free sets of exponent \texorpdfstring{$0.7580758$}{0.7580758}\\[3pt]\large Ordered intervals, composite digit codes, and a binary recursion}
\author{Eric Naslund\textsuperscript{*}\\[3pt]
\normalsize\href{mailto:naslund.math@gmail.com}{\texttt{naslund.math@gmail.com}}}
\date{October 6, 2026}
\begin{document}
\maketitle
\begin{abstract}
Let $D(N)$ be the largest cardinality of a subset of $\{1,\ldots,N\}$
containing no two elements whose difference is a nonzero perfect square. We
give a computer-assisted construction proving
$D(N)\ge N^{0.7580758318008816-o(1)}$, improving the exponent
$0.752796455874514\ldots$ of Krachun's ranked-block construction. The residues
we use carry subintervals of $[0,1]$ that are ordered along every modular
square difference. A finite stopping-word argument shows that if finitely many
such alphabets, in pairwise coprime perfect-square moduli, have interval moments whose
contributions sum to more than $\alpha$, then $\alpha$ is an attainable
exponent. Krachun's prime chains reproduce his exponent in this framework. The
gain comes from an alphabet modulo a power of $4$, built by a $25$-state
recursion, and from two composite alphabets, at the primes $5,43$ and at
$19,23$, in which the first differing digit is read separately at each prime. The
numerical conclusion is computer-assisted. The theorem, including every finite
certificate, is also formalized in Lean~4.
\end{abstract}
\vspace{1.5\baselineskip}
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\small\setlength{\parindent}{0pt}\vspace{2pt}%
\textsuperscript{*}\textbf{AI use:} This paper was written by GPT-6-Astra and revised by Claude Opus 5.5 under my prompting and supervision.
I have had the AI write a clearer introduction and outline of the key ideas, and produce a dependency graph which can be found on page~\pageref{fig:dependency}.
However, I recommend that readers explore the paper through a dialogue with AI, rather than a traditional reading.
Upload the paper to your preferred AI assistant and ask questions such as: ``What are the key ideas that improve on the previous best bound?'' or ``Where is the meat of the argument?''
In this manner, one can ask the paper questions, and understand the mathematics and the contribution through conversation, rather than directly reading the text.\vspace{2pt}
\end{minipage}}
\end{center}
\vfill
\noindent\rule{0.25\linewidth}{0.4pt}\par\smallskip
\noindent{\footnotesize Copyright \textcopyright\ 2026 Eric Naslund.
Licensed under the \href{https://www.apache.org/licenses/LICENSE-2.0}{Apache License, Version 2.0}.\par}
\clearpage
\setcounter{tocdepth}{1}
\begingroup
\small
\tableofcontents
\endgroup
\bigskip

\section{Introduction}\label{sec:intro}
\subsection{The problem and the quantitative landscape}
A set $A$ of integers is \emph{square-difference-free} if $a-a'$ is never a
nonzero perfect square for $a,a'\in A$. For a positive integer $N$, put
\[
 D(N)=\max\bigl\{\card A:A\subseteq\{1,\ldots,N\}\text{ is square-difference-free}\bigr\}.
\]
Furstenberg and S\'ark\"ozy independently proved that $D(N)=o(N)$
\cite{Furstenberg1977,Sarkozy1978}. The quantitative upper-bound problem has
subsequently developed through the work of Pintz, Steiger and Szemer\'edi,
Bloom and Maynard, and Green and Sawhney
\cite{PintzSteigerSzemeredi1988,BloomMaynard2022,GreenSawhney2025}.
In particular, Green and Sawhney prove a bound of the form
$D(N)\ll N\exp(-c\sqrt{\log N})$ for an absolute $c>0$
\cite[Theorem 1.1]{GreenSawhney2025}.
Our concern is the complementary construction problem. We call
$\alpha\ge0$ an \emph{attainable exponent} if for every $\varepsilon>0$ the
bound $D(N)\ge N^{\alpha-\varepsilon}$ holds for all sufficiently large $N$,
and we write $D(N)\ge N^{\alpha-o(1)}$ for this. The goal is to make $\alpha$
as large as possible. All logarithms in this paper are natural.

Ruzsa's construction gives exponent
\[
 \frac12+\frac{\log7}{2\log65}=0.733077\ldots
\]
by restricting alternating base-$65$ digits and leaving the others free
\cite{Ruzsa1984}. Beigel and Gasarch, and independently Lewko, use a
12-element square-difference-free residue set modulo $205$ to obtain
\[
 \frac12+\frac{\log12}{2\log205}=0.7334117970\ldots
 \qquad\cite{BeigelGasarch2008,Lewko2015}.
\]
For this construction family, a conjectural restriction on the size of
square-difference-free sets modulo square-free moduli gives a natural
three-quarter barrier; it is not an upper bound on $D(N)$
\cite{Ruzsa1984,Lewko2015}.
Further bounds for square-avoiding sets modulo square-free integers are
studied by Gabdullin \cite{Gabdullin2018}. These address local square
avoidance, whereas the local objects here permit directed square differences.
Computational exploration by Georgiev, G\'omez-Serrano, Tao and Wagner
recovered the modulus-$205$ construction without improving its exponent
\cite[Section 16]{GeorgievEtAl2025}; this records a search outcome, not
an optimality theorem.

In the function-field analogue, Naslund \cite[Conjecture~13]{Naslund2022}
conjectured that square-difference-free sets of polynomials of degree below
$n$ over $\mathbb F_q$ have size at most $q^{3n/4}$, for odd $q$ and $4\mid n$.
Jones \cite{JonesPalomar2026} disproves this at $q=3$ for every $n\ge8$
divisible by $4$, with a Lean-verified construction registered in Palomar.

Krachun crossed the integer three-quarter barrier by allowing modular square
differences, provided that a rank orients them consistently, and then
combining the ranks over coprime moduli \cite{Krachun2026}. His exponent is
\begin{equation}\label{eq:krachun}
 \alpha_{\mathrm K}=
 \frac{10+\sum_{i=1}^{10}\log p_i/\log t_i}
      {1+2\sum_{i=1}^{10}\log p_i/\log t_i}
 =0.752796455874514\ldots,
\end{equation}
where
\[
 (p_i,t_i)=(3,2),(7,3),(11,4),(19,5),(23,5),
 (31,7),(43,7),(59,9),(71,9),(103,11).
\]
Here $t_i$ is the length of a \emph{Paley chain} modulo the prime $p_i$: a
list of residues in which every later entry minus every earlier one is a
nonzero square modulo $p_i$ (Definition~\ref{def:chain}). Bounds for such
chains explain some limitations of this particular family
\cite{Satake2021,Krachun2026}; those limitations do not apply to the alphabets
of this paper.

\subsection{The main result}
\begin{theorem}\label{thm:main}
Let $\al=473797394875551/625000000000000=0.7580758318008816$. For every
$\varepsilon>0$ there is $N_0(\varepsilon)$ such that, for every integer
$N\ge N_0(\varepsilon)$, there exists a square-difference-free
$A\subseteq\{1,\ldots,N\}$ with
\[
 \card A\ge N^{\al-\varepsilon}.
\]
In fact $D(N)\ge N^{\al}/Q$ for every $N\ge1$, where $Q$ is a fixed
(astronomically large) constant.
\end{theorem}

Relative to \eqref{eq:krachun}, the increase in the exponent is approximately
$0.0052793759$. The comparison concerns Krachun's construction just
described, not an upper bound on the true optimum.

\subsection{From Ruzsa to Krachun to ordered intervals}\label{subsec:overview}
All three constructions start from residues modulo a perfect square and build
integers from them digit by digit. They differ in how they treat square
differences between the residues.

\emph{Free digits (Ruzsa).} Let $m$ be square-free and let $S$ be a set of
residues modulo $m$, no two of which differ by a square modulo $m$. Ruzsa
takes the integers whose base-$m$ digits lie in $S$ at even positions and are
arbitrary at odd positions. Equivalently, the $m\card S$ residues $s+mu$
($s\in S$, $0\le u<m$) modulo $m^2$ have no square difference at all, and
integers are formed from them in base $m^2$. The \emph{free digits} $u$ cost
nothing, which is the source of the term $1/2$ in the exponent
$\frac12+\log\card S/(2\log m)$ (Remark~\ref{rem:ruzsa}). This exceeds $3/4$
only if $\card S>m^{1/2}$, which the conjectural restriction mentioned above
would rule out.

\emph{Ranks (Krachun).} Krachun keeps the free digits but allows the
restricted digits to have square differences, provided these can be oriented
by a rank. A \emph{ranked alphabet} modulo a perfect square $b$
(Definition~\ref{def:ranked}) is a set of $q$ residues with a rank $r$ taking
$h$ values, its \emph{height}, such that $r(y)>r(x)$ whenever $y-x$ is a
nonzero square modulo $b$. Choosing integer representatives that decrease as the rank increases
removes every square difference, at the price of a factor $h$ in the length of
the ambient interval (Lemma~\ref{lem:reverse}). Ranks of alphabets in coprime
moduli add rather than multiply (Lemma~\ref{lem:crt}), so each alphabet can be
assessed on its own: if one takes words of length $k_i$ over the $i$th
alphabet with $h_i^{k_i}$ roughly equal to a common $H$, the glued alphabet has
height about $nH$, while its size and modulus are the products of the
$q_i^{k_i}$ and the $b_i^{k_i}$. Letting $H\to\infty$ in this bookkeeping, an
alphabet with $h\ge2$ has \emph{contribution}
\begin{equation}\label{eq:rankcontribution}
 \varphi=\frac{\log q-\alpha\log b}{\log h}
\end{equation}
at a target exponent $\alpha$, and $\alpha$ is attainable when the
contributions of the alphabets sum to more than $\alpha$
(Remark~\ref{rem:ranked} and Theorem~\ref{thm:criterion}).
Krachun's alphabets are Paley chains: for a prime $p\equiv3\pmod4$ and a chain
$s_0,\ldots,s_{t-1}$ modulo $p$, the residues $s_j+pu$ modulo $p^2$, with
rank $j$, give $q=pt$, $b=p^2$ and $h=t$ (Lemma~\ref{lem:chain}). Their
contributions are $1-(2\alpha-1)\log p/\log t$, and the condition that the
ten contributions sum to more than $\alpha$ is exactly $\alpha<\alpha_{\mathrm
K}$ (Remark~\ref{rem:krachun}).

\emph{Ordered intervals (this paper).} We replace the rank of a residue $x$ by
an interval $I_x=[a_x,a_x+w_x]\subseteq[0,1]$ and require that $I_x$ lie
entirely before $I_y$ whenever $y-x$ is a nonzero square
(Definition~\ref{def:interval}). A rank of height $h$ is the special case of
$h$ consecutive intervals of equal length. What governs an alphabet is now its
\emph{moment} $Z_f=\sum_xw_x^f$: the alphabet has contribution $f$ at exponent
$\alpha$ if $Z_f\ge b^\alpha$, and the \emph{interval-moment criterion}
(Theorem~\ref{thm:criterion}) shows that $\alpha$ is attainable whenever such
contributions sum to more than $\alpha$. Its proof reduces to Krachun's
mechanism: it selects words of digits whose intervals all have nearly the same
width, rounds the intervals to ranks, and glues the components. For equal
widths the criterion returns \eqref{eq:rankcontribution} exactly, so intervals
change nothing for Paley chains. They pay off for alphabets whose square
differences are far from forming a chain: residues with no square difference
between them may share most of an interval, and only residues on long directed
paths of square differences need narrow intervals. Intervals are what make the
binary component below possible, and they improve the composite codes
slightly. The composite codes, on the other hand, would already help within
Krachun's framework.

\emph{Composite codes at two primes.} Instead of one prime at a time, we
restrict the digits at two primes jointly (Section~\ref{subsec:composite}). A
point of such a code is a pair of words of base-$p$ digits, one for each
prime, and two points are related only if, \emph{at each prime separately},
their words agree or differ first (that is, at the least significant
differing digit) by a nonzero square (Definition~\ref{def:lowcode}). Lemma~\ref{lem:odd} adds free digits and
yields an alphabet modulo the square of $F=\prod_pp^{e_p}$, where $e_p$ is the
number of restricted digits at $p$. The relation at one prime can thus be
oriented by the other. At $(5,43)$, the prime $5\equiv1\pmod4$, useless for
chains because its square relation is symmetric, separates points without
ordering them, as in Ruzsa's construction, while $43$ orients the remaining
relations, as in Krachun's (Remark~\ref{rem:composite}). Our $(5,43)$ code is
built from a $17$-point code with one digit at each prime, whose longest
directed path has $11$ points; even as a ranked alphabet it contributes
$0.0255$ at $\al$, against $0.0023$ for Krachun's chain modulo $43$. At
$(19,23)$ the corresponding code has $27$ points, two more than a product of
maximal chains, and contributes $0.0597$ as a ranked alphabet, against
$0.0557$ for the chain modulo $19$ (the chain modulo $23$ contributes nothing
at $\al$). Our $(19,23)$ code takes three digits at $19$ and two at $23$.

