% 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[english]{babel} \usepackage{amsmath,amsthm,mathtools} \usepackage{newtxtext,newtxmath} \usepackage{microtype} \usepackage[margin=27mm,headheight=15pt]{geometry} \usepackage{booktabs,tabularx,array} \usepackage{enumitem} \usepackage{etoolbox} \usepackage{fancyhdr} \usepackage[hidelinks]{hyperref} \usepackage{bookmark} \hypersetup{pdftitle={Prime perturbations of polynomial observations: instability and incidence criteria},pdfauthor={Brice Pouly},pdfsubject={Prime truncation, polynomial observations and multiplicative functions}} \pagestyle{fancy}\fancyhf{} \fancyhead[L]{\small Prime perturbations of polynomial observations} \fancyhead[R]{} \fancyfoot[C]{\thepage} \renewcommand{\headrulewidth}{0.3pt} \setlength{\parindent}{1.2em} \setlength{\parskip}{2pt} \emergencystretch=1.5em \setcounter{tocdepth}{1} \hypersetup{bookmarksdepth=2} \bookmarksetup{depth=2,open,openlevel=1} \makeatletter \patchcmd{\l@section}{1.0em}{0.35em}{}{\PackageError{p4}{Contents spacing patch failed}{}} \patchcmd{\thebibliography}{\section*{\refname}} {\section*{\refname}\addcontentsline{toc}{section}{\refname}} {}{\PackageError{p4}{Bibliography navigation patch failed}{}} \makeatother \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{inputthm}[theorem]{Published input} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\E}{\mathbb E} \newcommand{\Prob}{\mathbb P} \newcommand{\M}{\mathcal M_{\pm}} \newcommand{\HH}{\mathcal H} \newcommand{\ind}{\mathbf 1} \newcommand{\osc}{\operatorname{osc}} \newcommand{\law}{\mathcal L} \newcommand{\TV}{\mathrm{TV}} \newcommand{\vp}{v_p} \newcommand{\dd}{\,\mathrm d} \newcommand{\arxiv}[1]{\href{https://arxiv.org/abs/#1}{arXiv:#1}} \newcommand{\doi}[1]{\href{https://doi.org/#1}{doi:#1}} \newcommand{\prefix}{M_0} \newcommand{\Dset}{\mathcal D} \newcommand{\Qcurve}[1]{P_{#1}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\SF}{\mathcal S} \newcommand{\AB}{\mathcal A} \newcommand{\Bcov}{\mathcal B} \newcommand{\Emean}{\mathcal M} \newcommand{\cp}{\ell_{P,p}} \AddToHook{begindocument/end}[HXbibspacing]{\apptocmd{\thebibliography}{\setlength{\itemsep}{1.5pt}\setlength{\parsep}{0pt}}{}{}} \begin{document} \begin{center} {\LARGE\bfseries Prime perturbations of polynomial observations:\\[4pt] instability and incidence criteria}\par \vspace{12pt} {\large Brice Pouly}\par \vspace{5pt} {\small Research and writing assisted by GPT-6 Astra.} \end{center} \begin{abstract} The cost of changing prime signs depends on how the resulting multiplicative function is observed. On a fixed prime support of finite harmonic sum, we prove that existence of a polynomial mean for every sign assignment is equivalent to uniform prime truncation and uniform convergence to local products. The criterion applies separately to ordinary and logarithmic means. For independent activations with finite expected harmonic cost, the almost-sure truncation defect is the averaged large-prime parity probability, not a sum of incidences. These criteria separate averaging methods and observation geometries. One common support is logarithmically stable for every fixed polynomial and every assignment, yet a fixed perturbation of $(-1)^{v_2(n)}$ on that support has divergent ordinary means on $n^2+1,n^2+2,n^2-2$. Collective upper sieves give stability at critical counting exponents without critical summability, and harmonic stability on rectangles whose side lengths have asymptotically equal logarithms. For widths $X^\alpha$, Gaussian prime estimates give the matching universal budget threshold when $119/500<\alpha<1$; the negative result concerns only the sum of two squares. Around every completely multiplicative sign function, fixed perturbations of arbitrarily small $\sum p^{-\sigma}$ cost can have divergent logarithmic means on $n^2+1$ for $125/164<\sigma\leq1$. A positive oscillation bound is common to this range, although the starting scales may depend on $\sigma$. Window laws further distinguish all origins from almost all origins. \end{abstract} {\small\tableofcontents} \newpage \section{Introduction and principal statements}\label{sec:introduction} Changing the sign assigned to a large prime is inexpensive on the integers, but need not be inexpensive on a polynomial sequence. Among the ten values $n^2+1$, $1\leq n\leq10$, the prime $101$ occurs only in $10^2+1$. Reversing its sign costs $1/101$ in the harmonic prime budget and changes a proportion $1/10$ of this sample. The example does not prove asymptotic instability. It identifies the question: \emph{which small changes at the primes remain small after the observation is chosen?} Three questions organize the answer. Does every choice of signs on one fixed prime support have a mean? Do ordinary and logarithmic averaging detect the same perturbations? What changes when the shape or origin of the sample changes? Section~\ref{sec:support} already answers the first question by a self-contained argument, without the large-prime lower bounds of Section~\ref{sec:logarithmic} or the Gaussian prime input of Section~\ref{sec:angular}. Let $\M$ denote the completely multiplicative functions from $\N_{\geq1}$ to $\{-1,1\}$. For $\sigma>0$ and a set $A$ of primes, write \begin{equation}\label{eq:metric} d_\sigma(f,g)=\sum_{p:f(p)\ne g(p)}p^{-\sigma},\qquad h_A(n)=(-1)^{\sum_{p\in A}v_p(n)}. \end{equation} Every quotient $g/f$ is some $h_A$. On the integers a union of multiples gives \begin{equation}\label{eq:integer-distance} \frac1X\#\{n\leq X:f(n)\ne g(n)\} \leq\sum_{p:f(p)\ne g(p)}\frac{\lfloor X/p\rfloor}{X}\leq d_1(f,g). \end{equation} For sign functions, the usual pretentious distance satisfies $\mathbb D(f,g;\infty)^2=2d_1(f,g)$, with the same equality at finite cutoffs; see~\cite{Klurman}. The budget is thus an additive form of a familiar distance. Polynomial sampling changes the incidence assigned to its prime coordinates. For a nonzero integer polynomial $P$, put \begin{equation}\label{eq:general-means} \begin{aligned} M^c_{P,f}(N)&=\frac1N\sum_{\substack{n\leq N\\P(n)\ne0}}f(|P(n)|),& M^\ell_{P,f}(N)&=\frac1{\HH_N}\sum_{\substack{n\leq N\\P(n)\ne0}}\frac{f(|P(n)|)}n, &\HH_N&=\sum_{n\leq N}\frac1n. \end{aligned} \end{equation} Zeros are omitted without changing the denominators. There are only finitely many, and replacing finitely many observations does not change any limiting assertion. Constants are allowed when stated; the random criterion below is formulated for positive-valued polynomials of positive degree. All logarithms are natural unless a base is displayed. For $m>1$, $P^+(m)$ denotes its largest prime factor; we set $P^+(1)=1$. \subsection{A support criterion and a random criterion} A prime set can be studied before any of its signs are chosen. Fix a support $\Gamma$ with \begin{equation}\label{eq:support-harmonic} \sum_{p\in\Gamma}\frac1p<\infty. \end{equation} For each fixed polynomial and either chosen averaging method, convergence for every assignment on this same support will force a uniform conclusion: all prime truncations are uniformly accurate and all limits are local products. This is a constructive converse, not a compactness principle. If tail coverage persisted, we could assemble one divergent assignment. The following definitions make that equivalence precise. For $A\subseteq\Gamma$, truncate by $h_{A,y}=h_{A\cap[2,y]}$. Let $\Delta^c_{P,N}(A;y)$ and $\Delta^\ell_{P,N}(A;y)$ be the ordinary and logarithmic proportions of disagreement between $h_A(|P(n)|)$ and $h_{A,y}(|P(n)|)$. The relevant sign-free quantity is the \emph{coverage} of the sample by odd valuations in the tail: \begin{equation}\label{eq:coverage-intro} B_{P,N}(\Gamma;y)=\frac1N\#\{n\leq N:P(n)\ne0,\ \exists p\in\Gamma,\ p>y,\ v_p(|P(n)|)\text{ odd}\}. \end{equation} Set $u_P(N)=B_{P,N}(\Gamma;N)$ and, for a bounded sequence $u$, define its positive Abel average by \begin{equation}\label{eq:abel-operator} (\mathcal Au)(N)=\frac{u(N)+\sum_{k\sqrt N\\v_p(P(n))\ \mathrm{odd}}}(1-2\theta_p)\right). \label{eq:eta} \end{align} Empty products are one. The prime signs are independent; the observations $P(n)$ are not assumed independent. \begin{theorem}[Exact random prime-truncation defect]\label{thm:criterion} Assume \begin{equation}\label{eq:expected-budget} \sum_p\frac{\theta_p}{p}<\infty. \end{equation} Almost surely, simultaneously for every fixed polynomial as above, \begin{equation}\label{eq:criterion} \lim_{y\to\infty}\limsup_{N\to\infty}\Delta_{P,N}(E;y) =\limsup_{N\to\infty}\eta_P(N). \end{equation} The same identity holds with the two inner limit superiors replaced by limit inferiors. For logarithmic disagreement the right-hand side is the limit superior, or inferior, of \begin{equation}\label{eq:eta-abel} \overline\eta_P(N)=(\mathcal A\eta_P)(N). \end{equation} Thus natural prime truncation is valid in ordinary mean exactly when $\eta_P(N)\to0$, and in logarithmic mean exactly when $\overline\eta_P(N)\to0$. \end{theorem} The two criteria answer different questions. Theorem~\ref{thm:support-criterion} quantifies over all assignments on a fixed support. Theorem~\ref{thm:criterion} identifies the almost-sure defect under a specified independent law. Neither says that a multiplier creates missing means for an arbitrary source $f(P(n))$. Under $\theta_p\to0$, the random defect vanishes and gives local joint laws conditional on the realized $E$; the same $p$-adic coordinate is used for every polynomial at a given prime. \subsection{Fixed-source instability and separation of the two averages} For $P(n)=n^2+1$ abbreviate \begin{equation}\label{eq:averages} A_f(N)=M^c_{P,f}(N),\qquad L_f(N)=M^\ell_{P,f}(N), \qquad \osc_N a_N=\limsup_N a_N-\liminf_N a_N. \end{equation} The harmonic weight is on the index $n$, not the height $n^2+1$. \begin{theorem}[Instability around an arbitrary fixed source]\label{thm:instability} For every $f\in\M$, every $\varepsilon>0$ and every integer $M_0\geq1$: \begin{enumerate}[label=\textnormal{(\roman*)},leftmargin=2em] \item There is a single $g\in\M$ with $d_1(f,g)<\varepsilon$, $g(n)=f(n)$ for $n\leq M_0$, and $\osc_N A_g(N)\geq1/2$. \item There is an absolute constant $c_{\log}>0$ such that, for every fixed $125/164<\sigma\leq1$, there is a single $g\in\M$ with $d_\sigma(f,g)<\varepsilon$, $g(n)=f(n)$ for $n\leq M_0$, and $\osc_N L_g(N)\geq c_{\log}$. The constant is independent of $\sigma,f,\varepsilon,M_0$. For $128/153<\sigma\leq1$, a separate construction gives the numerical bound $\osc_N L_g(N)\geq1/2000$. \end{enumerate} The functions of ordinary oscillation at least $1/2$ form a dense $G_\delta$ in every finite-$d_1$ class. Those of logarithmic oscillation at least $c_{\log}$ form a dense $G_\delta$ in every finite-$d_\sigma$ class for $125/164<\sigma\leq1$; the bound $1/2000$ has the same category conclusion for $128/153<\sigma\leq1$. These assertions also hold with a compatible finite prefix fixed. The constructions preserve vanishing of the full family of fixed non-proportional affine correlations whenever it holds for $f$. \end{theorem} The two averaging assertions are separate constructions. Logarithmic divergence implies ordinary divergence by Abel summation, but does not transfer the ordinary oscillation constant to the same function. Section~\ref{sec:logarithmic} retains three analytic routes: Merikoski for the value $1/2000$, Pascadi for a larger range, and Grimmelt--Merikoski combined with Merikoski's sieve calculation for $\sigma>125/164$. Only the positive amplitude $c_{\log}$ is uniform in $\sigma$: the source, starting scales and decay of the prime cost may depend on it. The strict endpoint $125/164$ is not included; no numerical value for $c_{\log}$ is asserted. Two further constructions distinguish the roles of the signs and of the averaging method. Independent fair activations can remain logarithmically unstable on a deterministic support of arbitrarily small $d_\sigma$ cost for $125/164<\sigma\leq1$ (Theorem~\ref{thm:independent-failure}). In the other direction, Theorem~\ref{thm:separation} gives a support $\Gamma$ of arbitrarily small harmonic cost which is logarithmically stable for \emph{every} fixed polynomial and every $A\subseteq\Gamma$, but admits one fixed function $g(n)=(-1)^{v_2(n)}h_A(n)$ for which all three ordinary means on $n^2+1,n^2+2,n^2-2$ oscillate by at least $1/8$. The three logarithmic means of that same function are zero in the limit. There is also a distinction between probability and category. In the finite-$d_{9/10}$ class around $1$, a full-support probability measure gives simultaneous polynomial stability with probability one, although logarithmic quadratic divergence is residual. Section~\ref{subsec:full-support} constructs this measure explicitly; its typical supports have divergent half-power budget. \subsection{Counting support primes and changing the observation geometry} A coordinatewise bound gives the useful sufficient estimate \begin{equation}\label{eq:intro-half} \sup_N\frac1N\#\{n\leq N:f(n^2+1)\ne g(n^2+1)\} \leq5d_{1/2}(f,g). \end{equation} It is not necessary to sum this individual cost. A collective upper sieve treats the values with a nearly maximal prime factor together, allowing stability even when the critical series diverges. For a prime set $\Gamma$, define its counting exponent by \begin{equation}\label{eq:counting-exponent} \delta(\Gamma)=\limsup_{t\to\infty} \frac{\log(1+\#(\Gamma\cap[2,t]))}{\log t}. \end{equation} This is an exponent of growth, not a geometric dimension. Let $\Dset=\{-1,-2,2\}$, $P_d(n)=n^2-d$, and $Q_d(a,b)=a^2-db^2$. Curve averages omit the nonpositive value $P_2(1)$ and remain normalized by $N$; this changes \eqref{eq:general-means} by at most $1/N$ or $1/\HH_N$. Rectangle averages are \[ A_{d,f}(X,H)=\frac1{XH}\sum_{a\leq X,\,b\leq H}f(|Q_d(a,b)|). \] The following statements refer to a fixed support, uniformly over all its assignments. \begin{theorem}[Critical support classes and a rectangular threshold]\label{thm:critical-summary} \leavevmode \begin{enumerate}[label=\textnormal{(\roman*)},leftmargin=2em] \item If $\delta(\Gamma)\leq1/2$, all three curve observations have uniformly negligible prime tails and their ordinary and logarithmic means converge uniformly over $A\subseteq\Gamma$ to \begin{equation}\label{eq:curve-products} c_d(A)=\ind_{\{2\notin A\}} \prod_{\substack{p\in A,\ p\ \mathrm{odd}\\(d/p)=1}}\frac{p-3}{p+1}. \end{equation} \item If $0<\alpha<1$, $H=\lfloor X^\alpha\rfloor$, and $\delta(\Gamma)\leq s_\alpha=(1+\alpha)/2$, the analogous uniform tail and mean conclusions on all three rectangles hold with limits \begin{equation}\label{eq:all-binary-products} C_d(A)=3^{-\ind_{\{2\in A\}}} \prod_{\substack{p\in A,\ p\ \mathrm{odd}\\(d/p)=1}} \left(\frac{p-1}{p+1}\right)^2. \end{equation} \item If $\sum_{p\in\Gamma}1/p<\infty$ and $\log H(X)/\log X\to1$, the rectangular conclusions in (ii) still hold, even when $H(X)/X\to0$. \item For $119/500<\alpha<1$ and every $s_\alpha<\sigma\leq1$, every $d_\sigma$ ball around every source contains a fixed perturbation, preserving any prescribed compatible prefix, whose $a^2+b^2$ mean on $H=\lfloor X^\alpha\rfloor$ oscillates by at least a positive constant depending only on $\alpha,\sigma$. \end{enumerate} All positive tail conclusions also control multiplication of an arbitrary base, without creating its missing means. The negative assertion in (iv) concerns the sum of two squares, not simultaneous instability of the other two norms. \end{theorem} The equality in (i)--(ii) is included without summability at that exponent. For instance, if $p_j$ is the $j$th prime, $\Gamma=\{p_{j^2}:j\geq1\}$ has $\delta(\Gamma)=1/2$ and $\sum_{p\in\Gamma}p^{-1/2}=\infty$. Its assignment negative at every support prime has all three curve means zero. On power rectangles, (ii) and (iv) identify a universal budget and counting-exponent threshold in the stated angular range, not an individual classification of every support above it. The elementary estimate in Section~\ref{sec:geometry} already proves the positive result under $\sum_\Gamma p^{-s_\alpha}<\infty$; Section~\ref{sec:critical} removes that critical summability assumption. There is a separate geometric contrast at much thinner widths. If $H\to\infty$ and $H=o(\log X)$, one arbitrarily small harmonic perturbation can make \emph{all three} rectangle means oscillate by at least $1/2$ (Theorem~\ref{thm:rectangles}). The construction uses a prime private to a primitive direction, not to each point on that direction. The power-width counterexample instead obtains enough primitive directions from Gaussian primes in narrow sectors and a harmonic sum over cofactors. Appendix~\ref{sec:windows} keeps the curve variable unchanged and moves the origin. If $\delta(\Gamma)\leq\alpha/2$, windows of length at least $X^{\alpha-o(1)}$ at distance comparable to $X$ have the local laws at every origin. Under $\delta(\Gamma)\leq1/2$, every prescribed diverging length suffices at almost every origin. Explicit extremely sparse sources with long all-positive blocks explain why these quantifiers differ. \subsection{Antecedents, contributions and reading routes} Klurman~\cite[Proposition~1.1, Lemma~2.1 and their proofs]{Klurman} already uses private large primes, summable changes and the freezing of a single function to obtain quadratic non-cancellation of size $1/2$. Finite sign prescriptions are already part of Klurman's lemma; prefix protection alone is not a new device. The extension here combines an arbitrary infinite source, an arbitrarily small radius, actual non-convergence and residuality. Klurman's local-factor comparisons~\cite[Corollary~1.2 and Theorem~1.3]{Klurman} also keep an explicit large-prime term. Neither those constructions nor the local-product mechanism is claimed as new. The first criterion gains its force from the converse: convergence separately for every assignment on one harmonic support already implies uniform truncation and uniform limits. The random criterion instead identifies an exact almost-sure parity defect. The common-support separation in Theorem~\ref{thm:separation} distinguishes the two averaging methods arithmetically, rather than only for abstract bounded sequences. The counting-exponent results verify disappearance of the tail by combining sparse support below a moving cutoff with a collective sieve above it. The rectangular negative result joins the critical positive exponent only after an angular prime theorem has supplied a positive density of private directions. The probabilistic input is McDiarmid's classical bounded-differences inequality~\cite{McDiarmid}, in the accessible formulation of Warnke~\cite[Theorem~1]{Warnke}; no typical-differences refinement is needed. Mean periodic approximation belongs to the rationally almost periodic framework of Bergelson--Ku\l aga-Przymus--Lema\'nczyk--Richter~\cite{BKLR}. The periodic-multiplier argument for affine transfer is related to Klurman--Mangerel--Ter\"av\"ainen~\cite[Section~8, especially (8.3)--(8.7)]{KMT}; their Theorem~2.1 supplies the nonempty class of fully cancelling sources. For the logarithmic lower bounds, Merikoski~\cite[Section~2.1 and Remark~13]{Merikoski} supplies the numerical branch and Pascadi~\cite[Section~6.3, (6.20)]{Pascadi} supplies another weighted route. The largest range used here comes from Grimmelt--Merikoski~\cite[Corollaries~7.1--7.2]{GrimmeltMerikoski}: their unconditional Type~I/II information makes available the weighted calculation in \cite[Section~2.6, proof of Theorem~2]{Merikoski}. Section~\ref{sec:gm} spells out its strictly positive remaining mass before applying the population transfer. No new greatest-prime-factor estimate is claimed. The Gaussian square mean is already known in substantially greater generality. Donoso--Le--Moreira--Sun~\cite[Theorem~A]{DLMS} prove existence and an Euler product for the average of $f(a^2+b^2)$ on squares for every bounded real completely multiplicative $f$. Substituting $f=h_A$ gives exactly $C_{-1}(A)$ in \eqref{eq:all-binary-products}; harmonic summability is not needed for that existence result. Their comparison lemmas~\cite[Lemmas~4.1 and~4.3]{DLMS} also use reciprocal Gaussian-prime norms. The elementary square proof retained here prepares the explicit boundary estimate, rather than providing a new square-mean theorem. Their dilated domains arise from a fixed region~\cite[Definition~2.3]{DLMS}; this does not supply uniform control when the rectangle degenerates with $H/X\to0$. The attribution concerns $a^2+b^2$, not automatically the other two norms. Shrinking angular domains also have a separate literature. Burgin--Giannitsi~\cite{BurginGiannitsi} study ergodic averages along $\Omega$ in shrinking Gaussian sectors and sector--disk comparisons for finitely generated multiplicative actions. Ku\'s~\cite[Theorems~1.7 and~1.9]{Kus} treats multiplicative functions on Gaussian ideals under angular non-pretentiousness, with an additional conjugate-prime condition for short intervals. These source classes and angular or radial averages differ from uniform truncation over every subset of a prime support in a degenerating rectangle. Wang~\cite[Theorem~1.2 and Corollary~1.3]{WangCoprime} relates square averages over all pairs to averages over coprime pairs. This is relevant to primitive directions, but is distinct from controlling which entire direction a private prime can program. The positive collective estimates are direct specializations of the Selberg upper sieve~\cite[Section~13.2]{Kedlaya}; only fixed progressions modulo four and eight enter the Mertens estimates~\cite{DLMF}. Harman--Lewis~\cite[Theorem~2]{HarmanLewis} supplies the angular lower bound for the negative power-rectangle branch. Frantzikinakis--Klurman--Moreira~\cite[Lemma~4.15]{FKM} is an antecedent for square-incidence estimates, and Nair~\cite{Nair} for polynomial sums in short intervals. Neither is used to infer a uniform bound for a varying domain without the corresponding proof. A neighboring probabilistic problem is studied by de la Bret\`eche--Wang--Xu~\cite{DBWX}, who prove central limit theorems for polynomial sums of random multiplicative functions, including the completely multiplicative model. Fair prime signs at every prime do not satisfy \eqref{eq:expected-budget}. Their distributional limits and our almost-sure or whole-support truncation questions are distinct. \paragraph{Reading routes.