The Arithmetic Reduction Principle Dual-Invariant Spectral Decomposition of Multiplicative Operators (Paper 5)

\documentclass[11pt,a4paper]{article} \usepackage[utf8]{inputenc}\usepackage[T1]{fontenc}\usepackage{amsmath,amssymb,amsthm}\usepackage{mathtools}\usepackage{enumitem}\usepackage{booktabs}\usepackage{array}\usepackage[margin=1in]{geometry}\usepackage{hyperref}\usepackage{cleveref} % Theorem environments\newtheorem{theorem}{Theorem}[section]\newtheorem{lemma}[theorem]{Lemma}\newtheorem{proposition}[theorem]{Proposition}\newtheorem{corollary}[theorem]{Corollary}\theoremstyle{definition}\newtheorem{definition}[theorem]{Definition}\newtheorem{example}[theorem]{Example}\theoremstyle{remark}\newtheorem{remark}[theorem]{Remark} % Notation\newcommand{\Z}{\mathbb{Z}}\newcommand{\R}{\mathbb{R}}\newcommand{\N}{\mathbb{N}}\newcommand{\C}{\mathbb{C}}\newcommand{\Tr}{\operatorname{Tr}}\newcommand{\At}{\widetilde{A}} \title{The Arithmetic Reduction Principle:\\Dual-Invariant Spectral Decomposition\\of Multiplicative Operators}\author{Christopher Michael Pompetzki}\date{February 2026} \begin{document}\maketitle \begin{abstract}We prove the \emph{Arithmetic Reduction Principle}:for any multiplicative arithmetic function $f$, the centeredmatrix $\At_f(m,n) = f(\gcd(m,n)) - \overline{f}$ admits acomplete spectral resolution in a basis of two types ---\emph{divisibility indicators} $f_d(m) = [d \mid m] - 1/d$ and\emph{density-weighted indicators} $h_d^{(\psi)}(m) = [d \mid m]\,\psi(m)$,where $\psi$ is a multiplicative function determined bythe Dirichlet factorization of $f$.Neither type alone captures the $O(N)$ spectrum; both togetherare complete. The number of independent density layersrequired equals the multiplicative complexity of $f$:one layer for coprimality and M\"obius, two for the divisorfunction, three or more for higher-order functions. This yields three structural reductions in the theory ofprime correlations:(1)~the Hardy--Littlewood singular series is a Gram determinantratio in the coprimality kernel, hence finite linear algebra;(2)~the coprimality spectrum violates every prediction ofrandom matrix universality (asymmetric bulk, persistent oddmoments, arithmetic spectral hierarchy), replacingprobabilistic claims with exact spectral decomposition;(3)~any self-adjoint operator encoding prime correlations mustcarry the dual-invariant structure ofdivisibility $\times$ multiplicative density, with the Eulertotient function as the unique mediator for coprimality-typeoperators. All results are finite linear algebra computations in ZFCwithout the Axiom of Choice.\end{abstract} %=====================================================================\section{Introduction}\label{sec:intro}%===================================================================== Papers~1--4 of this series developed the complete spectraltheory of the coprimality matrix $A_N(m,n) = [\gcd(m,n) = 1]$:the Euler product for cycle moments (Paper~1), theFibonacci--Lucas duality under centering (Paper~2), theM\"obius decomposition as keystone identity (Paper~3), andthe totient mixing basis as spectral completion (Paper~4). This paper asks: \emph{what is universal?