\emph{The prime $2$.} Modulo $4^m$, a square difference is either divisible by
$4$, in which case the factor $4$ can be cancelled, or odd and then congruent
to $1$ modulo $8$ (Lemma~\ref{lem:dyadic}). Alphabets modulo $4^m$ can
therefore be built recursively, one base-$4$ digit at a time. A \emph{state}
of the recursion records two subintervals of $[0,1]$, its \emph{windows},
which hold the intervals of the even and of the odd residues
(Section~\ref{sec:binary}). Each state is built from up to four child states,
its \emph{branches}, placed in its windows by affine maps of different
scales, so the widths are unequal by construction. A policy with $25$ states
and $94$ branches makes the moment grow geometrically with $m$. The earlier
constructions use only odd moduli; here the prime $2$ contributes $0.155$, the
largest gain over Krachun's construction.

\begin{table}[!ht]
\centering\small
\begin{tabular}{lcc}\toprule
& \multicolumn{2}{c}{Contribution at $\alpha=\al$}\\\cmidrule(l){2-3}
Alphabet & with ranks \eqref{eq:rankcontribution} & with intervals\\\midrule
Paley chains at $3,7,11,31,59,103$ & $0.50854$ & $0.50854$\\
Paley chains at $19$ and $43$ & $0.05805$ & not used\\
Composite code at $(5,43)$ & $0.02550$ & $0.02650$\\
Composite code at $(19,23)$ & $0.06201$ & $0.06812$\\
Binary recursion modulo $4^{10^{20}}$ & --- & $0.15492$\\\midrule
Total of Krachun's chains (positive terms) & $0.56659$ & \\
Total of the components of this paper & & $0.75808$\\\bottomrule
\end{tabular}
\caption{Where the improvement comes from. An exponent $\alpha$ is attainable
when the contributions of the alphabets used sum to more than $\alpha$; all
entries are evaluated at the exponent $\al$ of Theorem~\ref{thm:main}.
The eight of Krachun's chains with positive contributions supply only $0.5666$
(his chains at $23$ and $71$ contribute $-0.0056$ and $-0.0013$ there; all ten
give $0.5597$). We replace the chains at $19$ and $43$ by the two composite
codes, gaining $0.0366$, and add the binary component, gaining $0.1549$. In the
first column a composite code is ranked by the number of points on the longest
directed path of square differences ending at each point; these paths have
$1331$ and $686$ points. The second column uses the certified unequal widths of
Section~\ref{sec:parameters}. The binary recursion has unequal widths by
construction, and we know no ranked counterpart. The first-column values for
the composite codes are illustrations, not used in the proof.}\label{tab:budget}
\end{table}

Table~\ref{tab:budget} shows where the improvement comes from. At our exponent
$\al$, Krachun's chains fall short by about $0.19$. The binary component
supplies most of the difference, and the composite codes the rest. Of the
composite codes' gain of $0.0366$, $0.0295$ is available with ranks; unequal
widths add $0.0010$ and $0.0061$. The nine alphabets of
Section~\ref{sec:parameters} have contributions, exact rationals, that sum to
$\al+15468032462\cdot10^{-25}$. The exponent $\al$ is the $16$-decimal
truncation of the largest exponent these alphabets can certify when the depth
of the binary recursion may grow (Definition~\ref{def:admissible} and
Section~\ref{subsec:closing}).

\FloatBarrier

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\node[iv,thm,text width=2.6cm] (main) at (10.3,-11.80) {Thm.~\ref{thm:main}\\ $D(N)\ge N^{\al-o(1)}$};
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\draw[e] (logenc.east) -- (wch.west);
\draw[e] (wch) -- (pch);
\draw[e] (w215) -- (p215);
\draw[e] (w437) -- (p437);
\draw[e] (wbin) -- (pbin);
\draw[e] (wch.south west) -- (wbud.north east);
\draw[e] (crit.south) |- (main.west);
\draw[e] (wbud.south) -- (main.north west);
\draw[e] (pch.south) -- (main.north);
\draw[e] (p215.south) -- (main.north);
\draw[e] (p437.south) -- (main.north east);
\draw[e] (pbin.south) -- (main.north east);
\end{tikzpicture}
\caption{Dependencies between the numbered results. Each framed column is one
section, read from top to bottom; an arrow $X\to Y$ means that $Y$ uses $X$,
and arrows implied by a path of drawn arrows are omitted. Thick borders mark
the new ingredients; the remaining statements restate known arguments or
elementary facts, with proofs. Dashed nodes are certificates, established by
exact computation (Section~\ref{sec:parameters}). All four component
certificates use the rational logarithm bounds of Lemma~\ref{lem:logenc}, and
the budget certificate uses all four; to keep the figure legible only one arrow
of each kind is drawn. The numbers in the component propositions are their
contributions (Table~\ref{tab:powers}), which sum to more than $\al$.
Lemma~\ref{lem:crtsq} is also used in Lemma~\ref{lem:odd}. Definitions,
remarks and the one-line Lemmas~\ref{lem:sqdiv} and \ref{lem:finitewidth} are
not shown. Abbreviations: Lem.\ lemma, Cor.\ corollary, Prop.\ proposition,
Thm.\ theorem, Cert.\ certificate.}\label{fig:dependency}
\end{figure}

\subsection{Organization and dependency graph}\label{subsec:roadmap}
The proof of Theorem~\ref{thm:main} has four steps.
\begin{itemize}
\item Section~\ref{sec:ranked} develops the mechanism of Ruzsa and Krachun:
ranked alphabets, their integer realization, the gluing of coprime
components, and amplification to every $N$. Nothing in it is new in
substance; it fixes notation and proofs.
\item Section~\ref{sec:criterion} introduces ordered interval alphabets and
proves the interval-moment criterion. This is the main new general tool.
\item Sections~\ref{sec:odd} and \ref{sec:binary} build alphabets at the odd
primes and at the prime $2$, and reduce the criterion's hypotheses for them to
finitely many checkable inequalities.
\item Section~\ref{sec:parameters} supplies nine explicit alphabets and checks
those inequalities. These certificates are the only statements established by
computation; Section~\ref{sec:verification} explains how they are checked.
\end{itemize}
Figure~\ref{fig:dependency} shows how the numbered results depend on one
another. Its columns are Sections~\ref{sec:ranked}--\ref{sec:binary} and its
bottom band is Section~\ref{sec:parameters}, each read from top to bottom; an
arrow $X\to Y$ means that the statement or proof of $Y$ uses $X$. Thick borders mark
the new ingredients. A reader who wants the method should follow the two left
columns down to Theorem~\ref{thm:criterion} (about five pages). A reader who
wants the new alphabets should read Sections~\ref{subsec:composite} and
\ref{sec:binary}, where Lemmas~\ref{lem:odd}, \ref{lem:binary} and
\ref{lem:growth} carry the weight. A reader who wants to check the numbers
should read the bottom band, Section~\ref{sec:parameters}, and
Appendix~\ref{app:reproduction}. Definitions and remarks are not shown in the
figure; each definition is stated where its term first occurs, and
Appendix~\ref{app:index} lists every numbered statement with the statements it
uses.


\subsection{Scope, computer assistance and formalization}\label{subsec:scope}
Theorem~\ref{thm:main} is a computer-assisted theorem. Its analytic and
combinatorial implications are proved in the text; the specific finite
rational inequalities collected in the certificates of
Section~\ref{sec:parameters} are established by the supplied executable
programs. No optimization output, floating-point eigenvalue, or claim that a
search was exhaustive is an assumption of the proof.

The \LaTeX\ source contains a complete Python attachment, including all
witnesses and checking programs. Compiling that one source file writes the
attachment and embeds it in the PDF. Section~\ref{sec:verification}
specifies its data and acceptance conditions, and
Appendix~\ref{app:reproduction} gives extraction and execution instructions.
The large integer arrays are distributed as an attachment, not as thousands of
typeset rows. Standard Python 3 and exact integer and rational arithmetic
suffice.

A separate Lean~4 development proves Theorem~\ref{thm:main} itself
\cite{deMouraUllrich2021,Mathlib2020}. It proves Theorem~\ref{thm:criterion},
the lifting and recursion lemmas, and every finite condition of the
certificates in Section~\ref{sec:parameters}, which Lean's kernel checks with
its own exact certificates; its proofs use only Lean's standard axioms. It
bypasses Lemma~\ref{lem:paths} by recording interval starts explicitly, and
Lemma~\ref{lem:minusone} by checking the orientation of every chain pair. The
development is publicly available at
\url{https://github.com/enaslund/sarkozy-lower-bound-0.758}. The English argument here is
complete on its own and uses no Lean theorem as a premise.

\section{Ranked alphabets and their integer realization}\label{sec:ranked}
This section develops the mechanism behind the constructions of Ruzsa and
Krachun. The integer realization and the gluing of ranks over coprime moduli
are Krachun's Lemmas~4 and~5 \cite{Krachun2026}; we use the opposite rank
convention and include short proofs.

\begin{definition}[Square residues and square differences]\label{def:sqrel}
For an integer $b\ge1$ let $\QR(b)=\{z^2\bmod b:z\in\Z\}$ be the set of
square residues modulo $b$, zero included. Residues are represented
canonically in $\{0,\ldots,b-1\}$. For distinct residues $x,y$ we write
$x\to_b y$, and say that $y-x$ is a \emph{square difference} modulo $b$, if
$y-x\in\QR(b)$.
\end{definition}
The relation is directed: it may hold in one direction, in both, or in
neither.
For composite $b$, a nonzero square residue may vanish modulo some prime
factor of $b$; invertibility is never assumed.

\begin{lemma}[Squares under the Chinese remainder theorem]\label{lem:crtsq}
Let $M_1,\ldots,M_n$ be pairwise coprime with product $M$. Reduction
$x\mapsto(x\bmod M_i)_i$ is a bijection from residues modulo $M$ to tuples of
residues, and if $y-x\in\QR(M)$ then $y_i-x_i\in\QR(M_i)$ for every $i$.
\end{lemma}
\begin{proof}
The first statement is the Chinese remainder theorem. If $y-x\equiv z^2
\pmod M$, reduce this congruence modulo each $M_i$. Some coordinates may
satisfy $y_i=x_i$.
\end{proof}

\begin{definition}[Ranked alphabet]\label{def:ranked}
A \emph{ranked alphabet} modulo $b$ is a finite set $C$ of residues modulo $b$
together with a function $r:C\to\{0,\ldots,h-1\}$, its \emph{rank}, such that
$x\to_b y$ implies $r(x)<r(y)$ for all $x,y\in C$. The integer $h\ge1$ is its
\emph{height}.
\end{definition}
Residues without square differences between them may share a rank. In
particular a ranked alphabet of height one has no square differences at all.

\begin{lemma}[Reverse-rank realization]\label{lem:reverse}
A ranked alphabet $C$ of size $q$ modulo $M$ with height $h$ gives the
square-difference-free set $\{T(x)=1+x+M(h-1-r(x)):x\in C\}$ of size $q$ in
$\{1,\ldots,Mh\}$.
\end{lemma}
\begin{proof}
The numbers $T(x)$ lie in the stated range and are distinct modulo $M$.
If $T(y)-T(x)=z^2>0$, then $x\ne y$ and reduction modulo $M$ gives
$x\to_My$, so $r(y)\ge r(x)+1$. But then
$T(y)-T(x)=(y-x)-M(r(y)-r(x))\le(M-1)-M<0$.
\end{proof}
This is the only place where integers, rather than residues, appear.

\begin{lemma}[Gluing coprime components]\label{lem:crt}
Let $C_i$ be ranked alphabets modulo pairwise coprime $M_i$, with $q_i$
elements and heights $h_i$ ($1\le i\le n$). The residues modulo
$M=\prod_iM_i$ whose reductions lie in the $C_i$ form a ranked alphabet with
$\prod_iq_i$ elements, rank $r=\sum_ir_i$, and height $1+\sum_i(h_i-1)$.
\end{lemma}
\begin{proof}
By Lemma~\ref{lem:crtsq} there are $\prod_iq_i$ such residues, and a square
difference modulo $M$ is a square difference or an equality in every
coordinate. Equal coordinates keep their rank and unequal ones raise it; at
least one coordinate is unequal. The rank range follows.
\end{proof}
Heights add while sizes multiply; this is what lets each component be
assessed separately.