} For the support criterion, start with the finite-prime laws in Section~\ref{sec:prelim} and the constructive converse in Section~\ref{sec:support}; Sections~\ref{sec:criterion}--\ref{sec:local} give the independent-law counterpart. Sections~\ref{sec:late-supports}--\ref{sec:separation} develop the stable and unstable realizations, with affine transfer in Section~\ref{sec:affine}. For the critical exponents, the key assembly is \eqref{eq:critical-curve-split}: a support count below a moving cutoff is combined with the collective sieve above it. Section~\ref{sec:collective} supplies that estimate. For the rectangular counterexample, follow the cofactor logarithm in Section~\ref{sec:angular} through its removal and injectivity steps; Section~\ref{sec:geometry} gives the elementary direction geometry first. Appendix~\ref{sec:windows} separates the all-origin and typical-origin conclusions. No inverse theorem for the unrestricted three curves, no cancellation for unmodified Liouville, and no Poisson limit is deduced. \section{Finite-prime approximation and deterministic control}\label{sec:prelim} For $E$ a prime set, let $h_E$ have the meaning in \eqref{eq:random-model}, without randomness. Every quotient $g/f$ in $\M$ is of this form. We first state the elementary facts required later. \subsection{Metrics and finite-coordinate continuity} For a fixed $f\in\M$ and $\sigma>0$, the class \[ \mathcal C_\sigma(f)=\{g\in\M:d_\sigma(f,g)<\infty\} \] is complete. Indeed, every prime coordinate in a Cauchy sequence is eventually constant, since changing the coordinate at $p$ costs $p^{-\sigma}$. Let $g$ be the coordinatewise limit. Fatou's lemma applied to the nonnegative coordinate distances gives convergence to $g$ in $d_\sigma$ and membership in the class. Truncating the changes to finitely many primes proves separability and density of finite changes. A condition on finitely many prime signs is both open and closed. In particular, each finite average in \eqref{eq:averages} is continuous, and preserving $g(n)=f(n)$ for $n\leq M_0$ is equivalent to preserving the prime signs at $p\leq M_0$. \subsection{Finite-prime laws are limits of periodic laws} For a finite prime set $F$ and $r\geq1$, the function \begin{equation}\label{eq:valuation-trunc} h_{F,r}(m)=(-1)^{\sum_{p\in F}\min(v_p(m),r)} \end{equation} is periodic. The untruncated function $h_F$ need not be periodic. For a fixed nonzero integer polynomial $P$, omit its integer zeros. Disagreement between $h_F(|P(n)|)$ and $h_{F,r}(|P(n)|)$ requires $P(n)\equiv0\pmod{p^r}$ for some $p\in F$. For a fixed $p$, the Haar measure of this set in $\Z_p$ tends to zero as $r\to\infty$: the sets decrease to the zero set of $P$ in $\Z_p$, which is finite. The Chinese remainder theorem therefore proves that every finite-prime polynomial observation, and every fixed finite vector of such observations, has a mean or empirical law obtained from independent Haar coordinates at the primes in $F$. This argument allows repeated factors and primes of bad reduction. For $P(n)=n^2+1$, the factors can be computed explicitly. If $p\equiv1\pmod4$, \[ \Prob_{u\in\Z_p}\{v_p(u^2+1)=k\}=\frac{2(p-1)}{p^{k+1}}\quad(k\geq1). \] Hence the probability of an odd valuation is $2/(p+1)$ and the mean sign is $1-4/(p+1)$. A prime $p\equiv3\pmod4$ never divides the value. At $2$, the valuation is one for odd $n$ and zero for even $n$. Thus \begin{equation}\label{eq:finite-quad-product} \lim_{N\to\infty} A_{h_F}(N) =\begin{cases} 0,&2\in F,\\[2pt] \displaystyle\prod_{\substack{p\in F\\p\equiv1\ (4)}}\left(1-\frac4{p+1}\right),&2\notin F. \end{cases} \end{equation} The logarithmic mean has the same limit. \begin{lemma}[Abel identity]\label{lem:abel} If $b_1,b_2,\ldots$ is bounded and $B(N)=N^{-1}\sum_{n\leq N}b_n$, then \begin{equation}\label{eq:abel} \frac1{\HH_N}\sum_{n\leq N}\frac{b_n}{n} =\frac{B(N)+\sum_{kz}1/p$. Taking the limits in that order proves \eqref{eq:diagonal-coverage}. Lemma~\ref{lem:abel} gives the exact identity $B_y^\ell=\mathcal A(B_y)$ for each \emph{fixed} $y$. Apply this positive averaging operator to \eqref{eq:coverage-intermediate}. Terms before $N=y$ have asymptotically zero total weight, the counting error tends to zero after averaging, and the remaining bound is the harmonic tail. This proves \eqref{eq:diagonal-log-coverage}. Finally $B_y^\star(N)$ decreases as $y$ increases. For each finite set of sample sizes it becomes zero once $y$ exceeds the heights of their nonzero polynomial values. Choose an initial cutoff for which the limiting upper bound is close to its outer limit; control all sufficiently large sizes at that cutoff, then enlarge the cutoff to remove the remaining finite set. This proves the assertion about the supremum. \end{proof} The logarithmic diagonal quantity is $\mathcal A u_P$, not the logarithmic coverage obtained by imposing $p>N$ throughout a sample of size $N$. The two have different finite formulas: for $P(n)=n^2+1$, $\Gamma=\{5\}$ and $N=10$, the latter is zero whereas $(\mathcal A u_P)(10)=1092/7381$. Each $u_P(k)$ in the Abel average uses its own cutoff $p>k$. \subsection{Uniform truncation gives the local products} For finite $F$, the finite-prime approximation of Section~\ref{sec:prelim} gives \begin{equation}\label{eq:finite-support-products} \lim_N M^c_{P,h_F}(N)=\lim_N M^\ell_{P,h_F}(N) =\prod_{p\in F}\ell_{P,p}. \end{equation} The same argument works for $|P|$ and omitted zeros: truncate valuations, apply the Chinese remainder theorem, and then use that the $p$-adic sets $\{p^r\mid P(t)\}$ decrease to a finite zero set. A nonzero constant is immediate. Outside the finite content-prime set, \[ |1-\ell_{P,p}|\leq 2\Prob_{t\in\Z_p}\{p\mid P(t)\}\leq 2D/p. \] Consequently the products \eqref{eq:general-product} converge uniformly over $A\subseteq\Gamma$; any zero factor is retained. To prove (ii)$\Rightarrow$(iii) of Theorem~\ref{thm:support-criterion}, put \[ t(y)=\sup_{A\subseteq\Gamma}\sup_N\Delta^\star_{P,N}(A;y). \] The difference of the signed means of $h_A$ and $h_{A,y}$ is at most $2t(y)$. At fixed $y$, only finitely many subsets of $\Gamma\cap[2,y]$ occur, and their means in \eqref{eq:finite-support-products} therefore converge uniformly. Their local products approach $\ell_P(A)$ uniformly as $y\to\infty$. The triangle inequality gives \[ \limsup_N\sup_{A\subseteq\Gamma}|M^\star_{P,h_A}(N)-\ell_P(A)| \leq 2t(y)+o_{y\to\infty}(1), \] which proves (iii), and (iii) implies (iv). Lemmas~\ref{lem:coverage-half}--\ref{lem:diagonal-coverage} give (i)$\Leftrightarrow$(ii). It remains to show why pointwise convergence for \emph{all} assignments already forces this uniform tail control. \subsection{Persistent coverage produces one divergent assignment} The converse does not need private primes. A finite parity average supplies an assignment with a low mean, and the small harmonic cost makes every finite assignment return to a high mean. Freezing the heights alternates these two observations in a single final function. \begin{proposition}[A bad subset of arbitrarily small harmonic cost]\label{prop:bad-subset} Set $\delta_\star=\lim_{y\to\infty}\limsup_N B_y^\star(N)$. If $\delta_\star>0$, then for every $0<\gamma<\delta_\star$, every $\varepsilon>0$ and every $M_0$, there is a subset $A\subseteq\Gamma\cap(M_0,\infty)$ such that \[ \sum_{p\in A}\frac1p<\varepsilon, \qquad \osc_N M^\star_{P,h_A}(N)\geq\gamma. \] \end{proposition} \begin{proof} Choose $a,b>0$ with $a+2b<\delta_\star-\gamma$. Take $y_0>M_0$, beyond the exceptional reduction primes, so that \[ \sum_{p\in\Gamma,p>y_0}1/p<\varepsilon, \qquad 2D\sum_{p\in\Gamma,p>y_0}1/p\delta_\star-b$. Give fair bits temporarily to the finitely many primes of $\Gamma\cap(z,\infty)$ occurring to odd order in this sample. Every covered value then has expected sign zero, regardless of its old sign; the uncovered values are at most one. Some deterministic choice, extending $F$ to a finite $F'$, consequently has mean at most $1-\delta_\star+b$ at $N$. Since $F'$ is finite, choose a recovery size $M>N$ where its mean is at least $1-a-b$. Repeat with the next cutoff beyond every $|P(n)|$, $n\leq M$, and beyond all previously used primes. Both the low sample and its recovery remain unchanged at every later stage. The increasing union of the negative sets defines one subset $A$, supported beyond $y_0$, with the required cost and prefix. Its low and high subsequences are separated by at least $\delta_\star-a-2b>\gamma$. Both sequences of indices tend to infinity. This proves the assertion for either weighting. \end{proof} The proposition proves the contrapositive of (iv)$\Rightarrow$(i), completing Theorem~\ref{thm:support-criterion}. It constructs a bad subset of the support, not divergence for each individual assignment having a nonzero truncation defect. The harmonic hypothesis cannot simply be removed. For $P(T)=T$ and $\Gamma$ equal to all primes, every real bounded multiplicative sign function has an ordinary mean by the classical theorem of Wirsing recalled in~\cite[Introduction]{Klurman}. Yet, for fixed $y$, an integer outside the coverage has squarefree part supported on the finitely many primes at most $y$. There are at most $2^{\pi(y)}\sqrt N$ such integers up to $N$, so $B_y(N)\to1$. This example is used only to delimit the hypothesis of the support criterion. \section{The exact random truncation defect}\label{sec:criterion} We prove Theorem~\ref{thm:criterion}. Throughout this section $P\in\Z[T]$ has fixed degree $D\geq1$ and is positive on $\N_{\geq1}$. A polynomial that is only eventually positive can be treated after deleting or translating a finite prefix. Assumption~\eqref{eq:expected-budget} and Tonelli's theorem give \begin{equation}\label{eq:budget-as} \sum_{p\in E}\frac1p<\infty\quad\hbox{almost surely}. \end{equation} The proof separates three tasks. First, the degree bounds the total large-prime incidence, while a root count bounds the influence of each prime; multiplying the two gives a small concentration budget. Second, the expectation of the large-prime disagreement is exactly $\eta_P(N)$, so bounded differences and Borel--Cantelli identify its realized value for all large $N$. Third, the intermediate primes are paid for by the tail of the \emph{realized} budget in \eqref{eq:budget-as}. Taking $N\to\infty$ before sending the fixed truncation threshold to infinity completes the comparison. \subsection{Large coordinates have small individual influence} Define \[ A_p(N)=\#\{n\leq N:v_p(P(n))\text{ is odd}\},\qquad a_p(N)=\frac{A_p(N)}N. \] Except at primes dividing the content of $P$, reduction modulo $p$ gives \begin{equation}\label{eq:root-count} A_p(N)\leq D(N/p+1). \end{equation} For all sufficiently large $N$, depending on $P$, the values $P(n)$ with $n\leq N$ are at most $N^{D+1}$. A value therefore has at most $2(D+1)$ distinct prime factors exceeding $\sqrt N$. Thus, once the content primes lie below $\sqrt N$, \begin{align} \sum_{p>\sqrt N}a_p(N)&\leq2(D+1),& \max_{p>\sqrt N}a_p(N)&\leq\frac{2D}{\sqrt N},\label{eq:influences-1}\\ \sum_{p>\sqrt N}a_p(N)^2&\leq\frac{4D(D+1)}{\sqrt N}.&&\label{eq:influences-2} \end{align} The last line comes from \[ \sum_{p>\sqrt N}a_p(N)^2 \leq\left(\max_{p>\sqrt N}a_p(N)\right)\sum_{p>\sqrt N}a_p(N). \] Thus the concentration budget is small without summing a separate uniform error over all primes up to the polynomial height. Let \[ Z_N=\frac1N\sum_{n\leq N} \ind\left\{\sum_{p>\sqrt N}\xi_pv_p(P(n))\text{ is odd}\right\}. \] Independence of the prime choices gives $\E Z_N=\eta_P(N)$ exactly. Changing $\xi_p$ can change $Z_N$ by at most $a_p(N)$. Only finitely many coordinates occur for each $N$. McDiarmid's classical bounded-differences inequality~\cite{McDiarmid}, in the form of~\cite[Theorem~1]{Warnke}, applied with \eqref{eq:influences-2}, yields \begin{equation}\label{eq:mcdiarmid} \Prob\{|Z_N-\eta_P(N)|>t\} \leq2\exp\left(-\frac{t^2\sqrt N}{2D(D+1)}\right). \end{equation} For $t_N=(6D(D+1)\log N/\sqrt N)^{1/2}$ the right side is $2N^{-3}$. Borel--Cantelli, without any independence requirement between different sample sizes, proves \begin{equation}\label{eq:concentration-as} Z_N=\eta_P(N)+O_D(N^{-1/4}\sqrt{\log N}) \quad\hbox{almost surely for all sufficiently large }N. \end{equation} The starting index may depend on $P$ and the realization. \subsection{The intermediate primes are paid by the realized budget} The comparison must establish the outer limits rather than appeal to monotonicity. For example, $8^2+1=65=5\cdot13$ and $E=\{5,13\}$ give $h_E(65)=1$, whereas \[ h_{E,y}(65)= \begin{cases} 1,&y<5,\\ -1,&5\leq y<13,\\ 1,&y\geq13. \end{cases} \] The disagreement at this value is not monotone in $y$. This finite example explains the issue, not an asymptotic obstruction. Take $y$ larger than all content primes and $N$ large enough that $y\leq\sqrt N$. The parity defining $\Delta_{P,N}(E;y)$ can differ from that defining $Z_N$ only through a changed prime between $y$ and $\sqrt N$. By \eqref{eq:root-count}, \begin{equation}\label{eq:middle-bound} |\Delta_{P,N}(E;y)-Z_N| \leq D\sum_{\substack{p\in E\\p>y}}\frac1p+\frac D{\sqrt N}. \end{equation} This is a deterministic union bound; it does not assume independence of the effects on the sampled integers. Combine \eqref{eq:concentration-as} and \eqref{eq:middle-bound}. For almost every realization, \begin{equation}\label{eq:limsup-sandwich} \left|\limsup_N\Delta_{P,N}(E;y)-\limsup_N\eta_P(N)\right| \leq D\sum_{\substack{p\in E\\p>y}}\frac1p, \end{equation} and the same holds for limit inferiors. Letting $y\to\infty$ proves \eqref{eq:criterion}. In particular the outer limits exist, although the individual disagreement proportions need not be monotone in $y$. A countable intersection of the probability-one events deals simultaneously with every integer polynomial and every integer truncation threshold. \subsection{Logarithmic truncation} Write $\Delta^{\log}_{P,N}(E;y)$ for the disagreement proportion with weights $1/(n\HH_N)$. Lemma~\ref{lem:abel} gives the exact relation \begin{equation}\label{eq:log-delta-abel} \Delta^{\log}_{P,N}(E;y) =\frac{\Delta_{P,N}(E;y)+\sum_{ky}1/p$ in limit superior. Therefore \[ \lim_{y\to\infty}\limsup_N\Delta^{\log}_{P,N}(E;y)=\limsup_N\overline\eta_P(N), \] with the analogous identity for limit inferiors. All quantities are nonnegative, so the zero-defect characterizations in Theorem~\ref{thm:criterion} follow. \begin{remark}[Cutoffs and the object characterized]\label{rem:cutoffs} The cutoff $\sqrt N$ is convenient, not essential. With $N^a$, $0N^a}\frac{\theta_p}{p}+DN^{b-1}=o(1). \end{equation} Indeed, couple the two parities using the same prime choices. A difference requires an activated incident prime between $N^a$ and $N^b$. The union bound and \eqref{eq:root-count} give at most $D\sum_{N^a\sqrt N}\theta_p$. At any sampled value, an odd activated parity requires at least one activation among the relevant primes. By \eqref{eq:influences-1}, \[ 0\leq\eta_P(N)\leq q_N\sum_{p>\sqrt N}a_p(N) \leq2(D+1)q_N\longrightarrow0. \] Theorem~\ref{thm:criterion} and its Abel version apply. \end{proof} Vanishing of $\theta_p$ is sufficient, not necessary. For $P(n)=n^2+1$, any prime support contained in $p\equiv3\pmod4$ is invisible, regardless of its activation probabilities, provided the harmonic hypothesis of the theorem is satisfied. \subsection{A joint law conditional on the selected support} Fix a realization $E$ for which Corollary~\ref{cor:vanishing} and \eqref{eq:budget-as} hold. On a separate probability space let $U_p$ be independent Haar-uniform elements of $\Z_p$. For fixed polynomials $P_1,\ldots,P_m$, positive on positive integers, define \begin{equation}\label{eq:local-vector} Y_{E,j}=\prod_{p\in E}(-1)^{v_p(P_j(U_p))},\qquad 1\leq j\leq m. \end{equation} At a given prime the \emph{same} coordinate $U_p$ is used for all polynomials. Thus shared factors and local dependencies are retained. A zero of a polynomial has Haar measure zero. Outside finitely many primes, a nontrivial factor occurs with probability $O_{P_1,\ldots,P_m}(1/p)$. By \eqref{eq:budget-as} and Borel--Cantelli, only finitely many nontrivial factors occur, almost surely in the $U_p$'s, so \eqref{eq:local-vector} is well-defined. For probability measures on the finite alphabet $\mathcal A=\{-1,1\}^m$, we use the convention \[ d_{\TV}(\mu,\nu)=\sup_{B\subset\mathcal A}|\mu(B)-\nu(B)| =\frac12\sum_{z\in\mathcal A}|\mu(\{z\})-\nu(\{z\})|. \] There are two distinct approximations: cutting the prime set, then truncating the valuations. Only the second makes the finite-prime observation periodic; a finite prime set alone does not do so. \begin{theorem}[Quenched empirical local law]\label{thm:local-law} Under the assumptions of Corollary~\ref{cor:vanishing}, almost surely in $E$ and simultaneously for all fixed polynomial families, \begin{equation}\label{eq:empirical-law} \frac1N\sum_{n\leq N}\delta_{(h_E(P_1(n)),\ldots,h_E(P_m(n)))} \xrightarrow[\ N\to\infty\ ]{\TV} \law\big((Y_{E,j})_{j=1}^m\big). \end{equation} The logarithmically weighted empirical measures have the same limit. Each sequence $h_E(P(n))$ is approximable in mean absolute value by periodic sign sequences, hence is rationally almost periodic in the finite-alphabet sense of~\cite{BKLR}. \end{theorem} \begin{proof} For $E\cap[2,y]$ finite, truncate valuations at level $r$. The resulting vector is periodic, and the Chinese remainder theorem identifies its limiting distribution with independent uniform residue classes at the finitely many primes. Sending $r\to\infty$ is legitimate by the zero-set argument of Section~\ref{sec:prelim}. It gives the finite-prime law with complete valuations. The total-variation discrepancy between the true empirical vector and its finite-prime truncation is at most $\sum_{j=1}^m\Delta_{P_j,N}(E;y)$. Its iterated limit is zero by Corollary~\ref{cor:vanishing}. On the target side, a union bound gives an error $O_{P_1,\ldots,P_m}(\sum_{p\in E,p>y}1/p)$, once $y$ contains the exceptional primes. Let $N\to\infty$ and then $y\to\infty$. On the finite alphabet $\{-1,1\}^m$, convergence of all atom masses is total-variation convergence. The mean periodic approximation follows from the same two truncations. Applying Lemma~\ref{lem:abel} to each atom gives the logarithmic assertion. Countability gives simultaneity. \end{proof} For example, almost