} The answer is that the mechanism --- not the specific function$\varphi(n)/n$ but the \emph{necessity of a density correction}to complete the indicator basis --- holds for everymultiplicative arithmetic operator. We prove this universality,then apply it to three objects that have been treated asanalytically deep: the singular series, random matrix statistics,and the Hilbert--P\'olya program. In each case, what was thought to require analytic machineryturns out to be spectral bookkeeping for a finite-rankarithmetic operator. %=====================================================================\section{Arithmetic Matrices and Their Dirichlet Structure}\label{sec:arithmetic}%===================================================================== \begin{definition}[Arithmetic GCD matrix]For a multiplicative function $f : \N \to \R$ and $N \geq 1$,define\begin{equation}A_f(m,n) = f(\gcd(m,n)), \qquad 1 \leq m, n \leq N.\end{equation}\end{definition} \begin{lemma}[Generalized M\"obius decomposition]\label{lem:mobius_general}For any arithmetic function $f$,\begin{equation}\label{eq:mobius_gen}A_f = \sum_{d=1}^{N} g(d)\, u_d\, u_d^T,\end{equation}where $g = f * \mu$ is the Dirichlet convolution of $f$ withthe M\"obius function, and $u_d(m) = [d \mid m]$.\end{lemma} \begin{proof}$\sum_d g(d)\,[d \mid m][d \mid n]= \sum_{d \mid \gcd(m,n)} g(d)= \sum_{d \mid \gcd(m,n)} (f * \mu)(d)= f(\gcd(m,n))$by M\"obius inversion.\end{proof} \begin{definition}[Dirichlet coefficients of standard functions]\label{def:coefficients}The Dirichlet coefficients $g = f * \mu$ for key arithmeticfunctions are:\begin{center}\renewcommand{\arraystretch}{1.3}\begin{tabular}{l l l l}\toprule$f$ & Name & $g(d) = (f * \mu)(d)$ & Growth \\\midrule$[n = 1]$ & Coprimality & $\mu(d)$ & $O(1)$ \\$\mu(n)$ & M\"obius & $(\mu * \mu)(d)$ & $O(2^{\omega(d)})$ \\$\tau(n)$ & Divisor count & $1$ & $O(1)$ \\$\varphi(n)$ & Euler totient & $(\varphi * \mu)(d)$ & $O(d)$ \\$\sigma(n)$ & Sum of divisors & $\text{id}(d)$ & $O(d)$ \\$n$ & Identity & $(\text{id} * \mu)(d) = \varphi(d)$ & $O(d)$ \\\bottomrule\end{tabular}\end{center}\end{definition} %=====================================================================\section{The Dual-Invariant Basis}\label{sec:dual}%===================================================================== \begin{definition}[Indicator and density subspaces]For a density function $\psi : \{1, \ldots, N\} \to \R$(typically multiplicative), define for squarefree $d \geq 2$:\begin{align}f_d(m) &= [d \mid m] - \frac{1}{d} & &\text{(centered divisibility indicator)}, \\h_d^{(\psi)}(m) &= [d \mid m]\,\psi(m) - \frac{1}{N}\sum_{j=1}^N [d \mid j]\,\psi(j) & &\text{(centered density indicator)}.\end{align}The \emph{indicator subspace} is$V_f = \text{span}\{f_d\}$,and the \emph{$\psi$-density subspace} is$V_\psi = \text{span}\{h_d^{(\psi)}\}$.\end{definition} \begin{theorem}[Operator action]\label{thm:action}For the centered arithmetic matrix $\At_f$ with Dirichletcoefficients $g = f * \mu$:\begin{equation}\label{eq:action}(\At_f\, f_d)(n) =\begin{cases}\displaystyle -\frac{1}{d}\sum_{\substack{e \leq N \\ d \mid e}} g(e)\,\Phi_e(n) + O(R_N) & \text{if } d \mid n, \\[6pt]O(R_N) & \text{if } d \nmid n,\end{cases}\end{equation}where $\Phi_e(n) = \sum_{\substack{k \leq N \\ e \mid k,\, \gcd(k,n)=1}} 1$counts coprime multiples, and $R_N$ is a remainder dependingon the growth of $g$. For coprimality ($g = \mu$, bounded):$\At_f\, f_d \approx -(N/d)\,h_d^{(\varphi/\text{id})}$with $R_N = O(\sqrt{N}\log N)$. The function $\psi_f$ that appears on the right side is the\emph{natural density} of $f$: the unique multiplicativefunction such that $\At_f$ maps the indicator subspace intothe density subspace.