% Hexagon source adaptation of the published Zenodo edition. % Changes are limited to packaging, bibliography rendering, navigation and archive links. % Mathematical formulas and their order are preserved; see the local integrity record. \PassOptionsToPackage{expansion=false,nopatch=footnote}{microtype} \documentclass[11pt,a4paper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern,microtype} \usepackage[margin=28mm,headheight=14pt]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools,aliascnt} \usepackage{booktabs,array,longtable} \usepackage{xcolor} \usepackage{enumitem,etoolbox,needspace} \usepackage{fancyhdr} \usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black,breaklinks=true]{hyperref} \usepackage[nameinlink,noabbrev]{cleveref} \hypersetup{pdftitle={Arithmetic coding against unknown small-prime signs},pdfauthor={Brice Pouly},pdfsubject={Multiplicative coding, explicit analytic input, persistent prescriptions and entropy}} \pagestyle{fancy}\fancyhf{}\fancyhead[L]{\small Arithmetic coding against unknown small-prime signs}\fancyhead[R]{\small Brice Pouly}\fancyfoot[C]{\thepage} \renewcommand{\headrulewidth}{0.3pt} \setlength{\parskip}{3pt}\setlength{\parindent}{1.2em} \setcounter{tocdepth}{1} \hypersetup{bookmarksdepth=1} % Use the existing heading destinations, not a second anchor below the heading. % The second argument is a stable descriptive key retained by the source tooling. \makeatletter \newcommand{\subbookmark}[2]{% \begingroup\def\Hy@bookmarksdepth{2}% \Hy@writebookmark{}{\thesubsection\space #1}{\@currentHref}{2}{toc}% \endgroup} \patchcmd{\thebibliography}{\section*{\refname}}{\section*{\refname}\Hy@writebookmark{}{References}{\@currentHref}{1}{toc}}{}{\PackageError{N5}{Cannot install the references bookmark}{Check the bibliography heading.}} \makeatother \setlist[enumerate]{itemsep=2pt,topsep=4pt} \newtheorem{theorem}{Theorem}[section] \newaliascnt{lemma}{theorem} \newtheorem{lemma}[lemma]{Lemma}\aliascntresetthe{lemma} \newaliascnt{proposition}{theorem} \newtheorem{proposition}[proposition]{Proposition}\aliascntresetthe{proposition} \newaliascnt{corollary}{theorem} \newtheorem{corollary}[corollary]{Corollary}\aliascntresetthe{corollary} \theoremstyle{definition} \newaliascnt{definition}{theorem} \newtheorem{definition}[definition]{Definition}\aliascntresetthe{definition} \theoremstyle{remark} \newaliascnt{remark}{theorem} \newtheorem{remark}[remark]{Remark}\aliascntresetthe{remark} \crefname{theorem}{theorem}{theorems}\Crefname{theorem}{Theorem}{Theorems} \crefname{lemma}{lemma}{lemmas}\Crefname{lemma}{Lemma}{Lemmas} \crefname{proposition}{proposition}{propositions}\Crefname{proposition}{Proposition}{Propositions} \crefname{corollary}{corollary}{corollaries}\Crefname{corollary}{Corollary}{Corollaries} \newcommand{\N}{\mathbb N}\newcommand{\Z}{\mathbb Z}\newcommand{\F}{\mathbb F} \newcommand{\M}{\mathcal M_{\pm}}\newcommand{\Dset}{\mathcal D} \newcommand{\one}{\mathbf 1}\newcommand{\wt}{\operatorname{wt}} \newcommand{\core}{\operatorname{core}}\newcommand{\bl}{\operatorname{bl}} \newcommand{\TV}{d_{\mathrm{TV}}}\newcommand{\Ent}{\mathsf H} \newcommand{\GF}{\mathrm{GF}}\newcommand{\dd}{\mathrm d} \DeclareMathOperator{\rank}{rank}\DeclareMathOperator{\dist}{dist} \newcommand{\floor}[1]{\left\lfloor #1\right\rfloor}\newcommand{\ceil}[1]{\left\lceil #1\right\rceil} \numberwithin{equation}{section} \title{\vspace{-1.2em}Arithmetic coding against\\ unknown small-prime signs} \author{Brice Pouly\\[3pt]\small With AI assistance} \date{} \usepackage{bookmark} \begin{document} \maketitle\thispagestyle{plain} \begin{abstract} We encode binary messages in completely multiplicative sign functions and read them along the three sequences $n^2+1$, $n^2+2$ and $n^2-2$. Two observations with the same exact cofactor remove the unknown smaller-prime signs when multiplied. Allocating shared large primes before pairing embeds ordinary binary error-correcting codes in each trace. Under an explicit large-factor mass hypothesis, a computable schedule gives one infinite encoding with positive prefix rate and a linear errors-and-erasures allowance. A first-occurrence filter makes its raw readings depend causally on a linear message prefix, while encoding and decoding remain polynomial in the numerical observation length. Its supporting primes have arbitrarily small harmonic cost. A finite construction, independent of the analytic hypothesis, stores $273$ bits in $8736$ readings of each trace and corrects every pattern of $499$ reading errors. Complementary constructions give persistent sign prescriptions, uniform laws for the active coordinates of words, and compact multiplicative subgroups whose nontrivial quotients have positive quadratic entropy. The word-law and entropy conclusions are not asserted for the polynomially decoded encoding. A separate matrix construction gives another causal positive-rate map. Fixed-scale balancing provides an unconditional quarter-sign bound against every small-prime assignment. All coding statements concern deliberately constructed families, not recovery of an arbitrary multiplicative source. The sparse constructions at superlinear cutoffs keep their large-factor mass input explicit. \end{abstract} \begingroup \small \setlength{\parskip}{0pt} \makeatletter \renewcommand*\l@section[2]{\vspace{2pt}\@dottedtocline{1}{0em}{2.1em}{#1}{#2}} \makeatother \tableofcontents \endgroup \newpage % Original source: 01_introduction.tex \section{Encoding a message without knowing the small signs}\label{sec:intro} A completely multiplicative function with values in $\{-1,1\}$ is determined by its signs at the primes. This makes it natural to place information at large primes. It does not make that information directly readable: an observation of $h(aq)$ is the intended large-prime sign $h(q)$ multiplied by an unknown cofactor sign $h(a)$. The difficulty is not ordinary bit noise. A single change at a small prime can alter many readings at once, and the receiver is not given those changes. Here is the cancellation that will be used throughout the main construction: \begin{equation}\label{eq:example} 2067^2+1=10\cdot427249,\qquad 2073^2+1=10\cdot429733. \end{equation} The two displayed large factors are prime. For every completely multiplicative sign function, \[ h(2067^2+1)h(2073^2+1)=h(427249)h(429733). \] Prescribing the latter two signs as $(-1)^b$ and $1$ makes the product reveal the bit $b$, irrespective of $h(2)$ and $h(5)$. The same argument works for any exact common cofactor. It does not require the cofactors attached to different pairs to be independent, known, or distinct. One pair transmits one bit, but provides no protection against a reading error. We therefore put an error-correcting code on the pair products. The arithmetic tasks are to find enough pairs, to allocate a prime shared by several traces only once, and to preserve every earlier assignment while extending the same infinite message. These are separate from the classical coding tasks. \subsection{The observation model and the principal theorem} Write \[ \Dset=\{-1,-2,2\},\qquad Q_d(n)=n^2-d, \] and let $\M$ be the group of completely multiplicative functions $\N_{\ge1}\to\{-1,1\}$. We retain the label $(n,d)$ whenever $Q_d(n)>0$; repeated integer values at distinct labels remain distinct observations. Logs are natural unless their base is indicated. The distance between two prime assignments is the possibly infinite quantity \begin{equation}\label{eq:prime-distance} d_\sigma(f,g)=\sum_{q\ {\rm prime}:\,f(q)\ne g(q)}q^{-\sigma},\qquad 0<\sigma\le1. \end{equation} An error changes a received sign, whereas an erasure marks a missing reading. Their numbers refer to labels, not to the number of modified prime signs. The analytic input is a uniform population statement. For a fixed rational $1<\gamma<6/5$, set \begin{align} R_\gamma(X)&=\max\{2X,\floor{X^\gamma}\},\label{eq:Rgamma}\\ M_d^\gamma(X)&=\#\{X/2R_\gamma(X)\}. \end{align} Here $P^+(1)=1$. The hypothesis used below is \begin{equation*}\tag{$\GF_\gamma$}\label{hyp:GF} \text{there exist }a>0,X_a\text{ such that } \min_{d\in\Dset}M_d^\gamma(X)\ge aX \quad\text{for every integer }X\ge X_a. \end{equation*} Section~\ref{sec:analytic} gives a sufficient weighted input and its counting consequence; Section~\ref{sec:provenance} discusses a possible analytic route to that input. One large prime at each scale cannot replace \eqref{hyp:GF}. \Needspace{4\baselineskip} \begin{theorem}[Causal prefix dependence with polynomial decoding]\label{thm:main} Fix rational numbers $1<\vartheta<\gamma<6/5$ and assume \eqref{hyp:GF}. For every rational $\varepsilon>0$ and integer $B\ge1$, there exist a decidable set of primes $E\subset(B,\infty)$ and a continuous injective homomorphism \[ \Omega:\F_2^{\N_{\ge1}}\longrightarrow\M, \qquad \Omega_{x+y}=\Omega_x\Omega_y, \] computable relative to $x$, such that $\Omega_x(q)=1$ off $E$ and $\sum_{q\in E}q^{-1}<\varepsilon$. There are constants $r,c>0$ and $N_0$ such that, for all $N\ge N_0$, each one of the three traces up to $N$ recovers the first $\floor{rN}$ message bits under the following conditions: all prime signs up to $N^\vartheta$ may have been reassigned arbitrarily; later message bits are unknown; and the