\begin{definition}[Square base and words]\label{def:words}
A \emph{square base} is an integer $b=s^2>1$. A \emph{word} of length $k$ is a
sequence $(x_0,\ldots,x_{k-1})$ of residues modulo $b$; it \emph{encodes} the
residue $X=\sum_{j<k}x_jb^j$ modulo $b^k$, so the first letter is the least
significant digit.
\end{definition}
Encoding is injective, by uniqueness of base-$b$ expansions.

\begin{lemma}\label{lem:sqdiv}
If $s\ge1$ and $z\in\Z$ satisfy $s^2\mid z^2$, then $s\mid z$.
\end{lemma}
\begin{proof}
For every prime $\ell$, $2v_\ell(s)\le2v_\ell(z)$, where $v_\ell$ denotes
the $\ell$-adic valuation.
\end{proof}

\begin{lemma}[First differing letter]\label{lem:firstdiff}
Let $b=s^2>1$, and let distinct words $(x_j)$ and $(y_j)$ of length $k$
encode $X$ and $Y$. If $Y-X\in\QR(b^k)$ and $j$ is the first position with
$x_j\ne y_j$, then $x_j\to_by_j$.
\end{lemma}
\begin{proof}
Let $Y-X\equiv z^2\pmod{b^k}$. The integer $Y-X$ is divisible by $b^j$, and
$j<k$, so $s^{2j}=b^j$ divides $z^2$ and $s^j\mid z$ by
Lemma~\ref{lem:sqdiv}. Dividing the congruence by $b^j$ and reducing modulo
$b$ gives $y_j-x_j\equiv(z/s^j)^2\pmod b$.
\end{proof}
Since $b$ is a square, a square difference of words is decided by the first
letter where they differ. This is why all bases in this paper are squares.

\begin{lemma}[Ranked words]\label{lem:words}
If $C$ is a ranked alphabet modulo a square base $b$, of size $q$ and height
$h$, then its words of length $k$ form a ranked alphabet modulo $b^k$ of size
$q^k$ and height $h^k$, ranked by $R(X)=\sum_jr(x_j)h^{k-1-j}$.
\end{lemma}
\begin{proof}
Let $X\to_{b^k}Y$, with first differing position $j$. By
Lemma~\ref{lem:firstdiff}, $r(y_j)\ge r(x_j)+1$, which raises $R$ by at least
$h^{k-1-j}$; the later positions lower it by at most
$\sum_{i>j}(h-1)h^{k-1-i}<h^{k-1-j}$.
\end{proof}

\begin{proposition}[Amplification]\label{prop:amplify}
Let $\alpha\ge0$. If a ranked alphabet of size $Q$ modulo a square base $M$
has height $h$ and $(Mh)^\alpha\le Q$, then $D(N)\ge N^\alpha/Q$ for every
$N\ge1$.
\end{proposition}
\begin{proof}
Put $L=Mh>1$. By Lemma~\ref{lem:words}, words of length $t$ form a ranked
alphabet of size $Q^t$ modulo $M^t$ with height $h^t$, and
Lemma~\ref{lem:reverse} turns it into a square-difference-free set of size
$Q^t$ in $\{1,\ldots,L^t\}$ (for $t=0$ this is $\{1\}$). Each $N\ge1$ has
$t\ge0$ with $L^t\le N<L^{t+1}$; then $N^\alpha\le L^{(t+1)\alpha}\le Q^{t+1}$,
so $D(N)\ge Q^t\ge N^\alpha/Q$.
\end{proof}

\begin{remark}[Ruzsa's construction]\label{rem:ruzsa}
Let $m$ be square-free and let $S$ be a set of residues modulo $m$ with
$s'-s\notin\QR(m)$ for distinct $s,s'\in S$. The $m\card S$ residues $s+mu$
($s\in S$, $0\le u<m$) modulo $m^2$ have no square differences. Indeed, if two
of them differ by a square modulo $m^2$, their low digits agree, by reduction
modulo $m$; the difference is then $m(u'-u)$ with $0<\card{u'-u}<m$, and a
prime $p\mid m$ with $p\nmid u'-u$ exists because $m$ is square-free. Modulo
$p^2$ that difference has $p$-adic valuation exactly one, which no square
residue has. So these residues form a ranked alphabet of height one, and
Proposition~\ref{prop:amplify}, with $h=1$, gives the exponent
$\log(m\card S)/\log m^2=\frac12+\log\card S/(2\log m)$. Its words are exactly
Ruzsa's integers with alternating restricted and free base-$m$ digits.
\end{remark}

\section{Ordered interval alphabets and the moment criterion}\label{sec:criterion}
This section introduces the main new general tool. Ranks are replaced by
intervals, and the cardinality and height of an alphabet by a single moment.

\begin{definition}[Ordered interval alphabet, moment, test]\label{def:interval}
An \emph{ordered interval alphabet} modulo $b$ is a finite set $C$ of residues
modulo $b$, each $x\in C$ carrying an interval $I_x=[a_x,a_x+w_x]\subseteq[0,1]$
of \emph{width} $w_x>0$, such that $x\to_b y$ implies $a_x+w_x\le a_y$ for all
$x,y\in C$. For $f\ge0$ its \emph{moment} is
\[
 Z_f(C)=\sum_{x\in C}w_x^f .
\]
If $b$ is a square base and $Z_f(C)\ge b^\alpha$, we say that $C$
\emph{passes the test at $\alpha$ with contribution $f$}.
\end{definition}
Along a cycle of square differences the positive widths would have to sum to
at most zero, so such an alphabet has no cycles. Residues without a square
difference between them may have overlapping intervals.

\begin{definition}[Admissible contributions and exact crossing]\label{def:admissible}
Let $C$ be an ordered interval alphabet in a square base $b$ with all widths
below one. A power $f\ge0$ is \emph{admissible at $\alpha$} if $C$ passes the
test at $\alpha$ with contribution $f$. Since $Z_f$ is strictly decreasing in
$f$, the admissible powers form an interval $[0,f^*(\alpha)]$ when
$Z_0(C)=\card C\ge b^\alpha$, and no power is admissible otherwise;
$f^*(\alpha)$ is the \emph{largest admissible contribution}. For finitely many
fixed alphabets, $\alpha\mapsto\sum_if_i^*(\alpha)-\alpha$ is strictly
decreasing; its zero $\alpha^*$ is their \emph{exact crossing}, and every
$\alpha<\alpha^*$ admits admissible contributions summing to more than $\alpha$.
\end{definition}

\begin{remark}[Ranks are equal widths]\label{rem:ranked}
A ranked alphabet of size $q$ and height $h\ge2$ in a square base $b$ becomes
an ordered interval alphabet with $I_x=[r(x)/h,(r(x)+1)/h]$. All its widths
are $1/h$, so $Z_f=qh^{-f}$, and it passes the test at $\alpha$ with
contribution $f$ exactly when $f\le\varphi$, the quantity of
\eqref{eq:rankcontribution}. Since equal widths are one possible choice of
intervals, optimizing the widths can only increase the largest admissible
contribution (Definition~\ref{def:admissible}).
\end{remark}


\begin{theorem}[Interval-moment criterion]\label{thm:criterion}
Let $C_1,\ldots,C_n$ $(n\ge1)$ be ordered interval alphabets in pairwise
coprime square bases $b_i$, with all widths in $[\sigma,\rho]$ for some
$0<\sigma\le\rho<1$. If $\alpha\ge0$, $f_i\ge0$,
\[
 Z_{f_i}(C_i)\ge b_i^\alpha\quad(1\le i\le n)
 \qquad\text{and}\qquad \Phi:=\sum_{i=1}^nf_i>\alpha,
\]
then $\alpha$ is attainable. More precisely, $D(N)\ge N^\alpha/Q$ for all
$N\ge1$ and a constant $Q$ depending only on the alphabets.
\end{theorem}
The proof occupies the rest of this section. It selects one word length in
each component (Lemma~\ref{lem:stopping}), turns the selected words into one
finite ranked alphabet (Proposition~\ref{prop:block}), and amplifies that
alphabet (Proposition~\ref{prop:amplify}).

\begin{lemma}[Rounding intervals to ranks]\label{lem:round}
If every interval of an ordered interval alphabet has width at least
$\gamma>0$, then $r(x)=\lfloor a_x/\gamma\rfloor$ is a rank of height at most
$\lceil1/\gamma\rceil$.
\end{lemma}
\begin{proof}
If $x\to_by$, then $a_y\ge a_x+w_x\ge a_x+\gamma$, so $r(y)\ge r(x)+1$.
Moreover $0\le a_x\le1-\gamma$.
\end{proof}

\begin{definition}[Word intervals]\label{def:wordint}
Let $C$ be an ordered interval alphabet modulo a square base $b$. For $x\in C$
let $F_x(t)=a_x+w_xt$, the increasing affine map from $[0,1]$ onto $I_x$. The
word $(x_0,\ldots,x_{k-1})$ encoding $X$ receives the interval
$I_X=F_{x_0}\circ\cdots\circ F_{x_{k-1}}([0,1])$, of width
$W_X=\prod_jw_{x_j}$.
\end{definition}
Thus the first letter chooses a subinterval of $[0,1]$, the second a
subinterval of that, and so on.

\begin{lemma}[Interval words]\label{lem:intwords}
With the word intervals of Definition~\ref{def:wordint}, the words of length
$k$ over $C$ form an ordered interval alphabet modulo $b^k$, with moment
$Z_f(C)^k$.
\end{lemma}
\begin{proof}
Let $X\to_{b^k}Y$, with first differing position $j$. By
Lemma~\ref{lem:firstdiff}, $x_j\to_by_j$, so $I_{x_j}$ lies before $I_{y_j}$.
The words' intervals are the images of subintervals of $I_{x_j}$ and
$I_{y_j}$ under the common increasing map $F_{x_0}\circ\cdots\circ F_{x_{j-1}}$,
which preserves their order. The moment factors because widths multiply.
\end{proof}

\begin{lemma}[Stopping and selecting a length]\label{lem:stopping}
Fix an ordered interval alphabet in a square base $b$ with widths in
$[\sigma,\rho]$, $\rho<1$, a power $f\ge0$, and $q=b^\alpha$, and suppose
$Z=Z_f\ge q$. For every integer $K\ge1$ and $\delta=\rho^K$ there are
$1\le k\le K$ and a set $D$ of words of length $k$ with
\[
 \sigma\delta<W_X\le\delta\ \ (X\in D),\qquad
 q^k\le(K+1)\card D\,\delta^f .
\]
\end{lemma}
\begin{proof}
Grow words from the empty word, of width one. A child word stays
\emph{active} while its width exceeds $\delta$ and otherwise becomes
\emph{terminal}; only active words are extended. Every word of length $K$ has
width at most $\rho^K=\delta$, so nothing is active at length $K$. A terminal
word's parent was wider than $\delta$, so its own width exceeds $\sigma\delta$.
Let $A_j$ be the sum of $W^f$ over active words of length $j$, and $T_j$ the
sum over terminal words of length $j+1$. Multiplicativity of widths gives
\[
 A_0=1,\qquad A_K=0,\qquad A_{j+1}+T_j=ZA_j\quad(0\le j<K).
\]
If $T_j<q^{j+1}/(K+1)$ for every $j<K$, induction with $Z\ge q$ gives
$A_j\ge q^j(K+1-j)/(K+1)$, contradicting $A_K=0$. Choose $j$ with
$T_j\ge q^{j+1}/(K+1)$, let $k=j+1$, and let $D$ be the terminal words of
length $k$. Since $W\le\delta$ and $f\ge0$, $T_j\le\card D\,\delta^f$.
\end{proof}
The selected words have widths within a factor $\sigma$ of each other, so
little is lost when their intervals are rounded to ranks.

\begin{lemma}[Common width bounds]\label{lem:finitewidth}
A finite family of ordered interval alphabets whose widths all lie in $(0,1)$
admits common bounds $0<\sigma\le w\le\rho<1$.
\end{lemma}
\begin{proof}
Take the minimum and maximum of the finitely many widths together with $1/2$.
\end{proof}
No numerical value of $\sigma$ is needed, even for the very large binary
alphabet of Section~\ref{sec:binary}.