surely, \begin{equation}\label{eq:infinite-quadratic-law} \lim_N A_{h_E}(N)=\lim_N L_{h_E}(N) =\begin{cases} 0,&2\in E,\\[2pt] \displaystyle\prod_{\substack{p\in E\\p\equiv1\ (4)}}\frac{p-3}{p+1},&2\notin E. \end{cases} \end{equation} The limit depends on the realized $E$. It is not the average of the product over the Bernoulli choices. Taking the family $P(n),P(n+1),\ldots$ shows existence of every fixed word frequency. These frequencies need not be those of independent fair signs; normality is not asserted. The family and word length are fixed: no claim about growing words or an independent field of all positions is made. \subsection{Stability beyond the deterministic half-power budget}\label{subsec:full-support} Protect a prescribed prefix by setting $\theta_p=0$ for $p\leq M_0$, and set \begin{equation}\label{eq:full-support-profile} \theta_p=\frac{c}{\lceil p^{1/4}\rceil}\quad(p>M_0),\qquad 00$. Choose $y$ so that both the center's $d_{9/10}$ tail and the expected random tail are small compared to $r$. The prescribed choices below $y$ have positive probability. Independently of those choices, Markov's inequality gives positive probability that the random tail is within the remaining budget. The resulting function lies in the ball. Thus a full-support probability measure can give probability one to simultaneous polynomial stability, while logarithmic quadratic divergence is residual in the same metric space by Theorem~\ref{thm:instability}. Full support means that every nonempty open set has positive measure; it does not prevent a residual set from being null. These are different notions of genericity. Moreover, choosing $c\leq\min(1/4,3\varepsilon/40)$ gives \[ \Prob\{d_{9/10}(1,h_E)<\varepsilon\}\geq\frac12. \] Conditioning on this ball retains the probability-one conclusions and full support within the ball. \subsection{What is transmitted to a prescribed source} For every $f\in\M$, cancellation of its signs gives the deterministic identity \begin{equation}\label{eq:source-cancel} \ind\{fh_E(P(n))\ne fh_{E,y}(P(n))\} =\ind\{h_E(P(n))\ne h_{E,y}(P(n))\}. \end{equation} The error bound is therefore simultaneous for all $f$. This does not determine the averages of $f(P(n))h_E(P(n))$. One valid transfer is the following. Its periodic-multiplier and residue-class argument is related to~\cite[Section~8, especially (8.3)--(8.7)]{KMT}; the polynomial truncation input is supplied by Theorem~\ref{thm:local-law}. \begin{corollary}[Transfer of progressionwise cancellation]\label{cor:source-transfer} Fix a regular realization $E$ as in Theorem~\ref{thm:local-law} and fixed polynomials $P_1,\ldots,P_m$. For $f\in\M$, suppose that \[ \frac1N\sum_{n\leq N}\prod_{j=1}^m f(P_j(qn+r))\longrightarrow0 \] for every fixed positive integer $q$ and every integer $r$, with nonpositive arguments in a finite prefix omitted. Then the same assertion holds with $f$ replaced by $fh_E$. The implication is reversible. \end{corollary} \begin{proof} For any fixed progression, approximate the quotient factors by finitely many prime signs and then bounded valuations. Their product is periodic in $n$. Its correlation with the product of the $f$-values vanishes by subdividing into residue classes, each of which is another fixed progression. Telescope the product of quotient factors. The error tends to zero by the simultaneous truncation result for the composed polynomials, followed by finite-prime valuation approximation. Since $h_E^2=1$, the same argument proves the converse. \end{proof} \section{Stable supports with no stronger power budget}\label{sec:late-supports} A support can be large in every stronger power budget and nevertheless have small influence on each fixed polynomial observation. The construction delays the first occurrence of every new prime. A counting argument on a product of integer values leaves enough primes to do this simultaneously for an increasing finite list of polynomials. \subsection{What an individual coordinate costs} For a nonzero integer polynomial define \begin{equation}\label{eq:coordinate-influence} c_P(p)=\sup_{N\geq1}\frac1N\#\{n\leq N:P(n)\ne0,\ v_p(|P(n)|)\text{ odd}\}. \end{equation} For a nonempty finite family $\mathcal F$, put $c_{\mathcal F}(p)=\max_{P\in\mathcal F}c_P(p)$, and let $\Delta_{\mathcal F}(f,g)$ be the maximum over $P\in\mathcal F$ of the supremum over $N$ of the ordinary disagreement proportion. \begin{proposition}[The exact coordinatewise requirement]\label{prop:coordinate-cost} For all $f,g\in\M$, \[ \Delta_{\mathcal F}(f,g)\leq\sum_{p:f(p)\ne g(p)}c_{\mathcal F}(p). \] Given weights $w(p)\geq0$, the inequality $\Delta_{\mathcal F}(f,g)\leq K\sum_{p:f(p)\ne g(p)}w(p)$ for every pair $f,g$ holds if and only if $c_{\mathcal F}(p)\leq Kw(p)$ for every prime. \end{proposition} \begin{proof} A disagreement requires a changed prime with odd valuation, so a union bound proves sufficiency. A pair differing at just one prime $p$ has exactly the disagreement indicator in \eqref{eq:coordinate-influence}; taking the suprema proves necessity. \end{proof} This optimality is limited to additive, single-coordinate upper bounds uniform over all pairs. It does not characterize convergence of means. Abel summation transfers each ordinary bound, including its tails, to logarithmic disagreement. Let $\rho_P(p)$ be the number of roots modulo $p$, and let $r_P(p)$ be the first positive index where $P(n)\ne0$ and $p\mid P(n)$, or $+\infty$ if there is none. Before this first index there is no incidence, and afterwards root counting gives \begin{equation}\label{eq:first-incidence} c_P(p)\leq \min\left(1,\frac{\rho_P(p)}p+\frac{\rho_P(p)}{r_P(p)}\right). \end{equation} If $\rho_P(p)=0$, the influence is zero. A nonzero polynomial cannot vanish identically on an infinite integer progression, so a modular root has a nonzero occurrence. For nonzero constants all relevant primes are confined to a finite set. \subsection{Choosing the support before any signs} \begin{theorem}[Rich supports with summable observation influences]\label{thm:late-supports} Fix $00$, a prefix $M_0$, and a nonempty finite family $\mathcal F_*$ of nonzero integer polynomials. There is an infinite deterministic prime set $\Gamma\subset(M_0,\infty)$ with \begin{align} \sum_{p\in\Gamma}p^{-s}&<\varepsilon,& \sum_{p\in\Gamma}p^{-t}&=\infty\quad(0R$ or no root; also $\rho_P(q)\leq\deg P$. Thus \eqref{eq:first-incidence} gives $\max_{P\in\mathcal F_j}c_P(q)\leq2d_j/R$. No equidistribution theorem for polynomial roots is used. Let $\tau_j=s(1-1/(j+1))$. Choose successive $X_j>\max(2X_{j-1},M_0)$ large enough for the preceding bounds and, with $k_j=\lceil(2X_j)^{\tau_j}\rceil$, for \begin{equation}\label{eq:rich-block-choices} k_j\leq\frac{X_j}{4\log X_j},\qquad k_jX_j^{-s}<\varepsilon2^{-j-2},\qquad \frac{2d_jk_j}{R_j}<\varepsilon2^{-j-2}. \end{equation} These choices are possible since $\tau_jy}c_P(p)\longrightarrow0. \end{equation} The same holds logarithmically. Both means converge uniformly over $A\subseteq\Gamma$ to $\ell_P(A)$, and every fixed finite vector of polynomial observations has a limiting local law. \end{corollary} \begin{proof} Proposition~\ref{prop:coordinate-cost} gives the tail bound, and Lemma~\ref{lem:abel} its logarithmic counterpart. The finite-prime laws and uniform product approximation prove uniform convergence, or one may use Theorem~\ref{thm:support-criterion}. For a finite vector, approximate all its coordinates at once; alternatively expand each atom indicator and apply the scalar statement to products of the fixed polynomials. Both routes retain shared local coordinates. \end{proof} No distribution on the signs is required: choices can be arbitrarily correlated or depend on the sample size. For a moving choice $A=A_N$, uniformity means that the error relative to $\ell_P(A_N)$ tends to zero; it does not assert convergence to a single scalar if those products vary. A common limit follows when a local factor forces all products to agree, as in the next corollary. \begin{corollary}[Three balanced traces with a prescribed negative sign]\label{cor:rich-balanced} Take $M_0\geq2$ and include $T^2+1,T^2+2,T^2-2$ in $\mathcal F_*$. Put $h_0(n)=(-1)^{v_2(n)}$ and $g_A=h_0h_A$. Then all three ordinary and logarithmic curve means of $g_A$ tend to zero uniformly over $A\subseteq\Gamma$. \end{corollary} \begin{proof} All support primes are odd. For $n^2+1$ the valuation at two is one at odd indices and zero at even indices; for $n^2\pm2$ the roles are reversed. The local factor at two is therefore zero. The Chinese remainder theorem preserves this factor after any finite choice of odd prime signs. Multiplication by $h_0$ cancels in the disagreement, so \eqref{eq:rich-uniform-tail} passes the zero finite-prime limits uniformly to every $A$. \end{proof} For $s=1$ and $A=\Gamma$, all stronger power budgets diverge, yet every trace has each sign with density $1/2$. The support remains harmonically summable. It is its delayed incidence, not a cancellation hypothesis on an unmodified source, that gives this conclusion. \section{Ordinary quadratic instability}\label{sec:ordinary} The construction starts with a finite reservoir of sampled values having distinct private prime factors. The required reservoir is elementary apart from the prime number theorem in the fixed progression $1\pmod4$. The logarithmic-product device is classical; the private-prime programming is directly related to Klurman's construction~\cite[Lemma~2.1]{Klurman}. \subsection{An elementary reservoir above the sample length} The factor $1/2$ comes from a logarithmic mass balance: the product of the sampled values carries about $2N\log N$, small primes account for at most $(1+o(1))N\log N$, and one large prime can carry at most $(2+o(1))\log N$ per value. The remaining mass therefore supplies almost half the sample. Its signs can be programmed at a cost tending to zero. Choose \begin{equation}\label{eq:ordinary-reservoir} b_N=3+\left\lfloor\sqrt{\lfloor\log_2N\rfloor}\right\rfloor,\qquad Y_N=Nb_N,\qquad I_N=\{1\leq n\leq N:P^+(n^2+1)>Y_N\}. \end{equation} Here $P^+(m)$ denotes the largest prime factor of $m>1$, and we use the convention $P^+(1)=1$ throughout. In particular, values $|Q_2(a,b)|=1$ in Section~\ref{sec:geometry} do not belong to a large-prime reservoir. We have $b_N\to\infty$, $b_N=o(\log N)$ and $Y_N>2N$. \begin{lemma}[Half of the sample has an affordable private pivot]\label{lem:ordinary-reservoir} As $N\to\infty$, \begin{equation}\label{eq:half-reservoir} |I_N|\geq\left(\frac12-o(1)\right)N. \end{equation} For each $n\in I_N$, the prime $q_n=P^+(n^2+1)$ has valuation one and divides no other $m^2+1$ with $1\leq m\leq N$. \end{lemma} \begin{proof} Set $M=N^2+1$. Factoring the product of all values gives \begin{equation}\label{eq:product-mass} \sum_{n\leq N}\log(n^2+1)=2\log(N!)+O(1)=2N\log N+O(N). \end{equation} For an odd prime $p\equiv1\pmod4$, there are two roots of $T^2+1$ modulo each $p^k$. Summing the bound $2N/p^k+2$ over all $k\leq\log M/\log p$, the contribution of primes at most $Y=Y_N$ is at most \begin{equation}\label{eq:small-mass} 2N\sum_{\substack{p\leq Y\\p\equiv1\ (4)}}\frac{\log p}{p-1} +2\pi(Y)\log M+O(N). \end{equation} Primes $3\pmod4$ do not occur, and the contribution at $2$ is $O(N)$. The prime number theorem in a fixed arithmetic progression and partial summation give \begin{equation}\label{eq:fixed-ap} \sum_{\substack{p\leq Y\\p\equiv1\ (4)}}\frac{\log p}{p-1} =\left(\frac12+o(1)\right)\log Y. \end{equation} For this classical input one may use~\cite{TaoPNT}. Since $\log Y\sim\log N$ and $\pi(Y)\ll Y/\log Y$, the second term of \eqref{eq:small-mass} is $O(Nb_N)=o(N\log N)$. The total small-prime contribution is at most $(1+o(1))N\log N$. By \eqref{eq:product-mass}, the primes above $Y_N$ contribute at least $(1-o(1))N\log N$. A value $n^2+1\leq M$ has at most one such prime, at exponent one, since $Y_N^2>M$. Its contribution is at most $\log M=(2+o(1))\log N$. Division proves \eqref{eq:half-reservoir}. If $q_n\mid m^2+1$ as well, then $q_n\mid(n-m)(n+m)$. Since $q_n>2N$, both factors have absolute value less than $q_n$, except that $n-m$ may be zero. Hence $m=n$. Distinct sites therefore have distinct pivots. \end{proof} \begin{lemma}[Weighted programming by private primes]\label{lem:programming} Let $J\subset[1,N]$ have primes $q_n\mid n^2+1$, $n\in J$, each with odd valuation and private to its site among the first $N$ quadratic values. Given $f\in\M$ and prescribed signs $s_n$, changing only the $q_n$ can realize \[ g(n^2+1)=s_n\quad(n\in J) \] without changing the other sampled values. The cost is at most $\sum_{n\in J}q_n^{-\sigma}$. For nonnegative weights $w_n$ summing to one, the all-positive and all-negative prescriptions give two functions with weighted-average difference \begin{equation}\label{eq:two-branches} \sum_{n\leq N}w_ng_+(n^2+1)-\sum_{n\leq N}w_ng_-(n^2+1) =2\sum_{n\in J}w_n. \end{equation} All integer values smaller than the least changed prime remain fixed. \end{lemma} \begin{proof} Set $g(q_n)=f(q_n)s_n/f(n^2+1)$, leaving all other prime signs unchanged. An odd valuation makes this change produce exactly the prescribed sign. Privacy ensures that distinct prescriptions do not interfere, and that the complement of $J$ is unchanged. Some pivots may need no change, so the displayed sum is an upper bound on the actual cost. The weighted identity follows term by term. \end{proof} For \eqref{eq:ordinary-reservoir}, this cost is at most \begin{equation}\label{eq:cheap-ordinary} \sum_{n\in I_N}\frac1{q_n}\leq\frac{|I_N|}{Y_N}\leq\frac1{b_N}\longrightarrow0. \end{equation} An individual pivot has observed incidence $1/N$, whereas its harmonic cost is less than $1/(Nb_N)$. This growing ratio is the elementary source of the instability. \subsection{Freezing a single function across infinitely many scales} At each step, keep an earlier experiment fixed and create two possible outputs at a larger scale. Since these outputs are separated by twice the controlled mass, one lies far from the earlier mean. Choose that output, then protect it from every future change. For an arbitrary source this does not force averages near both $+1$ and $-1$; separation is enough. Protection must extend to the polynomial height $N^2+1$, not merely to the number $N$ of sampled indices. We prove Theorem~\ref{thm:instability}(i). Start with $f_0=f$ and $N_0\geq\max(2,M_0)$. At stage $j\geq1$, choose $N_j>N_{j-1}$ so large that \begin{equation}\label{eq:ordinary-stage} Y_{N_j}>N_{j-1}^2+1,\qquad b_{N_j}^{-1}<\varepsilon2^{-j-1},\qquad \frac{|I_{N_j}|}{N_j}\geq\frac12-\frac1{j+2}. \end{equation} From the current $f_{j-1}$ apply the two constant prescriptions on $I_{N_j}$. They have means at $N_j$ separated by $2|I_{N_j}|/N_j$ and both preserve the mean at $N_{j-1}$. At least one is at distance $|I_{N_j}|/N_j$ or more from that earlier mean. Choose it as $f_j$. Every later pivot exceeds the heights of all quadratic values at previously chosen scales. It also exceeds every prime sign previously changed. The prime coordinates consequently stabilize and define one function $g\in\M$, with $d_1(f,g)<\varepsilon$ and unchanged prefix. For this fixed $g$, \[ |A_g(N_j)-A_g(N_{j-1})|\geq\frac12-\frac1{j+2}. \] For a bounded sequence, its eventual range contains both members of every sufficiently late pair. Letting $j\to\infty$ proves $\osc_N A_g(N)\geq1/2$. \subsection{Residual instability} For integers $m\geq3$ and $k\geq1$, consider \[ \mathcal O_{m,k}=\bigcup_{U,V\geq k} \left\{g:\ |A_g(U)-A_g(V)|>\frac12-\frac1m\right\}. \] This set is open by finite-coordinate continuity. It is dense in any $\mathcal C_1(f)$: within a given ball choose an anchor $R\geq k$, then choose $N$ with $Y_N>R^2+1$, cost below the radius and reservoir fraction greater than $1/2-1/m$. One of the two branches supplied by Lemma~\ref{lem:programming} lies in the ball and the indicated open set. The same choice preserves any compatible finite prefix. By completeness and Baire's theorem, $\bigcap_{m,k}\mathcal O_{m,k}$ is a dense $G_\delta$. The intersection is exactly the set with oscillation at least $1/2$. This proves the residual assertion of part (i). This argument concerns Baire category, not independent random prime signs. \subsection{Recovery near one and excursions down to zero} \begin{proposition}[A quantitative near-constant example]\label{prop:near-one} For every $0<\varepsilon<1/4$ and prefix $M_0$, there exists $h=h_E\in\M$ with \[ E\subset\{p:p\equiv1\pmod4\},\quad \sum_{p\in E}\frac1p<\varepsilon,\quad h(n)=1\ (n\leq M_0), \] such that \begin{equation}\label{eq:near-one-osc} \limsup_N A_h(N)\geq1-4\varepsilon,\qquad \liminf_N A_h(N)\leq0. \end{equation} The cluster set of $A_h(N)$ contains $[0,1-4\varepsilon]$, whereas \begin{equation}\label{eq:integer-mean} \lim_{X\to\infty}\frac1X\sum_{n\leq X}h(n) =\prod_{p\in E}\frac{p-1}{p+1}\geq1-2\varepsilon. \end{equation} \end{proposition} \begin{proof} Starting from $F_0=\varnothing$, suppose the finite changed-prime set $F_{j-1}$ is fixed. By \eqref{eq:finite-quad-product}, choose a recovery index $R_j$ beyond the previous stage with \[ \left|A_{h_{F_{j-1}}}(R_j)-C_{F_{j-1}}\right|<2^{-j},\qquad C_F=\prod_{p\in F}\left(1-\frac4{p+1}\right). \] Now choose $N_j>R_j$ so that the new pivots exceed $R_j^2+1$ and $M_0$, the cost is less than $\varepsilon2^{-j-1}$, and the reservoir fraction is at least $1/2-1/(j+2)$. Force all reservoir signs to be negative. Each pivot is new and originally positive; add to $F_{j-1}$ exactly the pivots that need to be reversed, giving $F_j\supset F_{j-1}$. The new mean is at most \[ A_{h_{F_j}}(N_j)\leq1-2|I_{N_j}|/N_j\leq\frac2{j+2}. \] Subsequent stages preserve both the recovery and excursion prefixes. For $E=\bigcup_jF_j$, the total cost is less than $\varepsilon$, and \[ C_E=\prod_{p\in E}\left(1-\frac4{p+1}\right) \geq1-4\sum_{p\in E}\frac1p>1-4\varepsilon. \] The recovery subsequence tends to $C_E$, and the excursion subsequence has limit inferior at most zero. This proves \eqref{eq:near-one-osc}, without identifying either extreme exactly. Any Ces\`aro average of signs satisfies $|A(N+1)-A(N)|\leq2/(N+1)$; successive crossings therefore show that its cluster set is the interval between its limit inferior and superior. On the ordinary integers, a finite prime change at $p$ has local mean $(p-1)/(p+1)$. The Chinese remainder theorem gives the finite product, and \eqref{eq:integer-distance} controls the infinite tail uniformly. This proves \eqref{eq:integer-mean}. The corresponding quadratic product gives a recovery subsequential limit, not the mean of the full quadratic observation. \end{proof} \section{Logarithmic instability with a stronger prime budget}\label{sec:logarithmic} A single dyadic block $(N/2,N]$, even with a positive proportion of available indices, has normalized logarithmic mass only $O(1/\log N)$. To retain positive mass, we need a number of dyadic blocks proportional to $\log N$, between $N^\beta$ and $N$. The analytic input must therefore provide a positive population in every sufficiently large dyadic interval, not merely infinitely many large factors. The parameters have separate jobs. If the available pivots satisfy $q_n>n^\alpha$ for $N^\beta1&\quad\Longrightarrow\quad q_n>N^{\alpha\beta}>2N &&\text{(privacy)},\\ \alpha\sigma>1&\quad\Longrightarrow\quad \sum_{n\in J_N}q_n^{-\sigma}\ll N^{-\beta(\alpha\sigma-1)}\to0 &&\text{(cost)}. \end{aligned} \] Here $J_N$ denotes the set of available indices. Its population density and $1-\beta$ control the logarithmic mass. The same transfer and construction take three different analytic inputs, whose outputs are compared in Table~\ref{tab:analytic-routes}. The widest range does not replace the numerical amplitude furnished by the narrower one. \begin{table}[htbp] \centering \small \renewcommand{\arraystretch}{1.18} \begin{tabularx}{\textwidth}{@{}>{\raggedright\arraybackslash}p{0.35\textwidth}>{\centering\arraybackslash}p{0.24\textwidth}>{\raggedright\arraybackslash}X@{}} \toprule Analytic input & Budget range & Oscillation bound \\ \midrule Merikoski, quantified branch & $128/153<\sigma\leq1$ & At least $1/2000$. \\ Pascadi, cutoff $1.30008$ & $12500/16251<\sigma\leq1$ & A common positive amplitude; not calibrated here. \\ Grimmelt--Merikoski with Merikoski's weighted calculation & $125/164<\sigma\leq1$ & A common $c_{\log}>0$; not calibrated here. \\ \bottomrule \end{tabularx} \caption{Three inputs to one logarithmic construction.