received trace has $t$ errors and $s$ erasures with \begin{equation}\label{eq:main-radius} 2t+s0} \] depends only on $x_1,\ldots,x_{\floor{3N/4}}$, for every $N\ge1$. The encoding tables, the requested encoded readings, and the decoder are computable in time polynomial in the numerical parameter $N$, for fixed construction parameters. The threshold $N_0$ is common to all messages and reassignments. \end{theorem} The proof supplies the stronger finite fact that only the two reserved signs of a used pair matter. Every other prime sign may be arbitrary. Neither a factorisation oracle nor an enumeration of all small-prime assignments is used by the decoder. The causal statement concerns message dependence, not bounded computational delay: the public arithmetic tables can require finite anticipation of later epochs. Nor does it restrict an adversary's later, possibly adaptive, prime reassignment. Polynomial in $N$ is not polynomial in the bit length $\log N$ of a single index. A binary code of length $L$ uses $2L$ arithmetic readings on one trace. The words transmitted on the three traces encode the same prefix, not three unrelated messages. In the finite implementation of Section~\ref{sec:finite}, $L=4368$, the message has $273$ bits, and the guaranteed binary distance after pairing is $1000$. Consequently every $499$ reading errors can be corrected from any one trace. This finite assertion does not use \eqref{hyp:GF} and does not determine $r,c$ or $N_0$. The small harmonic support has an elementary additional consequence. For every two messages and every $T\ge1$, \begin{equation}\label{eq:ordinary-closeness} \#\{m\le T:\Omega_x(m)\ne\Omega_y(m)\} \le T\sum_{q\in E}q^{-1}<\varepsilon T. \end{equation} Thus a code can be almost constant on ordinary integers while carrying a positive amount of recoverable information on the quadratic observations. The latter are not uniform samples of the former. \subsection{Three complementary ways to use the large primes} Exact cancellation is the route to polynomial decoding. Other uses of the same arithmetic resource retain properties which are not part of \cref{thm:main}. Persistent prescriptions in Section~\ref{sec:prescriptions} balance the large signs before an arbitrary small-prime assignment is known. Their finite discrepancy estimate is uniform in the observation window. Under the stated mass input, a single computable prescription keeps both signs present on each trace at all large scales. On $n^2+1$ alone, exact alternation is possible and has a wider analytic input, associated with the weighted estimate at exponent $13/10$. Sections~\ref{sec:masks}--\ref{sec:groups} instead make the signs on an arithmetically determined set of active word coordinates nearly uniform. A finite information budget pays for every small-prime assignment and every currently visible message. One deterministic selection then gives a compact multiplicative subgroup in an arbitrarily small prime ball; each fixed nontrivial quotient has positive entropy on the quadratic traces but zero entropy on the ordinary integers. The identity is excluded from the positive-entropy assertion. These groups are not asserted to have the polynomial decoder of \cref{thm:main}. Section~\ref{sec:clock} gives an alternative positive-rate construction by selecting a matrix whose row lengths are fixed by first admissible occurrences. It has causal raw dependence and computable selection and decoding, but no polynomial bound for those algorithms is proved. This is not a necessary computational price of causality: the first-occurrence filter in Section~\ref{sec:schedule} already gives that property to the polynomially decoded paired construction. The alternative retains a different, concentration-based organisation of the information budget. Finally, Appendix~\ref{app:balance} works at the smaller cutoff $2N$, without \eqref{hyp:GF}. It gives a quarter of each sign, up to $O(N/\sqrt{\log N})$, against every small-prime assignment. It also describes the small-dimensional cofactor code near which an almost-positive source must lie. A small set of possible tails is not an empty set of possible tails. \subsection{Relation to earlier methods and a route through the proof} Large prime factors as private coordinates of polynomial observations occur in earlier work on multiplicative correlations, including Klurman~\cite[Section~2, Lemma~2.1]{Klurman}. In that lemma, the small-prime-power data are fixed first, and the remaining prime values are then chosen to produce a large quadratic mean. Here the arithmetic families are completely multiplicative and sign-valued, and the message is read despite small signs chosen after the encoding. The order of choices and the information recovered differ; we do not identify the two statements. The required simultaneous allocation is important here: a prime is private to one location on a fixed trace, but can occur on all three traces. The same paper's proof of Proposition~1.1 also uses widely spaced scales and a summable reciprocal-prime cost; Appendix~\ref{app:balance} identifies these antecedents locally. The local threshold $2n$ and first appearance of a large factor are also classical in the literature on primitive divisors of quadratic values, as recorded by Everest--Harman~\cite[Proposition~1.3]{EverestHarman}. The contribution of the paired construction lies in combining this arithmetic resource with disjoint two-reading cancellation, compatible allocation between traces, and one persistent causal encoding. A dense elimination of the small-sign variables would not by itself preserve the same reading-error budget; the disjoint pairs do. The binary codes are classical concatenated codes: a short linear inner code, Reed--Solomon evaluation~\cite{ReedSolomon} outside, and Forney's generalised minimum-distance decoding~\cite{Forney}. We use the exposition in Guruswami's notes~\cite[Sections~1 and~4]{Guruswami6}, \cite[Sections~3.3 and~5]{Guruswami7}, and give the errors-and-erasures argument in the precise form needed here. We do not claim a new rate--distance tradeoff for abstract binary codes. The balancing argument uses floating variables of Beck--Fiala type~\cite{BeckFiala,Nikolov}. For the ordered lists arising here, it also belongs to the classical discrepancy problem for three permutations~\cite{Bohus,NewmanNikolov}; Section~\ref{sec:prescriptions} gives the reduction. The effective selections use the classical conditional-expectation principle; constructive statistical selection has a broader antecedent in Galatolo--Hoyrup--Rojas~\cite{GHR}. Section~\ref{sec:selection} compares the exact finite-cost selections in the companion papers~\cite[Section~5]{PoulySelective} and~\cite[Section~7]{PoulyCorrelation} with the integer bad-event potential used here. The proof below is autonomous. The recovery problem in~\cite[Theorems~1.1--1.2]{PoulyReconstruction} concerns arbitrary multiplicative sources and the complete set of compatible readings. Here the sender chooses a family with additional coded redundancy. A linear corruption allowance for this family does not give the same allowance for every source. For the principal theorem, Sections~\ref{sec:arithmetic}--\ref{sec:schedule} are a complete route: count candidate primes, control their first occurrences, allocate and pair them, construct a classical code, and prove causal message dependence, with bounded multiplicative gaps under \eqref{hyp:GF}. Section~\ref{sec:finite} turns the same finite mechanism into inspectable arithmetic data. The later sections can be read independently after the elementary geometry and support estimates. % Original source: 02_arithmetic.tex \section{Arithmetic resources and the mass input}\label{sec:arithmetic} The coding argument needs many independent prime assignments, not independence of the observed integers. Two elementary facts identify which prime assignments can be used and the cost of their cofactors. \begin{lemma}[Uniqueness on one trace]\label{lem:private} Let $X\ge5$ and $1\le n\le X$. If a prime $q>2X$ divides $Q_d(n)>0$, it is the unique prime factor greater than $2X$, has valuation one, and \[ Q_d(n)=qa,\qquad 1\le a< X/2. \] For a fixed $d$, the same $q$ divides at most one of these values. Across the three traces it divides at most three labelled observations. \end{lemma} \begin{proof} Two factors greater than $2X$, including a repeated one, have product at least $(2X+1)^2>X^2+2$. The cofactor satisfies \[ a\le\frac{X^2+2}{2X+1}2X$. \end{proof} The local version of this geometry has a classical primitive-divisor interpretation. For a fixed integer $b$ with $-b$ not a square and $n>|b|$, Everest--Harman~\cite[Proposition~1.3]{EverestHarman} record that $n^2+b$ has a primitive divisor exactly when $P^+(n^2+b)>2n$; it is then a unique prime. Here a primitive divisor is coprime to every earlier nonzero term. Our common window threshold $2X$, and especially $R_\gamma(X)$, impose additional requirements. The cited local criterion is not the population hypothesis \eqref{hyp:GF}, and it does not separate primes shared by different polynomials. For example, the prime $41$ occurs in $9^2+1=2\cdot41$, $11^2+2=3\cdot41$ and $17^2-2=7\cdot41$. It must receive only one sign. Treating those