\begin{proposition}[One finite ranked block]\label{prop:block}
Under the hypotheses of Theorem~\ref{thm:criterion} there is a ranked
alphabet with $Q$ elements modulo a square base $M$, of height $h$, such that
$(Mh)^\alpha\le Q$.
\end{proposition}
\begin{proof}
Fix $K$ (chosen below) and put $\delta=\rho^K$. Apply
Lemma~\ref{lem:stopping} in each component with this $K$, obtaining lengths
$k_i$ and word sets $D_i$ with widths in $(\sigma\delta,\delta]$ and
$b_i^{\alpha k_i}\le(K+1)\card{D_i}\delta^{f_i}$. By Lemma~\ref{lem:intwords}
the selected words form ordered interval alphabets modulo $b_i^{k_i}$, and
Lemma~\ref{lem:round} with $\gamma=\sigma\delta$ ranks them below
$H=\lceil1/(\sigma\delta)\rceil$. Lemma~\ref{lem:crt} glues them into a
ranked alphabet with $Q=\prod_i\card{D_i}$ residues modulo the square base
$M=\prod_ib_i^{k_i}$, of height $h=1+n(H-1)\le\kappa/\delta$, where
$\kappa=(n+1)(1+\sigma^{-1})$ (using $\delta\le1$). Multiplying the length
estimates gives $M^\alpha\le Q\,\delta^\Phi(K+1)^n$. Since $\alpha\ge0$,
\[
 (Mh)^\alpha\le Q\cdot \kappa^\alpha(K+1)^n\rho^{K(\Phi-\alpha)} .
\]
Because $\Phi>\alpha$ and $\rho<1$, the factor
$\kappa^\alpha(K+1)^n\rho^{K(\Phi-\alpha)}$ tends to zero as $K\to\infty$ (the
ratio of consecutive terms tends to $\rho^{\Phi-\alpha}<1$). Fix $K$ with
that factor at most one.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:criterion}]
Proposition~\ref{prop:block} and then Proposition~\ref{prop:amplify}.
All alphabets, then $K$, and hence $M$, $h$ and $Q$ are fixed before $N$
varies. For every $\varepsilon>0$, eventually $N^\varepsilon\ge Q$.
\end{proof}

\begin{remark}[Relation to entropy]\label{rem:entropy}
For a probability vector $\pi$ on the letters of an alphabet, put
\[
 H(\pi)=-\sum_x\pi_x\log\pi_x,\qquad R(\pi)=-\sum_x\pi_x\log w_x.
\]
With $\nu_x=w_x^f/Z_f$, the inequality $\log u\le u-1$ gives
$\sum_x\pi_x\log(\nu_x/\pi_x)\le0$, that is
$H(\pi)-fR(\pi)\le\log Z_f$, with equality when $\pi=\nu$
\cite{CoverThomas2006}. Words in which each letter $x$ occurs with frequency
$\pi_x$ then give an alternative proof of Theorem~\ref{thm:criterion}. Its
exponent is $U/(V+1)$, where $U=\sum_iH_i/R_i$ and
$V=\sum_i\log b_i/R_i$, and the budget $\sum_if_i>\alpha$ is exactly
$U-\alpha V>\alpha$. The quantity $R$ is
the \emph{rank cost} of a component, and its contribution can be read as
$f=(H-\alpha\log b)/R$, which generalizes \eqref{eq:rankcontribution}. The
stopping proof above needs neither this identity nor Stirling's formula.
\end{remark}

\section{Odd primes: Paley chains and composite codes}\label{sec:odd}
\subsection{Squares modulo odd prime powers}
\begin{lemma}[Square differences modulo $p^{2e}$]\label{lem:padic}
Let $p$ be prime and $e\ge1$, and let $x\not\equiv y\pmod{p^{2e}}$ satisfy
$y-x\equiv z^2\pmod{p^{2e}}$. Let $d\in(0,p^{2e})$ represent $y-x$ and
$k=v_p(d)$. Then $k$ is even, $k<2e$, and $d/p^k$ is a nonzero square
modulo $p$.
\end{lemma}
\begin{proof}
Let $v=v_p(z)$. If $2v\ge2e$ then $d\equiv0$, which is impossible. Otherwise
$d=z^2+tp^{2e}$ for an integer $t$, and $v_p(z^2)=2v<2e\le v_p(tp^{2e})$, so
$k=2v$. Writing $z=p^vz'$ with $p\nmid z'$ and dividing by $p^{2v}$ gives
$d/p^{2v}\equiv z'^2\pmod p$.
\end{proof}
In terms of base-$p$ digits: if two integers agree in all digits below
position $k<2e$ and differ at position $k$, a square difference modulo
$p^{2e}$ forces $k$ to be even and the digit difference at position $k$ to be
a nonzero square modulo $p$. We call the digits in even positions \emph{low
digits} and those in odd positions \emph{free digits}.

\begin{lemma}\label{lem:minusone}
Let $p\equiv3\pmod4$ be prime. Then $-1\notin\QR(p)$, and for every residue
$d\not\equiv0$ exactly one of $d,-d$ is a nonzero square modulo $p$.
\end{lemma}
\begin{proof}
An element $g$ with $g^2\equiv-1$ would have order four in the cyclic group
$(\Z/p\Z)^\times$ of order $p-1\not\equiv0\pmod4$. The nonzero squares have
index two in that group, so $-d=(-1)d$ is a square exactly when $d$ is not.
\end{proof}
Thus for $p\equiv3\pmod4$ every two distinct residues modulo $p$ are related
in exactly one direction: square differences modulo $p$ form a tournament,
the \emph{Paley tournament}. For $p\equiv1\pmod4$, in contrast, $-1$ is a
square, and $x\to_py$ holds exactly when $y\to_px$.

\subsection{Paley chains}
\begin{definition}[Paley chain]\label{def:chain}
Let $p\equiv3\pmod4$ be prime. A \emph{chain} of length $t$ modulo $p$ is a
list of distinct residues $s_0,\ldots,s_{t-1}$ such that $s_j-s_i$ is a
nonzero square modulo $p$ whenever $i<j$.
\end{definition}

\begin{lemma}[Prime-chain alphabet]\label{lem:chain}
For a chain $s_0,\ldots,s_{t-1}$ modulo $p\equiv3\pmod4$, the $pt$ residues
$s_j+pu$ $(0\le u<p)$ modulo $p^2$, with interval $[j/t,(j+1)/t]$ on every
residue $s_j+pu$, form an ordered interval alphabet with moment
$Z_f=pt^{1-f}$. Equivalently, they form a ranked alphabet of height $t$ with
rank $j$.
\end{lemma}
\begin{proof}
If two residues have different low digits $s_i\ne s_j$, a square difference
modulo $p^2$ reduces to a nonzero square modulo $p$, which by
Lemma~\ref{lem:minusone} and the chain property points from $s_i$ to $s_j$
with $i<j$; the intervals are ordered. If the low digits agree, a nonzero
difference has $p$-adic valuation one, which Lemma~\ref{lem:padic} forbids.
There are $p$ residues with each interval, which gives the moment.
\end{proof}
This is the local mechanism of \cite{Krachun2026}; the free digit $u$
multiplies the number of residues by $p$ without any ordering cost, as in
\cite{Ruzsa1984}.

\begin{corollary}[Chain contribution]\label{cor:chainpower}
Let $t\ge2$ and $f_p^*=1-(2\alpha-1)\log p/\log t$. For $0\le f\le f_p^*$ the chain
alphabet of Lemma~\ref{lem:chain} passes the test at $\alpha$ with
contribution $f$, that is $pt^{1-f}\ge(p^2)^\alpha$.
\end{corollary}
\begin{proof}
$t^{1-f}$ decreases in $f$, and $pt^{1-f_p^*}=p\cdot p^{2\alpha-1}=p^{2\alpha}$.
\end{proof}
A chain is useful only when $f_p^*>0$, that is $t>p^{2\alpha-1}$.

\begin{remark}[Krachun's exponent]\label{rem:krachun}
With $\Lambda=\sum_{i=1}^{n}\log p_i/\log t_i$, the budget
$\sum_{i=1}^nf_{p_i}^*>\alpha$ for $n$ chains reads
$n-(2\alpha-1)\Lambda>\alpha$, that is
\[
 \alpha<\frac{n+\Lambda}{1+2\Lambda}.
\]
For Krachun's ten chains this is $\alpha<\alpha_{\mathrm K}$ of
\eqref{eq:krachun}. So Theorem~\ref{thm:criterion}, applied to Paley chains
alone, reproduces Krachun's exponent. Unequal widths cannot help a chain when
$0\le f\le1$: residues with different low digits differ by a square modulo
$p^2$ in one direction (a nonzero square modulo $p$ lifts to one modulo $p^2$),
so choosing one residue for each low digit gives a directed path, and the
widest residues $w_j$ of the $t$ low digits satisfy $\sum_jw_j\le1$. The
moment is then at most $p\sum_jw_j^f\le pt^{1-f}$, by concavity of
$w\mapsto w^f$. A larger exponent needs alphabets that are not chains.
\end{remark}

\subsection{Composite codes}\label{subsec:composite}
\begin{definition}[Low code and the prime-specific relation]\label{def:lowcode}
Fix a finite set $\mathcal P$ of odd primes and \emph{depths} $e_{p}\ge1$, the
numbers of restricted digits at each $p\in\mathcal P$, and put
$F=\prod_{p\in\mathcal P}p^{e_p}$. A \emph{low point} $\mathbf a$ is a tuple of
digit strings $\mathbf a_p=(a_{p,0},\ldots,a_{p,e_p-1})$ with $0\le a_{p,j}<p$,
one for each prime; we call $\mathbf a_p$ the \emph{low word} of $\mathbf a$
at $p$ (it is not read in a square base, unlike the words of
Definition~\ref{def:words}). For distinct low points write $\mathbf a\sto\mathbf b$ if, \emph{at
every prime $p$}, either $\mathbf a_p=\mathbf b_p$ or the first index $j$ with
$a_{p,j}\ne b_{p,j}$ satisfies
\begin{equation}\label{eq:lowedge}
 b_{p,j}-a_{p,j}\in\QR(p)\setminus\{0\}.
\end{equation}
A \emph{low code} is a finite set $S$ of low points, each carrying an interval
$I_{\mathbf a}\subseteq[0,1]$ of width $w_{\mathbf a}>0$, such that
$\mathbf a\sto\mathbf b$ implies $\sup I_{\mathbf a}\le\inf I_{\mathbf b}$.
A \emph{directed path} of $\sto$ is a sequence
$\mathbf a^{(0)}\sto\mathbf a^{(1)}\sto\cdots$ of points.
\end{definition}
The first differing index may depend on $p$, and no condition is imposed at a
prime whose word is unchanged. A relation needs agreement at every prime, so a
pair of points is unrelated as soon as one prime separates it.

\begin{remark}[How two primes cooperate]\label{rem:composite}
Take depth one at the primes $5$ and $43$, so a low point is a pair
$(a_5,a_{43})$. The nonzero squares modulo $5$ are $1$ and $4$, so the
$5$-digits of related points are equal or differ by $\pm1$, in either
direction. Points whose $5$-digits differ by $\pm2$ are never related,
whatever their $43$-digits, so their intervals may overlap freely, as in
Ruzsa's construction. Points whose $5$-digits are related are related in the
direction given by their $43$-digits, which Lemma~\ref{lem:minusone} orients,
as in Krachun's. A code must avoid pairs with equal $43$-digits and $5$-digits
differing by $\pm1$, which would be related in both directions. The
$17$-point code of Section~\ref{subsec:odd215} uses all five $5$-digits; a
chain modulo $43$ has at most $7$ elements, by an exhaustive search.
\end{remark}

\begin{lemma}[Prime-coordinate free-digit lift]\label{lem:odd}
Let $S$ be a low code. For $\mathbf a\in S$ and free digits
$u_{p,j}\in\{0,\ldots,p-1\}$ put
\begin{equation}\label{eq:oddlift}
 x_p=\sum_{j<e_p}\bigl(a_{p,j}p^{2j}+u_{p,j}p^{2j+1}\bigr)\pmod{p^{2e_p}},
\end{equation}
and combine the $x_p$ by the Chinese remainder theorem. Giving every such
residue the interval of its low point yields an ordered interval alphabet in
the square base $F^2$, with $F\card S$ residues and moment
$Z_f=F\sum_{\mathbf a\in S}w_{\mathbf a}^f$. The depths $e_p$ may differ
between primes.
\end{lemma}
\begin{proof}
Distinctness and the count follow from unique $p$-adic expansions and the
Chinese remainder theorem. Let lifted residues $x\ne y$ have a square
difference modulo $F^2$. By Lemma~\ref{lem:crtsq}, every prime coordinate with
$x_p\ne y_p$ has a square difference modulo $p^{2e_p}$. By
Lemma~\ref{lem:padic} its first differing $p$-adic digit sits at an even
position $2j$, which carries a low digit, and the digit difference there is a
nonzero square modulo $p$. All lower positions agree, so the low words agree
before index $j$: this is the first differing low index, and
\eqref{eq:lowedge} holds. If all low words agreed, some coordinate would
differ first at an odd (free) position, which Lemma~\ref{lem:padic} forbids.
Hence the low points $\mathbf a,\mathbf b$ of $x,y$ differ and
$\mathbf a\sto\mathbf b$, so the intervals are ordered. Each
low point has $F$ lifted copies, which gives the moment.
\end{proof}
Requiring a single common first-difference index at all primes would impose
extra relations that this proof does not need; reading the first difference
separately at each prime is what lets a low code exceed a product of chains.