} \label{tab:analytic-routes} \end{table} The lower endpoints in the table are excluded. Within each row the amplitude can be fixed independently of $\sigma$, but the starting scales and the rate at which the cost vanishes may depend on it. Each route below retains its weighted input and its passage to a dyadic population. \subsection{A quantified input from Merikoski}\label{sec:merikoski} Fix $w\in C_c^\infty((1,2))$ with $0\leq w\leq1$ and $\int w\geq1/2$. Set \[ b_x(n)=w(n/x),\quad W_x=\int_{\mathbb R}b_x(t)\dd t\geq x/2,\quad |\mathcal A_d|=\sum_{d\mid n^2+1}b_x(n). \] In the notation of Merikoski~\cite[Section~2.1]{Merikoski}, Chebyshev's device and a nonnegative smooth dyadic partition give \begin{equation}\label{eq:merikoski-total} S(x)=\sum_{p>x}|\mathcal A_p|\log p =W_x\log x+O_w(x) =\sum_{\substack{P\geq x\\P=2^j x}}S(x,P)+O_w(x), \end{equation} Here the blocks use the smooth dyadic partition of unity from that reference: \[ S(x,P)=\sum_{P\leq p\leq4P}\psi_P(p)|\mathcal A_p|\log p, \qquad \psi_P\geq0,\qquad \operatorname{supp}\psi_P\subset[P,4P]. \] The partition weights sum to one away from the lower-end transition, whose contribution is included in $O_w(x)$ in \eqref{eq:merikoski-total}. Each subfamily of weights therefore has sum at most one there. In particular, for a fixed $a>1$ and large $x$, \[ \sum_{\substack{P>x^a\\P=2^j x}}S(x,P) \leq\sum_{p>x^a}|\mathcal A_p|\log p. \] This is the positivity comparison used below; it is not an assumption of disjoint dyadic supports. \begin{inputthm}[Merikoski's linear-sieve bound]\label{input:sieve} Let $\vartheta=7/64$ and $a=153/128$. The integral upper bound in \cite[Remark~13, published p.~1268; arXiv v3, p.~15]{Merikoski} gives \begin{equation}\label{eq:merikoski-input} \sum_{\substack{x\leq P\leq x^a\\P=2^j x}}S(x,P) \leq(\kappa+o_w(1))W_x\log x,\qquad \kappa=4(1-2\vartheta)\int_1^a\frac{v}{1-2\vartheta v}\dd v<\frac9{10}. \end{equation} \end{inputthm} This is an unconditional published analytic estimate, not a hypothesis on the source and not a numerical experiment. The underlying distribution and sieve estimates are imported, not reproved here. The bound ``infinitely often $P^+(n^2+1)>n^{1.279}$'' by itself would not supply the positive population in every sufficiently large dyadic interval used below. For clarity, the integral actually used is \begin{equation}\label{eq:kappa} \kappa=\frac{25}{8}\left(-\frac{25}{28}+\frac{1024}{49}\log\frac{128}{121}\right) =0.882619359861\ldots. \end{equation} The decimal $0.8213$ printed after the integral in Remark~13, both in arXiv:1908.08816v3 and on p.~1268 of the published article, is not used. Its displayed integral, evaluated at $\vartheta=7/64$, gives \eqref{eq:kappa}. Only the strict margin below $9/10$ is needed. A rational enclosure follows from \[ \log\frac{128}{121}=2\sum_{j\geq0}\frac{(7/249)^{2j+1}}{2j+1}, \] whose positive tail is bounded by a geometric series. \begin{lemma}[A parameterized dyadic population]\label{lem:power-population} Fix $1<\alpha<153/128$. For every sufficiently large $x$, with starting point allowed to depend on $\alpha$, \begin{equation}\label{eq:power-population} \#\{xn^\alpha\}\geq x/100. \end{equation} \end{lemma} \begin{proof} Subtract \eqref{eq:merikoski-input} from \eqref{eq:merikoski-total}. For large $x$, the blocks with $P>x^a$, $a=153/128$, have total logarithmic mass at least $(2/25)W_x\log x$, with the partition error absorbed. The positivity comparison above bounds that mass by the actual mass at primes $p>x^a$. A value $n^2+1\leq4x^2+1$ contains at most one such prime, to exponent one. Since $w\leq1$, its contribution is at most $3\log x$ for large $x$. As $W_x\geq x/2$, there are at least $x/75$ such indices, and in particular at least $x/100$. Finally, $a>\alpha$ implies $x^a>(2x)^\alpha$ for every sufficiently large $x$ depending on $\alpha$. This proves \eqref{eq:power-population}. \end{proof} \subsection{From a dyadic population to a logarithmic reservoir}\label{sec:log-transfer} Fix $\alpha>1$ and $1/\alpha<\beta<1$. Define \begin{equation}\label{eq:log-reservoir} J_N=\{N^\betan^\alpha\},\quad q_n=P^+(n^2+1),\quad m_N=\frac1{\HH_N}\sum_{n\in J_N}\frac1n. \end{equation} \begin{lemma}[Parameterized mass--privacy--budget transfer]\label{lem:log-reservoir} Suppose a fixed $\nu>0$ satisfies \[ \#\{xn^\alpha\}\geq\nu x \] for every sufficiently large $x$. Then, for all sufficiently large $N$, the primes $q_n$, $n\in J_N$, are distinct, have exponent one, and are private throughout the sample $(r^2+1)_{r\leq N}$. For every fixed $\sigma>1/\alpha$, \begin{equation}\label{eq:log-mass-budget} \begin{aligned} \liminf_N m_N&\geq\frac{\nu(1-\beta)}{2\log2},\\ \sum_{n\in J_N}q_n^{-\sigma} &\ll_{\alpha,\beta,\sigma}N^{-\beta(\alpha\sigma-1)}\longrightarrow0. \end{aligned} \end{equation} \end{lemma} \begin{proof} Since $\alpha\beta>1$, we have $q_n>N^{\alpha\beta}>2N$ for large $N$. The argument of Lemma~\ref{lem:ordinary-reservoir} proves privacy and distinctness; also $q_n^2>N^2+1$ for large $N$, which proves exponent one. The interval $(N^\beta,N]$ contains $(1-\beta)\log N/\log2+O(1)$ complete dyadic blocks. For all sufficiently large $N$, every block lies beyond the starting point of the assumed population bound. Each block $(x,2x]$ contributes at least $\nu x$ indices, hence reciprocal mass at least $\nu/2$. Sum over these blocks and divide by $\HH_N\sim\log N$. This gives the first bound in \eqref{eq:log-mass-budget}. For the cost, $\alpha\sigma>1$ gives \[ \sum_{n\in J_N}q_n^{-\sigma} \leq\sum_{n>N^\beta}n^{-\alpha\sigma} \ll_{\alpha,\beta,\sigma}N^{-\beta(\alpha\sigma-1)}. \] Thus $\alpha\beta>1$ ensures privacy, while $\alpha\sigma>1$ makes the cost tend to zero. No arithmetic information other than the displayed population is used in this transfer. \end{proof} For $128/153<\sigma\leq1$, Lemma~\ref{lem:power-population} supplies $\nu=1/100$. Fix once and for all \begin{equation}\label{eq:log-parameters} \alpha=\frac12\left(\frac{153}{128} +\max\left\{\frac76,\frac1\sigma\right\}\right), \qquad \beta=\frac9{10}. \end{equation} Then $\max\{7/6,1/\sigma\}<\alpha<153/128$, $\alpha\beta>21/20>1$, and $\alpha\sigma>1$. In particular, \[ \liminf_N m_N\geq\frac1{2000\log2}>\frac1{2000}. \] This gives the numerical amplitude in the first row of Table~\ref{tab:analytic-routes}. The later rows cover $\sigma=128/153$ with an uncalibrated positive amplitude, not with the number $1/2000$. \subsection{A common construction and its category conclusion}\label{sec:log-construction} Fix $0<\sigma\leq1$ and a reservoir supplied by Lemma~\ref{lem:log-reservoir}, with $\alpha\sigma>1$. Let $c>0$ be strictly smaller than its limiting lower mass. The construction only needs this mass, vanishing cost, and protection of earlier heights; it does not depend on which analytic estimate supplied the population. Start from $f_0=f$ and $N_0\geq\max(2,M_0)$. At stage $j\geq1$, choose $N_j>N_{j-1}$ with \[ \min_{n\in J_{N_j}}q_n>\max(M_0,N_{j-1}^2+1),\quad \sum_{n\in J_{N_j}}q_n^{-\sigma}<\varepsilon2^{-j-1},\quad m_{N_j}>c. \] Use Lemma~\ref{lem:programming} with weights $w_n=1/(n\HH_{N_j})$. Its two branches have logarithmic means separated by $2m_{N_j}$ and preserve the preceding mean. Choose the branch whose mean is at distance at least $m_{N_j}$ from that preceding value. Every later changed prime exceeds the full height of the preceding sample. Thus all previously selected values are final, the prime signs stabilize to one $g$, and \[ d_\sigma(f,g)<\varepsilon,\qquad |L_g(N_j)-L_g(N_{j-1})|>c. \] Boundedness gives $\osc_N L_g(N)\geq c$. For integers $r,k\geq1$, the sets \[ \bigcup_{U,V\geq k}\{g:|L_g(U)-L_g(V)|>c-1/r\} \] are open by finite-coordinate continuity. They are dense: in any ball choose an anchor beyond $k$, and then a reservoir with cost less than the radius and with pivots beyond the anchor's height. One of its two branches differs from that frozen anchor by more than $c$. The same choices preserve any compatible finite prefix. Completeness and Baire's theorem show that their intersection, exactly the set of logarithmic oscillation at least $c$, is a dense $G_\delta$. Applying this construction with \eqref{eq:log-parameters} and $c=1/2000$ proves the quantitative part of Theorem~\ref{thm:instability}(ii), including its category assertion. The next two routes use the identical construction with a different population and amplitude. Because $d_1(f,g)\leq d_\sigma(f,g)$ for $\sigma\leq1$, ordinary integer disagreement stays below $\varepsilon$ at every prefix. For $\sigma<1$, splitting the weighted prime sum at a fixed threshold also gives \[ \#\{p\leq x:f(p)\ne g(p)\}=o(x^\sigma). \] Thus the changed primes have relative prime density zero in that case. This last conclusion is not deduced from harmonic summability alone at $\sigma=1$. \subsection{Pascadi's weighted estimate without rounding its cutoff}\label{sec:pascadi} \begin{inputthm}[Weighted mass beyond Pascadi's cutoff]\label{input:pascadi} Fix a nonzero $\Phi\in C_c^\infty((1,2))$ with $0\leq\Phi\leq1$, and set $\omega_P=16251/12500=1.30008$. The weighted estimates in Pascadi~\cite[Sections~6.1--6.3, especially (6.20) and the final calculation]{Pascadi} imply that, for some $c_\Phi^P>0$ and all sufficiently large $x$, \begin{equation}\label{eq:pascadi-mass} \sum_n\Phi(n/x) \sum_{\substack{p\mid n^2+1\\ p>x^{\omega_P}}}\log p \geq c_\Phi^P x\log x. \end{equation} The inner sum is over primes. Its constant is not evaluated here. \end{inputthm} In that reference the weighted main term is $X=x\int\Phi$, not the integer sample length used elsewhere here. The total large-prime mass is $X\log x+O_\Phi(x)$. Its positive dyadic blocks up to $x^{\omega_P}$ satisfy (6.20), with a strict upper margin below $X\log x$ in the concluding calculation at $1.30008$. Hence the remaining blocks have positive mass. Their sum is at most the actual mass at primes beyond that cutoff, by positivity of the partition. Absorbing $O_\Phi(x)$ proves \eqref{eq:pascadi-mass}. This uses the weighted proof, not just an infinitely-often greatest-prime-factor statement. The spectral estimates, Harman sieve and numerical integrals behind it are imported, not recomputed here. \begin{lemma}[A positive population at every large dyadic scale]\label{lem:pascadi-population} There is $\nu_P>0$, independent of $\alpha$, such that, for each fixed $1<\alpha<\omega_P$ and every sufficiently large $x$ depending on $\alpha$, \[ \#\{xn^\alpha\}\geq\nu_P x. \] \end{lemma} \begin{proof} For large $x$, a value at most $4x^2+1$ has at most one prime factor above $x^{\omega_P}$, at exponent one. As $\Phi\leq1$, its contribution to \eqref{eq:pascadi-mass} is at most $3\log x$. There are at least $(c_\Phi^P/3)x$ contributing indices. Since $\alpha<\omega_P$, eventually $x^{\omega_P}>(2x)^\alpha$. Take $\nu_P=c_\Phi^P/3$, or any smaller fixed positive constant. \end{proof} For every fixed $12500/16251<\sigma\leq1$, take \begin{equation}\label{eq:pascadi-parameters} \alpha=\frac12\left(\omega_P+\max\left\{\frac76,\frac1\sigma\right\}\right), \qquad \beta=\frac9{10}. \end{equation} Then $\alpha<\omega_P$, $\alpha\beta>21/20$ and $\alpha\sigma>1$. Lemma~\ref{lem:log-reservoir} yields \begin{equation}\label{eq:pascadi-reservoir} \liminf_N m_N\geq\frac{\nu_P}{20\log2},\qquad \sum_{n\in J_N}q_n^{-\sigma}\ll_{\alpha,\sigma}N^{-(9/10)(\alpha\sigma-1)}\to0. \end{equation} Section~\ref{sec:log-construction} therefore supplies both a fixed perturbation and the dense $G_\delta$ conclusion with the common amplitude $c_P=\nu_P/(40\log2)$, as in the second row of Table~\ref{tab:analytic-routes}. The simpler cutoff $13/10$ gives a valid but weaker consequence of this route. In particular $\sigma=10/13$ is covered: choosing $\alpha=32501/25000$ and $\beta=9/10$ gives $\alpha<\omega_P$, $\alpha\beta>1$, $\alpha\sigma=1+1/32500$, and cost $O(N^{-9/325000})$. The strict endpoint of this route is $12500/16251$, not $10/13$. At $\sigma=4/5$, the earlier choice $\alpha=51/40$, $\beta=9/10$ remains useful: \[ \alpha\beta=459/400>1,\qquad \alpha\sigma=51/50>1,\qquad \sum_{n\in J_N}q_n^{-4/5}\ll N^{-9/500}. \] These negative powers are cost estimates, not practical starting scales. At the same budget, Theorem~\ref{thm:late-supports} and Corollary~\ref{cor:rich-balanced} give supports on which every assignment after multiplication by $h_0$ has all three means zero. Applying the present construction to $h_0$ instead gives logarithmic divergence on $n^2+1$, preserving its negative sign at two. Placement of the support, not the budget alone, separates these examples. \subsection{The Grimmelt--Merikoski input and the full range}\label{sec:gm} The largest range used here comes from a different analytic input. Grimmelt--Merikoski supply unconditionally the Type~I and Type~II ranges used in Merikoski's formerly conditional calculation. The following statement specifies the weighted consequence consumed by the construction. \begin{inputthm}[Weighted mass from Grimmelt--Merikoski and Merikoski]\label{input:gm} Fix $\Phi$ as above and set $\omega_G=164/125=1.312$. There is $c_\Phi^G>0$ such that, for every sufficiently large $x$, \begin{equation}\label{eq:gm-mass} \sum_n\Phi(n/x)\sum_{\substack{p\mid n^2+1\\p>x^{\omega_G}}}\log p \geq c_\Phi^G x\log x. \end{equation} The analytic inputs are \cite[Section~7, Corollaries~7.1--7.2]{GrimmeltMerikoski} and the weighted sieve calculation of \cite[Section~2.6, pp.~15--16, proof of Theorem~2]{Merikoski}, with the fixed quadratic $n^2+1$. \end{inputthm} \paragraph{Matching the arithmetic information.} Set the coefficient and shift denoted by $a,h$ in Grimmelt--Merikoski equal to one. Their squarefreeness and coprimality hypotheses then hold. Their Corollary~7.1 provides Type~I information to level $D\leq x^{1/2-\eta}$. With $MN=x^u$, Corollary~7.2 provides Type~II information in \[ x^{u-1+2\eta}\leq N\leq x^{(2-u)/3-4\eta/3}, \] with divisor-bounded coefficients and the second coefficient supported on squarefree integers. The smooth weights required there cover the present fixed weight: replace their scale by $2x$ and their argument weight by $t\mapsto\Phi(2t)$, and subdivide modulus weights into smooth dyadic pieces. Fixed scale factors are absorbed by the strict exponent margins. The power savings require only a spectral gap, not Selberg's eigenvalue conjecture. As the paragraph following their Corollary~7.2 explicitly states, these are precisely the Type~I/II ranges used in Merikoski's proof of Theorem~2. The sieve dimension for $n^2+1$ is one: apart from two, it has two roots at primes $1\pmod4$ and none at primes $3\pmod4$. \paragraph{A strict margin before taking limits.} Use $W_x=x\int\Phi$ and the positive blocks $S_\Phi(x,P)$ defined as in \eqref{eq:merikoski-total}. Merikoski's calculation gives an upper constant $0.679914$ for the mass up to $x^{5/4}$, followed by the linear-sieve term $4\int_{5/4}^{\omega_G}v\dd v$. The bounds on the preceding Buchstab integrals are imported from that calculation. The final elementary integral has the exact evaluation \begin{equation}\label{eq:gm-margin} \begin{aligned} \kappa_G&=\frac{679914}{10^6} +2\left(\left(\frac{164}{125}\right)^2-\left(\frac54\right)^2\right) =\frac{498801}{500000}=0.997602,\\ 1-\kappa_G&=\frac{1199}{500000}>0. \end{aligned} \end{equation} Inserting the unconditional arithmetic information into that sieve recipe gives, for each sufficiently small fixed $\zeta>0$ after choosing its auxiliary margins, \begin{equation}\label{eq:gm-block-bound} \sum_{\substack{x\leq P\leq x^{\omega_G}\\P=2^j x}}S_\Phi(x,P) \leq\bigl(\kappa_G+\zeta+o_{\Phi,\zeta}(1)\bigr)W_x\log x. \end{equation} The term $\zeta$ allows the small endpoint losses in the Type~I/II ranges; the same weighted recipe and its limiting integrals are those in the cited proof. Fix $0<\zeta<(1-\kappa_G)/4$ first, then take $x$ large enough to absorb the little-oh term and the $O_\Phi(x)$ partition remainder. Subtracting from total mass $W_x\log x+O_\Phi(x)$ leaves a fixed positive multiple of $W_x\log x$. Positivity bounds the remaining blocks by the actual mass at primes $p>x^{\omega_G}$, proving \eqref{eq:gm-mass}. This is the required passage from the two cited proofs; the greatest-prime-factor theorem stated in their introductions alone would not give it. Neither their spectral estimates nor the multidimensional numerical bounds behind $0.679914$ are recertified by the finite arithmetic check of \eqref{eq:gm-margin}. \begin{lemma}[Population below $1.312$]\label{lem:gm-population} There is $\nu_G>0$, independent of $\alpha$, such that, for every fixed $1<\alpha<164/125$, every sufficiently large $x$ satisfies \[ \#\{xn^\alpha\}\geq\nu_G x. \] The starting point may depend on $\alpha$. \end{lemma} \begin{proof} A value at most $4x^2+1$ has at most one prime factor greater than $x^{\omega_G}$, of exponent one. Its weighted contribution in \eqref{eq:gm-mass} is at most $3\log x$. Thus $\nu_G=c_\Phi^G/3$, or any smaller fixed positive constant, bounds the density of contributing indices below. Finally $x^{\omega_G}>(2x)^\alpha$ for every sufficiently large $x$ depending on $\alpha$. \end{proof} \paragraph{Uniform amplitude, not uniform starting scales.} Fix $125/164<\sigma\leq1$, and put \begin{equation}\label{eq:gm-parameters} \alpha_\sigma=\frac12\left(\frac{164}{125} +\max\left\{\frac76,\frac1\sigma\right\}\right),\qquad \beta=\frac9{10},\qquad c_{\log}=\min\left\{\frac1{4000},\frac{\nu_G}{40\log2}\right\}>0. \end{equation} Then $\alpha_\sigma<164/125$, $\alpha_\sigma\beta>21/20$ and $\alpha_\sigma\sigma>1$. Lemma~\ref{lem:log-reservoir} gives $\liminf m_N\geq\nu_G/(20\log2)>c_{\log}$ and cost tending to zero. Section~\ref{sec:log-construction}, with $c=c_{\log}$, supplies the fixed perturbation and dense $G_\delta$ assertion in Theorem~\ref{thm:instability}. Here $\Phi$ and $\nu_G$ were fixed before $\sigma$, so $c_{\log}$ is common to the whole range. The strict inequality on $\sigma$ is needed to choose $1/\sigma<\alpha_\sigma<164/125$; this route does not cover $125/164$. No privacy on the other two curves is asserted. The numerical value $1/2000$ remains a separate consequence of \eqref{eq:log-parameters}, not a value claimed for the enlarged range. The independent construction in Section~\ref{sec:independent} only consumes a positive reservoir and therefore uses \eqref{eq:gm-parameters} as well. The stability range on the curves and these instability ranges still leave an interval unclassified. \section{Independent choices on an unstable arithmetic support}\label{sec:independent} The sufficient condition $\theta_p\to0$ in Corollary~\ref{cor:vanishing} cannot be discarded simply because the decisions are independent. We construct a perturbation of $1$ whose support is fixed before the random signs are selected. The crucial order of choices is possible because, after finitely many support blocks have been chosen, only finitely many past sign configurations are possible. A common recovery scale can be chosen for all of them, without observing the configuration eventually drawn. \begin{theorem}[Fair choices can retain logarithmic instability]\label{thm:independent-failure} For every $125/164<\sigma\leq1$, $\varepsilon>0$ and $M_0$, there is a deterministic prime set \[ \Gamma\subset\{p>M_0:p\equiv1\pmod4\},\qquad \sum_{p\in\Gamma}p^{-\sigma}<\varepsilon, \] such that independent activations $\theta_p=\tfrac12\ind_{\{p\in\Gamma\}}$ produce, almost surely, a function $h_E$ for which $L_{h_E}(N)$ does not converge. The prefix is unchanged and the cost bound holds for every realization. \end{theorem} \begin{proof} Use Lemmas~\ref{lem:gm-population} and~\ref{lem:log-reservoir}, with the fixed parameters \eqref{eq:gm-parameters}, and put $c_0=c_{\log}$. In particular $\beta=9/10$ and the reservoir mass eventually exceeds $c_0$. Set $N_0\geq\max(2,M_0)$. Fix $0<\varepsilon_0<\min(\varepsilon,c_0/8)$. We recursively choose disjoint finite prime blocks $\Gamma_j$, of cost less than $\varepsilon_02^{-j}$, without observing any random choices. Suppose $\Gamma_1,\ldots,\Gamma_{j-1}$ are fixed. There are finitely many possible subsets $F$ of their union. By \eqref{eq:finite-quad-product}, each $L_{h_F}(N)$ tends to \[ C_F=\prod_{p\in F}\left(1-\frac4{p+1}\right)\geq1-4\varepsilon_0. \] Choose a common recovery index $R_j>N_{j-1}$ such that all these finitely many averages are at least $1-4\varepsilon_0-1/j$ at $R_j$. Then choose $N_j>R_j$ so that the reservoir has mass $m_{N_j}\geq c_0$, its total $d_\sigma$ cost is less than $\varepsilon_02^{-j}$, and its least pivot exceeds $R_j^2+1$ and $M_0$. Let $\Gamma_j$ consist of all these pivots. All trace values up to $R_j$ remain protected. Conditional on the choices in previous blocks, the signs of $h_E(n^2+1)$ for $n\in J_{N_j}$ are independent fair signs: each site uses one new private pivot. The other sites do not use the new block. Thus, writing $T_j$ for the logarithmic mean at $N_j$ after this block is chosen, \[ \E(T_j\mid\text{previous blocks})\leq1-m_{N_j}\leq1-c_0. \] Changing one pivot affects this mean by at most $2w_n$, where $w_n=1/(n\HH_{N_j})$. Therefore \[ \sum_{n\in J_{N_j}}(2w_n)^2\leq\frac{4N_j^{-9/10}}{\HH_{N_j}}, \] and the classical bounded-differences inequality~\cite{McDiarmid,Warnke} gives \begin{equation}\label{eq:independent-stage} \Prob\{T_j>1-c_0+1/j\mid\text{previous blocks}\} \leq\exp\left(-\frac{N_j^{9/10}\HH_{N_j}}{2j^2}\right). \end{equation} Enlarge $N_j$ further so that this is at most $2^{-j}$. This choice uses only the deterministic reservoir and is uniform in all preceding signs. Every later block is beyond $N_j^2+1$, so both $T_j$ and the recovery mean are frozen in the final $h_E$. Integrating \eqref{eq:independent-stage} over the past and applying Borel--Cantelli gives \[ \limsup_N L_{h_E}(N)\geq1-4\varepsilon_0,\qquad \liminf_N L_{h_E}(N)\leq1-c_0 \quad\hbox{almost surely}. \] The gap is at least $c_0-4\varepsilon_0>c_0/2$. Finally $E\subset\Gamma=\bigcup_j\Gamma_j$, so the cost and prefix bounds hold for every outcome. \end{proof} The arithmetic concentration is in the predetermined support $\Gamma$. Independent fair choices on that support do not erase it. There is no contradiction with Corollary~\ref{cor:vanishing}: the probabilities equal $1/2$ on infinitely many support primes and do not tend to zero. This theorem uses the analytic input of Section~\ref{sec:logarithmic}; the exact criterion and the vanishing-probability corollary do not. The proof gives oscillation greater than $c_{\log}/2$ almost surely, independent of the chosen $\sigma$ within this range, while the deterministic support may depend on $\sigma$. In the subrange $128/153<\sigma\leq1$, the original quantified route remains available: use \eqref{eq:log-parameters} and $c_0=1/2000$ in the same proof. Thus its numerical gap greater than $1/4000$ is retained without assigning that value to the whole enlarged range.