three occurrences as independent prime bits would change the source model. At cutoff $R_\gamma(X)$ the exact cofactors belong to \begin{equation}\label{eq:cofactor-count} 1\le a\le A_\gamma(X):=\floor{\frac{X^2+2}{R_\gamma(X)+1}} =O_\gamma(X^{2-\gamma})=o(X). \end{equation} This is the gain used in pairing. A linear population of readings split among $o(X)$ cofactor classes loses only $o(X)$ readings when at most one is left unpaired in each class. \subsection{From a weighted deficit to a population}\label{sec:analytic} \subbookmark{Weighted deficit and counting mass}{nav:weighted} The exact role of \eqref{hyp:GF} is worth isolating. Fix a nonnegative smooth function $b$, supported in $(1,2)$, with $b\le1$ and $I_b=\int b>0$. For each of the three fixed $d$, put \[ \mathcal A_{d,q}(x)=\sum_n b(n/x)\one_{q\mid n^2-d},\qquad W=xI_b. \] A sufficient analytic input at an exponent $u>1$ is \begin{equation}\label{eq:weighted-input} \sum_{q>x^u}\mathcal A_{d,q}(x)\log q\ge\eta_d W\log x \quad(x\ge x_d),\qquad \eta_d>0. \end{equation} All sums over $q$ in this section are over primes. \begin{lemma}[Weighted mass implies the required counting mass]\label{lem:weighted} If \eqref{eq:weighted-input} holds for every $d\in\Dset$, then \eqref{hyp:GF} holds for every fixed $1<\gamma\max\{2X,X^\gamma\}. \] Taking the minimum of the three positive constants and slightly decreasing it for integer endpoints proves the claim. No constant is required to be uniform in a moving polynomial. \end{proof} This elementary last step requires a weighted lower bound of order $x\log x$. The inequality $P^+(\prod_{xx^u$ alone does not provide it. \subsection{A possible analytic route to the mass hypothesis}\label{sec:provenance} \subbookmark{Possible analytic route}{nav:analytic} This subsection describes a possible route to \eqref{hyp:GF}, not an independent proof of it. The simultaneous sparse constructions retain that hypothesis. The weighted Chebyshev argument in de la Bret\`eche--Drappeau~\cite[Theorem~1.2 and Section~8.4]{BD}, together with the sieve main-term calculation of Deshouillers--Iwaniec~\cite[Section~8]{DI}, motivates the estimates below. The references identify the published editions; the locators for de la Bret\`eche--Drappeau refer to arXiv:1703.03197v3. We write $D_{\rm s}$ for the sieve level, to distinguish it from the nonsquare integer defining the polynomial. Section~8.4 of~\cite{BD} obtains, with $P=x^t$ and spectral parameter $s=7/64$, an error estimate of the shape \[ R_H\ll x^{\epsilon+O(\eta)} \bigl(x^{-1}D_{\rm s}P+(D_{\rm s}P)^{1/2} +x^{1/2}P^sD_{\rm s}^{1-2s}\bigr). \] The admissible level relevant to the proposed transfer is \begin{equation}\label{eq:sieve-level} D_{\rm s}=x^{\delta_s(t)-\eta},\qquad \delta_s(t)=\frac{1-2st}{2(1-2s)},\qquad 1\le t\le6/5, \end{equation} with a sufficiently enlarged fixed loss in $\eta$ when inserted into the error estimate. In this interval $\delta_s(t)\le1/2$ and $\delta_s(t)<2-t$, so the ancillary constraints $D_{\rm s}\le x^{1/2}$ and $D_{\rm s}\ll x^2/P$ are compatible. The smoothed upper-bound sieve estimate needed in addition is \begin{equation}\label{eq:sieve-deficit} \limsup_{x\to\infty} \frac{1}{W\log x} \sum_{xx}\mathcal A_{d,q}(x)\log q =W\log x+O_{d,b}(x). \end{equation} This is not the logarithm of the full product: the full weighted logarithmic sum is \[ \sum_n b(n/x)\log(n^2-d)=2W\log x+O_{d,b}(x). \] The Chebyshev decomposition removes the contribution of primes at most $x$ and accounts for repeated prime powers before reaching \eqref{eq:weighted-total}. Combining \eqref{eq:weighted-total} with the smoothed bound \eqref{eq:sieve-deficit} and the strict margin \eqref{eq:margin} would yield \eqref{eq:weighted-input} at $u=6/5$. The numerical margin does not prove the required smoothed main-term estimate for each polynomial. That estimate, rather than the numerical comparison, is the additional analytic obligation in this proposed route. For $n^2+1$ alone, the larger sufficient weighted input at $u=13/10$ is located in Pascadi~\cite[Sections~6.1 and~6.3, equation~(6.20)]{Pascadi}. The final sieve calculation has a strict deficit even at $1.30008$. Its use here is precisely \eqref{eq:weighted-input}, followed by \cref{lem:weighted}, not an inference from the infinitude formulation of the theorem. Section~\ref{sec:single} records the construction under this single-trace input. No transfer of Pascadi's range to the other two polynomials is assumed. \paragraph{Larger prime-factor exponents versus the mass used here.} Two preprints from 2025 refine the greatest-prime-factor problem. Grimmelt--Merikoski~\cite[Theorem~1.1]{GrimmeltMerikoski} obtain exponent $1.312$ for $an^2+h$ in their stated positive-parameter range, with a real-character hypothesis for the moving-parameter uniformity; fixed admissible positive $a,h$ satisfy their hypothesis for sufficiently large scales. This includes the fixed cases $n^2+1$ and $n^2+2$. Li~\cite[Theorem~1.1]{Li} states exponent $1.317$ in a further refinement, in the specifically identified second version of that preprint. These are context, not inputs whose full analytic proofs or numerical sieve calculations are reverified here. Their positive-$h$ statements are not automatically assertions for $n^2-2$. Nor can their existence conclusions alone replace the simultaneous weighted population \eqref{hyp:GF}. We therefore leave the range and conditional status of the coding theorems unchanged. \subsection{A common sparse support}\label{sec:sparse} The later constructions use the cutoff \[ R_\vartheta(n)=\max\{2n,\floor{n^\vartheta}\},\qquad 1<\vartheta<2. \] An occurrence $(n,d)$ is \emph{admissible} if $q\mid Q_d(n)$ and $q>R_\vartheta(n)$. The same elementary root argument as in \cref{lem:private} shows that a prime has at most one admissible occurrence on each trace, even over the entire infinite sequence: both indices of two putative occurrences would be less than $q/2$. For $L\ge2$ let $E_L$ be the set of primes having an admissible occurrence with $n\ge L$. One can assign each prime a single such occurrence, so \begin{equation}\label{eq:support-cost} \sum_{q\in E_L}q^{-\sigma} <3\sum_{n\ge L}n^{-\vartheta\sigma} \le\frac{3(L-1)^{1-\vartheta\sigma}}{\vartheta\sigma-1} \quad(\vartheta\sigma>1), \end{equation} and $\#(E_L\cap[1,y])\le3y^{1/\vartheta}$. This is a bound on the entire possible support, before any message or sign is chosen. Membership is decidable. For a given prime $q$, inspect the centred roots of $n^2\equiv d\pmod q$ with $1\le nR_\vartheta(n)$. Rational exponents can be tested by integer powers. Thus neither decidability nor the choice of $L$ in \eqref{eq:support-cost} uses an unknown threshold in \eqref{hyp:GF}. \subsection{First occurrences control every raw reading}\label{sec:first-occurrence} \subbookmark{First occurrences and raw readings}{nav:first} To limit the message bits that a raw reading can consult, the first admissible occurrence is more useful than the epoch at which a prime is reserved. For the cutoff $R_\vartheta$ above, define \begin{equation}\label{eq:nu-def} \nu(q)=\min\{n\ge2:\text{ for some }d\in\Dset,\ q\mid Q_d(n),\ q>R_\vartheta(n)\} \end{equation} whenever this set is nonempty. This minimum is taken across all three traces. It is a common-cutoff refinement of the first-appearance geometry discussed after \cref{lem:private}, not a new primitive-divisor criterion. \begin{lemma}[First occurrence and all later uses]\label{lem:first-occurrence} If $\nu(q)$ is defined and $q$ divides a positive value $Q_d(n)$ with $n\ge2$, then \begin{equation}\label{eq:clock-causal} \nu(q)\le n. \end{equation} There are at most $3(s-2)_+$ primes with $\nu(q)R_\vartheta(n)$, then $n$ participates in the defining minimum. Otherwise $R_\vartheta(\nu(q))R_\gamma(X)$ and an occurrence is already known at an index at most $X$. Every smaller index $n\le X$ satisfies $q>R_\vartheta(n)$ when $\vartheta<\gamma$. Thus $\nu(q)$ can be found by inspecting the three traces only up to $X$. In particular, the test $\nu(q)0$ and $P\le X/4$, then, for all sufficiently large $X$, \[ m=O(\log X),\qquad 2^m=O(P),\qquad L\ge P/2,\qquad b\ge P/32,\qquad D\ge P/32. \] The implied constants and the threshold may depend on $\beta$. \end{lemma} \begin{proof} Minimality in \eqref{eq:code-parameters} gives $m=O(\log P)$ and $2^m=O(P)$ (the finitely many cases $m=2$ can be absorbed). Also $P-8mX^\gamma$, so \begin{equation}\label{eq:moving-cutoff} q>X^\gamma\ge(N/C)^\gamma>N^\vartheta \end{equation} for all sufficiently large $N$, because $\gamma>\vartheta$ is fixed. The first $b_X$ bits are consequently recoverable even after arbitrary changes of all prime signs at most $N^\vartheta$. Cofactor signs need not be preserved; they cancel by \eqref{eq:cancellation}. Equations \eqref{eq:epoch-rate}--\eqref{eq:moving-cutoff} and the finite decoder prove the recovery assertion of \cref{thm:main}, with, for instance, any fixed \[ 00$ with $\gamma\sigma>1$, the support satisfies \begin{equation}\label{eq:epoch-cost} \sum_{q\in E}q^{-\sigma} <3\sum_{j\ge0}(2^jX_0)^{1-\gamma\sigma} =\frac{3X_0^{1-\gamma\sigma}}{1-2^{1-\gamma\sigma}}. \end{equation} Increasing $X_0$ makes this smaller than a prescribed positive budget and places every