A low code is most conveniently given by its widths alone. The following
lemma supplies the intervals.

\begin{lemma}[Path budgets]\label{lem:paths}
Let a finite directed acyclic graph have positive vertex weights $w_x$.
There are intervals in $[0,1]$ of these widths, ordered along every edge,
if and only if every directed path has total weight at most one.
\end{lemma}
\begin{proof}
Ordered intervals along a path have disjoint interiors, so their widths sum to
at most one. Conversely, let $d_x$ be the largest weight of a path ending at
$x$, including $x$, and put $a_x=d_x-w_x\ge0$. Then $a_x+w_x=d_x\le1$, and an
edge $x\to y$ gives $d_y\ge d_x+w_y$, that is $a_y\ge a_x+w_x$.
\end{proof}
This is the weighted-chain viewpoint underlying the chain polytope of a
finite poset \cite{Stanley1986}; the elementary proof above supplies the part
used here.

\begin{corollary}[Odd test on low data]\label{cor:oddtest}
Let $S$ be a finite set of low points with positive widths $w_{\mathbf a}$.
Suppose either that $S$, with given intervals of these widths, is a low code,
or that $\sto$ is acyclic on $S$ and every directed path of $\sto$ has total
width at most one. Then $S$ is a low code with intervals of these widths, and
its lift (Lemma~\ref{lem:odd}) passes the test at $\alpha$ with contribution
$f$ if and only if
\[
 \log\sum_{\mathbf a\in S}w_{\mathbf a}^f\ \ge\ (2\alpha-1)\log F .
\]
\end{corollary}
\begin{proof}
In the second case Lemma~\ref{lem:paths} gives the intervals. The lift has
moment $Z_f=F\sum_{\mathbf a}w_{\mathbf a}^f$, and
$(F^2)^\alpha=F\cdot F^{2\alpha-1}$.
\end{proof}
The number $F\card S$ of lifted residues enters only through $F$: a low code
helps when $\sum_{\mathbf a}w_{\mathbf a}^f$ exceeds $F^{2\alpha-1}$, roughly
$F^{0.516}$.

\begin{remark}[Free-digit copies need equal widths]\label{rem:fiber}
For a lift that keeps all free digits, the maps that add a fixed
$c_{p,j}$ to each free digit $u_{p,j}$ modulo $p$, digit by digit, preserve
square differences (they preserve the first differing $p$-adic position and
the low digits) and act transitively on the copies of each low point. Averaging widths over them keeps all path budgets and, by concavity of
$w\mapsto w^f$, does not decrease the moment. Thus giving different widths to
the copies of one low point cannot help; improvements must change the low
code itself.
\end{remark}

\section{The prime 2: a recursion with parity windows}\label{sec:binary}
Modulo $4^m$ there are no free digits in the sense of
Section~\ref{sec:odd}. Instead, square differences have a recursive structure.

\begin{lemma}[Square differences modulo $4^m$]\label{lem:dyadic}
Let $m\ge2$. (i) If $4d\equiv z^2\pmod{4^m}$ then $z=2z'$ and
$d\equiv z'^2\pmod{4^{m-1}}$. (ii) A residue $\equiv2$ or $3\pmod4$ is never
a square modulo $4^m$. (iii) An odd square residue modulo $4^m$ is
$\equiv1\pmod8$.
\end{lemma}
\begin{proof}
(i) Since $4\mid z^2$, $z$ is even; divide by $4$. (ii) Squares modulo $4$ are
$0$ and $1$. (iii) $(2k+1)^2=4k(k+1)+1$ and $8\mid4^m$.
\end{proof}
Write a residue $Y$ modulo $4^m$ as $Y=r+4x'$ with $r\in\{0,1,2,3\}$ and
$x'$ a residue modulo $4^{m-1}$. By the lemma, two residues with the same $r$
differ by a square exactly when their $x'$ do, modulo $4^{m-1}$ (the converse
of (i) is clear: $4z'^2=(2z')^2$); residues whose $r$ differ by $2$ never
differ by a square; and an odd square difference moves from a residue class
$k$ modulo $8$ to the class $k+1$, cyclically. An alphabet modulo $4^m$ is
therefore made of up to four alphabets modulo $4^{m-1}$, its
\emph{branches}, placed so that the residue classes modulo $8$ that occur
occupy successive parts of $[0,1]$, in cyclic order; since the order is
cyclic, some class must be absent. The classes modulo $8$ of the
residues in a branch are determined by the parities of its residues $x'$,
which is why the recursion tracks parities.

\subsection{States and transitions}
\begin{definition}[Parity-window policy]\label{def:policy}
A \emph{state} $s$ has an \emph{even window} $E_s$ and an \emph{odd window}
$O_s$: closed subintervals of $[0,1]$ of positive length, each possibly
absent, at least one present. State $s$ is \emph{valid at depth $m$} if it
carries an ordered interval alphabet modulo $4^m$ in which every residue's
interval lies in the window of the residue's parity.

A \emph{policy} assigns to each state at most one \emph{branch} for each
$r\in\{0,1,2,3\}$. A branch specifies a child state $j$, a scale $u>0$, a
shift $\theta\in\R$ and bits $\tau,\eta\in\{0,1\}$ (\emph{swap} and
\emph{reflect}). A child residue $x$ modulo $4^{m-1}$ becomes
\begin{equation}\label{eq:binarydigit}
 x'=((-1)^\eta x+\tau)\bmod 4^{m-1},\qquad y=r+4x'\pmod{4^m};
\end{equation}
its interval $[a,a+w]$ is first reflected to $[1-a-w,1-a]$ if $\eta=1$ and
then mapped by $z\mapsto\theta+uz$. The parity of $x'$ is that of $x$ toggled by
$\tau$. For transformed parity $\pi\in\{0,1\}$ let $J_{r+4\pi}$ be the image
of the child's window of parity $\pi\oplus\tau$ under the same maps; $J_k$ is
indexed by the residue class $k$ of the parent residue modulo $8$. The policy
is \emph{valid} if, whenever the windows involved are present,
\begin{align}
 J_{r+4\pi}&\subseteq E_s\quad(r\text{ even}),\qquad
 J_{r+4\pi}\subseteq O_s\quad(r\text{ odd}),\label{eq:containment}\\
 \sup J_k&\le\inf J_{k+1\bmod8}\qquad(0\le k<8).\label{eq:cyclic}
\end{align}
\end{definition}
Condition \eqref{eq:cyclic} includes the comparison from class $7$ to class
$0$; it can hold only because some classes are absent. Reflection and swap
let a branch reuse a child state in another orientation or parity.

\begin{lemma}[Binary transition]\label{lem:binary}
Let $m\ge2$. If the children are valid at depth $m-1$ and the policy is
valid, then the branches define a valid state at depth $m$.
\end{lemma}
\begin{proof}
Different branches have different residues modulo $4$, and
\eqref{eq:binarydigit} is injective within a branch, so all residues are
distinct. Containment \eqref{eq:containment} puts every interval in the window
of its parity. Let $Y-X$ be a nonzero square modulo $4^m$.
\emph{Same branch:} $Y-X=4(y'-x')$, and by Lemma~\ref{lem:dyadic}(i)
$y'-x'$ is a square modulo $4^{m-1}$. Without reflection the child's order
applies directly. With reflection, $y'-x'$ is the negative of the difference
of the child residues, so the child's square difference points the other way
and the reflected intervals reverse it back. The shift cancels in differences
and the positive scale preserves order.
\emph{Residues differing by $2$ modulo $4$:} excluded by
Lemma~\ref{lem:dyadic}(ii).
\emph{Odd difference:} by Lemma~\ref{lem:dyadic}(iii), $Y-X\equiv1\pmod8$, so
the classes of $X,Y$ modulo $8$ are $k,k+1$ cyclically; their intervals lie in
$J_k$ and $J_{k+1}$, which \eqref{eq:cyclic} orders.
\end{proof}

\subsection{Moment growth and a finite depth}
\begin{definition}[Growth certificate]\label{def:growthcert}
Let $0\le f\le1$. A \emph{growth certificate} for a policy is a vector $v>0$
indexed by states and a number $a>1$ such that
\begin{equation}\label{eq:rows}
 \sum_{\text{branches }s\to j}u^fv_j\ge av_s\qquad\text{for every state }s.
\end{equation}
Let $d_s$ be the length of the wider present window of $s$ and
$c=\min_sd_s/v_s>0$. The \emph{branch matrix} $B_f$ has entries
$(B_f)_{s,j}=\sum_{\text{branches }s\to j}u^f$, so \eqref{eq:rows} reads
$B_fv\ge av$; this forces $a$ to be at most the \emph{Perron root} of
$B_f$, its largest eigenvalue.
\end{definition}
This positive-vector condition is a familiar device for bounding the growth of
a nonnegative matrix \cite{Seneta2006}; here only the following elementary
induction is needed.

\begin{lemma}[Moment growth]\label{lem:growth}
Under a valid policy with a growth certificate, every state is valid at every
depth $m\ge1$, with moment $Z_{s,m}\ge c\,a^{m-1}v_s$.
\end{lemma}
\begin{proof}
At depth one, take the single residue $0$ or $1$ whose parity window is the
wider one, with that whole window as its interval. A singleton has no square
differences, and its moment is $d_s^f\ge d_s\ge cv_s$ because $0<d_s\le1$ and
$f\le1$. Reflection and shift preserve widths, scaling multiplies a moment by
$u^f$, and branches are disjoint, so $Z_{s,m}=\sum_{s\to j}u^fZ_{j,m-1}$.
Induct using \eqref{eq:rows}; validity comes from Lemma~\ref{lem:binary}.
\end{proof}

\begin{corollary}[Root at a fixed depth]\label{cor:bindepth}
Suppose some \emph{root} state has both windows equal to $[0,1]$ and
$v_{\mathrm{root}}=1$. Then for every $m\ge1$ there is an ordered interval
alphabet modulo $4^m$, with all widths at most $1/2$, whose moment satisfies
$Z_f\ge 2^{-f}c\,a^{m-1}$. In particular,
\begin{equation}\label{eq:binarymargin}
 \log c+(m-1)\log a-f\log2-\alpha m\log4>0
 \quad\Longrightarrow\quad Z_f>(4^m)^\alpha .
\end{equation}
\end{corollary}
\begin{proof}
The root alphabet of Lemma~\ref{lem:growth} has no window restriction.
Halving every width at its left endpoint keeps all orders, makes every width
at most $1/2$, and multiplies the moment by $2^{-f}$.
\end{proof}
The depth $m$ is any fixed integer, chosen before $N$ varies. A very large
depth affects only constants, and the alphabet is never listed explicitly.
The binary component contributes $f$ whenever some growth certificate has
$a>4^\alpha$ and $m$ is large enough. As $m$ grows, its admissible
contributions therefore approach the power $f$ at which the Perron root of
$B_f$ equals $4^\alpha$.