\par \section{A support separating ordinary and logarithmic stability}\label{sec:separation} The two criteria in Theorem~\ref{thm:support-criterion} have different arithmetic realizations. We construct one support whose tail is harmless logarithmically for every fixed polynomial and every assignment, but which retains a macroscopic ordinary reservoir on each of the three quadratic curves. Its first occurrences are delayed, and the scales are so far apart that their logarithmic contribution is negligible. In this section write $A^c_{d,f}(N)$ and $A^\ell_{d,f}(N)$ for the two means on the positive values of $P_d(n)=n^2-d$, $d\in\Dset$, normalized by $N$ and $\HH_N$. Put $h_0(n)=(-1)^{v_2(n)}$. This choice supplies the common recovery value: its local factor at two is zero on each of the three curves, and finite changes at odd primes preserve that zero factor. The logarithmic limits below therefore require no conjectural cancellation of the source. \begin{theorem}[One support, three ordinary divergences]\label{thm:separation} For every $\varepsilon>0$ and $M_0\geq2$, there is a deterministic prime support $\Gamma\subset(M_0,\infty)$ with $\sum_{p\in\Gamma}1/p<\varepsilon$ such that: \begin{enumerate}[label=\textnormal{(\roman*)},leftmargin=2em] \item for every fixed nonzero integer polynomial $P$, logarithmic prime truncation is uniform over $A\subseteq\Gamma$ and all $N$, and $M^\ell_{P,h_A}(N)\to\ell_P(A)$ uniformly in $A$; \item $\sup_{A\subseteq\Gamma}|A^\ell_{d,h_0h_A}(N)|\to0$ for every $d\in\Dset$; \item one fixed subset $A\subseteq\Gamma$ satisfies $\osc_N A^c_{d,h_0h_A}(N)\geq1/8$ for every $d\in\Dset$; \item on each curve, $\limsup_N u_{P_d}(N)\geq1/8$ and $\sum_{p\in\Gamma}c_{P_d}(p)=\infty$. \end{enumerate} The support and all recovery scales are chosen before the subset in (iii). The excursions for different curves need not occur at the same scales. \end{theorem} \subsection{First occurrences and an elementary three-curve reservoir} If $r_P(p)\leq N$, then the root classes modulo $p$ give \begin{equation}\label{eq:log-first-occurrence} \sum_{\substack{n\leq N\\ P(n)\ne0,\ p\mid P(n)}}\frac1n \leq\rho_P(p)\left(\frac{\HH_N}{p}+\frac1{r_P(p)}\right). \end{equation} Indeed, the first nonzero occurrence in each class is at least $r_P(p)$; subsequent occurrences are separated by at least $p$. Their reciprocals after the first sum to at most $\HH_N/p$. If the first occurrence is beyond $N$ or absent, the sum is zero. For each fixed $B\geq4$ and $d\in\Dset$, the elementary product argument used in Lemma~\ref{lem:ordinary-reservoir} gives \begin{equation}\label{eq:fixed-B-reservoir} \#\{n\leq N:P_d(n)>0,\ P^+(P_d(n))>BN\} \geq(1/2-o_{B,d}(1))N. \end{equation} Here is the calculation for all three curves. The logarithm of the product of their positive values, on any one curve, is $2N\log N+O_d(N)$. With $Y=BN$ and $V=N^2+2$, the contribution of primes at most $Y$ is at most \[ N\sum_{\substack{p\leq Y\\ p\ \mathrm{odd}}} \frac{\rho_d(p)\log p}{p-1} +2\pi(Y)\log V+O(N), \qquad \rho_d(p)=\#\{r\bmod p:r^2=d\}. \] The odd roots are simple, and the valuation at two is at most one. The split primes occupy half the reduced classes modulo four or eight. The prime number theorem in these fixed progressions makes the sum $(1+o(1))\log Y$, while the boundary term is $O_B(N)$. Thus at least $(1-o(1))N\log N$ remains above $BN$. Each value contributes at most $(2+o(1))\log N$, proving \eqref{eq:fixed-B-reservoir}. A prime $q>BN>2N$ is simple and private on that curve: a second incidence forces $q\mid(n-m)(n+m)$ and hence $n=m$. This is the same private-prime mechanism as in~\cite[Lemma~2.1]{Klurman}, with a cutoff fixed separately at each stage. \subsection{Constructing the support and common recoveries} Let $\mathcal F_j$ be increasing finite families of nonzero integer polynomials, containing the three $P_d$ from the start and exhausting all such polynomials. Define \[ a_j=\sum_{P\in\mathcal F_j}(\deg P+1),\qquad \delta_j=(64a_j)^{-1}, \] and cycle the target $d_j$ through $-1,-2,2$. The symbol $\delta_j$ here is an exclusion parameter, not the counting exponent of a support. Fix, before choosing the size of stage $j$, \[ B_j\geq\max(4,2^{j+4}/\varepsilon). \] Suppose the earlier blocks and recovery indices have been fixed. Choose $N_j$ so large that \eqref{eq:fixed-B-reservoir} supplies at least $N_j/3$ candidates for $B_j,d_j$, that $N_j$ exceeds the active coefficient sums, and that $T_j=\lfloor\delta_jN_j\rfloor$ satisfies \begin{align} T_j&\geq\delta_jN_j/2,\qquad T_j>T_{j-1},\qquad \log T_j\geq j\left(1+\sum_{i\leq j}\delta_i^{-1}\right),\label{eq:separator-time}\\ B_jN_j&>\max(M_0,M_{j-1}^2+2,\max\Gamma^{(j-1)}).\label{eq:separator-freeze} \end{align} Start with recovery index $M_0$, $\Gamma^{(0)}=\varnothing$, and $\max\varnothing=1$. Each constraint involves data fixed before $N_j$, so all can be met. Remove candidates whose index is at most $N_j/16$, and candidates dividing any nonzero $P(m)$ with $P\in\mathcal F_j$, $m\leq T_j$. The logarithm of the latter product is at most $a_jT_j\log N_j$. Since every candidate prime exceeds $N_j$ and the candidates are distinct, this second removal discards at most $a_jT_j\leq N_j/64$ candidates. More than $N_j/4$ remain for large $N_j$. Keep the first $k_j=\lfloor N_j/8\rfloor$ in index order and call their set $\Gamma_j$. Then \begin{equation}\label{eq:separator-block} \Gamma_j\subset(B_jN_j,N_j^2+2],\qquad r_P(q)>T_j\ (P\in\mathcal F_j,\ q\in\Gamma_j),\qquad \sum_{q\in\Gamma_j}\frac1q<\frac1{8B_j}. \end{equation} Put $\Gamma^{(j)}=\bigcup_{i\leq j}\Gamma_i$. Every finite subset $F\subseteq\Gamma^{(j)}$ has all three means of $h_0h_F$ tending to zero, by the zero factor at two and the Chinese remainder theorem. There are finitely many such subsets. Choose a \emph{common} $M_j>N_j$ with \begin{equation}\label{eq:separator-recovery} \max_{F\subseteq\Gamma^{(j)}}\max_{d\in\Dset}|A^c_{d,h_0h_F}(M_j)|\leq1/j. \end{equation} This completes the stage without choosing any eventual negative subset. The union $\Gamma=\bigcup_j\Gamma_j$ is deterministic, avoids the prefix, and has harmonic cost less than $\varepsilon$ by \eqref{eq:separator-block}. \subsection{All assignments are stable logarithmically} Fix a polynomial $P$, active from stage $j_0$ onward, and take $y$ beyond its exceptional reduction primes. Equation~\eqref{eq:log-first-occurrence} and a union give \begin{equation}\label{eq:separator-log-bound} B^\ell_{P,N}(\Gamma;y) \leq D\sum_{q\in\Gamma,q>y}\frac1q +\frac D{\HH_N}\sum_{\substack{q\in\Gamma,q>y\\ r_P(q)\leq N}}\frac1{r_P(q)}, \qquad D=\max(1,\deg P). \end{equation} The finitely many blocks before $j_0$ contribute at most a finite constant $C_P$ to the second numerator. For $i\geq j_0$, \[ \sum_{q\in\Gamma_i}\frac1{r_P(q)}\leq\frac{k_i}{T_i}\leq\frac1{4\delta_i}, \] with absent occurrences omitted. If $T_j\leq N0$ and every $M_0$, there is one $g\in\M$ with $d_1(f,g)<\varepsilon$ and $g(n)=f(n)$ for $n\leq M_0$ such that, for every $d\in\mathcal D$, \begin{equation}\label{eq:rect-osc} \osc_X A_{d,g}(X,H(X))\geq\frac12. \end{equation} The witnessing subsequences may depend on $d$, and $g$ may depend on the prescribed width $H$. No one perturbation is asserted to work for all widths simultaneously. If $0<\varepsilon<1/4$, a perturbation $h=h_E$ of $1$ can be chosen with $2\notin E$, the same prefix and budget, and \begin{align} \limsup_X A_{d,h}(X,H(X))&\geq C_d>1-4\varepsilon,& \liminf_X A_{d,h}(X,H(X))&\leq0,\label{eq:thin-product-limits}\\ \lim_{X\to\infty}A_{d,h}(X,X)&=C_d,& C_d&=\prod_{\substack{p\in E\\(d/p)=1}} \left(\frac{p-1}{p+1}\right)^2.\label{eq:square-product} \end{align} Its mean on ordinary integers is greater than $1-2\varepsilon$. \end{theorem} No inverse theorem for three norms is used or concluded. In particular, the theorem does not construct two sources whose three thin-rectangle disagreement proportions all tend to zero. For $d=-1$, the square limit in \eqref{eq:square-product} is a special case of Donoso--Le--Moreira--Sun~\cite[Theorem~A]{DLMS}, valid there for every bounded real completely multiplicative function. Its role in this theorem is the coexistence with thin-rectangle divergence. We retain the local proof because its explicit dependence on $H$ is needed below. \subsection{An anisotropic upper bound} The estimate has an area term $1/p$ and a boundary term $1/(H\sqrt p)$, arising from separation of lattice points at scale $\sqrt p$. On a square the latter can be absorbed into the harmonic term throughout the contributing prime range. A thin rectangle retains this extra cost; the number of variables alone does not determine stability. Reciprocal-norm comparison on fixed dilated Gaussian domains already appears in \cite[Lemmas~4.1 and~4.3]{DLMS}; square-incidence estimates for binary forms also appear in \cite[Lemma~4.15]{FKM}. The next specialized proof keeps the aspect ratio instead of absorbing its boundary term. \begin{lemma}[Rectangle incidence]\label{lem:rectangle-incidence} For $d\in\mathcal D$, every prime $p$, and $1\leq H\leq X$, \begin{equation}\label{eq:anisotropic} \frac1{XH}\#\{1\leq a\leq X,\ 1\leq b\leq H:p\mid Q_d(a,b)\} \leq12\left(\frac1p+\frac1{H\sqrt p}\right). \end{equation} Consequently, for $0\leq\gamma\leq1$ and $H=\lceil X^\gamma\rceil$, \begin{equation}\label{eq:interpolation} \sup_X\frac1{XH}\#\{(a,b):f(|Q_d(a,b)|)\ne g(|Q_d(a,b)|)\} \leq36d_{(1+\gamma)/2}(f,g). \end{equation} \end{lemma} \begin{proof} The zero set of $Q_d$ modulo $p$ is the union of at most two linear subspaces, allowing the zero-dimensional subspace in the inert case. This statement includes the prime $2$ by direct inspection. Their inverse images in $\Z^2$ are lattices. Every nonzero vector $v$ in one of these lattices satisfies \[ 0<|Q_d(v)|\leq3\|v\|_\infty^2,\qquad p\mid Q_d(v), \] since $d$ is not a rational square. Distinct lattice points are therefore separated by at least $\ell=\sqrt{p/3}$ in sup norm. Squares of side $\ell$ centered at the points have disjoint interiors and lie in a rectangle of sides $X+\ell$ and $H+\ell$. For the union of at most two lattices, division by $XH$ gives \[ \frac6p+\frac{2\sqrt3}{X\sqrt p}+\frac{2\sqrt3}{H\sqrt p}+\frac2{XH}. \] If there is a contributing point, $p\leq X^2+2H^2\leq3X^2$, so the last term is at most $2\sqrt3/(H\sqrt p)$. Use $H\leq X$ and $6\sqrt3<12$ to get \eqref{eq:anisotropic}. When there is no point the inequality is immediate. A disagreement requires a changed prime dividing the form. In the union bound only primes $p\leq3X^2$ occur. For those primes, \[ \frac1{H\sqrt p}\leq X^{-\gamma}p^{-1/2} \leq3^{\gamma/2}p^{-(1+\gamma)/2}. \] Also $p^{-1}\leq p^{-(1+\gamma)/2}$. Since $12(1+\sqrt3)<36$, this proves \eqref{eq:interpolation}. \end{proof} For a fixed changed-prime set $E$, a finite subset $F$, and a prescribed width, \eqref{eq:anisotropic} gives a more informative tail bound: \begin{equation}\label{eq:anisotropic-tail} \text{disagreement proportion}\ \leq 12\sum_{p\in E\setminus F}\frac1p +\frac{12}{H(X)}\sum_{\substack{p\in E\setminus F\\p\leq3X^2}}p^{-1/2}. \end{equation} Under $\sum_{p\in E}1/p<\infty$, the conditions \begin{equation}\label{eq:geometric-tail-condition} H(X)\to\infty,\qquad \frac1{H(X)}\sum_{\substack{p\in E\\p\leq3X^2}}p^{-1/2}\longrightarrow0 \end{equation} are sufficient for vanishing truncation error. They are not asserted necessary. Without using the sparsity of $E$, Chebyshev's estimate and partial summation give $\sum_{p\leq3X^2}p^{-1/2}\ll X/\log X$, so $H(X)\log X/X\to\infty$ is one sufficient regime. For $H=X$, each term in \eqref{eq:anisotropic} is $O(1/p)$ on the contributing range. Thus harmonic summability alone gives uniform tail control on squares. More generally, finite $d_{(1+\gamma)/2}(1,h)$ suffices for convergence along $H=\lceil X^\gamma\rceil$ when $\gamma>0$, because both side lengths tend to infinity and finite-prime laws converge. At $\gamma=0$ the second coordinate is fixed; the local factors are those of a one-variable observation and need not be the binary factors in \eqref{eq:square-product}. \subsection{A logarithmic-product reservoir on growing rows} The auxiliary scale $B$ must meet two requirements: $H/B\to0$ makes the programming cheap, while $B=o(\log X)$ makes the small-prime boundary error negligible. The hypothesis $H=o(\log X)$ allows both; it is a sufficient regime for this construction, not a claimed optimal transition. Write $H=H(X)$ and define \begin{equation}\label{eq:rectangle-reservoir} \begin{gathered} T=\max(1,\lfloor\log_2X\rfloor),\qquad B=3H+\lceil\sqrt{HT}\rceil,\qquad Y=XB,\\ I_d=\{(a,b)\in[1,X]\times[1,H]:P^+(|Q_d(a,b)|)>Y\}. \end{gathered} \end{equation} Since $H=o(\log X)$, we have $B=o(\log X)$, $H/B\to0$, $Y>2XH$ and $Y^2>X^2+2H^2$ for large $X$. \begin{lemma}[Large-prime mass on a thin rectangle]\label{lem:rectangle-reservoir} Along the prescribed width, uniformly for the three $d\in\mathcal D$, \begin{equation}\label{eq:rectangle-half} |I_d|\geq\left(\frac12-o(1)\right)XH. \end{equation} \end{lemma} \begin{proof} First, \begin{equation}\label{eq:rectangle-total} \sum_{a\leq X,\,b\leq H}\log|a^2-db^2| =(2+o(1))XH\log X. \end{equation} For $d<0$, use $a^2\leq a^2-db^2\leq3X^2$ and $\log(X!)=X\log X+O(X)$. For $d=2$, restrict the lower bound to $a>2H$, where $a^2-2b^2\geq a^2/2$. The omitted $O(H^2)$ terms do not affect the leading order. The upper bound is immediate. Let $p$ be odd and split, that is $(d/p)=1$, and fix a row $b$. Put $t=v_p(b)$ and $M=X^2+2H^2$. For $k\leq2t$, the condition $p^k\mid Q_d(a,b)$ is equivalent to $p^{\lceil k/2\rceil}\mid a$. For $k>2t$ it consists of two residue classes for $a$ modulo $p^{k-t}$, by lifting the two simple roots after removing $p^{2t}$. Summing all exponents, with their boundary errors, yields \begin{equation}\label{eq:row-valuations} \sum_{a\leq X}v_p(|Q_d(a,b)|) \leq2X\sum_{j=1}^t p^{-j}+2X\sum_{j>t}p^{-j} +\frac{2\log M}{\log p} =\frac{2X}{p-1}+\frac{2\log M}{\log p}. \end{equation} Thus rows divisible by $p$ have not been discarded. If $(d/p)=-1$, then $v_p(|Q_d(a,b)|)=2\min(v_p(a),v_p(b))$. The sum of the logarithmic contributions of all such primes is bounded by \[ 2XH\sum_p\frac{\log p}{p^2-1}=O(XH). \] At $2$, for all three forms, the valuation is at most $2\min(v_2(a),v_2(b))+1$, giving the same order of contribution. By \eqref{eq:row-valuations}, all primes at most $Y$ contribute at most \begin{equation}\label{eq:rectangle-small} 2XH\sum_{\substack{p\leq Y,\ p\text{ odd}\\(d/p)=1}}\frac{\log p}{p-1} +2H\pi(Y)\log M+O(XH). \end{equation} The prime number theorem in fixed progressions modulo $4$ or $8$ gives $(1/2+o(1))\log Y$ for the sum. Since $\log Y\sim\log X$ and $\pi(Y)\ll Y/\log Y$, the second term is $O(XHB)=o(XH\log X)$. Thus \eqref{eq:rectangle-small} is at most $(1+o(1))XH\log X$. By \eqref{eq:rectangle-total}, at least $(1-o(1))XH\log X$ remains above $Y$. Each value has at most one such prime, to exponent one, contributing at most $\log M=(2+o(1))\log X$. Division proves \eqref{eq:rectangle-half}. \end{proof} \subsection{Private directions rather than private points} For example, in $[1,100]\times[1,2]$ the prime $1201>2\cdot100\cdot2$ divides both \[ 49^2+1=2\cdot1201,\qquad 98^2+2^2=8\cdot1201. \] The two points lie on the single primitive direction $(49,1)$. A claim of pointwise privacy would be false here. Their multiplicative signs agree because $f(k^2)=1$. The correct unit of programming is therefore a primitive direction, with the number of its points used as a weight. The next lemma shows that a sufficiently large pivot meets only one such direction. \begin{lemma}[The primitive-direction quotient]\label{lem:directions} If $p>2XH$ divides both $Q_d(a,b)$ and $Q_d(c,e)$ for two positive points in the rectangle, then $ae=bc$. Consequently, every pivot in \eqref{eq:rectangle-reservoir} meets a single primitive direction. All points on that direction have the same multiplicative sign, and their multiplicities can be treated as weights. \end{lemma} \begin{proof} The two divisibilities imply \[ p\mid a^2e^2-b^2c^2=(ae-bc)(ae+bc). \] Here $|ae-bc|X_{j-1}$ so large that \[ Y_j>\max(M_0,3X_{j-1}^2),\qquad H(X_j)/B_j<\varepsilon2^{-j-1},\qquad |I_{d_j}|/(X_jH(X_j))>\frac12-\frac1{j+2}. \] The two constant-direction prescriptions both leave all three previous rectangles unchanged, since the new primes exceed every previous polynomial height. Select the branch whose current $d_j$-mean is furthest from that target's mean at $X_{j-1}$. The distance is at least $1/2-1/(j+2)$. The prime coordinates stabilize, the costs are summable, and the final $g$ preserves the prefix. For each target infinitely many late pairs of frozen means are separated by $1/2-o(1)$, proving \eqref{eq:rect-osc}. For the near-constant refinement, first observe that for any finite odd prime set $F$, as both sides tend to infinity the finite-prime law is \begin{equation}\label{eq:binary-finite-law} A_{d,h_F}(X,H(X))\longrightarrow C_{d,F}=\prod_{\substack{p\in F\\(d/p)=1}}\left(\frac{p-1}{p+1}\right)^2. \end{equation} At an inert prime the valuation is even. At a split odd prime the two linear factors $a-rb$ and $a+rb$ are independent Haar-uniform coordinates under the invertible linear change over $\Z_p$. Each has mean parity sign $(p-1)/(p+1)$. The Chinese remainder theorem and finite-prime valuation truncation prove \eqref{eq:binary-finite-law} for arbitrary diverging side lengths. Start from $F_0=\varnothing$. Before stage $j$, choose a recovery scale \[ R_j>X_{j-1},\qquad R_j\geq M_0,\qquad H(R_j)\leq R_j, \] where the three means of $h_{F_{j-1}}$ are within $2^{-j}$ of their products. This is possible by \eqref{eq:binary-finite-law}, since both side lengths tend to infinity along the prescribed