reserved prime above $B$. For $\sigma=1$ this gives the support assertion of \cref{thm:main}; the same construction permits the stronger budget $1/\gamma<\sigma\le1$. To prove continuity in $d_\sigma$, choose finitely many reserved primes whose complement has total weight less than $\eta$. Their signs depend on a common finite message prefix. Two messages agreeing on that prefix have $d_\sigma$-distance less than $\eta$. The image is thus compact and, by injectivity, homeomorphic to the Cantor group $\F_2^{\N_{\ge1}}$. Formula \eqref{eq:ordinary-closeness} follows because every integer disagreement has a divisor in the set of disagreeing primes. \subsection{What the algorithm must compute} \subbookmark{Computation and causal dependence}{nav:causal} Membership of a prime $q$ in $E$ is decidable: only candidate scales with $X1/10$. Formula \eqref{eq:epoch-cost} bounds the future cost by \[ \frac{3X_*^{-1/6}}{1-2^{-1/6}}<\frac{30}{1920}=\frac1{64}. \] Consequently the complete possible support, including the uncomputed continuation, has \begin{equation}\label{eq:pilot-cost} \sum_{q\in E}\frac1q <\frac{27697555482416339}{10^{18}}+\frac1{64} =0.043322555482416339<\frac1{20}. \end{equation} The bound holds whether or not later candidates succeed. Infinitely many successes and positive rate at all sufficiently large $N$ require \eqref{hyp:GF}. This pilot uses $\gamma=7/6$ and therefore supports the asymptotic resistance range $1<\vartheta<7/6$, under that input. The general theorem uses a different chosen $\gamma$ when $\vartheta$ is larger. No future epoch at or beyond $X_*$ was executed for the finite instance. The long gap also prevents interpolating the pilot's numerical rate across intermediate scales. % Original source: 07_prescriptions.tex \section{Persistent prescriptions before the small signs are known}\label{sec:prescriptions} Encoding a message is one use of a large-prime assignment. Another is to force both signs to remain present, irrespective of how the small signs are later chosen. A finite balancing rule must now be made consistent over every observation window and every scale. Exact cofactors again supply the order needed for that consistency. \subsection{One trace: exact alternation}\label{sec:single} Fix $1<\vartheta<2$ and retain the occurrences \[ n\ge L,\qquad n^2+1=aq,\qquad q>R_\vartheta(n). \] On this one trace, a prime has only one retained occurrence in the entire sequence. For each exact cofactor $a$, list the retained indices increasingly as $n_{a,1}1$ the support costs at most $(L-1)^{1-\vartheta\sigma}/(\vartheta\sigma-1)$. Membership and the parity of the rank $j$ in \eqref{eq:alternation} are decidable by finite arithmetic searches. This constructs the same prescription at all sizes, rather than a new optimised table for each $N$. \Needspace{4\baselineskip} \begin{corollary}[Single-trace range]\label{cor:single} Assume the weighted input \eqref{eq:weighted-input} for $d=-1$ at $u=13/10$. For rational parameters $1<\vartheta<13/10$ and $1/\vartheta<\sigma\le1$, there is a computable prescription of arbitrarily small $\sigma$-cost, above any prescribed prime prefix, such that every sufficiently large interval $(N/2,N]$ contains at least $c_\vartheta N$ occurrences of each sign after any reassignment of the prime signs up to $R_\vartheta(N)$. \end{corollary} \begin{proof} The single-trace version of \cref{lem:weighted}, using an intermediate exponent between $\vartheta$ and $13/10$, gives $M\gg N$ on this interval. The defect in \eqref{eq:single-balance} is $O(N^{2-\vartheta})=o(N)$. Increasing $L$ pays the desired support cost and protects the prefix, without changing the positive asymptotic density constant. \end{proof} The weighted provenance is Pascadi's calculation specified in Section~\ref{sec:provenance}. This wider range is a one-trace statement. It does not prescribe the other two traces, nor follow from the three-trace input at $\gamma<6/5$. \subsection{Why three independent alternations fail} A prime shared between traces cannot be given separate signs. This obstruction already occurs in the following lists, valid for cutoff exponent $7/6$: \[ \begin{array}{c|c|c} (d,a)&n&q\\\hline (-2,3)&7,11,17&17,41,97\\ (2,2)&6,8,12,14&17,31,71,97. \end{array} \] The first alternation would require $\tau(17)=\tau(97)$; the second would require $\tau(17)=-\tau(97)$. No initial phase choice fixes their five-edge cycle. Instead of perfect alternation, we permit a small discrepancy in every list interval. \begin{lemma}[Ordered-list rounding]\label{lem:lists} Let a finite family of ordered lists have $m$ distinct elements, with no element repeated in a list and each element belonging to at most $t$ lists. A deterministic rational procedure chooses signs such that every consecutive interval $J$ of every list satisfies \[ \left|\sum_{q\in J}\tau(q)\right| \le8t\bigl(1+\ceil{\log_2(m+1)}\bigr). \] \end{lemma} \begin{proof} Begin with values $x_q=0$ in $[-1,1]$. A coordinate reaching either endpoint remains fixed. At the beginning of a phase let $u$ coordinates be free. Filter each list to these coordinates, and partition it into consecutive groups of $2t$, discarding the last incomplete group. Preserve each complete group's sum throughout the phase. The number $a$ of equations satisfies $2ta\le tu$, hence $a\le\floor{u/2}$. As long as more than $\floor{u/2}$ coordinates are free, rational elimination finds a nonzero direction preserving all the group sums. Move along it until a coordinate reaches the boundary. Eventually at least half of the phase's free coordinates have become fixed. For any original list interval, its intersection with the free list consists of whole groups and at most two partial boundary groups. The whole-group sums are preserved. There are fewer than $4t$ boundary coordinates, each changing by less than two, so its sum changes by less than $8t$ in this phase. After at most $1+\ceil{\log_2(m+1)}$ phases all coordinates are fixed. Summing the phase defects proves the bound. \end{proof} This is a floating-variable discrepancy argument of Beck--Fiala type~\cite{BeckFiala,Nikolov}, not a new general discrepancy theorem. For the arithmetic application, the sharper comparison is with discrepancy of permutations. Bohus~\cite{Bohus} gives a logarithmic bound and an efficient colouring algorithm for a fixed number of orders. Newman--Nikolov~\cite{NewmanNikolov} give three permutations with discrepancy of order at least $\log m$. That lower bound concerns general permutations; it proves no optimality for the particular arithmetic lists used here. \subsection{Fixing the blocks before the windows} Use the common support $E_L$ of Section~\ref{sec:sparse}. For each $(d,a)$, order its admissible primes by magnitude. They are also ordered by their corresponding indices. Divide the prime axis once and for all into blocks \[ (A_j,A_{j+1}],\qquad A_j=2^{2^j},\qquad A_{j+1}=A_j^2. \] On a finite prime block, the cofactor lists for one trace are disjoint. Concatenate them and append primes absent from that trace to obtain a permutation of the block's support. Repeating this for the three traces gives three permutations, and every original list interval is still an interval of its corresponding permutation. This is the precise reduction to the classical setting just cited. The arithmetic task is to fix these blocks irreversibly and retain their control when the observation window changes. On every block, form all the restricted lists and apply \cref{lem:lists} with $t=3$. Each block is finite; choosing deterministic rules in rational elimination defines a computable global table $\tau:E_L\to\{-1,1\}$. No later calculation recolours an earlier block. For $0\le u\le N$, let $J_d(N,u)$ be the indices $uR_\vartheta(N)$, and write $Q_d(n)=q_na_n$. Define \[ B_{d,a}(N,u)=\sum_{\substack{n\in J_d(N,u)\\a_n=a}}\tau(q_n), \qquad \Lambda_d(N,u)=\sum_a|B_{d,a}(N,u)|. \] Put $V_N=N^2+2$ and $A(N)=\floor{V_N/(R_\vartheta(N)+1)}$. For $N\ge5$, the interval of possible primes $(R_\vartheta(N),V_N]$ meets at most two prime blocks: $V_N0,\ h(Q_d(n))=v\} &\ge\frac{|J_d(N,u)|-\Lambda_d(N,u)}2. \end{aligned} \end{equation} The unselected observations can only increase the two sign counts. The bound uses the actual $\Lambda_d$; replacing it by \eqref{eq:three-defect} may give a negative, hence uninformative, lower bound at small sizes. \begin{theorem}[Persistent simultaneous prescription]\label{thm:persistent} Fix rational $1<\vartheta<\gamma<6/5$, $1/\vartheta<\sigma\le1$, $\varepsilon>0$ and $B\ge1$, and assume \eqref{hyp:GF}. There is a decidable prime set $E\subset(B,\infty)$ with $\sum_{q\in E}q^{-\sigma}<\varepsilon$, and a computable sign table $\tau$ fixed independently of the input source, with the following property. For every sufficiently large $N$, every source $h$ satisfying \[ h(q)=\tau(q)\quad(q\in E,\ q>R_\vartheta(N)) \] has at least $cN$ occurrences of each sign on $(N/2,N]$ on each of the three traces. The positive constant $c$ is independent of the input and of the reassigned signs. \end{theorem} \begin{proof} Choose $L$ by \eqref{eq:support-cost}. For large $N$, all indices in $(N/2,N]$ exceed $L$, and \eqref{hyp:GF} supplies at least $aN$ primes above $R_\gamma(N)>R_\vartheta(N)$. Hence $|J_d(N,N/2)|\ge aN$. Apply \eqref{eq:persistent-finite}, using the $o(N)$ defect in \eqref{eq:three-defect}, and take $c=a/4$ after increasing the threshold. \end{proof} For an arbitrary input $f_0$, overwrite its signs on $E$ by $\tau$ and keep all others. This operation protects $[1,B]$, is continuous in the product topology of prime signs, is idempotent, and is computable relative to $f_0$. It is a prescription operation, not multiplication by a fixed factor independent of $f_0$. All but fewer than $\varepsilon T$ integers in each ordinary prefix $[1,T]$ retain their signs. The same finite estimate also clarifies its limitation. For an unchanged source $h$, let $U_d$ count the relevant primes at which $h(q)\ne\tau(q)$. Each such prime changes at most one selected reading on trace $d$, so \[ \min_{v=\pm1}\#\{u0,\ h(Q_d(n))=v\}+U_d \ge\frac{|J_d(N,u)|-\Lambda_d(N,u)}2. \] This is a necessary test on an arbitrary source. It does not prohibit that source from disagreeing with a linear number of prescribed prime signs: the entire prescribed support itself has small harmonic cost. % Original source: 08_masks.tex \section{Uniform signs on the active coordinates of words}\label{sec:masks} Balancing each sign count does not force complicated words: an alternating sequence has balanced signs and zero entropy. To control words, we retain coordinates having a sufficiently large prime factor and mask the rest. The reference law keeps the actual arithmetic mask, rather than replacing its positions by independent random choices. \subsection{The mask and the collision bound} Fix $1<\vartheta<2$ and the support $E_L$ of Section~\ref{sec:sparse}. A finite set of additional primes and finitely many prescribed signs can also be allowed; their support cost is paid separately. Write the resulting support as $E$. At the $N$ origins $n=2,\ldots,N+1$, consider words of length $H$ on each trace: \[ W_h^H(n)=\bigl(h(Q_d(n+a))\bigr)_{0\le aR,\quad q_i(n)\in E. \] Its cofactor is at most $A$. Let $S(n)\subseteq[m]$ be the set of active coordinates. It depends on the arithmetic, not on any sign choices. \begin{lemma}[Distinct active primes in a word]\label{lem:word-primes} If $H\le N$ and $R\ge(H^2+4)^2$, the active primes within one word are all distinct. Across the $N$ origins, one active prime affects at most $3H$ words. \end{lemma} \begin{proof} Two distinct coordinates correspond to distinct monic irreducible polynomials $T^2-d$ and $(T+t)^2-e$, with $|t|\eta\} \le2\exp\left(-\frac{2\eta^2}{\sum_j c_j^2}\right), \] for independent inputs and coordinate oscillations at most $c_j$, therefore gives \begin{equation}\label{eq:mask-concentration} \Pr\{|F_{S,J,\kappa}|>\eta\} \le2\exp\left(-\frac{N\eta^2}{108H^2}\right). \end{equation} This is McDiarmid's bounded-differences inequality~\cite{McDiarmid}; the stated form is Theorem~2 of Raginsky's notes~\cite{Raginsky}. Independence of overlapping words is not required. There are at most $2^{\pi(A)}$ heads $\kappa$, and \[ \sum_{S\subseteq[m]}(2^{|S|}-1)=3^m-2^m<2^{6H} \] mask-character tests. Although all small-prime assignments number $2^{\pi(A)}$, their cost in the exponent of the union bound is only $\pi(A)$. For $H\le N$, the cofactor bound $A=O(N^{2-\vartheta})$ makes this cost sublinear; the mask-character tests add a cost proportional to $H$. This is the room needed to choose one table before any of the small signs. To accommodate $k_0$ arbitrarily prescribed prime-sign bits, put $\ell_N=\ceil{\log_2(N+1)}$ and \begin{equation}\label{eq:scalar-eta} \eta_{N,H,k_0}^2= \frac{108H^2}{N}\bigl(\pi(A)+7H+4+4\ell_N+k_0\bigr). \end{equation} Define the bad event $B_{N,H}$ to mean that one of the tests exceeds this threshold. If $H>N$, the collision condition fails, or $\eta\ge1$, set $B_{N,H}=\varnothing$. The union bound and $e^{-u}\le2^{-u}$ imply \begin{equation}\label{eq:bad-prob} \Pr(B_{N,H})\le2^{-k_0-H-3}(N+1)^{-4}. \end{equation} The union is taken before selecting the table. It thus covers a head chosen after the entire table is known. Arbitrary unrelated weights at all $N$ origins would have a much larger information cost and are not covered by this argument. \subsection{Selecting one computable table}\label{sec:selection} \subbookmark{One computable table}{nav:selector} An almost-sure existence statement alone does not supply a computable prescription. The method of conditional expectations has the broader constructive statistical context of~\cite{GHR}. More specifically, the selections in~\cite[Section~5, Proposition~5.1]{PoulySelective} and~\cite[Section~7]{PoulyCorrelation} use exact rational costs, irreversible decisions, and positive tails controlled after conditioning. Their potentials are weighted sums of normalised nonnegative costs. Here we instead count bad events and keep the conditional expectation strictly below one. Once a finite collection of indicators is determined, its sum is an integer smaller than one and must vanish. We give this specialised argument in full, including its tail budget. \begin{lemma}[Effective elimination of all bad events]\label{lem:selection} Suppose the events $B_{N,H}$ depend on finitely many binary coordinates, can be decided by finite exact computation, and satisfy \eqref{eq:bad-prob} under the fair product measure. After any prescribed values of $k_0$ coordinates, the remaining coordinates can be chosen computably so that no bad event occurs. \end{lemma} \begin{proof} Consider the nonnegative integer-valued potential \[ U=\sum_{N,H\ge1}\one_{B_{N,H}}. \] Its expectation is at most $2^{-k_0}/8$. Conditioning on any $k_0$ specified fair bits multiplies this upper bound by at most $2^{k_0}$, leaving at most $1/8$. For an integer $r\ge1$, the sum of the probabilities outside the finite rectangle $H\le r$, $N\le2^r$ is at most $2^{-k_0-r-2}$. Indeed, the $H$-tail is geometric, and \[ \sum_{N>2^r}(N+1)^{-4}\le2^{-3r}/3. \] After $s$ further bits are fixed, each branch for the next bit has unconditional probability $2^{-(k_0+s+1)}$. Positivity bounds its conditional omitted expectation by $2^{s-r-1}$. At step $s$, take $r=2s+6$. Compute exactly the two conditional expectations of the finite-rectangle sum and choose the smaller, with a fixed tie rule. The full conditional expectation then increases by at most $2^{-s-7}$, since the omitted expectation is nonnegative and bounded by that quantity in both branches. These losses sum to $1/64$. Any finite subcollection of the bad indicators becomes determined once all its finitely many bits have been fixed. Its value is bounded by the full conditional expectation, hence by $1/8+1/64<1$. It is an integer, so it is zero. No bad event survives. Each computation is finite: enumerate the finitely many unchosen bits involved in the rectangle and average the exact indicators. This proves computability, not a practical complexity bound. \end{proof} In the arithmetic application, event membership is exact: after finite factorisations and enumerations, compare the square of each rational $F_{S,J,\kappa}$ with the rational threshold in \eqref{eq:scalar-eta}. The algorithm does not have to recognise the onset of an asymptotic estimate. \subsection{Recovering the masked law from its characters} For the selected table, let $p_S(w)$ be the empirical joint mass of mask $S$ and signs $w$ for a head $\kappa$. Fourier inversion on $\{-1,1\}^S$ gives \[ p_S(w)-\rho_{N,H}(S)2^{-|S|} =2^{-|S|}\sum_{\varnothing\ne J\subseteq S} F_{S,J,\kappa}\prod_{i\in J}w_i. \] Summing absolute values over $w$ and $S$, and dividing by two, yields \begin{equation}\label{eq:masked-tv} \sup_\kappa\TV\bigl(\mathcal L_N(S,W_\kappa^H|_S),U(\rho_{N,H})\bigr) \le\Delta_{N,H,k_0}, \end{equation} where, when the collision condition holds and $H\le N$, \[ \Delta_{N,H,k_0}= \min\left\{1,\frac{3^{3H}-2^{3H}}2\eta_{N,H,k_0}\right\}. \] Outside this range we use the trivial bound $\Delta=1$. Here $\TV$ is one half of the $\ell^1$ distance. The same bound applies to projection onto one trace. Since $A=O_\vartheta(N^{2-\vartheta})$ for $H\le N$, the error tends to zero at each fixed $H$: \begin{equation}\label{eq:masked-fixed} \Delta_{N,H,k_0}=O_{\vartheta,H,k_0}(N^{-(\vartheta-1)/2}). \end{equation} It also tends to zero uniformly for \begin{equation}\label{eq:log-horizon} 1\le H\le c\log N,\qquad c<\frac{\vartheta-1}{6\log3}, \end{equation} because the upper bound is $O_{\vartheta,k_0,c}(H N^{-(\vartheta-1)/2+3c\log3})$. The collision condition holds eventually throughout this range. These are statements about a single table and the real masks at all the indicated sizes, not about independently chosen tables or model masks. \subsection{Entropy without subtracting the mask entropy} \subbookmark{Entropy transfer}{nav:entropy} We measure entropy in bits, writing $h_2(t)=-t\log_2t-(1-t)\log_2(1-t)$ with the usual endpoint conventions. \begin{lemma}[Entropy transfer]\label{lem:entropy} Let $Y\in\{-1,1\}^m$ and let $S\subseteq[m]$ be any possibly dependent mask. If the law of $(S,Y_S)$ is at total variation distance $D_0$ from $\rho(S)2^{-|S|}$, with the same mask marginal $\rho$, then \begin{equation}\label{eq:entropy-transfer} \Ent(Y)\ge\mathbb E|S|-mD_0-h_2(D_0). \end{equation} \end{lemma} \begin{proof} For each mask $s$, couple $Y_s$ conditional on $S=s$ to a uniform word of length $|s|$ with disagreement probability $\delta_s$ equal to their total variation distance. Revealing whether they disagree and, on disagreement, the uniform word, proves \[ \Ent(Y_s\mid S=s)\ge|s|-|s|\delta_s-h_2(\delta_s). \] The average of $\delta_s$ is $D_0$. Average the inequality, apply concavity of $h_2$, and use \[ \Ent(Y)\ge\Ent(Y\mid S)\ge\Ent(Y_S\mid S). \] This proves \eqref{eq:entropy-transfer}. In particular, no independence of the mask and the full word was used. \end{proof} When only $D_0\le\Delta$ is known, we use $m\Delta+h_2(\Delta)$ once $\Delta\le1/2$; otherwise the trivial entropy lower bound zero is sufficient. Monotonicity of $h_2$ on all of $[0,1]$ is not assumed. \begin{theorem}[An entropy-resistant prescription]\label{thm:scalar-entropy} Fix rational $1<\vartheta<\gamma<6/5$, $1/\vartheta<\sigma\le1$, and assume \eqref{hyp:GF}. Above any fixed prefix, an arbitrarily small $\sigma$-cost support admits a computable prescription $\tau$ such that, for some $\delta>0$ and every fixed $H$, \[ \liminf_{N\to\infty}\ \inf_{h\in\mathcal H_N} \frac{\Ent(\mu_{N,h}^{H,d})}{H}\ge\delta\quad(d\in\Dset), \qquad \liminf_{N\to\infty}\ \inf_{h\in\mathcal H_N} \frac{\Ent(\mu_{N,h}^{H})}{H}\ge3\delta, \] where $\mathcal H_N$ consists of all sources agreeing with $\tau$ at $q\in E$, $q>R_\vartheta(N)$. Every coherent subsequential limit of these empirical word laws, including those obtained from moving choices $h_N\in\mathcal H_N$, has metric entropy at least $\delta$ on each trace and $3\delta$ jointly. \end{theorem} \begin{proof} Use the support bound \eqref{eq:support-cost} and the selection above. Under \eqref{hyp:GF}, for each trace at least $aN$ indices in $(N/2,N]$ have a prime greater than $R_\gamma(N)$. For $H=O(\log N)$, this exceeds $R_\vartheta(N+H)$ eventually. All these indices exceed $L$, so their primes lie in $E$. For each shift $0\le a'0$, independent of $B,\varepsilon$, with the following properties. For every fixed $x\ne y$ and every trace $d$, \begin{equation}\label{eq:group-separation} \liminf_{N\to\infty}\frac1N \#\{2\le n\le N+1:\Psi_x(Q_d(n))\ne\Psi_y(Q_d(n))\}\ge c. \end{equation} For each nonzero message, every limiting frequency law on a quadratic trace has metric entropy at least $\delta$, and every joint limiting law has entropy at least $3\delta$. These entropy and sign-count guarantees also hold uniformly for visible messages after arbitrary reassignment of all prime signs other than the prescribed ones in $E$ above $R_\vartheta(N)$. \end{theorem} \begin{proof} Choose $L$ to pay the support cost, and choose the matrix as above. The mass and entropy arguments in \cref{thm:scalar-entropy} apply uniformly to every visible message. At horizon one, the reference masked law assigns exactly half of its active mass to each sign. Thus, on each trace, the number of negative active readings of any such source is at least \[ \delta N/2-N\Delta_{N,1,k_0}\ge\delta N/4 \] for every sufficiently large $N$. The same lower bound holds for positive readings. Take $c=\delta/4$. If $x\ne y$, the message $x+y$ has a first nonzero coordinate, which is visible eventually. Applying the preceding sign bound to the quotient \eqref{eq:matrix-group} proves \eqref{eq:group-separation} and injectivity. For continuity, choose finitely many primes whose omitted $\sigma$-weight is less than $\eta$, and then a message prefix long enough to determine all of their rows. Agreement on this prefix implies $d_\sigma<\eta$. The domain is compact and the target metric space is Hausdorff, so the image is compact and homeomorphic to the domain. The diameter bound follows from the common support. \end{proof} The identity $\Psi_0=1$ necessarily has zero quadratic entropy and is excluded from the positive-entropy assertion. A computable message gives a computable source; an arbitrary message gives a source computable relative to that message. The uncountable image is not asserted to consist entirely of computable functions. \subsection{Ordinary frequencies have entropy zero} A common harmonic support gives more than ordinary closeness. The following fact applies to each member of all the constructed groups, including the paired code. \begin{proposition}[Ordinary entropy under a summable prime support]\label{prop:ordinary-entropy} Suppose $f\in\M$ and $\sum_{q:f(q)=-1}q^{-1}<\infty$. Then the ordinary sequence $f(n)$ has frequencies for every finite sign word. The resulting stationary measure has metric entropy zero. \end{proposition} \begin{proof} First truncate the negative-prime set to a finite subset. The upper density of disagreement is bounded by the reciprocal sum of the omitted primes, since every disagreement is a multiple of at least one of them. For the remaining primes, truncate valuations at a level $J$ by assigning a fixed value to multiples of $q^J$. This adds at most $\sum q^{-J}$ to the disagreement density and gives a periodic sequence, of period dividing $\prod q^J$. Thus $f$ admits periodic approximations arbitrarily close in upper density. For a fixed word length $H$, the discrepancy between empirical word laws is at most $H$ times the single-site discrepancy. The limiting laws of the periodic approximations are Cauchy, and the empirical law of $f$ converges to their limit. Take a periodic approximation of period $T$ and error density at most $\eta<1/2$. A word of $f$ is determined by the phase of the periodic sequence and the vector of disagreements. Averaging over origins and passing to a stationary limit gives \[ \Ent(\mu_f^H)\le\log_2 T+Hh_2(\eta). \] Divide by $H$, let $H\to\infty$, and then let $\eta\downarrow0$. The entropy rate is zero. \end{proof} This is an elementary specialization of the theory of rationally almost periodic sequences. For a finite alphabet, Bergelson--Ku\l{}aga-Przymus--Lema\'nczyk--Richter~\cite[Theorem~1.7 in the cited author version]{Bergelson} prove genericity for an ergodic measure with rational discrete spectrum, a conclusion stronger than the entropy-zero consequence used here. The proof above is included to make its summable-prime application self-contained, not as a new entropy principle. A broader multiplicative-function context is provided by Frantzikinakis--Lema\'nczyk--de la Rue~\cite{FLR}; no result from that broader theory is needed for this proof. The proposition does not assert zero topological entropy. For a prime set $A$ of finite harmonic sum, the ordinary density of negative values of $\prod_{q\in A}(-1)^{v_q(n)}$ also exists and equals \[ \frac12\left(1-\prod_{q\in A}\frac{q-1}{q+1}\right). \] For finite $A$, the local factor is the mean of $(-1)^{v_q(n)}$, computed by a geometric series; the Chinese remainder theorem multiplies the local means. The uniform tail bound $\sum_{q\in A,\ q>Y}q^{-1}$ passes to infinite $A$. This ordinary calculation does not imply convergence of a quadratic mean. \subsection{Two topologies and the order of quantifiers} \subbookmark{Topologies and quantifiers}{nav:topologies} Define the aligned observation pseudometric \[ D_d(f,g)=\limsup_{N\to\infty}\frac1N \#\{2\le n\le N+1:f(Q_d(n))\ne g(Q_d(n))\}. \] It satisfies the triangle inequality. By \eqref{eq:group-separation}, the compact prime-metric group contains an uncountable set separated by at least $c$ in every $D_d$. After identifying pairs of zero pseudodistance, this set is nonseparable in the observation metric. No countable collection of prototypes can approximate all its members within a fixed radius smaller than $c/2$. Translation by any reference $f_0$ preserves all pairwise observation distances and ordinary agreements. It does not give each translated member positive quadratic entropy: the translate contains $f_0$ itself, and $f_0$ is arbitrary. The positive-entropy assertion applies to the nontrivial quotients, or to the nonidentity members over base $1$. For the unit messages $e_j$, continuity gives $d_\sigma(\Psi_{e_j},1)\to0$, while $D_d(\Psi_{e_j},1)\ge c$ for every $j$. These are aligned sequence distances, not distances between word laws. Two different sequences can have identical word frequencies. There is no scale which separates all nonzero infinite messages simultaneously. At any fixed $N$, choose a nonzero message whose first $\bl((N+1)^2+2)$ bits vanish. Every raw reading up to $N+1$ then equals $1$. Thus the infimum over nonzero messages of the normalised finite discrepancy is zero at every $N$, even though every fixed nonzero message has positive limiting lower discrepancy. Visibility is indispensable. At a fixed large $N$, the visible prefix has length $b_N=\vartheta\log_2N+O(1)$. Distinct such prefixes remain at least $cN$ apart even after independently changing their cofactor heads. Consequently they are uniquely recoverable under $2t+sR_\vartheta(N),\ q\in E,\ \nu(q)\ge s_N\}. \] Write $M_{d,N}=|J_{d,N}|$, $M_N=\min_dM_{d,N}$ and \[ A_N=\floor{\frac{N^2+2}{R_\vartheta(N)+1}}=O(N^{2-\vartheta}). \] There are at most $3(s-2)_+$ primes whose first occurrence is less than $s$, because each earlier index supplies at most three. Each such prime removes at most one candidate on each trace. Under \eqref{hyp:GF}, choose a rational \[ 0\vartheta$ gives \begin{equation}\label{eq:clock-mass} M_N\ge