\section{The nine components and the proof of Theorem~\ref{thm:main}}\label{sec:parameters}
We now fix all data. Every terminating decimal in this section denotes an
exact rational. Each \emph{certificate} below is a finite list of exact
rational or integer inequalities; Section~\ref{sec:verification} describes
the programs that check them. Table~\ref{tab:powers} summarizes the
components. All decisive numerical comparisons use the following rational
bounds for logarithms.

\begin{lemma}[Rational logarithm bounds]\label{lem:logenc}
For $1\le x\le2$ and $z=(x-1)/(x+1)$,
\[
 \log x=2\sum_{j=0}^{n-1}\frac{z^{2j+1}}{2j+1}+R_n,
 \qquad 0\le R_n\le\frac{2z^{2n+1}}{(2n+1)(1-z^2)}.
\]
Consequently, let $\lambda^-(x)\le\log x\le\lambda^+(x)$ be the resulting
rational bounds, extended to all positive rationals by writing $x=2^ky$ with
$1\le y\le2$ and adding $k$ times the bounds for $\log2$ (with lower and upper
bounds of $\log 2$ interchanged when $k<0$). Then for rationals $x>0$, $L>0$ and
$f\ge0$ the inequality $\lambda^+(L)\le f\lambda^-(x)$ implies $L\le x^f$.
\end{lemma}
\begin{proof}
Subtracting the series for $\log(1-z)$ from that of $\log(1+z)$ gives the
expansion; all terms are positive. The tail is bounded by replacing every
denominator by $2n+1$ and summing the geometric series in $z^2\le1/9$.
For the consequence, $\log L\le\lambda^+(L)\le f\lambda^-(x)\le f\log x$,
and the exponential function is increasing.
\end{proof}
Section~\ref{sec:arithmetic} describes how these bounds are implemented.

\begin{table}[!h]
\centering\small
\begin{tabular}{lll}\toprule
Component & Square base $b_i$ & Exact contribution $f_i$\\\midrule
Chain at $3$ ($t=2$) & $3^2$ & $0.1819189685063644313459576$\\
Chain at $7$ ($t=3$) & $7^2$ & $0.0857695922261077385453455$\\
Chain at $11$ ($t=4$) & $11^2$ & $0.1072043074619092931875010$\\
Chain at $31$ ($t=7$) & $31^2$ & $0.0891366647927413831505830$\\
Chain at $59$ ($t=9$) & $59^2$ & $0.0421426390460550181119771$\\
Chain at $103$ ($t=11$) & $103^2$ & $0.0023654891206769647316637$\\
Low code at $5,43$, depths $(3,3)$ & $(5^3\cdot43^3)^2=215^6$ & $0.0264983230432418544045862$\\
Low code at $19,23$, depths $(3,2)$ & $(19^3\cdot23^2)^2$ & $0.0681150585658841330427119$\\
Binary root, depth $m=10^{20}$ & $4^{10^{20}}$ & $0.1549247890379023302829202$\\\midrule
Total & & $0.7580758318008831468032462$\\\bottomrule
\end{tabular}
\caption{The nine components, their bases and their contributions. The total
exceeds $\al=0.7580758318008816$ by $15468032462\cdot10^{-25}$.}\label{tab:powers}
\end{table}

\subsection{Prime chains}
\begin{certificate}[Six prime chains]\label{cert:chains}
The following lists are chains modulo the indicated primes, all
$\equiv3\pmod4$: for each of the $244$ ordered pairs of distinct entries,
$s_j-s_i$ is a square modulo $p$ exactly when $j>i$.
\begin{center}
\begin{tabular}{rrl}\toprule
$p$&$t$&Chain $s_0,\ldots,s_{t-1}$\\\midrule
3&2&$0,1$\\
7&3&$0,1,2$\\
11&4&$0,1,4,5$\\
31&7&$0,1,8,5,2,9,10$\\
59&9&$0,1,28,17,22,4,26,20,29$\\
103&11&$0,1,29,92,2,61,17,30,93,18,19$\\\bottomrule
\end{tabular}
\end{center}
For each chain, the contribution $f_p$ in Table~\ref{tab:powers} satisfies
$f_p>0$ and $(1-f_p)\lambda^-(t)-(2\al-1)\lambda^+(p)>0$ with the rational
bounds of Lemma~\ref{lem:logenc}; the smallest of these six margins exceeds
$2.4\cdot10^{-26}$.
\end{certificate}

\begin{proposition}[Chain components]\label{prop:chains}
For each of the six primes there is an ordered interval alphabet modulo $p^2$,
with widths in $[1/11,1/2]$, passing the test at $\al$ with contribution
$f_p$.
\end{proposition}
\begin{proof}
The certified margin gives $(1-f_p)\log t>(2\al-1)\log p$, that is
$f_p<f_p^*$. Apply Lemma~\ref{lem:chain} to the chains of
Certificate~\ref{cert:chains} and Corollary~\ref{cor:chainpower}.
\end{proof}

\subsection{The composite code at 5 and 43}\label{subsec:odd215}
The low code at the primes $5$ and $43$ is the three-letter product $S^3$ of
the following $17$-point code $S$ of depth one, given as pairs
$(a_5,a_{43})$:
\[
\begin{gathered}
(0,18),(3,25),(1,27),(4,29),(0,24),(0,22),(3,42),(2,5),(1,19),\\
(2,40),(3,22),(4,35),(3,3),(0,33),(4,39),(1,28),(2,20).
\end{gathered}
\]
A point of $S^3$ has the word $(x_5,y_5,z_5)$ at the prime $5$ and the word
$(x_{43},y_{43},z_{43})$ at $43$, built from three points of $S$. As explained
in Remark~\ref{rem:composite}, the prime $5$ separates points and the prime
$43$ orients every related pair. The widths are not products of widths of
$S$; they were optimized for the full relation $\sto$ of
Definition~\ref{def:lowcode}, whose longest directed paths have $1331$
points.

\begin{certificate}[The $(5,43)$ low code]\label{cert:odd215}
The witness consists of $4{,}913=17^3$ distinct low points with depths
$(3,3)$, listed in a topological order, and integer width numerators over the
denominator $10^{14}$. Each of the $6{,}031{,}254$ related pairs
$\mathbf a\sto\mathbf b$ has $\mathbf a$ before $\mathbf b$ in the list, and
the largest total width of a directed path of $\sto$ is
$99999974386028/10^{14}<1$. With $f_{215}=0.0264983230432418544045862$ (we
index the two composite components by $215=5\cdot43$ and $437=19\cdot23$),
\begin{equation}\label{eq:odd215margin}
 \log\sum_{\mathbf a\in S^3}w_{\mathbf a}^{f_{215}}-(2\al-1)\log(215^3)>6.919\cdot10^{-18}.
\end{equation}
\end{certificate}

\begin{proposition}[The $(5,43)$ component]\label{prop:odd215}
There is an ordered interval alphabet in the square base $215^6$, with all
widths in $(0,1)$, passing the test at $\al$ with contribution $f_{215}$.
\end{proposition}
\begin{proof}
The certificate shows that $\sto$ is acyclic on the code and bounds every
directed path by one. Corollary~\ref{cor:oddtest}, with $F=215^3$ and
\eqref{eq:odd215margin}, gives the lifted alphabet and the test.
\end{proof}

\subsection{The composite code at 19 and 23}
The low code at the primes $19$ and $23$ has \emph{unequal depths}: three low
digits at $19$ and two at $23$, so $F=19^3\cdot23^2=3{,}628{,}411$. Its
$3{,}645$ points form a product $B\times\{0,1,6,17,7\}$. Here
$B$ is a $729$-point low code with depths $(2,2)$, and the list
$0,1,6,17,7$ is a chain modulo $19$ that supplies the third $19$-digit. Two points
that agree in their first two $19$-digits are compared at $19$ through this
chain. All other comparisons at $19$ are decided by the first two digits.
The code $B$ descends from the $27$-point depth-one code
\[
\begin{gathered}
(10,6),(14,6),(2,6),(10,19),(0,16),(0,19),(11,16),(0,20),(11,19),\\
(11,20),(10,9),(16,10),(16,19),(10,22),(17,10),(16,22),(17,19),(14,9),\\
(17,22),(14,22),(2,9),(2,22),(16,12),(0,5),(1,12),(11,5),(17,12)
\end{gathered}
\]
(pairs $(a_{19},a_{23})$), which exceeds the $25$ points of a product of
two maximal chains. The code $B$ was obtained from it by conditional
transformations of the second letters, by exchanges of points, and
finally by eighteen rounds of exhaustive single-point exchanges that
maximize the moment. None of this history is used: only the final points
and intervals enter the proof. The longest directed paths of $\sto$ in this
code have $686$ points.

\begin{certificate}[The $(19,23)$ low code]\label{cert:odd437}
The witness consists of $3{,}645$ distinct low points with depths $(3,2)$,
listed in a topological order, with integer interval starts and widths over
the denominator $10^{16}$. Each of the $2{,}702{,}580$ related pairs
$\mathbf a\sto\mathbf b$ has $\mathbf a$ before $\mathbf b$ in the list and
$\sup I_{\mathbf a}\le\inf I_{\mathbf b}$, and the largest right endpoint is
$9999999218567850/10^{16}<1$. With $f_{437}=0.0681150585658841330427119$,
\begin{equation}\label{eq:odd437margin}
 \log\sum_{\mathbf a}w_{\mathbf a}^{f_{437}}-(2\al-1)\log(19^3\cdot23^2)>5.927\cdot10^{-18}.
\end{equation}
\end{certificate}

\begin{proposition}[The $(19,23)$ component]\label{prop:odd437}
There is an ordered interval alphabet in the square base $(19^3\cdot23^2)^2$,
with all widths in $(0,1)$, passing the test at $\al$ with contribution
$f_{437}$.
\end{proposition}
\begin{proof}
The certificate shows that the code, with its certified intervals, is a low
code. Corollary~\ref{cor:oddtest}, with depths $(e_{19},e_{23})=(3,2)$,
$F=19^3\cdot23^2$ and \eqref{eq:odd437margin}, gives the lifted alphabet and
the test.
\end{proof}

\subsection{The binary component}
\begin{certificate}[The binary policy]\label{cert:binary}
The witness specifies $25$ states, $94$ branches with rational windows,
scales and shifts, and a positive rational vector $v$ with denominator
$10^{45}$. The root is state $1$ (states are numbered from $0$), its windows are both
$[0,1]$, and $v_1=1$. The exact parameters are
\begin{gather*}
 f_B=0.1549247890379023302829202,\qquad m=10^{20},\\
 a=\frac{1430135321718334301514215293356149372101}{5\cdot10^{38}},
\end{gather*}
and $c$ is the exact rational $\min_sd_s/v_s>0.3551$. The policy is valid:
$304$ containment and cyclic-order comparisons \eqref{eq:containment},
\eqref{eq:cyclic} hold, including the wrap from class $7$ to class $0$. The
growth rows and margins satisfy
\begin{align}
 \min_s\Bigl(\sum_{s\to j}u^{f_B}v_j-av_s\Bigr)
       &>3.27\cdot10^{-19},\label{eq:rowmargin}\\
 \log a-\al\log4&>4.91\cdot10^{-19},\label{eq:growthmargin}\\
 \log c+(m-1)\log a-f_B\log2-\al m\log4
       &>46.91.\label{eq:depthmargin}
\end{align}
\end{certificate}
The tiny margins in \eqref{eq:rowmargin} and \eqref{eq:growthmargin} are
deliberate: $a$ lies between $4^{\al}$ and the Perron root of the branch
matrix $B_{f_B}$ (Definition~\ref{def:growthcert}), and the large depth $m$ converts the small gap
\eqref{eq:growthmargin} into the comfortable margin \eqref{eq:depthmargin}.

\begin{proposition}[The binary component]\label{prop:binary}
There is an ordered interval alphabet modulo $4^{10^{20}}$, with all widths in
$(0,1/2]$, passing the test at $\al$ with contribution $f_B$.
\end{proposition}
\begin{proof}
By Certificate~\ref{cert:binary}, the policy is valid and $(v,a)$ is a growth
certificate. Lemmas~\ref{lem:binary} and \ref{lem:growth} and
Corollary~\ref{cor:bindepth}, with \eqref{eq:depthmargin}, give the
alphabet and its moment.
\end{proof}

\subsection{Closing the bound}\label{subsec:closing}
\begin{certificate}[Coprimality and contribution budget]\label{cert:budget}
The prime supports $\{2\}$, $\{3\}$, $\{7\}$, $\{11\}$, $\{31\}$,
$\{59\}$, $\{103\}$, $\{5,43\}$ and $\{19,23\}$ of the nine components
are pairwise disjoint, so the nine
square bases are pairwise coprime. The exact sum of the contributions in
Table~\ref{tab:powers} is
\[
 \sum_if_i=0.7580758318008831468032462=\al+\frac{15468032462}{10^{25}}.
\]
\end{certificate}
The surplus is tiny by design. The contributions of the odd components lie
about $10^{-18}$ below their largest admissible values at $\al$
(Definition~\ref{def:admissible}). For the binary component the relevant
threshold is the power $f$ at which the Perron root of $B_f$ equals
$4^{\al}$, the limit of its admissible contributions as the depth grows
(Section~\ref{sec:binary}); $f_B$ lies about $10^{-18}$ below it. With this
convention, $\al$ is the $16$-decimal truncation of the exact crossing
$\alpha^*=0.75807583180088164538\ldots$ of the nine alphabets. With the same codes, chains and policy geometry, any exponent below
$\alpha^*$ could be certified by retuning the contributions, $a$ and $m$;
none above $\alpha^*$ can be without changing a code or the policy.