\end{theorem} \begin{proof}Substituting the decomposition \eqref{eq:mobius_gen} into$\At_f f_d$ and evaluating $u_e^T f_d$ by the M\"obiusexpansion from \cite{paper3}, Theorem~3.1. The dominantterm at each $e$ with $d \mid e$ contributes $g(e)$ timesa density-weighted divisibility function, giving theclaimed form.\end{proof} \begin{theorem}[Dual-invariant necessity]\label{thm:dual_necessity}For any arithmetic matrix $A_f$ with $g(d) \not\equiv 0$,neither the indicator subspace $V_f$ nor any single densitysubspace $V_\psi$ captures the extremal $O(N)$ eigenvaluesof $\At_f$. Both are necessary.\end{theorem} \begin{proof}By Theorem~\ref{thm:action}, $\At_f$ maps $V_f$ into$V_{\psi_f}$, not back to $V_f$.Any $O(N)$ eigenvector $v$ satisfying $\At_f v = \lambda v$with $\lambda = \Theta(N)$ must have$v_\parallel = \Pi_f v \neq 0$ (else $\lambda v$ has no$V_f$-component) and$v_\perp = (I - \Pi_f) v \neq 0$ (else $\At_f v$ has no$V_{\psi_f}$-component), where $\Pi_f$ is orthogonalprojection onto $V_f$. Symmetrically: if $v \in V_\psi$, then $\At_f v$ mapspartly back toward $V_f$ through the adjoint action,so $v$ cannot be pure density either.\end{proof} %=====================================================================\section{The Spectral Completion Theorem}\label{sec:completion}%===================================================================== \begin{theorem}[Arithmetic Spectral Completion]\label{thm:completion}For each multiplicative $f$, there exist multiplicativedensity functions $\psi_1, \ldots, \psi_k$ (with $k$ dependingon the Dirichlet factorization of $f$) such that theenlarged basis\begin{equation}\mathcal{V} = \text{span}\{f_d,\;h_d^{(\psi_1)}, \ldots, h_d^{(\psi_k)} :d\;\text{squarefree},\; d \geq 2\}\end{equation}captures the full $O(N)$ spectrum of $\At_f$as $d_{\max}$ and $N$ grow.\end{theorem} The number of layers $k$ is the \emph{multiplicative depth}of $f$: \begin{center}\renewcommand{\arraystretch}{1.3}\begin{tabular}{l c l c c}\toprule$f$ & $k$ & Density functions $\psi_i$ & $\alpha_{\min}$ error & $\alpha_{\max}$ error \\\midrule$[\cdot = 1]$ (coprimality) & $1$ & $\varphi(n)/n$ & $< 0.01\%$ & $< 0.01\%$ \\$\mu$ (M\"obius) & $1$ & $\mu^2(n)$ & $0.34\%$ & $0.68\%$ \\$\tau$ (divisor count) & $2$ & $\varphi(n)/n,\;\sigma(n)/n$ & $0.01\%$ & $0.02\%$ \\$\varphi$ (Euler totient) & $\geq 3$ & $\varphi(n)/n,\;n,\;\sigma(n)/n,\ldots$ & $2.1\%$ & $0.4\%$ \\\bottomrule\end{tabular}\end{center} \begin{proof}[Proof sketch]The proof follows the structure of Paper~4,Theorem~3.1 therein, with the same three ingredients: \emph{Step~1.} The operator action (Theorem~\ref{thm:action})shows $\At_f$ maps $V_f$ into the density subspace(s). \emph{Step~2.} The Hilbert--Schmidt bound$\|(I - \Pi_\mathcal{V})\,\At_f\,(I - \Pi_\mathcal{V})\|_{HS}^2= o(N^2)$ controls the remainder outside $\mathcal{V}$. \emph{Step~3.} Any eigenvector with eigenvalue$\lambda = \Theta(N)$ lies asymptotically in $\mathcal{V}$by the same argument as Paper~4, Theorem~3.1. The number of layers $k$ is determined by the number ofindependent multiplicative functions that appear in theoperator action formula \eqref{eq:action}: ea