aN-3\ceil{\alpha N}\ge aN/2. \end{equation} Thus one can insist that every remaining row starts recently enough to see a common prefix of length proportional to $N$, without exhausting the supply. \subsection{Paying for linearly many message bits} Put $K_N=k(N)$ and $b_N=k(s_N)$. Call $z\in\F_2^{K_N}$ visible when $z_{\le b_N}\ne0$. Every row in $J_{d,N}$ has length between $b_N$ and $K_N$, so it sees a nonzero projection of $z$. Choose rows independently and uniformly for the existence argument. For a fixed visible $z$ and a small-prime assignment $\kappa$ up to $A_N$, the sum \[ S_{d,z,\kappa}=\sum_{n\in J_{d,N}} \kappa(a_n)(-1)^{\langle v(q_n),z\rangle} \] is a sum of $M_{d,N}$ independent fair signs. The primes are distinct on this trace, although primes shared between traces keep the same row. The elementary bound $\cosh t\le e^{t^2/2}$ and exponential Markov give \[ \Pr\{|S_{d,z,\kappa}|>M_{d,N}/2\}\le2e^{-M_{d,N}/8}. \] Taking the union over three traces, messages and heads yields \begin{equation}\label{eq:clock-prob} \Pr(B_N)\le6\,2^{K_N+\pi(A_N)}e^{-M_N/8}. \end{equation} Unlike Section~\ref{sec:groups}, the number of visible message bits is now linear. A vanishing discrepancy error would not automatically pay this cost. The constant tolerance $|S|\le M/2$ leaves a positive distance, which is all the present decoding statement needs. \subsection{A computable selection without an analytic stopping test} Allow $k_0$ predetermined coefficients. Define a bad event only at sizes with $b_N\ge1$ and satisfying the finite test \begin{equation}\label{eq:clock-test} M_N\ge8\bigl(K_N+\pi(A_N)+k_0+8+4\bl(N+1)\bigr). \end{equation} Otherwise put $B_N=\varnothing$. From \eqref{eq:clock-prob}, \[ \Pr(B_N)\le2^{-k_0-5}(N+1)^{-4}. \] The potential $U=\sum_{N\ge2}\one_{B_N}$ has conditional expectation below $1/16$ after any values of the $k_0$ prescribed bits. After $s$ further coefficients have been fixed, the omitted expectation of the events $N>2^R$ in either next-bit branch is at most \[ \frac{2^{s-4-3R}}3. \] This follows by summing the displayed probability bound and dividing by the branch probability $2^{-(k_0+s+1)}$. Taking $R=s+4$ makes this less than $2^{-s-6}$. At each step minimise the exact finite conditional expectation over $N\le2^R$. The increases sum to at most $1/32$. As in \cref{lem:selection}, every determined finite sum of bad indicators is an integer smaller than $1/16+1/32<1$, hence zero. This defines one computable matrix. Its algorithm takes the chosen rationals $\alpha,\rho$ as parameters; it does not need a numerical threshold at which \eqref{eq:clock-mass} starts. The existence of admissible positive parameters, rather than their numerical calibration, uses \eqref{hyp:GF}. \begin{theorem}[Causal prefix dependence and positive rate]\label{thm:clock} Under \eqref{hyp:GF}, the first-occurrence construction gives a continuous injective homomorphism $x\mapsto\Phi_x$ on an arbitrarily small harmonic support above any fixed prefix. All raw readings at indices at most $N$ depend on at most $K_N=\rho N+O(1)$ message bits. For every sufficiently large $N$, each trace alone determines the first $b_N=\rho\alpha N+O(1)$ message bits after arbitrary reassignment of all prime signs other than those prescribed at $q\in E$, $q>R_\vartheta(N)$, and after any errors and erasures with \[ 2t+s0$. Both the selection and the decoder are computable; no polynomial bound is asserted for them. \end{theorem} \begin{proof} Since $K_N=\rho N+O(1)$ and $\pi(A_N)=o(N)$, \eqref{eq:clock-mass} eventually implies \eqref{eq:clock-test}; indeed, $8\rho=a/80}} \one_{h(Q_d(n))=-1}. \] A positive distance between different members of our code says nothing by itself about this minimum. Even a translate by $\beta_p(n)=(-1)^{v_p(n)}$ with $p\le B$ contains $\beta_p$ itself, while all its members have sign $-1$ at $p$. The code's prescribed large signs are additional constraints which the free minimum is not required to obey. The same distinction applies to statistical complexity. Positive entropy of all limiting quadratic laws is weaker than complete Bernoulli normality, and does not assert a unique limiting law. Conversely, a code's aligned Hamming separation does not require different word laws. These observation types were kept separate in the constructions because they require different estimates. Several natural refinements are left as separate questions by these proofs: a quantitative calibration of the weighted mass input and its threshold; better rate and distance constants for the same arithmetic resource; and the addition of the masked word-law and quotient-entropy guarantees to the already causal, polynomially decoded paired encoding. None is a premise of the finite theorem or of the conditional implications proved above. \subsection*{Assistance and verification} GPT-6 Astra assisted the research notes and initial consolidation; GPT-6 Astra Pro assisted the subsequent proof exposition, reviews, code and finite checks, bibliography and manuscript revisions. The author is responsible for the article. No complete independent human verification, Lean formalisation or independent external replication is claimed. Appendix~\ref{app:implementation} identifies the reproducible finite checks and their limits; they do not establish \eqref{hyp:GF}. \appendix % Original source: 12_fixed_scale.tex \section{Fixed-scale balancing without the superlinear mass input}\label{app:balance} The superlinear cutoff makes the number of exact cofactors small and their support summable. At the smaller cutoff $2N$, a different count of square classes gives a useful unconditional result with an explicit leading constant. It does not produce the same all-scale sparse encoding. For $N\ge5$ let \[ J_d(N)=\{n\le N:Q_d(n)>0,\ P^+(Q_d(n))>2N\},\qquad M_d(N)=|J_d(N)|, \] and let $\mathcal Q_N$ be the set of all primes greater than $2N$ occurring in these observations. Write $Q_d(n)=q_na_n=q_nb_ns_n^2$, where $b_n=\core(a_n)$ is squarefree. By \cref{lem:private}, $a_n0}}\log Q_d(n)=2N\log N+O(N). \] For an odd split prime $q$, the congruence $n^2\equiv d\pmod{q^j}$ has exactly two roots for every $j\ge1$. With $V=N^2+2$, summing their counts gives \[ \sum_{n\le N}v_q(Q_d(n))\le\frac{2N}{q-1}+\frac{2\log V}{\log q}. \] Odd inert primes do not divide these polynomial values. The prime $2$ contributes $O(N)$ to the logarithmic mass. The split primes for these fixed $d$ are unions of residue classes modulo $4$ or $8$. The fixed-progression weighted Mertens formula~\cite[Section~27.11, equation~(27.11.11)]{DLMF} gives \[ \sum_{\substack{q\le x\\(d/q)=1}}\frac{\log q}{q-1} =\frac12\log x+O(1). \] Replacing $q$ by $q-1$ in the denominator adds a bounded sum. Summing the error terms using $\pi(2N)\ll N/\log(2N)$ shows that primes at most $2N$ contribute at most $N\log N+O(N)$. Thus primes greater than $2N$ contribute at least $N\log N-O(N)$. Each corresponding observation contributes at most $\log V=2\log N+O(N^{-2})$, which proves the lemma. \end{proof} This elementary weighted argument at the cutoff $2N$ is not a proof of \eqref{hyp:GF} at a fixed exponent greater than one. In particular, its prime-counting error cannot simply be used at a power-sized larger cutoff. \subsection{Few square classes and a common assignment} Let $s_d(N)$ be the number of distinct squarefree cofactors $b_n$ on trace $d$. For every integer $U\ge1$, \begin{equation}\label{eq:core-count} s_d(N)\le U+\pi\bigl((N^2+2)/U\bigr). \end{equation} There are at most $U$ classes with $b\le U$. For each remaining class, choose one representative. Its cofactor $a\ge b>U$ gives $q<(N^2+2)/U$, and distinct classes on this trace give distinct primes. This proves \eqref{eq:core-count}. Taking $U$ of order $N/\sqrt{\log(2N)}$ yields \[ s_d(N)\ll N/\sqrt{\log(2N)}. \] The class sets \[ V_{d,b}=\{q_n:n\in J_d(N),\ \core(a_n)=b\} \] have incidence at most three per prime. We use the classical Beck--Fiala bound $2t-1$ for incidence at most $t$~\cite{BeckFiala,Nikolov}. Its short proof fixes the exact finite construction. Start at $x_q=0$ and preserve the sum of every row having more than $t$ free coordinates. If $u$ coordinates are free and $a$ rows are constrained, $(t+1)a\le tu$, hence $a0,\ h(Q_d(n))=v\} &\ge(M_d(N)-\Lambda_d)/2. \end{aligned} \] For the constructed table, $\Lambda_d\le5s_d(N)$. We have proved the following unconditional assertion. \begin{theorem}[A quarter of each sign]\label{thm:quarter} For every $N\ge5$, a deterministic finite procedure chooses one assignment on the primes in $\mathcal Q_N$ such that every completely multiplicative extension has, on each trace, at least \[ \frac N4-O\left(\frac{N}{\sqrt{\log(2N)}}\right) \] occurrences of each sign. The prime signs at most $2N$ may be chosen after the table and without restriction. \end{theorem} The table can change with $N$. Changing $u$ reserved prime signs of this table decreases either sign-count guarantee by at most $u$ on each trace, by uniqueness of their trace occurrences. \subsection{A necessary cofactor code for any almost-positive source} We now drop the balancing prescription. For every $q\in\mathcal Q_N$, choose one canonical observation $Q_{d_q}(n_q)=qa_q$. Let $K_N=|\mathcal Q_N|$, and