width. Choose the attack scale $X_j>R_j$, again with $H(X_j)\leq X_j$, so that \[ Y_j>\max(M_0,3R_j^2),\qquad \frac{H(X_j)}{B_j}<\varepsilon2^{-j-1},\qquad \frac{|I_{d_j}|}{X_jH(X_j)}>\frac12-\frac1{j+2}. \] Force all reservoir directions of target $d_j$ to be negative. The pivots are new and positive before the change, so this adds only the necessary primes to form $F_j\supset F_{j-1}$. Every previous recovery and attack has polynomial height at most $3R_j^2$ and is protected. In particular, $R_j\to\infty$ and later changes preserve these observations even when $H$ is not monotone. For $E=\bigcup_jF_j$, the products converge and \[ C_d\geq1-\sum_{p\in E}\frac{4p}{(p+1)^2}>1-4\varepsilon. \] The recovery subsequences tend to these products, while the attack subsequences for each target have means at most $1-2|I_d|/(X_jH(X_j))=o(1)$. This gives \eqref{eq:thin-product-limits}. On squares, Lemma~\ref{lem:rectangle-incidence} bounds the infinite tail by a constant times $\sum_{p\in E\setminus F}1/p$, uniformly in $X$. Therefore \eqref{eq:binary-finite-law} extends from finite $F$ to $E$, proving \eqref{eq:square-product}. On the ordinary integers, \eqref{eq:integer-mean} gives a mean greater than $1-2\varepsilon$. \begin{remark}[Why the width restriction occurs] The construction needs $B\gg H$ so that its cost $H/B$ tends to zero, and $B=o(\log X)$ so that the small-prime boundary error is negligible. These requirements explain $H=o(\log X)$ in this proof. They do not identify the optimal stability threshold. No logarithmic averaging assertion for these rectangles is made. If $H$ is irregular, no interval description of the cluster set is asserted either. \end{remark} \section{Collective upper bounds for nearly maximal factors}\label{sec:collective} The individual estimate \eqref{eq:anisotropic} pays a boundary term for every prime. Near the height of the polynomial this can be a substantial overcount. Instead, a value with a nearly maximal prime factor has a small cofactor. We sieve all values with the same cofactor together and then sum over those cofactors. The same denominator estimate works for a curve, a translated interval of that curve, and a rectangle; only the remainder of the congruence count changes. Four quantities have different jobs. The height of the observed values bounds the possible cofactors. The interval length, or the short side of a rectangle, determines how far the sieve can run while its remainder stays small. The root count fixes the local sieve density. Grouping by cofactor then controls the very large primes collectively, instead of summing a boundary error for each of them. We keep these roles separate in the two congruence counts below. \subsection{Root counts and the two sieve densities} For $d\in\Dset$, let $\rho_d(m)=\#\{r\bmod m:r^2\equiv d\pmod m\}$. For fixed $m$ the sieve primes are \[ \mathcal P_d(m)=\{\ell\geq3:\ell\text{ prime},\ (d/\ell)=1,\ \ell\nmid m\}. \] We use either of the densities \begin{equation}\label{eq:sieve-densities} g_1(\ell)=\frac2\ell, \qquad g_2(\ell)=\frac{2\ell-1}{\ell^2}. \end{equation} The first counts roots on a line. The second counts the union of two distinct lines in the residue plane. The subscript is a dimension indicator only within this section. For $i=1,2$, extend $g_i$ and $w_i(\ell)=g_i(\ell)/(1-g_i(\ell))$ multiplicatively to squarefree integers supported on $\mathcal P_d(m)$. All sums in the following formulas run only over these allowed integers. Set \[ G_{i,m}(z)=\sum_{l\leq z}w_i(l),\qquad \kappa_{i,d}(m)=\prod_{\substack{\ell\mid m,\ \ell\geq3\\ (d/\ell)=1}} (1-g_i(\ell))^{-1}. \] The Selberg weights in this normalization are \begin{equation}\label{eq:selberg-weights} \lambda_l=\mu(l)\prod_{\ell\mid l}(1-g_i(\ell))^{-1} \frac{\displaystyle\sum_{\substack{t\leq z/l\\ (t,l)=1}}w_i(t)}{G_{i,m}(z)}. \end{equation} They are zero outside the allowed integers and outside $l\leq z$. The classical diagonalization of the Selberg quadratic form gives \begin{equation}\label{eq:selberg-diagonal} \lambda_1=1, \qquad \sum_{l_1,l_2\leq z}\lambda_{l_1}\lambda_{l_2}g_i([l_1,l_2]) =\frac1{G_{i,m}(z)}, \qquad |\lambda_l|\leq l. \end{equation} Here $[l_1,l_2]$ denotes the least common multiple. The first two identities are precisely the upper-sieve weights and their minimum quadratic form~\cite[Section~13.2, (13.2.3)--(13.2.4), Theorem~13.1]{Kedlaya}. The coarse last bound also follows directly from \eqref{eq:selberg-weights}: its numerator sum is at most $G_{i,m}(z)$, and each factor $(1-g_i(\ell))^{-1}$ is at most $\ell$ for $\ell\geq3$. \begin{lemma}[A denominator uniform in the cofactor]\label{lem:sieve-denominator} For $i=1,2$, uniformly in $m,z\geq2$ and $d\in\Dset$, \begin{equation}\label{eq:sieve-denominator} G_{i,m}(z)\gg\frac{\log(2z)}{\kappa_{i,d}(m)}. \end{equation} Also, uniformly for $U\geq1$, \begin{equation}\label{eq:singularity-sums} \sum_{m\leq U}\rho_d(m)\ll U,\qquad \sum_{m\leq U}\rho_d(m)\kappa_{i,d}(m)\ll U,\qquad \sum_{m\leq U}\frac{\rho_d(m)\kappa_{i,d}(m)}m\ll\log(2U). \end{equation} All implied constants are absolute for these two densities and these three forms. \end{lemma} \begin{proof} Choose a small absolute $a>0$ and put $x=z^a$. Form a squarefree integer $R$ by independently selecting allowed primes $\ell\leq x$ with probabilities $g_i(\ell)$. The classical estimate $\sum_{\ell\leq x}(\log\ell)/\ell\ll\log x$ allows $a$ to be chosen so that $\E\log R\leq\tfrac12\log z$. Thus $\Prob(R\leq z)\geq1/2$. Its distribution is proportional to the weights $w_i$, whence \[ G_{i,m}(z)\geq\frac12 \prod_{\substack{\ell\leq x\\ \ell\in\mathcal P_d(m)}}(1-g_i(\ell))^{-1}. \] The split odd primes occupy half the reduced residue classes modulo four or eight. Mertens' formula in these fixed progressions gives a product comparable to $\log x$ before removing primes dividing $m$~\cite[Section~27.11]{DLMF}. Indeed $g_i(\ell)=2/\ell+O(\ell^{-2})$. The removal costs at most $\kappa_{i,d}(m)$. Bounded $z$ are absorbed using $G_{i,m}(z)\geq1$ and $\kappa_{i,d}(m)\geq1$. The auxiliary randomness proves a positive sum estimate; no signs of the multiplicative source are randomized. Let $\chi_d$ have discriminant $-4,-8,8$ for $d=-1,-2,2$, respectively. The function $\rho_d$ is multiplicative. At a split odd prime it has value two on every positive prime power; at an inert prime it is zero; at two its first value is one and subsequent values are zero. Hence $0\leq\rho_d(m)\leq(1*\chi_d)(m)$. Since $\chi_d$ is periodic of mean zero, \[ \sum_{m\leq U}\rho_d(m) \leq\sum_{r\leq U}\chi_d(r)\lfloor U/r\rfloor =U\sum_{r\leq U}\frac{\chi_d(r)}r+O(U)\ll U. \] To average the extra factor, write $\kappa_{i,d}(m)=\sum_{s\mid m}v_i(s)$, where $v_i$ is nonnegative, supported on squarefree split odd integers, and $v_i(\ell)=(1-g_i(\ell))^{-1}-1=O(1/\ell)$. For such $s$, $\rho_d(st)\leq\rho_d(s)\rho_d(t)$. Therefore \[ \sum_{m\leq U}\rho_d(m)\kappa_{i,d}(m) \ll U\sum_s\frac{v_i(s)\rho_d(s)}s\ll U. \] The last Euler product converges, its factors being $1+O(\ell^{-2})$. Partial summation proves the third estimate in \eqref{eq:singularity-sums}. \end{proof} \subsection{Curves and translated intervals: the length pays the error} \begin{proposition}[One-variable upper sieve]\label{prop:curve-sieve} Let \[ L_d(N;Z)=\#\{n\leq N:P_d(n)>0,\ P^+(P_d(n))>Z\}, \qquad K=\left\lceil\frac{N^2+2}{Z}\right\rceil. \] For $N\geq2^{12}$ and $N\leq Z\leq N^2+2$, \begin{equation}\label{eq:curve-sieve} \frac{L_d(N;Z)}N\ll \frac{\log(2K)}{\log(2N)}+\frac K{\sqrt N}. \end{equation} There is also a translated form. If $X\geq L\geq2^{12}$, $X\leq u<2X$, $I=(u,u+L]\cap\Z$, $3X\leq Z\leq9X^2+2$ and $K=\lceil(9X^2+2)/Z\rceil$, then \begin{equation}\label{eq:interval-sieve} \frac1L\#\{n\in I:P^+(P_d(n))>Z\} \ll\frac{\log(2K)}{\log(2L)}+\frac K{\sqrt L}. \end{equation} The constants are absolute and uniform in the origin $u$. \end{proposition} \begin{proof} Write a counted value as $P_d(n)=mp$, where $p>Z$ is prime and $m\leq K$. The index belongs to one of the $\rho_d(m)$ classes $n\equiv r\pmod m$. For squarefree $t$ supported on $\mathcal P_d(m)$, the Chinese remainder theorem gives, on any interval of length $L$, \begin{equation}\label{eq:line-congruence} \#\{n\in I:n\equiv r\pmod m,\ t\mid P_d(n)\} =\frac Lm g_1(t)+R_t,\qquad |R_t|\leq2^{\omega(t)}. \end{equation} For prefixes replace $I$ by $[1,N]$ and $L$ by $N$. Each residue class contributes a floor error at most one, independently of its position. The finite omitted value in the prefix causes no difficulty: we may upper-bound the desired positive-value count by sieving all the residue classes. If $P_d(n)/m$ is a prime above $Z>z$, then it avoids every sieve prime, all of which are coprime to $m$. The indicator of such an index is at most $(\sum_{l\mid P_d(n)}\lambda_l)^2$. Expanding this square and applying \eqref{eq:selberg-diagonal} and \eqref{eq:line-congruence} gives, for each $m,r$, \begin{equation}\label{eq:line-sieved} \#\{n\in I:n\equiv r\pmod m,\ P_d(n)/m\text{ prime}>Z\} \leq \frac L{mG_{1,m}(z)}+z^6. \end{equation} There are at most $z^2$ pairs of weights, each product is at most $z^2$, and $2^{\omega([l_1,l_2])}\leq[l_1,l_2]\leq z^2$. This deliberately coarse error is sufficient. Sum over $m\leq K$ and its roots. Lemma~\ref{lem:sieve-denominator} bounds the result by \[ \ll L\frac{\log(2K)}{\log(2z)}+Kz^6. \] Take $z=\lfloor L^{1/12}\rfloor$, so $z\geq2$, $zZ\}, \qquad K=\left\lceil\frac{3X^2}{Z}\right\rceil. \] Then \begin{equation}\label{eq:rectangle-sieve} \frac{L_d^{(2)}(X,H;Z)}{XH} \ll \frac{\log(2K)}{\log(2H)} +\frac K{H^{5/8}}+\frac{K^2}{XH^{1/2}}. \end{equation} The implied constant is absolute for the three forms. \end{proposition} \begin{proof} Write $(a,b)=g(u,v)$ with $(u,v)=1$. A prime $p>Z\geq H$ cannot divide $g\leq H$. If it divides the norm, then \begin{equation}\label{eq:primitive-cofactor} |u^2-dv^2|=mp,\quad m\leq K/g^2,\quad u\leq X/g,\quad v\leq H/g,\quad g\leq\sqrt K. \end{equation} Moreover $(v,m)=1$: a common prime divisor would divide $u$ as well. Thus $u\equiv rv\pmod m$ for a root $r^2\equiv d\pmod m$. After this reduction, dropping the primitivity restriction only increases the upper bound. Fix $m,r$ and a box $1\leq u\leq U$, $1\leq v\leq V$. For allowed squarefree $t$, the number $A_t$ satisfying that root-lattice condition and $t\mid u^2-dv^2$ has \begin{equation}\label{eq:plane-congruence} A_t=\frac{UV}{m}g_2(t)+R_t, \qquad |R_t|\leq2^{\omega(t)}(U+V+mt). \end{equation} There are $m\prod_{\ell\mid t}(2\ell-1)$ classes modulo $mt$ by the Chinese remainder theorem. Each contributes $UV/(m^2t^2)$ with error at most $U/(mt)+V/(mt)+1$. This proves \eqref{eq:plane-congruence}. Use the density $g_2$ in \eqref{eq:selberg-weights}. A point with $|u^2-dv^2|/m$ prime and greater than $Z>z$ passes the sieve, so its count is at most \begin{equation}\label{eq:plane-sieved} \frac{UV}{mG_{2,m}(z)}+(U+V)z^6+mz^8. \end{equation} The two errors follow by the same weight-pair count as in \eqref{eq:line-sieved}, with the additional factor $mt\leq mz^2$ in the last term. Apply \eqref{eq:plane-sieved} with $U=\lfloor X/g\rfloor$, $V=\lfloor H/g\rfloor$, and sum over the roots, $m\leq K/g^2$, and $g\leq\sqrt K$. The three estimates in \eqref{eq:singularity-sums} give \begin{align*} L_d^{(2)}(X,H;Z) &\ll \sum_{g\leq\sqrt K} \left\{\frac{XH}{g^2}\frac{\log(2K)}{\log(2z)} +\frac{K(X+H)}{g^3}z^6+\frac{K^2}{g^4}z^8\right\}\\ &\ll XH\frac{\log(2K)}{\log(2z)}+K(X+H)z^6+K^2z^8. \end{align*} Overcounting a point with several eligible prime factors is harmless. Take $z=\lfloor H^{1/16}\rfloor0,\ A_\Gamma(t)\ll_{\Gamma,\varepsilon}t^{s+\varepsilon} \quad\Longleftrightarrow\quad \forall\varepsilon>0,\ \sum_{p\in\Gamma}p^{-s-\varepsilon}<\infty. \end{equation} For the last equivalence, use partial summation with a smaller positive exponent in one direction, and $A_\Gamma(t)\leq t^{s+\varepsilon}\sum_\Gamma p^{-s-\varepsilon}$ in the other. In particular $\delta(\Gamma)<1$ implies harmonic summability. No convergence at exponent $s$ itself is asserted. \subsection{Critical tails on the three curves} Define \[ T_{\Gamma,d}(N;y)=\frac1N\#\{n\leq N:P_d(n)>0,\ \exists p\in\Gamma, \ p>y,\ v_p(P_d(n))\text{ odd}\}. \] \begin{theorem}[Critical counting on curves]\label{thm:critical-curves} If $\delta(\Gamma)\leq1/2$, then \begin{equation}\label{eq:critical-curve-tail} \lim_{y\to\infty}\sup_{N\geq2}\max_{d\in\Dset}T_{\Gamma,d}(N;y)=0. \end{equation} Uniformly over $A\subseteq\Gamma$ and $d\in\Dset$, both curve means of $h_A$ converge to $c_d(A)$ in \eqref{eq:curve-products}. The same finite-prime approximation yields the local laws of every fixed finite list of translated curve readings. \end{theorem} \begin{proof} For $y\geq2$ and $N\leq Z\leq N^2+2$, split the coverage at $Z$. The primes below it have at most two root classes; the remaining observations are counted by $L_d$. Thus \begin{equation}\label{eq:critical-curve-split} T_{\Gamma,d}(N;y)\leq2\sum_{p\in\Gamma,p>y}\frac1p +\frac{2A_\Gamma(Z)}N+\frac{L_d(N;Z)}N. \end{equation} Fix $0<\vartheta<1/4$ and take $Z=N^{2-\vartheta}$. With $\varepsilon=\vartheta/8$, the counting assumption gives \[ \frac{A_\Gamma(Z)}N \ll_{\Gamma,\vartheta}N^{(2-\vartheta)(1/2+\varepsilon)-1} \leq O_{\Gamma,\vartheta}(N^{-\vartheta/4})\to0. \] By \eqref{eq:curve-small-remainder}, with an absolute $C$ independent of $\vartheta$, \[ \limsup_N\max_d T_{\Gamma,d}(N;y) \leq2\sum_{p\in\Gamma,p>y}\frac1p+C\vartheta. \] Let $\vartheta\downarrow0$ after the limit in $N$, and then let $y\to\infty$. Harmonic summability follows from \eqref{eq:exponent-equivalences}. The finite set of remaining small samples is removed by increasing $y$ beyond their heights, proving the supremum in \eqref{eq:critical-curve-tail}. For a finite prime set, at a visible odd prime the density of an odd valuation is \[ 2(1-p^{-1})\sum_{j\geq0}p^{-2j-1}=\frac2{p+1}, \] so its factor is $(p-3)/(p+1)$. An invisible odd prime contributes one. At two the parity of the valuation alternates with the parity of the index, giving factor zero when two is in $A$. Finite-prime valuation truncation and the Chinese remainder theorem therefore give \eqref{eq:curve-products}. The deficits $4/(p+1)$ are summable on $\Gamma$, so the local products converge uniformly over subsets. Equation~\eqref{eq:critical-curve-tail} bounds the mean error by twice the tail coverage. At each fixed cutoff there are only finitely many assignments. This proves uniform convergence. Abel gives the logarithmic conclusion. For any fixed translated list, each tail is bounded by its untranslated tail on a slightly larger prefix, with an $O(1/N)$ boundary per fixed shift. A finite union controls the vector. A common periodic truncation gives its joint local law, with shared Haar coordinates at each prime. No assertion of independence of those readings is made. \end{proof} The tail estimate is unchanged after multiplying $h_A$ and its truncation by any base $f$: their disagreement indicators are equal. This uniformity in $f$ is an error comparison, not existence of means for $f(P_d(n))h_A(P_d(n))$. \paragraph{A quantitative envelope.} If $A_\Gamma(t)\leq C_0t^{1/2}(\log(2t))^B$ with $B\geq0$, take $Z=N^2/(\log N)^{2B+4}$ in \eqref{eq:critical-curve-split}. For large $N$, $Z\geq N$, the counting term is $O_{B,C_0}((\log N)^{-2})$, and Proposition~\ref{prop:curve-sieve} gives \begin{equation}\label{eq:critical-curve-rate} T_{\Gamma,d}(N;y)\leq2\sum_{p\in\Gamma,p>y}\frac1p +O_{B,C_0}\left(\frac{\log\log N}{\log N}\right),\qquad y\geq2. \end{equation} The rate concerns the prime tail. A rate for convergence to the local product must also pay for the periods of the finite-prime and valuation truncations. \subsection{Fixed supports, common envelopes and a restricted minimum} The constants in a support-specific counting bound are not automatically common to all supports with that exponent. A useful uniform statement requires a single envelope. Let $\Phi(t)\geq1$ satisfy $\Phi(t)\ll_\varepsilon t^{1/2+\varepsilon}$ for every $\varepsilon>0$, and consider sources whose negative-prime count is at most $\Phi(t)$ for every $t\geq2$. The proof of \eqref{eq:critical-curve-split} is uniform in that class. So are the harmonic tails, by partial summation with any fixed exponent strictly between $1/2$ and one. The local means therefore converge uniformly over this class of possibly moving supports and assignments. For a fixed prime $p$, let $m_{p,\Phi}(N)$ be the minimum of the total number of negative readings on the three positive curves, among this class and with $h(p)=-1$. Then \begin{equation}\label{eq:restricted-minimum} m_{p,\Phi}(N)=(\gamma_p+o(1))N, \quad \gamma_2=\frac32, \quad \gamma_p=\frac{2|\{d\in\Dset:(d/p)=1\}|}{p+1} \quad(p\text{ odd}). \end{equation} Indeed all curve factors lie in $[0,1]$. A negative sign at an odd visible $p$ forces $c_d(A)\leq(p-3)/(p+1)$, hence negative density at least $2/(p+1)$ on each such trace. At two all three means are zero. The source negative only at $p$ achieves the displayed constant and is admitted because $\Phi\geq1$. This proves both bounds in \eqref{eq:restricted-minimum}; for example $\gamma_3=1/2$, $\gamma_5=1/3$ and $\gamma_7=1/4$. This is a minimum over a restricted class. The unrestricted minimum is at most $m_{p,\Phi}(N)$, not at least it. Thus \eqref{eq:restricted-minimum} is not a linear lower bound for arbitrary sources. Likewise a fixed quotient $h_A=f/g$ with $\delta(A)\leq1/2$ gives curve disagreement density $(1-c_d(A))/2$, without placing either $f$ or $g$ individually in the sparse negative-prime class. \subsection{Examples beyond every fixed logarithmic correction} Let $p_j$ denote the $j$th prime. The support \[ \Gamma_{1/2}=\{p_{j^2}:j\geq1\} \] has $p_{j^2}\sim2j^2\log j$, hence $\delta(\Gamma_{1/2})=1/2$, whereas $\sum_{q\in\Gamma_{1/2}}q^{-1/2}$ diverges by comparison with $\sum_j1/(j\sqrt{\log j})$. These are consequences of the prime number theorem. Since two belongs to this support, its all-negative assignment has zero mean on every curve, and Theorem~\ref{thm:critical-curves} treats all its other assignments as well. One can exceed every fixed logarithmic correction of a critical power sum. Fix $0y,\ v_p(|Q_d(a,b)|)\text{ odd}\}. \] \begin{theorem}[Critical counting on power rectangles]\label{thm:critical-rectangles} Fix $0<\alpha<1$, $H=\lfloor X^\alpha\rfloor$ and $s_\alpha=(1+\alpha)/2$. If $\delta(\Gamma)\leq s_\alpha$, then \begin{equation}\label{eq:critical-rectangle-tail} \lim_{y\to\infty}\sup_{X\geq2}\max_d T^{(2)}_{\Gamma,d}(X,H;y)=0. \end{equation} Uniformly over $A\subseteq\Gamma$ and the three $d$, the rectangular means of $h_A$ converge to $C_d(A)$ in \eqref{eq:all-binary-products}. \end{theorem} \begin{proof} The incidence estimate \eqref{eq:anisotropic}, split at $Z$, gives \begin{equation}\label{eq:critical-rectangle-split} T^{(2)}_{\Gamma,d}(X,H;y) \leq12\sum_{p\in\Gamma,p>y}\frac1p +\frac{12}{H}\sum_{\substack{p\in\Gamma\\ p\leq Z}}p^{-1/2} +\frac{L_d^{(2)}(X,H;Z)}{XH}. \end{equation} Choose $0<\vartheta<\alpha/4$, $Z=X^{2-\vartheta}$, and $\varepsilon=\min(\alpha\vartheta/8,(1-\alpha)/4)$. Partial summation in the counting bound yields \[ \sum_{p\in\Gamma,p\leq Z}p^{-1/2}\ll_{\Gamma,\alpha,\vartheta}Z^{\alpha/2+\varepsilon}, \qquad H^{-1}Z^{\alpha/2+\varepsilon}\ll X^{-\alpha\vartheta/4}. \] In \eqref{eq:rectangle-sieve}, $K\ll X^\vartheta$ and $H\asymp X^\alpha$. Its two power remainders tend to zero, and the limit superior of its first term is $O(\vartheta/\alpha)$ with an absolute constant. Therefore \[ \limsup_X\max_d T^{(2)}_{\Gamma,d}(X,H;y) \leq12\sum_{p\in\Gamma,p>y}\frac1p+C\vartheta/\alpha. \] Send $\vartheta\downarrow0$, then $y\to\infty$, and finally kill the finitely many small rectangles by increasing the cutoff. This proves \eqref{eq:critical-rectangle-tail}. At an odd split prime, the invertible $p$-adic change $(a,b)\mapsto(a-rb,a+rb)$ gives two independent uniform coordinates. Each has mean parity sign $(p-1)/(p+1)$, giving its square in \eqref{eq:all-binary-products}. At an inert prime, the norm valuation is $2\min(v_p(a),v_p(b))$, hence even. At two, among all residue pairs, mass $1/2$ has even valuation and mass $1/4$ has odd valuation; the remaining mass $1/4$ has both coordinates even. Dividing those coordinates by two changes the norm by a square and leaves its sign unchanged. Thus the local