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Propositions~\ref{prop:chains}, \ref{prop:odd215}, \ref{prop:odd437} and
\ref{prop:binary} give nine ordered interval alphabets in pairwise coprime
square bases (Certificate~\ref{cert:budget}), each passing the test at $\al$
with the contribution of Table~\ref{tab:powers}. They are finite with widths
in $(0,1)$, so Lemma~\ref{lem:finitewidth} supplies common bounds
$\sigma,\rho$. The contributions sum to more than $\al$
(Certificate~\ref{cert:budget}). Theorem~\ref{thm:criterion} gives
$D(N)\ge N^{\al}/Q$ for all $N\ge1$, with all finite choices, including the
binary depth, made before $N$ varies.
\end{proof}

\section{Exact finite verification}\label{sec:verification}
\subsection{Data sufficient to define the construction}
The embedded archive contains a manifest and three witness files, identified
by the hashes in Appendix~\ref{app:hashes}, together with the checking
programs. The manifest fixes $\al$, the two odd contributions, the six chains
and the SHA-256 hashes of the three witnesses. The binary witness fixes its
own contribution, depth, vector and growth constant $a$. The rational chain
check produces the six chain contributions of Table~\ref{tab:powers}.
Numerical fields left over from discovery are ignored.

For an odd witness, \texttt{primes} lists the coordinate primes, and either
\texttt{depth} (a common depth) or \texttt{depths} (one per prime) gives the
number of low digits. The array \texttt{points} lists the low words, one word
per prime, least significant digit first. A positive integer
\texttt{width\_denominator} $D$ and positive integers
\texttt{width\_numerators} $n_j$ define $w_j=n_j/D$. The $(19,23)$ file also
gives \texttt{start\_numerators} $s_j$, defining $a_j=s_j/D$; the $(5,43)$
starts are defined by Lemma~\ref{lem:paths}. Thus the data specify every
interval without reference to the programs that found them.

In the binary witness, a state's window record $[a,b,c,d,q]$ means
$E_s=[a/q,b/q]$ and $O_s=[c/q,d/q]$; the pair $c=d=-1$ denotes an absent odd
window. Each branch lists its residue \texttt{r}, \texttt{child},
\texttt{swap} and \texttt{reflect} bits, and rational \texttt{scale} and
\texttt{shift}. Equations \eqref{eq:binarydigit}--\eqref{eq:rows} specify
precisely what these data mean.

\subsection{Complete geometric checks}
Below, an \emph{edge} is a related pair $\mathbf a\sto\mathbf b$ of low
points, or a pair $X\to Y$ of residues.

\paragraph{The $(5,43)$ code.} The checker verifies prime validity,
distinctness of points, all digit ranges and word lengths, positivity of the
widths, and every ordered pair of distinct points. For each pair it evaluates
\eqref{eq:lowedge} separately in each coordinate, and every edge must point
forward in the supplied list. It then computes the path numerators
\[
 d_j=n_j+\max\bigl(\{d_i:i\sto j\}\cup\{0\}\bigr)
\]
and checks $\max_jd_j\le D$. This is the recurrence of
Lemma~\ref{lem:paths}; induction in the checked topological order proves its
validity.

\paragraph{The $(19,23)$ code.} The checker computes the same full relation
with integer bitsets and per-prime depths. For one prime $p$ with depth $e$,
let $G(v)$ be the bitset of points whose coordinate word begins with the
prefix $v$. For a target word $b=(b_0,\ldots,b_{e-1})$ the allowable sources
have bitset
\begin{equation}\label{eq:prefix}
 G(b)\ \cup\ \bigcup_{j=0}^{e-1}\ \bigcup_{q\in\QR(p)\setminus\{0\}}
 G\bigl((b_0,\ldots,b_{j-1},b_j-q)\bmod p\bigr).
\end{equation}
The first term covers identical words; each remaining term fixes the first
difference and permits all later digits. Hence \eqref{eq:prefix} is exactly
the relation required at that prime. Intersecting over the two primes and
removing the target enumerates every incoming edge. The checker verifies that
every incoming index precedes its target. It then sorts points by right
endpoint, sweeps targets by left endpoint while maintaining the bitset of
points whose right endpoints exceed the current start, and requires an empty
intersection with each incoming bitset; this is equivalent to $s_i+n_i\le s_j$
on every edge $i\sto j$. Sizes, ranges, distinctness, starts, widths and
endpoints are checked separately. The counts in
Section~\ref{sec:parameters} are outputs of these complete checks.

\paragraph{The binary policy.} The geometry checker constructs every $J$
window with exact fractions and checks \eqref{eq:containment} and
\eqref{eq:cyclic}, together with unique branch residues, valid child indices,
positive scales and valid bits. Its 304 comparisons count two endpoint
containments per present placed window and one inequality per present
adjacent pair. Lemma~\ref{lem:binary} is what makes these finitely many
conditions sufficient at every depth.

\subsection{Rational arithmetic}\label{sec:arithmetic}
The bounds $\lambda^-(x)\le\log x\le\lambda^+(x)$ of Lemma~\ref{lem:logenc}
are implemented in three ways. The $(5,43)$ checker uses 32 terms with exact
fractions, and the binary checker uses 70. The $(19,23)$ checker uses 48 terms
with outward rounding at scale $S=2^{192}$: if $L_z\le Sz\le U_z$, it sets
$L_{z^2}=\lfloor L_z^2/S\rfloor$ and $U_{z^2}=\lceil U_z^2/S\rceil$,
multiplies successively with lower products rounded down and upper products
rounded up, rounds each division by $2j+1$ in the same directions, and adds the
scaled tail bound
$\bigl\lceil2U_{\mathrm{next}}S/((2n+1)(S-U_{z^2}))\bigr\rceil$, where
$U_{\mathrm{next}}$ bounds $Sz^{2n+1}$. Induction preserves enclosure through
every operation \cite{MooreKearfottCloud2009}.

A positive rational proposal $L$ for a power $x^f$ is accepted as a lower
bound only when
\begin{equation}\label{eq:poweraccept}
 \lambda^+(L)\le f\lambda^-(x),
\end{equation}
as in Lemma~\ref{lem:logenc}. Decimal arithmetic proposes candidates; it
never decides validity, and a poor proposal causes rejection rather than a
false positive. For the odd components the proposals are integer multiples of
$10^{-24}$; with accepted $L_{\mathbf a}\le w_{\mathbf a}^f$ the checkers prove
$\lambda^-\bigl(\sum_{\mathbf a}L_{\mathbf a}\bigr)-(2\al-1)\sum_pe_p\lambda^+(p)>0$, which gives
\eqref{eq:odd215margin} and \eqref{eq:odd437margin}. For the binary component,
accepted lower powers give lower bounds for the rows of \eqref{eq:rows}, and
\[
 \lambda^-(c)+(m-1)\lambda^-(a)-f_B\lambda^+(2)-\al m\,\lambda^+(4)>0
\]
proves \eqref{eq:depthmargin} without approximating the astronomically large
alphabet. The chain check proves $(1-f_p)\lambda^-(t)-(2\al-1)\lambda^+(p)>0$ for each
of the six rational contributions, and Certificate~\ref{cert:budget} is an
exact rational addition performed by the chain-contribution program. The main
verifier and the independent arithmetic check close the budget more
generously: they use the certified lower bounds for the six $f_p^*$ themselves
instead of their $25$-decimal truncations, and therefore report a slightly
larger surplus ($1.5468032465\cdot10^{-15}$).

\paragraph{How the parameters were chosen.} For fixed witnesses each
component has a largest admissible contribution at a given $\alpha$, and the
exponent is limited by the crossing at which these sum to $\alpha$. The
odd and binary contributions in Table~\ref{tab:powers} were proposed by
computing their largest admissible values with 60-digit decimal arithmetic and
subtracting $10^{-18}$. Each chain contribution is the $25$-decimal truncation
of the certified rational lower bound
$1-(2\al-1)\lambda^+(p)/\lambda^-(t)$ for $f_p^*$. The binary
vector $v$ is a Perron vector of the branch matrix rounded to denominator
$10^{45}$. The value $a$ lies strictly between $4^{\al}$ and the Perron
root. The depth $10^{20}$ makes \eqref{eq:depthmargin} positive despite the
gap \eqref{eq:growthmargin} of order $10^{-19}$. These choices only propose
data; the exact checks above decide everything.

\subsection{Independent arithmetic and the trust boundary}
A further program in the attachment rechecks the moments, the binary growth
and depth inequalities, and the global budget with a different expansion,
\[
 \log x=\sum_{k=1}^{n}\frac{t^k}{k}+R_n',\qquad
 t=1-1/x\in[0,1/2],\qquad
 0\le R_n'\le\frac{t^{n+1}}{(n+1)(1-t)},
\]
for $1\le x\le2$, with $230$ terms, outward-rounded integer bounds at scale
$10^{60}$, and power proposals with denominator $10^{40}$. It does not
replace the geometry checks; it provides independently implemented arithmetic
for the same pinned witnesses. The Lean development of
Section~\ref{subsec:scope} checks the same finite conditions a third time,
with separately generated certificates and Lean's kernel.

The complete supplied run accepts all the finite conditions of
Section~\ref{sec:parameters}. The correctness of the numerical theorem uses
the execution of these integer/rational programs, not merely their printed
messages; the implications of the conditions are proved in the text.

\section{What the construction leaves open}\label{sec:discussion}
The proof isolates finite quantities that future work can improve: low codes
and their interval moments, the grouping of primes into coprime components,
and the binary transition geometry. The exponent $\al$ is a certified
feasible exponent, not an asserted optimum for the method or for $D(N)$. The
possible limitations of Paley chains discussed in \cite{Krachun2026,Satake2021}
do not bound composite and binary interval alphabets. The very large fixed
depths make the construction useless for moderate $N$; the objective is the
asymptotic exponent.

Several features of the present components indicate where further gains could
come from, without restricting future approaches.
\begin{itemize}
\item For a single prime $p\equiv3\pmod4$ one can show that chains are
optimal: two residues with different first low digits are always comparable,
so the first low digits of any alphabet modulo $p^{2e}$ form a chain (an
acyclic tournament is transitive), and concavity of $w\mapsto w^f$ and the
multiplicativity of moments then give at most $(pt^{1-f})^e$. For the prime $2$,
one can show that the window recursion loses nothing: an odd residue is a
square modulo $4^m$ exactly when it is $\equiv1\pmod8$
(Lemma~\ref{lem:dyadic} and Hensel lifting), so cross-branch square relations
depend only on residue classes modulo $8$, and every alphabet modulo $4^m$ arises
by one transition from its four branches, with the convex hulls of the class
intervals as windows. The open
question there is the best growth rate of that recursion.
\item Gains beyond products of chains come from composite low codes whose
cardinality exceeds the product of chain lengths, such as the $27$-point
$(19,23)$ code above. In the searches behind this paper, such excess
appeared only for primes whose maximal chain length is small for their size,
most notably $23$. Moreover, such codes compose badly. For a product of two chains, the
prime-specific relation makes heights add, and the rank cost per letter tends
to the logarithm of a single chain length. For the $27$-point code $S$, the
rank cost of $S\times S$ (as a depth-two code) is about twice that of $S$, the
rate of a lexicographic product. A code family that kept the excess at every
level with additive heights would raise the $(19,23)$ contribution
substantially; none is known.
\item A different integer realization than Lemma~\ref{lem:reverse}, one
that avoids only actual squares rather than all modular square residues,
would change the overall trade-off; it would require a new amplification
argument.
\end{itemize}

\appendix

\begin{filecontents*}[overwrite]{square-difference-free-certificate.py}
#!/usr/bin/env python3
"""Exact certificates for Square-difference-free sets of exponent 0.7580758318008816.
Run with Python 3 without optimization; uses the standard library and no network.
Default: unpack to a temporary directory and run every check.
Use --extract DIRECTORY to inspect and retain the data, programs, and reports.
"""
import argparse, base64, hashlib, io, pathlib, subprocess, sys, tempfile, zipfile
ARCHIVE_SHA256 = '088f7876b76fd83745932beec88548df796ac02a2c3b66907608c79b721a8295'
PAYLOAD = """
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BAAASrkBAAAA
"""

def main():
    if not __debug__:
        raise SystemExit('Run without -O or -OO: all verification assertions must execute.')
    parser=argparse.ArgumentParser(description=__doc__)
    parser.add_argument('--extract',type=pathlib.Path)
    args=parser.parse_args()
    raw=base64.b64decode(PAYLOAD)
    if hashlib.sha256(raw).hexdigest()!=ARCHIVE_SHA256:
        raise SystemExit('Certificate archive checksum mismatch.')
    def verify(destination):
        destination.mkdir(parents=True,exist_ok=True)
        with zipfile.ZipFile(io.BytesIO(raw)) as archive:
            for entry in archive.infolist():
                relative=pathlib.PurePosixPath(entry.filename)
                if relative.is_absolute() or '..' in relative.parts:
                    raise ValueError('Unsafe archive member')
                target=destination.joinpath(*relative.parts)
                if target.exists():
                    raise FileExistsError('Use a fresh extraction directory: '+str(target))
                target.parent.mkdir(parents=True,exist_ok=True)
                target.write_bytes(archive.read(entry))
        subprocess.run([sys.executable,str(destination/'run.py')],check=True)
    if args.extract:
        verify(args.extract.resolve())
    else:
        with tempfile.TemporaryDirectory(prefix='sarkozy-paper-') as directory:
            verify(pathlib.Path(directory))
if __name__=='__main__':
    main()
\end{filecontents*}