form the binary matrix \[ T_N(q,\ell)=v_\ell(a_q)\pmod2, \qquad \ell0}}\one_{h(Q_d(n))=-1}. \end{equation} Consequently the number of large-prime tables admitting any extension with $E_N(h)\le t$ is at most \begin{equation}\label{eq:tail-ball} \min\left\{2^{K_N},\ 2^{r_N}\sum_{j=0}^{\min(t,K_N)}\binom{K_N}{j}\right\}. \end{equation} No disjointness of the balls is needed. Moreover, \[ N/2-O(N/\log N)\le\frac13\sum_dM_d(N)\le K_N\le3N. \] If $t=o(N)$, the logarithm of the ball volume is $o(N)$, by the usual estimate $\sum_{j\le t}\binom Kj\le(t+1)(eK/t)^t$ when $1\le t\le K/2$. Thus at most $2^{o(N)}$ large-prime tables admit such an extension, a fraction at most $2^{-(1/2-o(1))N}$ of all tables. This is a condition on the real prime signs of any almost-positive source. It is not sufficient for extension to all remaining observations, and the small set it identifies need not be empty. In particular, a low-dimensional cofactor code cannot itself exclude the constant table or supply the missing negative-prime constraint in $m_p(N)$. \subsection{One random source, or a computable sparse subsequence} There are two distinct ways to make the quarter-sign conclusion persistent. Neither should be confused with the superlinear-cutoff constructions in the main text. First, choose independent fair signs $F(q)$ once at all primes. For a fixed trace and a fixed head on primes below $N/2$, the sum on $J_d(N)$ is a sum of $M_d(N)$ independent fair large-prime signs. If $R_N'=\pi(\floor{N/2})$, exponential Markov and a union over the three traces and all heads give \[ \Pr\left\{\max_{d,\kappa}|S_{d,\kappa}|>u\right\} \le6\,2^{R_N'}e^{-u^2/(2N)}. \] Taking \[ u_N=\sqrt{2N(R_N'\log2+4\log N+\log6)} =O(N/\sqrt{\log N}) \] makes this probability at most $N^{-4}$. Borel--Cantelli and \cref{lem:half} show that, almost surely, this one $F$ has the quarter-sign guarantee at every sufficiently large $N$, even after an arbitrary reassignment at primes at most $2N$. A finite prime prefix may be prescribed before the experiment. No small harmonic support or computability of the resulting $F$ is asserted. Second, the deterministic tables of \cref{thm:quarter} can be embedded into one computable source close to a reference, but only on a selected sequence of sizes. Uniformly over sets $\mathcal A_N$ of at most $3N$ primes greater than $2N$, \begin{equation}\label{eq:sparse-subsequence-cost} \sum_{q\in\mathcal A_N}\frac1q \ll\frac{\log\log(3N)}{\log(2N)}\longrightarrow0. \end{equation} This sparse-scale summability mechanism is already used in the proof of Klurman's Proposition~1.1~\cite[Section~2]{Klurman}, including the reciprocal-prime estimate of order $\log\log N/\log N$. For the uniform form \eqref{eq:sparse-subsequence-cost}, split at $N(\log N)^2$. The upper portion costs at most $3/(\log N)^2$, and partial summation of $\pi(x)\ll x/\log x$ bounds the lower portion as displayed. Given a reference $f_0$, a protected prefix $B$ and a rational $\varepsilon>0$, find successively integers $N_j$ satisfying \[ 2N_j>\max\{B,N_{j-1}^2+2\},\qquad \sum_{q\in\mathcal Q_{N_j}}q^{-1}<\varepsilon2^{-j-1}. \] Finite search terminates by \eqref{eq:sparse-subsequence-cost}. Assign the corresponding table $\tau_{N_j}$ on each of the disjoint prime intervals $(2N_j,N_j^2+2]$, and keep $f_0$ elsewhere. Later stages cannot change earlier tables. The total harmonic cost is less than $\varepsilon$, and the quarter-sign conclusion holds at every $N_j$ against all head reassignments up to $2N_j$. To evaluate a fixed integer, search only until $2N_j$ exceeds it. The source is computable relative to $f_0$, but the guarantee is not asserted at intervening sizes. The quantifiers explain why these statements do not solve the free minimum: selecting a large-prime table which defeats every head is a $\max_\tau\min_\kappa$ construction. The free minimum also minimises over $\tau$, under its prescribed negative prime. Interchanging those operations would discard the additional information on which the construction depends. % Original source: 13_implementation.tex \section{Finite data and reproduction}\label{app:implementation} The publication package \path{N5_arithmetic_coding_publication_v1_3.zip} contains the integer data for \cref{thm:finite}, the LaTeX source, the frozen bibliography and the reproduction programs. Its edition identifier is \texttt{N5-AC-1.3-2026-10-12}. Its root files \texttt{VERSION}, \texttt{MANIFEST.json} and \texttt{SHA256SUMS.txt} identify the edition, manuscript hash, execution record and supplied files. The arithmetic implementation uses Python~3.10 or later, the standard library, exact integers, rational comparisons and binary finite-field arithmetic. \subsection{Files that define the finite instance}\label{sec:instance-files} \subbookmark{Finite-instance files}{nav:files} The file \texttt{data/pairs.csv} records the epoch, trace, pair index, the two integer indices, their common cofactor and reserved primes. The field basis, inner generator, outer parameters and message-prefix lengths are in \texttt{data/epochs.json}. The decoder reconstructs the code from these tables; no message or prime-head assignment is supplied to it. The module \path{code/arithmetic_code.py} implements the inner code, field operations, outer linear-system decoder and reliability thresholds. The scripts \texttt{build.py}, \texttt{verify.py} and \texttt{check\_theory.py} construct the instance, check its saved arithmetic data and exercise the finite lemmas. The first-occurrence filter in \texttt{causal\_filter.py} is enabled by default; \texttt{--unfiltered} selects the cancellation-only variant and does not promise causal dependence. Filter decisions are recorded in \texttt{data/causal\_filter.json}. Validation uses explicit exceptions, including under \texttt{python -O}. From the \texttt{code} directory, decode the $499$-error example on $n^2+1$ by \begin{verbatim} python decode_readings.py --data ../data --epoch 32768 \ --trace -1 --received ../data/received.csv \end{verbatim} Use \texttt{received\_gmd.csv} for the reliability-threshold example. For the reading-level mixed example, run \begin{verbatim} python decode_readings.py --data ../data --epoch 32768 \ --trace -1 \ --received ../verification/revision/data/received_mixed.csv \end{verbatim} Expected-message files are separate test fixtures and are never read by this decoder. \subsection{Rebuilding and checking the finite construction}\label{sec:rebuilding} \subbookmark{Reproduction commands}{nav:reproduce} From the archive root, \texttt{bash reproduce.sh} rebuilds into a new directory, refuses to overwrite an existing destination, and compares the outputs with the supplied data. The individual commands, from \texttt{code}, include \begin{verbatim} python build.py --out ../recomputed python verify.py --data ../recomputed python check_theory.py --out ../recomputed/theory_checks.json python check_revision.py --data ../recomputed \ --out ../revision_recomputed \end{verbatim} The checks cover $14664$ pair identities, $29328$ reserved primes, the inner distances, the first-occurrence filter and $98303$ positive raw labels through $32768$. They include $24$ reading-level decodings using two messages and three multiplicative head assignments at the two accepted epochs. All use the decoding module of Section~\ref{sec:codes}; this is not a second independent decoder. Counts for the small exhaustive tests and byte comparisons between normal Python and \texttt{python -O} are recorded in \path{verification/publication_v1_3/execution.json} and its logs, rather than interpreted as independent proofs. The diagnostic windows $65536$ and $131072$ remove respectively zero and ten candidate primes, with first-occurrence witnesses in \path{verification/revision/data/revision_checks.json}. They do not add accepted epochs to the pilot's continuation schedule. The future epoch at $1920^6$, all infinite selectors and the auxiliary balancing and matrix-selection pilots are outside this replay. The harmonic bound in Section~\ref{sec:finite} pays for an uncomputed continuation; it is not an execution of it. Checksums establish file identity, not mathematical correctness. The finite tests do not establish \eqref{hyp:GF} or enumerate all admissible corruption patterns. The \texttt{research} directory contains the revision responses and comparison history; earlier logs retain their own dates and scope. The current execution record distinguishes this replay from those earlier runs. \begin{thebibliography}{10} \bibitem{BeckFiala} J{\'o}zsef Beck and Tibor Fiala. \newblock {``Integer-making'' theorems}. \newblock {\em Discrete Applied Mathematics}, 3(1):1--8, 1981. \newblock \href{https://doi.org/10.1016/0166-218X(81)90022-6}{doi:10.1016/0166-218X(81)90022-6}. The classical floating-variable proof is also given in Nikolov's Lecture 3. \bibitem{Bergelson} Vitaly Bergelson, Joanna Ku{\l}aga-Przymus, Mariusz Lema{\'n}czyk, and Florian~K. Richter. \newblock Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics. \newblock {\em Ergodic Theory and Dynamical Systems}, 39(9):2332--2383, 2019. \newblock \href{https://doi.org/10.1017/etds.2017.130}{doi:10.1017/etds.2017.130}. 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