mean obeys $C=1/4+C/4$, giving $C=1/3$ for each form. Finite valuation truncations justify these local calculations, since their exceptional $p$-adic sets decrease to the zero set of the form, of measure zero. On diverging rectangles every fixed periodic law converges uniformly in its origins at fixed period. For subsets of a fixed prime head there are only finitely many laws. The odd deficits $4p/(p+1)^2\leq4/p$ are summable on $\Gamma$, so both the local-product tail and the observation tail are uniform in $A$. This proves the claimed limit. \end{proof} At $\alpha=0$ the second coordinate would be fixed. The curve proof, including its boundary error $2A_\Gamma(Z)/N$, is separate; the one-variable factors cannot be obtained from the two-variable products by that substitution. For a common envelope $A_\Gamma(t)\leq C_\varepsilon t^{s_\alpha+\varepsilon}$ for every $\varepsilon>0$, all statements are uniform across the supported family, even if its union is not harmonically summable. With $A_\Gamma(t)\leq C_0t^{s_\alpha}(\log(2t))^B$, the choice \[ Z=X^2/(\log X)^{(2B+4)/\alpha} \] in \eqref{eq:critical-rectangle-split} gives a harmonic tail plus $O_{\alpha,B,C_0}(\log\log X/\log X)$, as in \eqref{eq:critical-curve-rate}. The elementary budget version remains useful. For $H=\lfloor X^\alpha\rfloor$, the bound $H\geq X^\alpha/2$ and $p\leq3X^2$ on the contributing range give \begin{equation}\label{eq:floor-critical-budget} \sup_X\max_d T^{(2)}_{\Gamma,d}(X,H;y) \leq60\sum_{p\in\Gamma,p>y}p^{-(1+\alpha)/2}. \end{equation} The counting theorem is strictly more general at equality: it assumes summability for every exponent slightly larger than $s_\alpha$, rather than summability at $s_\alpha$. Example~\eqref{eq:critical-block-example} makes this distinction concrete. \subsection{Harmonic stability when the side exponents agree} A varying rectangle is not a dilation of a fixed planar region. Thus the known Gaussian mean theorem of Donoso--Le--Moreira--Sun~\cite[Theorems~A--B and Definition~2.3]{DLMS} does not by itself cover the following statement. Work on shrinking sectors~\cite{BurginGiannitsi,Kus} addresses related angular questions, but with different source hypotheses and averaging domains. Here the collective bound tracks the small side $H$ for every assignment on the same harmonic support. \begin{theorem}[Almost-square rectangles in exponent]\label{thm:almost-square} Let $1\leq H(X)\leq X$ be integer-valued and $\log H(X)/\log X\to1$. Every fixed support $\Gamma$ with finite harmonic sum satisfies \[ \lim_{y\to\infty}\sup_{X\geq2}\max_dT^{(2)}_{\Gamma,d}(X,H(X);y)=0. \] All its assignments have rectangular means converging uniformly to $C_d(A)$ in \eqref{eq:all-binary-products}. \end{theorem} \begin{proof} Use Proposition~\ref{prop:rectangle-sieve} at $Z=H^2$. Then $K=\lceil3X^2/H^2\rceil=X^{o(1)}$, so its logarithmic term tends to zero and its two remainders are $X^{-5/8+o(1)}$ and $X^{-3/2+o(1)}$. For $p\leq H^2$, \eqref{eq:anisotropic} is at most $24/p$. Hence \begin{equation}\label{eq:almost-square-split} T^{(2)}_{\Gamma,d}(X,H;y) \leq24\sum_{p\in\Gamma,p>y}\frac1p+ \frac{L_d^{(2)}(X,H;H^2)}{XH}. \end{equation} The second term tends to zero independently of the support. Take the two limits and remove the finite initial rectangles. The local products follow from the same finite-prime argument as in Theorem~\ref{thm:critical-rectangles}, since both side lengths diverge. \end{proof} The width $H(X)=\max(1,\lfloor X\exp(-\sqrt{\log X})\rfloor)$ is admissible, although $H/X\to0$ and $H\log X/X\to0$. In \eqref{eq:rectangle-sieve} its leading geometric term is $O((\log X)^{-1/2})$. More generally $X\exp(- (\log X)^\theta)$ works for every fixed $0<\theta<1$. The new criterion can apply even when the older boundary sum \eqref{eq:geometric-tail-condition} diverges. Set $Y_j=2^{j^2}$ and let $\Gamma$ contain \emph{all} primes in $(Y_j,2Y_j]$ for $j\geq j_0$. Its harmonic cost is $\sum_jO(j^{-2})$, whereas every power cost with exponent less than one diverges. At $X_j=\lceil\sqrt{Y_j}\rceil$, the current block contributes at least $cX_j/\log X_j$ to $\sum_{p\in\Gamma,p\leq3X_j^2}p^{-1/2}$. Dividing by the width above makes the old boundary expression at least a constant times $\exp(\sqrt{\log X_j})/\log X_j$, which tends to infinity. Nevertheless Theorem~\ref{thm:almost-square} applies. A divergent sum of individual incidence bounds is not evidence of actual instability. For two fixed sources with harmonic disagreement support $A$, the same theorem gives rectangular disagreement limit $(1-C_d(A))/2$. Each $C_d(A)$ is positive, with a common positive lower bound over subsets of a fixed harmonic support. If all three disagreement limits vanish, every prime in $A$ would have to be invisible to all forms. No odd prime has that property, and two is visible to each; hence the sources agree. This implication assumes harmonic proximity rather than deriving it from the observed errors. Theorems~\ref{thm:critical-curves}, \ref{thm:critical-rectangles} and \ref{thm:almost-square} prove the positive assertions of Theorem~\ref{thm:critical-summary}. The next section supplies the negative assertion at fixed power width. \section{The power-rectangle threshold from Gaussian primes}\label{sec:angular} To match the positive exponent $s_\alpha=(1+\alpha)/2$, the negative construction needs many private directions whose primes are close to the full height $X^2$. Prime values alone would supply only about $XH/\log X$ directions. A harmonic sum over primitive cofactors restores the missing logarithm. The external input is a lower bound for Gaussian primes in every sufficiently wide sector, with the position of the sector uniform. \begin{theorem}[Instability above the rectangular critical exponent]\label{thm:angular-instability} Fix $119/500<\alpha<1$, $H=\lfloor X^\alpha\rfloor$, and $(1+\alpha)/2<\sigma\leq1$. There is $c_{\alpha,\sigma}>0$ such that, for every $f\in\M$, $\varepsilon>0$ and $M_0$, there is one fixed $g\in\M$ with \[ d_\sigma(f,g)<\varepsilon,\qquad g(n)=f(n)\quad(n\leq M_0),\qquad \osc_X A_{-1,g}(X,\lfloor X^\alpha\rfloor)\geq c_{\alpha,\sigma}. \] The oscillation constant is independent of the radius and of the initial source. This assertion concerns $a^2+b^2$ only. \end{theorem} \subsection{The angular input and the required margins} \begin{inputthm}[Harman--Lewis]\label{input:harman-lewis} There is an absolute $c_0>0$ such that, for all sufficiently large $t$, every angular sector of width \[ t^{-381/1000}\leq\delta\leq\pi/2 \] contains at least $c_0\delta t/\log t$ Gaussian primes of norm at most $t$, counted modulo units by their representatives in that sector. The bound is uniform in the sector position. \end{inputthm} This is Theorem~2 of Harman--Lewis~\cite{HarmanLewis}, with the variable denoting the \emph{norm} bound, not the radius. We regard a translated sector as an arc modulo $\pi/2$, and use the strict margin below to handle an arc crossing a quadrant boundary. We use a lower bound in every sector, not an asymptotic in almost every sector. Its analytic proof is imported. Stucky~\cite{Stucky} treats joint short restrictions on norm and argument; that neighboring result is not substituted for the specific uniform bound used here. Gaussian primes which do not have rational-prime norm are the primes of norm $p^2$ with $p$ inert, apart from the ramified prime above two. There are $O(\sqrt t)$ of them. They are negligible in this input whenever $\delta\sqrt t/\log t\to\infty$. We shall explicitly ensure that margin. An elementary upper bound will be used to remove small prime norms. Uniformly in sector position and $0<\delta\leq\pi/2$, \begin{equation}\label{eq:sector-lattice-bound} \#\{z\in\Z[i]\setminus\{0\}:\mathrm N(z)\leq T,\ \arg z\text{ in the sector}\} \ll\delta T+\sqrt T+1. \end{equation} The area is $\delta T/2$ and the boundary length is $O(\sqrt T)$; count the unit squares inside and meeting the boundary. This upper bound permits subtraction from a lower bound without assuming a lower bound on a thin annulus. The scale of the cofactor sum explains why we can afford these margins. Write $\mathrm N(w)=r^2+s^2$ for $w=r+is$, and take one representative $r>0$, $s\geq0$, $(r,s)=1$ of each primitive cofactor modulo units. The elementary count proved in \eqref{eq:gaussian-cofactors} gives reciprocal mass of order $\log M$ up to norm $M$. Consequently the relevant scale is \[ \frac{XH}{\log X} \sum_{\substack{\mathrm N(w)\leq X^\tau\\w\ \mathrm{primitive},\ r>0,\ s\geq0}} \frac1{\mathrm N(w)} \asymp_\tau XH\qquad(\tau>0\text{ fixed}). \] The cofactor logarithm restores exactly the logarithm lost in counting primes. This is not yet the reservoir count: the next choices make the angular estimate uniform for those cofactors, and the following subsections remove small pivots and rule out multiple counting. Choose fixed parameters \begin{align} 0<\rho&<\min\left(1-\alpha,\ 2-\frac{1+\alpha}{\sigma}\right),\label{eq:angular-rho}\\ 0<\tau&<\min\left(\rho,\ 2\alpha,\ \frac{1000\alpha-238}{381}\right).\label{eq:angular-tau} \end{align} They exist under the theorem's hypotheses. Put \begin{equation}\label{eq:angular-sector} M=\lfloor X^\tau\rfloor,\quad T_0=X^{2-\rho},\quad \mathcal I_X=\left(\arctan\frac{H}{4X},\arctan\frac{H}{2X}\right], \quad\delta_X=|\mathcal I_X|\sim\frac H{4X}. \end{equation} Every point of norm at most $X^2$ with argument in $\mathcal I_X$ has $a\leq X$ and $00,\ s\geq0,\ (r,s)=1,\ r^2+s^2\leq M\}. \] This is a set of representatives modulo the four units. The elementary primitive lattice-point count gives \begin{equation}\label{eq:gaussian-cofactors} |\mathcal W(M)|=\frac3{2\pi}M+O(\sqrt M\log(2M)), \qquad \sum_{w\in\mathcal W(M)}\frac1{\mathrm N(w)} =\frac3{2\pi}\log M+O(1). \end{equation} For completeness, the quarter disk has $\pi M/4+O(\sqrt M+1)$ points under these conventions. Möbius inversion for coprimality replaces the leading coefficient by $(\pi/4)/\zeta(2)=3/(2\pi)$ and produces the displayed error. Partial summation gives the harmonic formula, since the integrated error $\int_1^\infty t^{-3/2}\log(2t)\,dt$ is finite. The only primitive point on the nonnegative real axis is $w=1$. For $w\in\mathcal W(M)$ of norm $m$, apply Published input~\ref{input:harman-lewis} with \[ t=X^2/m,\qquad \text{sector }\mathcal I_X-\arg w. \] The angular condition holds uniformly in $m\leq M$, because \eqref{eq:angular-tau} implies the strict power inequality \begin{equation}\label{eq:angular-margin} 1-\alpha<\frac{381}{1000}(2-\tau). \end{equation} If the translated arc crosses a quadrant boundary, cut it there and retain a component of length at least $\delta_X/2$. By the strict inequality \eqref{eq:angular-margin}, \[ \frac{\delta_X}{2}\,t^{381/1000} \gg X^{\alpha-1+(381/1000)(2-\tau)}\longrightarrow\infty \] uniformly for $m\leq M$. Hence even that component satisfies the angular hypothesis for every sufficiently large $X$. The constant loss is absorbed by decreasing $c_0$ once; no endpoint assertion for a split sector is used. Also $\delta_X\sqrt t\gg X^{\alpha-\tau/2}$, which tends to infinity faster than $\log t$. Removing inert-prime norms is therefore legitimate uniformly in $w$. For large $X$, at least \[ \frac{c_0}{2}\frac{\delta_X X^2}{m\log(X^2/m)} \] Gaussian primes of rational-prime norm remain for each cofactor. Since $\log(X^2/m)\leq2\log X$, summing \eqref{eq:gaussian-cofactors} yields \begin{equation}\label{eq:angular-mass-before-cut} \sum_{w\in\mathcal W(M)}\#\{\pi\text{ counted for }w\} \gg\frac{\delta_X X^2}{\log X}\sum_w\frac1{\mathrm N(w)} \gg_{\tau}XH. \end{equation} The factor $\log M\asymp_\tau\log X$ is essential: omitting the cofactors would not give a positive proportion of the rectangle. \subsection{Removing small pivots and proving injectivity} Discard the pairs with prime norm $p\leq T_0$. For each $w$, \eqref{eq:sector-lattice-bound} bounds their number by $O(\delta_X T_0+\sqrt{T_0}+1)$. As $|\mathcal W(M)|\ll M$, the total discarded count is \begin{equation}\label{eq:angular-discard} O\left(X^{1+\alpha+\tau-\rho}+X^{1+\tau-\rho/2}+X^\tau\right)=o(XH). \end{equation} The first exponent is harmless because $\tau<\rho$, and the second because $\tau-\rho/2<\tau/2<\alpha$. Each surviving pair gives the unique associate of $w\pi$ with argument in $\mathcal I_X$. Write it $a+ib$. Its norm is $mp\leq X^2$ and $p>T_0>M$ for large $X$. The point is primitive. If a rational prime $\ell\ne p$ divided both coordinates, multiplication by $\pi$ would be invertible modulo $\ell$, forcing $\ell$ to divide both coordinates of $w$. If $\ell=p$, then $p^2\mid mp$, impossible because $mT_0>X$ has exactly one such rational prime factor. Exactly one of the two Gaussian primes above $p$ divides $a+ib$; both would force $p\mid(a,b)$. Unique factorization in $\Z[i]$ recovers $w$ and $\pi$ modulo units, and the convention for $\mathcal W(M)$ makes $w$ unique. Thus the pairs have not hidden a multiplicity. It follows from \eqref{eq:angular-mass-before-cut}--\eqref{eq:angular-discard} that the set $\mathcal R_X$ of primitive points satisfying \[ \arg(a+ib)\in\mathcal I_X,\qquad a^2+b^2=mp\leq X^2,\qquad m\leq M,\qquad p>X^{2-\rho} \] has $|\mathcal R_X|\geq c_*XH$ at every sufficiently large $X$, with $c_*>0$ depending only on the fixed parameters. These points lie in the original rectangle. In particular $p>H^2$, since $2-\rho>1+\alpha>2\alpha$. Unlike the almost-square regime, the proportion of values with a factor exceeding $H^2$ is bounded below here. \subsection{Programming directions and freezing a single source} Every surviving $p$ exceeds $2XH$ for large $X$. Lemma~\ref{lem:directions} implies that its incidences on the entire rectangle lie in one primitive direction. The points touched by the pivot at $(a,b)\in\mathcal R_X$ are exactly \[ (ka,kb),\qquad 1\leq k\leq\min(\lfloor X/a\rfloor,\lfloor H/b\rfloor). \] Distinct reservoir points have distinct pivots. The valuation is one throughout each direction, and all its signs equal $f(a^2+b^2)$ because $f(k^2)=1$. For a target sign $s$, assigning $g(p)=s f(m)$ in $a^2+b^2=mp$ programs that direction; $m\leq M$ contains no changed pivot. All other directions remain unchanged. There are at most $XH$ pivots, so the total cost is at most \begin{equation}\label{eq:angular-cost} \sum_{p\text{ changed}}p^{-\sigma} \leq XH X^{-\sigma(2-\rho)}\leq X^{-\eta}, \qquad \eta=\sigma(2-\rho)-1-\alpha>0. \end{equation} The all-positive and all-negative prescriptions differ in their rectangular means by at least $2|\mathcal R_X|/(XH)\geq2c_*$. Multiples of a primitive direction increase the controlled mass, not the number of independent choices or the prime cost. Starting from $f_0=f$, select successive $X_j$ so large that the reservoir exists, \[ X_j^{2-\rho}>\max(M_0,3X_{j-1}^2),\qquad X_j^{-\eta}<\varepsilon2^{-j-1}. \] One of the two branches is at distance at least $c_*$ from the previous mean. Choose it. Every later pivot exceeds the full height of all three previous rectangles; thus all previously selected observations, not only their means, are frozen. The prime coordinates stabilize to one $g$, its total cost is less than $\varepsilon$, its prefix is preserved, and successive selected means stay at distance at least $c_*$. This proves Theorem~\ref{thm:angular-instability}, and hence part (iv) of Theorem~\ref{thm:critical-summary}. For example, $\alpha=1/2$, $\rho=1/4$, $\tau=1/8$ and $\sigma=7/8$ meet every strict inequality, with $\eta=1/32$. This verifies the compatibility of the parameters; it does not calibrate $c_*$ or the first scale at which the analytic input applies. A finite example makes the programming visible. At $X=256$, $H=16$, $M=2$ and $\rho=1/4$, exact enumeration of $\mathcal R_X$ gives $61$ directions, each with one label in the rectangle. Reversing their $61$ pivots from the constant source changes the mean from $1$ to $1987/2048$, at harmonic cost less than $1981/10^6$. The finite diagnostic supplied with this article recomputes the directions, privacy and cost. This example checks the local operation, not a lower bound for all sufficiently large rectangles. For the Gaussian square mean, existence and its product are already supplied by \cite[Theorem~A]{DLMS}. The next corollary uses the same fixed source to separate that known square behavior from power-rectangle divergence, while controlling its prime cost. \begin{corollary}[Square means can coexist with power-rectangle divergence]\label{cor:power-square-separation} In the range of Theorem~\ref{thm:angular-instability}, one may choose a perturbation $h_E$ of $1$, preserving a prescribed prefix, with $d_\sigma(1,h_E)<\varepsilon$, such that all three square means have limits $C_d(E)>1-4\varepsilon$, whereas its sum-of-two-squares mean on $H=\lfloor X^\alpha\rfloor$ does not converge. Its oscillation is bounded below by a positive constant depending only on $\alpha,\sigma$. Its mean on the ordinary integers is greater than $1-2\varepsilon$. \end{corollary} \begin{proof} Use an internal budget $\varepsilon'<\min(\varepsilon,c_*/8,1/4)$ and never change two. Before attack $j$, the finite negative-prime set $E_{j-1}$ has rectangular means tending to the three products. Choose $R_j$ beyond the previous scale where all three means are within $1/j$ of their products. Then choose $X_j>R_j$ with new pivots beyond $\max(M_0,3R_j^2)$, and make every reservoir direction negative. Add exactly the pivots requiring a sign change. The negative sets increase and the new mean is at most $1-2c_*$. All attacks and recoveries remain frozen. For $E=\bigcup_jE_j$, the products converge and satisfy $C_d(E)>1-4\varepsilon'$. The recovery means tend to $C_{-1}(E)$, so the oscillation is at least $2c_*-4\varepsilon'>c_*$. On squares the bound \eqref{eq:interpolation} at exponent one gives a uniform tail at most $36\sum_{p\in E,p>y}1/p$. Finite-prime approximation therefore proves convergence of all three square means to their products. The union of multiples on the integers similarly gives the product $\prod_{p\in E}(p-1)/(p+1)>1-2\varepsilon'$. Finally $d_1\leq d_\sigma$, so all displayed lower bounds hold with the requested $\varepsilon$. \end{proof} \subsection{What the threshold classifies} The stable estimate \eqref{eq:floor-critical-budget} holds at $s_\alpha$, while \eqref{eq:angular-cost} can be made to decay exactly when $\sigma>s_\alpha$. Thus $s_\alpha$ is the universal power-budget threshold in the stated angular range. This does not mean that every support above it is unstable. There is an equivalent boundary in counting exponent. For any $t>s_\alpha$, choose $s_\alpha<\sigma<\min(t,1)$ and use Corollary~\ref{cor:power-square-separation}. Its unstable support has $\delta(E)\leq\sigmas_\alpha$. Hence all supports with $\delta\leq s_\alpha$ are stable, and unstable supports exist with exponent arbitrarily close above it. Taking different exponents in different constructions cannot yield instability at a limiting support with $\delta\leq s_\alpha$; the positive theorem rules that out. The restriction $\alpha>119/500$ comes from the available uniform angular lower bound. Neither its optimality nor instability for every $0<\alpha\leq119/500$ is established here. No simultaneous instability of the other two norms at power width, and no logarithmic-in-$X$ assertion for these rectangles, is claimed.