\section{Reproducing the entire finite verification}\label{app:reproduction}
The attachment is a self-contained Python program carrying a compressed
archive of the exact data and all programs needed by the checks.
\begin{center}
\textattachfile[color=0 0 0.5,mimetype=text/plain,description={Exact witnesses and complete verification programs}]{square-difference-free-certificate.py}{\fbox{\texttt{square-difference-free-certificate.py}}}
\end{center}
A PDF viewer supporting embedded files can save this attachment.
Alternatively, compiling the single \texttt{square-difference-free-sets-of-exponent-0.7580758318.tex}
source with \texttt{pdflatex} writes that same Python file beside the build
outputs. The source contains the actual archive bytes, not a link to them.
Run
\begin{lstlisting}
python3 square-difference-free-certificate.py
\end{lstlisting}
The program checks the embedded archive hash, extracts into a temporary
directory, and runs the complete geometry and moment verifier, the rational
chain-contribution check, and the independent arithmetic check. It makes no
network requests and needs no third-party Python packages. To retain the
extracted data, source code and generated reports, choose a fresh output
directory:
\begin{lstlisting}
python3 square-difference-free-certificate.py --extract certificate-check
\end{lstlisting}
The run finishes with
\begin{lstlisting}
PAPER CERTIFICATE PASS: complete geometry, rational target powers, and independent arithmetic.
\end{lstlisting}
The wrapper and all entry points reject Python's \texttt{-O} and \texttt{-OO}
modes, because some routines use assertions as checks. Compilation of the PDF
does not execute these mathematical checks; they are a separate step.

The principal verification files in the extracted archive are:
\begin{itemize}\small\raggedright
\item \path{research-notes/improved-bound/verify_bound.py}: manifest,
components, chains, coprimality and the global budget (with the certified lower
bounds for the chain contributions);
\item \path{research-notes/odd-search/verify_candidate.py}: complete pairwise
low-code and longest-path checking for the $(5,43)$ code, exact rational
logarithms;
\item \path{research-notes/odd-push/verify_unequal.py}: full-edge bitset
checking for the $(19,23)$ code with per-prime depths, directed fixed-point
logarithms;
\item \path{research-notes/binary-search/check_geometry.py} and
\path{research-notes/binary-search/certify.py}: binary transition geometry,
growth rows and the fixed-depth moment;
\item \path{research-notes/lean-planning/rational_chain_powers.py}: the six
rational chain contributions (run with 25 decimals) and the exact rational
addition of Certificate~\ref{cert:budget};
\item \path{research-notes/binary-push/independent_arithmetic.py}: independent
arithmetic with the second logarithm series.
\end{itemize}
The archive preserves the repository file names for reproducibility; they do
not introduce dependencies on any other files. Discovery-only functions present
in some modules are not executed.

\section{Certificate identity}\label{app:hashes}
The following SHA-256 hashes fix the exact bytes of the data files in the
embedded archive. Hashes identify witnesses; the finite checks, not the
hashes, establish their mathematical properties.
\begin{description}
\item[Manifest:] \path{research-notes/capacity-bound/bound.json}\\
\nolinkurl{080735a362d5fb736ae932d4c775e74c6c05ce3a25119e17cc0f1b928022bf84}
\item[Odd $(5,43)$:] \path{research-notes/odd-search/q215-depth3-product.json}\\
\nolinkurl{dae38a756de83c99d65973ff85f23a7068ce6e85a7a4f12400a923f24cf46574}
\item[Odd $(19,23)$:] \path{research-notes/next-push/q437-unequal32s.json}\\
\nolinkurl{3ef1e78e59f6e90327bd739c8c114604a352bb4ae7f0faecc6d547b3cd0104bc}
\item[Binary:] \path{research-notes/next-push/binary-cert-0.7580758318008816.json}\\
\nolinkurl{1b388dd3cfda324a57d985fc96746a8d36b8e7d2e4ac25ecd264c854a8cb8af7}
\end{description}
The SHA-256 of the embedded ZIP archive is
\begin{center}\small
\nolinkurl{088f7876b76fd83745932beec88548df796ac02a2c3b66907608c79b721a8295}.
\end{center}
The attached program checks this identity before unpacking.

\section{Index of numbered statements}\label{app:index}
The table lists every numbered statement in order, what it provides, and the
statements it uses (definitions included). The last column gives the
corresponding node of the machine-readable dependency graph
\texttt{proof-structure/graph.json} distributed with the source files; a node
may cover several statements.

{\small
\begin{longtable}{@{}p{2.3cm}p{4.9cm}p{4.5cm}>{\footnotesize\ttfamily}p{2.9cm}@{}}
\toprule
Statement & Content & Uses & Node\\\midrule
\endhead
\bottomrule
\endfoot
Thm~\ref{thm:main} & $D(N)\ge N^{\al-o(1)}$ & \ref{thm:criterion}, \ref{lem:finitewidth}, \ref{prop:chains}, \ref{prop:odd215}, \ref{prop:odd437}, \ref{prop:binary}, \ref{cert:budget} & T.MAIN\\
Def~\ref{def:sqrel} & square residues and differences & --- & D.SQREL\\
Lem~\ref{lem:crtsq} & squares under CRT & \ref{def:sqrel} & N.CRT\\
Def~\ref{def:ranked} & ranked alphabet, height & \ref{def:sqrel} & D.RANKED\\
Lem~\ref{lem:reverse} & reverse-rank realization & \ref{def:ranked} & L.REVERSE\\
Lem~\ref{lem:crt} & gluing, ranks add & \ref{def:ranked}, \ref{lem:crtsq} & L.CRT\\
Def~\ref{def:words} & square base, words & --- & D.WORDS\\
Lem~\ref{lem:sqdiv} & $s^2\mid z^2\Rightarrow s\mid z$ & --- & N.SQDIV\\
Lem~\ref{lem:firstdiff} & first differing letter & \ref{def:words}, \ref{lem:sqdiv} & L.WORDS\\
Lem~\ref{lem:words} & ranked words & \ref{def:ranked}, \ref{lem:firstdiff} & L.WORDS\\
Prop~\ref{prop:amplify} & amplification to every $N$ & \ref{lem:words}, \ref{lem:reverse} & P.AMPLIFY\\
Rem~\ref{rem:ruzsa} & Ruzsa's construction (not used) & \ref{prop:amplify} & ---\\
Def~\ref{def:interval} & ordered interval alphabet, moment, test & \ref{def:sqrel} & D.INTERVAL\\
Rem~\ref{rem:ranked} & ranks are equal widths (not used) & \ref{def:ranked}, \ref{def:interval} & X.CONTRIB\\
Def~\ref{def:admissible} & admissible contribution, exact crossing & \ref{def:interval} & D.INTERVAL\\
Thm~\ref{thm:criterion} & interval-moment criterion & \ref{prop:block}, \ref{prop:amplify} & T.CRITERION\\
Lem~\ref{lem:round} & intervals to ranks & \ref{def:ranked}, \ref{def:interval} & L.ROUND\\
Def~\ref{def:wordint} & word intervals & \ref{def:words}, \ref{def:interval} & D.WORDS\\
Lem~\ref{lem:intwords} & interval words & \ref{def:wordint}, \ref{lem:firstdiff} & L.WORDS\\
Lem~\ref{lem:stopping} & stopping and selecting a length & \ref{def:wordint} & L.STOP\\
Lem~\ref{lem:finitewidth} & common width bounds & \ref{def:interval} & L.FINITEWIDTH\\
Prop~\ref{prop:block} & one finite ranked block & \ref{lem:stopping}, \ref{lem:intwords}, \ref{lem:round}, \ref{lem:crt} & P.BLOCK\\
Rem~\ref{rem:entropy} & entropy accounting (not used) & \ref{def:interval}, \ref{lem:intwords} & X.ENTROPY\\
Lem~\ref{lem:padic} & square differences modulo $p^{2e}$ & \ref{def:sqrel} & N.PADIC\\
Lem~\ref{lem:minusone} & $-1$ is a non-square, $p\equiv3\ (4)$ & \ref{def:sqrel} & N.QR3MOD4\\
Def~\ref{def:chain} & Paley chain & \ref{def:sqrel} & D.CHAIN\\
Lem~\ref{lem:chain} & prime-chain alphabet & \ref{def:chain}, \ref{lem:minusone}, \ref{lem:padic} & L.CHAIN\\
Cor~\ref{cor:chainpower} & chain contribution & \ref{lem:chain} & C.CHAINPOWER\\
Rem~\ref{rem:krachun} & Krachun's exponent (not used) & \ref{cor:chainpower}, \ref{thm:criterion} & ---\\
Def~\ref{def:lowcode} & low code, prime-specific relation & \ref{def:sqrel} & D.LOWCODE\\
Rem~\ref{rem:composite} & how two primes cooperate (not used) & \ref{def:lowcode}, \ref{lem:minusone} & ---\\
Lem~\ref{lem:odd} & composite-code lift, any depths & \ref{def:lowcode}, \ref{lem:padic}, \ref{lem:crtsq} & L.ODDLIFT\\
Lem~\ref{lem:paths} & path budgets & --- & L.PATHS\\
Cor~\ref{cor:oddtest} & odd test on low data & \ref{lem:odd}, \ref{lem:paths} & C.ODDTEST\\
Rem~\ref{rem:fiber} & fiber averaging (not used) & \ref{lem:odd}, \ref{lem:paths} & X.FIBER\\
Lem~\ref{lem:dyadic} & square differences modulo $4^m$ & \ref{def:sqrel} & N.DYADIC\\
Def~\ref{def:policy} & parity-window policy & \ref{def:interval} & D.POLICY\\
Lem~\ref{lem:binary} & binary transition & \ref{def:policy}, \ref{lem:dyadic} & L.BINSTEP\\
Def~\ref{def:growthcert} & growth certificate & \ref{def:policy} & D.GROWTHCERT\\
Lem~\ref{lem:growth} & moment growth & \ref{lem:binary}, \ref{def:growthcert} & L.BINGROWTH\\
Cor~\ref{cor:bindepth} & root at a fixed depth & \ref{lem:growth} & C.BINDEPTH\\
Lem~\ref{lem:logenc} & rational logarithm and power bounds & --- & N.LOGENC\\
Cert~\ref{cert:chains} & six chains and their contributions & \ref{def:chain}, \ref{cor:chainpower}, \ref{lem:logenc} & W.CHAINS\\
Prop~\ref{prop:chains} & chain components & \ref{lem:chain}, \ref{cor:chainpower}, \ref{cert:chains} & T.CHAINS\\
Cert~\ref{cert:odd215} & the $(5,43)$ low code & \ref{def:lowcode}, \ref{cor:oddtest}, \ref{lem:logenc} & W.ODD215\\
Prop~\ref{prop:odd215} & the $(5,43)$ component & \ref{cor:oddtest}, \ref{cert:odd215} & T.ODD215\\
Cert~\ref{cert:odd437} & the $(19,23)$ low code & \ref{def:lowcode}, \ref{cor:oddtest}, \ref{lem:logenc} & W.ODD437\\
Prop~\ref{prop:odd437} & the $(19,23)$ component & \ref{lem:paths}, \ref{cor:oddtest}, \ref{cert:odd437} & T.ODD437\\
Cert~\ref{cert:binary} & the binary policy & \ref{def:policy}, \ref{def:growthcert}, \ref{cor:bindepth}, \ref{lem:logenc} & W.BINARY\\
Prop~\ref{prop:binary} & the binary component & \ref{lem:binary}, \ref{lem:growth}, \ref{cor:bindepth}, \ref{cert:binary} & T.BINARY\\
Cert~\ref{cert:budget} & coprimality and budget & \ref{cert:chains}, \ref{cert:odd215}, \ref{cert:odd437}, \ref{cert:binary} & W.BUDGET\\
\end{longtable}
}

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\end{document}