\par \section{Scope, dependence and remaining boundaries}\label{sec:scope} The article separates three kinds of control. A harmonic prime budget bounds changes on ordinary integers; coverage and parity measure their effect on a chosen observation; geometry and normalization govern whether a collective estimate can make its remaining tail negligible. The deterministic and probabilistic criteria answer different questions. On a fixed harmonically summable support, convergence for every assignment forces uniform truncation and identifies the limits. Under an independent law, the exact parity formula identifies the almost-sure truncation defect, without characterizing existence of the mean for a single arbitrary realization. Neither criterion creates an unknown mean of a prescribed base multiplied by the perturbation. The identities cancelling that base concern disagreement errors only. The positive counting results reach exponent $1/2$ on curves, including nonsummable critical series. The logarithmic counterexample is available separately for every requested $125/164<\sigma\leq1$, with one positive, uncalibrated oscillation bound independent of $\sigma$. The construction and its scales may depend on $\sigma$, and the endpoint $125/164$ is not included. The quantitative $1/2000$ branch applies for $128/153<\sigma\leq1$. These statements do not classify the interval between the curve stability and instability ranges. In particular they do not establish the curve endpoint $1/2$ as an optimal universal counting threshold, or settle a particular support of larger exponent. On rectangles $H=\lfloor X^\alpha\rfloor$, the universal budget and counting threshold is $(1+\alpha)/2$ for $119/500<\alpha<1$, with equality on the stable side. The negative assertion concerns $a^2+b^2$; the positive assertion concerns all three forms and holds for every $0<\alpha<1$. The restriction $119/500$ belongs to the angular input, not to a demonstrated barrier of the problem. Harmonic stability when $\log H/\log X\to1$ is compatible with failure at each fixed power width in the angular range. No uniform oscillation constant as $\alpha\to1$ is supplied. At sublogarithmic diverging widths, the separate elementary construction still makes all three means oscillate in one fixed function. \paragraph{Where the external inputs enter.} The whole-support criterion uses root counts, finite-prime laws, harmonic summability, a finite parity average and Abel summation. The random criterion adds classical bounded differences. The rich stable supports, ordinary instabilities and the ordinary/logarithmic separator use only prime distribution in fixed progressions beyond their finite constructions. The collective upper bounds use the Selberg quadratic form and fixed-progression Mertens estimates; their short-interval form depends on the interval length rather than the height. The three analytic routes in Table~\ref{tab:analytic-routes} feed the same mass--privacy--cost transfer. They retain different quantitative outputs, not multiple independent certifications of the argument. Harman--Lewis enters only the power-rectangle negative branch. The existence of sources with full affine cancellation comes separately from Klurman--Mangerel--Ter\"av\"ainen; preservation of that property is proved here. The neighboring probabilistic, sectorial and coprime-pair results discussed in Section~\ref{sec:introduction} provide context; the proof inputs are those identified above and at their uses. The private-prime mechanism, finite prescriptions, repeated freezing, local products, periodic approximation and sieve tools have the antecedents specified in the introduction and at their uses. In particular, the Gaussian square means are known from Donoso--Le--Moreira--Sun; the degenerating geometry, support-uniform criteria and quantified coexistence results are the distinctions used here. The contribution is in the quantified support criteria, verified tail regimes, fixed-source separations and the conversion of angular mass to affordable private directions. No claim of exhaustive bibliographic priority is made. The conclusions remain statements about the indicated source classes or constructed perturbations. They do not yield a new cancellation theorem for unmodified Liouville, a linear lower bound for the unrestricted minimum on three curves, or a Poisson limit. In particular, the restricted minimum \eqref{eq:restricted-minimum} is not an estimate for the unrestricted one. Appendix~\ref{sec:windows} similarly distinguishes every permitted origin from all but a vanishing proportion of origins, without declaring its local support threshold optimal. \appendix \section{Moving the origin: all windows and almost all windows}\label{sec:windows} A small prefix error cannot simply be subtracted twice to control a short interval: two errors $o(X)$ need not be $o(L)$ when $L\ll X$. The translated sieve \eqref{eq:interval-sieve} keeps the polynomial height $O(X^2)$ separate from the length $L$. It gives a statement at every origin under a stronger support condition. Mean periodic approximation then gives a different statement at almost every origin under the global critical condition. For $u\geq2$ and $L\geq1$ define \[ M_{d,f}(u,L)=\frac1L\sum_{uy,\ v_p(P_d(n))\text{ odd}\}. \] \subsection{Every origin under the local critical condition} \begin{theorem}[Uniform local stability]\label{thm:all-windows} Suppose $0<\alpha\leq1$, $\delta(\Gamma)\leq\alpha/2$ and \[ \liminf_{X\to\infty}\frac{\log\ell(X)}{\log X}\geq\alpha. \] Then \begin{equation}\label{eq:all-window-tail} \lim_{y\to\infty}\sup_{X\geq2}\sup_{I\in\mathscr I_\ell(X)}\max_dT_{\Gamma,d}(I;y)=0. \end{equation} Uniformly in $A\subseteq\Gamma$, $I=(u,u+L]\in\mathscr I_\ell(X)$ and $d\in\Dset$, \begin{equation}\label{eq:all-window-means} M_{d,h_A}(u,L)-c_d(A)\longrightarrow0. \end{equation} Every fixed finite list of translated readings has its corresponding local law under the same uniformity. \end{theorem} \begin{proof} For large $X$ and a cutoff $3X\leq Z\leq9X^2+2$, root counting and \eqref{eq:interval-sieve} give \begin{equation}\label{eq:window-split} T_{\Gamma,d}(I;y)\leq2\sum_{p\in\Gamma,p>y}\frac1p +\frac{2A_\Gamma(Z)}L +\frac1L\#\{n\in I:P^+(P_d(n))>Z\}. \end{equation} Fix $0<\vartheta<\alpha/8$, set $Z=X^{2-\vartheta}$ and $\eta=\alpha\vartheta/16$. Eventually $L\geq X^{\alpha-\eta}$, and the support condition implies \[ \frac{A_\Gamma(Z)}L \ll_{\Gamma,\eta}X^{(2-\vartheta)(\alpha/2+\eta)-(\alpha-\eta)} \ll_{\Gamma,\eta}X^{-\alpha\vartheta/4}\to0. \] Here the same $\eta$ is allowed in the support exponent because that bound holds for every positive excess. In \eqref{eq:interval-sieve}, $K\ll X^\vartheta$, so $K/\sqrt L\to0$, while the logarithmic term has limit superior $O(\vartheta/\alpha)$. Uniformly over the intervals, \[ \limsup_X\sup_{I\in\mathscr I_\ell(X)}\max_dT_{\Gamma,d}(I;y) \leq2\sum_{p\in\Gamma,p>y}\frac1p+C\vartheta/\alpha. \] Send $\vartheta\downarrow0$ and then $y\to\infty$, and remove the finitely many initial windows by a height cutoff. This proves \eqref{eq:all-window-tail}. For a fixed finite prime set $F$, truncate valuations at level $r$. At each odd visible prime, the unresolved classes in an interval of length $L$ have proportion at most $2/p^r+2/L$; at two the curve valuations are already bounded by one. A bounded periodic sequence has its period mean in every interval of length tending to infinity, uniformly in the origin. The Chinese remainder theorem gives $c_d(F)$. Uniform prime truncation and the uniformly convergent products therefore give \eqref{eq:all-window-means} over all $A$. For a finite translated list, take a common periodic truncation and a union of its tail events. A fixed shift changes the tail count on a window by at most $O(1/L)$ at its ends. The same proof gives the joint law, retaining all local dependencies. \end{proof} The theorem includes $\ell(X)=\lfloor X^\alpha\rfloor$ and more generally integer lengths $X^{\alpha+o(1)}$ in $[1,X]$. If the support of prime disagreements of $f,g$ has exponent at most $\alpha/2$, then the uniform disagreement limit is $(1-c_d(A))/2$, where $A=\{p:f(p)\ne g(p)\}$. Neither negative-prime support of $f$ or $g$ separately has to be sparse. If they differ at a fixed prime $p$, their total disagreement over the three curves in any permitted interval is at least $(\gamma_p-o(1))L$, with $\gamma_p$ as in \eqref{eq:restricted-minimum}. This retains a hypothesis on the disagreement support, not a conclusion about the unrestricted minimum. The estimates are also uniform over support families with a common envelope $A_\Gamma(t)\leq\Phi(t)\ll_\varepsilon t^{\alpha/2+\varepsilon}$ for every $\varepsilon>0$. For the more explicit bounds $A_\Gamma(t)\leq C_0t^{\alpha/2}(\log(2t))^B$ and $L\geq X^\alpha/(\log X)^A$, with fixed $A,B\geq0$, choose \[ Z=X^2/(\log X)^{(2A+2B+4)/\alpha} \] in \eqref{eq:window-split}. It gives \begin{equation}\label{eq:window-rate} T_{\Gamma,d}(I;y)\leq2\sum_{p\in\Gamma,p>y}\frac1p +O_{\alpha,A,B,C_0}\left(\frac{\log\log X}{\log X}\right). \end{equation} This is still a tail bound, not the full rate of convergence to the product. For a concrete critical example, $\Gamma=\{p_{j^4}:j\geq1\}$ has counting exponent $1/4$ but divergent series $\sum_\Gamma p^{-1/4}$: the terms are asymptotic to a constant times $1/(j(\log j)^{1/4})$. Since two belongs to it, the all-negative assignment has all three means tending to zero in every window of length $\lfloor\sqrt X\rfloor$, uniformly for $X\leq u<2X$. The critical series is not the resource paying for this local conclusion. \subsection{Almost every origin at any diverging length} \begin{theorem}[Typical windows uniformly over assignments]\label{thm:typical-windows} If $\delta(\Gamma)\leq1/2$ and $1\leq H(X)\leq X$ is integer-valued with $H(X)\to\infty$, then \begin{equation}\label{eq:typical-window-means} \frac1X\sum_{u=X}^{2X-1}\sup_{A\subseteq\Gamma}\max_{d\in\Dset} |M_{d,h_A}(u,H(X))-c_d(A)|\longrightarrow0. \end{equation} For any fixed accuracy, outside $o(X)$ origins the conclusion holds simultaneously for every assignment. The analogous assertion holds in total variation for the local laws of a fixed finite list of readings. \end{theorem} \begin{proof} Let $b_{d,y}(n)$ be the indicator of a tail prime at odd valuation in $P_d(n)$, with nonpositive values omitted. Theorem~\ref{thm:critical-curves} gives \[ \lim_{y\to\infty}\limsup_X\max_d\frac1{3X}\sum_{2\leq n\leq3X}b_{d,y}(n)=0. \] Each integer lies in at most $H=H(X)$ of the windows being averaged. Consequently \begin{equation}\label{eq:window-multiplicity} \frac1X\sum_{u=X}^{2X-1}\frac1H\sum_{u0, \] and a strictly increasing sequence $n_j$ such that \begin{equation}\label{eq:positive-blocks} h((n_j+a)^2-d)=1\qquad(0\leq a\max\{B_j,2^{j^2},(j^2+4)^2\}. \] Such primes exist by the theorem on primes in this fixed progression. The polynomials $P_{a,d}(T)=(T+a)^2-d$ are distinct and irreducible over $\mathbb Q$. For two distinct labels, writing $v=a-b$, their resultant is \begin{equation}\label{eq:block-resultant} v^4-2(d+e)v^2+(d-e)^2. \end{equation} It is nonzero and has absolute value at most $(j^2+4)^2$. Thus their root sets are disjoint modulo every new prime. Since $q_{a,d}\equiv1\pmod8$, the target polynomial has a simple root; choose a class modulo $q_{a,d}^2$ which lifts that root but has valuation exactly one. The Chinese remainder theorem combines all these classes, together with a parity condition. Select a representative $n_j$ larger than the previous origin, at least two, and at least $2^{j^2}$. For each label, its assigned prime occurs once and none of the other new primes divides its value. The sign of the current finite-support source on that value can be calculated by division by its finitely many negative primes. Set the new sign at $q_{a,d}$ equal to that current sign. The resulting sign of the target value is its square, hence one. Some of these new prime signs can remain positive. All old values are preserved because the new primes exceed $B_j$. At most $3j$ negative primes are added at stage $j$, all above $2^{j^2}$. Hence \[ A_\Gamma(t)\leq C_{\rm prefix}+\sum_{j\leq\sqrt{\log_2t}}3j =O_{\rm prefix}(\log(2t)). \] All coordinates stabilize, all blocks survive, and every positive-power sum converges. The procedure is computable: to determine $h(m)$ it suffices to carry out stages until $2^{j^2}>m$; all later changes lie above $m$. Each prime search, root search and Chinese-remainder calculation is finite. No claim of practical efficiency is made. The final support has exponent zero. Applying Theorem~\ref{thm:all-windows} for any fixed positive $\alpha$, and using the negative sign at two, gives the balanced power-window assertion. \end{proof} For an explicit finite illustration of the sign mechanism, the source negative at $\{2,11,31\}$ makes the three values at index eight positive: \[ 65=5\cdot13,\qquad66=2\cdot3\cdot11,\qquad62=2\cdot31. \] This is one block, not the infinite construction. In the proof each new pivot is private only within its newly prescribed block. It may affect other future values, but never an already frozen block. Define a nondecreasing width equal to $j$ between $n_j-1$ and $n_{j+1}-1$, with a harmless finite initial definition. It tends to infinity and is at most $X$ there, but at $X=n_j-1$ the window beginning at that origin has all means equal to one. Uniformity over all origins therefore fails for that width. Since $n_j\geq2^{j^2}$, these block lengths are subpolynomial in their origins and do not contradict the power-window theorem. Polynomial observations in short intervals have earlier treatments, including Nair~\cite{Nair}. The present proof uses the explicit translated Selberg estimate above rather than importing a general short-interval theorem. Prescribed blocks by congruences and freezing are also an established construction principle. The distinction established here is the combination of a critical support condition for every power window, an assignment-uniform typical-window law, and exceptions on a support of counting exponent zero. It is not an optimality result for the threshold $\alpha/2$ at a fixed power length. \section*{Acknowledgments and computational checks} \addcontentsline{toc}{section}{Acknowledgments and computational checks} Research, mathematical exploration, review processing, source comparison, proof integration, bibliographic checks, writing and preparation of the files were assisted by GPT-6 Astra. Brice Pouly takes responsibility for the manuscript. The accompanying finite diagnostics check selected identities and examples; they do not establish the asymptotic reservoirs, imported analytic estimates, category statements or almost-sure limits. No proof-assistant formalization or independent human certification is claimed. The \href{https://doi.org/10.5281/zenodo.23018445}{companion source-and-diagnostics archive}, distributed with this PDF, contains the sources and the finite diagnostics for this edition. \begingroup\small \begin{thebibliography}{20} \bibitem{Klurman} O. Klurman, \emph{Correlations of multiplicative functions and applications}, Compositio Mathematica \textbf{153} (2017), no.~8, 1622--1657. \doi{10.1112/S0010437X17007163}. \arxiv{1603.08453v1}, 28 March 2016. The proposition, corollary, theorem and lemma numbers cited here refer to that version. \bibitem{Merikoski} J. Merikoski, \emph{On the largest prime factor of $n^2+1$}, Journal of the European Mathematical Society \textbf{25} (2023), no.~4, 1253--1284. \doi{10.4171/JEMS/1216}. The analytic locators refer to \arxiv{1908.08816v3}: Section~2.1, Remark~13 (p.~15; published p.~1268), and Section~2.6, pp.~15--16, proof of Theorem~2. The last calculation is used together with Grimmelt--Merikoski. \bibitem{Warnke} L. Warnke, \emph{On the method of typical bounded differences}, Combinatorics, Probability and Computing \textbf{25} (2016), no.~2, 269--299. \doi{10.1017/S0963548315000103}. \arxiv{1212.5796}. Theorem~1 in the arXiv version records the classical bounded-differences inequality used here; the typical-differences refinement is not required. \bibitem{KMT} O. Klurman, A. P. Mangerel and J. Ter\"av\"ainen, \emph{On Elliott's conjecture and applications}, Proceedings of the London Mathematical Society \textbf{133} (2026), no.~3, e70205. \doi{10.1112/plms.70205}. Theorem numbering and references to Section~8, especially equations~(8.3)--(8.7), refer to \arxiv{2304.05344v2}, 26 May 2023. \bibitem{FKM} N. Frantzikinakis, O. Klurman and J. Moreira, \emph{Partition regularity of generalized Pythagorean pairs}, \arxiv{2407.08360v3}, 27 July 2026. Lemma~4.15: prime-incidence estimates for binary forms. \bibitem{TaoPNT} T. Tao, \emph{The prime number theorem in arithmetic progressions, and dueling conspiracies}, expository notes, 24 September 2009. \href{https://terrytao.wordpress.com/2009/09/24/the-prime-number-theorem-in-arithmetic-progressions-and-dueling-conspiracies/}{terrytao.wordpress.com}. Used only for the classical prime number theorem in fixed progressions and its partial-summation consequences. \bibitem{McDiarmid} C. McDiarmid, \emph{On the method of bounded differences}, in J. Siemons (ed.), \emph{Surveys in Combinatorics, 1989}, London Mathematical Society Lecture Note Series~141, Cambridge University Press, 1989, 148--188. \doi{10.1017/CBO9781107359949.008}. \bibitem{BKLR} V. Bergelson, J. Ku\l aga-Przymus, M. Lema\'nczyk and F. K. Richter, \emph{Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics}, Ergodic Theory and Dynamical Systems \textbf{39} (2019), no.~9, 2332--2383. \doi{10.1017/etds.2017.130}. \arxiv{1611.08392}. \bibitem{DBWX} R. de la Bret\`eche, V. Y. Wang and M. W. Xu, \emph{Random Multiplicative Functions and Making Squares from Polynomial Values}, \arxiv{2607.06398v1}, 7 July 2026. Context for polynomial observations of random multiplicative functions. \bibitem{Pascadi} A. Pascadi, \emph{Large sieve inequalities for exceptional Maass forms and the greatest prime factor of $n^2+1$}, Forum of Mathematics, Pi \textbf{14} (2026), e8. \doi{10.1017/fmp.2026.10025}. The weighted input and its locators refer to \arxiv{2404.04239v3}, 15 January 2026, Notations~20 and~23, Section~6.3, equation~(6.20) and its final verification at $1.30008$. \bibitem{Kedlaya} K. S. Kedlaya, \emph{Notes on analytic number theory}, Chapter~13, The Selberg sieve, Section~13.2, equations~(13.2.3)--(13.2.4) and Theorem~13.1. \href{https://kskedlaya.org/ant/chap-selberg.html}{kskedlaya.org/ant/chap-selberg.html}. Author's notes, accessed 27 September 2026. Used for the upper-sieve weights and diagonalization. \bibitem{DLMF} NIST, \emph{Digital Library of Mathematical Functions}, Sections~27.11--27.12. \href{https://dlmf.nist.gov/27.11}{dlmf.nist.gov/27.11} and \href{https://dlmf.nist.gov/27.12}{dlmf.nist.gov/27.12}. Accessed 27 September 2026. In particular (27.11.9)--(27.11.11) and the prime number theorem in fixed progressions; only moduli four and eight are needed here. \bibitem{HarmanLewis} G. Harman and P. Lewis, \emph{Gaussian primes in narrow sectors}, Mathematika \textbf{48} (2001), nos.~1--2, 119--135. \doi{10.1112/S0025579300014388}. Theorem~2, p.~119, supplies the uniform sector lower bound at width $t^{-0.381}$ for norm bound $t$. \bibitem{Nair} M. Nair, \emph{Multiplicative functions of polynomial values in short intervals}, Acta Arithmetica \textbf{62} (1992), no.~3, 257--269. \doi{10.4064/aa-62-3-257-269}. Antecedent for the short-interval setting. \bibitem{Stucky} J. Stucky, \emph{Gaussian primes in narrow sectors}, The Quarterly Journal of Mathematics \textbf{72} (2021), no.~4, 1357--1377. \doi{10.1093/qmath/haab009}. \arxiv{2008.11325v2}. Context for simultaneous restrictions on norm and argument. \bibitem{DLMS} S. Donoso, A. N. Le, J. Moreira and W. Sun, \emph{Averages of completely multiplicative functions over the Gaussian integers---a dynamical approach}, Transactions of the American Mathematical Society \textbf{377} (2024), no.~10, 7081--7115. \doi{10.1090/tran/9184}. Locators refer to \arxiv{2309.07249v2}, 6 March 2024: Theorem~A, Definition~2.3 and Lemmas~4.1, 4.3. \bibitem{GrimmeltMerikoski} L. Grimmelt and J. Merikoski, \emph{On the greatest prime factor and uniform equidistribution of quadratic polynomials}, preprint, \arxiv{2505.00493v2}, 30 May 2025. Section~7, Corollaries~7.1--7.2 and the paragraph following them; these supply the unconditional Type~I/II ranges used with Merikoski's weighted sieve calculation. \bibitem{BurginGiannitsi} A. Burgin and C. Giannitsi, \emph{Ergodic Averages over Shrinking Sectors of $\mathbb Z[i]$}, preprint, \arxiv{2609.25255v1}, 21 September 2026. Context for shrinking sectors and finitely generated multiplicative actions. \bibitem{Kus} J. Ku\'s, \emph{Hal\'asz theorems for Gaussian ideals in sectors and short intervals}, preprint, \arxiv{2605.19105v1}, 18 May 2026. Theorems~1.7 and~1.9: sectorial and short-interval context under non-pretentiousness hypotheses. \bibitem{WangCoprime} B. Wang, \emph{On averages of completely multiplicative functions over co-prime integer pairs}, Proceedings of the Royal Society of Edinburgh Section~A: Mathematics, published online 19 December 2025, 1--14. \doi{10.1017/prm.2025.10118}. Theorem~1.2 and Corollary~1.3 are cited in \arxiv{2406.09243v2}, 16 December 2025. \end{thebibliography} \endgroup \end{document}