Arithmetic Spectral Decomposition of Centered Coprimality Matrices (Paper 3)

TeX code for @Grok: \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{\Tr}{\operatorname{Tr}} \newcommand{\one}{\mathbf{1}} \newcommand{\bw}{\mathbf{w}} \newcommand{\e}{\mathbf{e}} \title{Centered Coprimality Matrices and the Rank--Nullity Decomposition\\of Arithmetic Moment Sequences} \author{Christopher Michael Pompetzki} \date{February 2026} \begin{document} \maketitle \begin{abstract} We study the coprimality matrix $A_N$ after removal of its rank-one mean-field component. Defining the centered matrix $\widetilde{A}_N = A_N - c_N P$ with $c_N = \e^T A_N \e / N$ and $P = \e\e^T / N$, we prove that the centered trace moments $\mu_k = \lim_{N\to\infty} \Tr(\widetilde{A}_N^k)/N^k$ exist for all $k \geq 1$ and admit explicit Euler product representations involving two distinct families of local factors: cycle counts $C_k(p) = \Tr(M_p^k)$ and path counts $P_k(p) = \one^T M_p^k \bw$. We establish that $P_k(2) = F_{k+3}$ (Fibonacci numbers) while $C_k(2) = L_k$ (Lucas numbers), proving a duality between open and closed coprimality walks at the smallest prime. We prove the sign alternation $(-1)^k \mu_k > 0$ for all $k \geq 1$, in particular establishing that odd centered moments are strictly nonzero. The spectral decomposition $P_k(p) = \tfrac{p}{2}\,C_k(p) + \tfrac{\delta(p)}{2}\,D_k(p)$, where $D_k(p) = \lambda_+^k - \lambda_-^k$ and $\delta(p) = \sqrt{p-1}\,(p+2)/\sqrt{p+3}$, separates the moment sequence into eigenvalue data (even moments) and eigenvector data (odd moments). This decomposition is the rank--nullity theorem applied to the arithmetic moment sequence: the odd moments form a minimal spanning set for the spectral asymmetry of the coprimality transfer system. \end{abstract} %===================================================================== \section{Introduction} \label{sec:intro} %===================================================================== In \cite{paper1} we introduced the coprimality matrix $A_N$ and proved that its normalized cycle moments $m_k = \lim_{N\to\infty} N^{-k}\Tr(A_N^k)$ exist and factor as Euler products with local factors $C_k(p)/p^k$, where $C_k(p) = \Tr(M_p^k)$ for the $2 \times 2$ transfer matrix $M_p = \bigl(\begin{smallmatrix} p-1 & p-1 \\ 1 & 0 \end{smallmatrix}\bigr)$. The matrix $A_N$ has a dominant rank-one component: the projection $P = \e\e^T/N$ onto the all-ones vector carries an eigenvalue of order $N/\zeta(2)$, which dominates all trace moments. To access finer spectral structure, one must center: define $\widetilde{A}_N = A_N - c_N P$ with the exact centering $c_N = \e^T A_N \e / N$. This paper develops the spectral theory of $\widetilde{A}_N$. The central discovery is that centering forces the introduction of a second family of arithmetic counts --- \emph{path counts} $P_k(p) = \one^T M_p^k \bw$ --- which are distinct from cycle counts $C_k(p) = \Tr(M_p^k)$ due to the non-symmetry of $M_p$. The interaction between cycles and paths under centering produces nonvanishing odd moments and a rank--nullity decomposition of the full moment sequence. All results are proved in ZFC without the Axiom of Choice. All matrices are finite. All limits are of sequences of real numbers. %===================================================================== \section{Definitions and Notation} \label{sec:definitions} %===================================================================== \begin{definition}[Coprimality matrix] For $N \geq 1$, the coprimality matrix $A_N \in \R^{N \times N}$ is defined by $(A_N)_{mn} = [\gcd(m,n) = 1]$ for $1 \leq m,n \leq N$. \end{definition} \begin{definition}[Centering] Let $\e = (1,1,\ldots,1)^T \in \R^N$. Define: \begin{align} S_k &= \e^T A_N^k \e = \sum_{m,n=1}^N (A_N^k)_{mn}, \label{eq:Sk}\\ c_N &= S_1 / N, \label{eq:cN}\\ P &= \e\e^T / N, \label{eq:P}\\ \widetilde{A}_N &= A_N - c_N P. \label{eq:Atilde} \end{align} \end{definition} \begin{lemma}[Properties of centering] \label{lem:centering-properties} The following hold for each $N$: \begin{enumerate}[label=(\roman*)] \item $P^2 = P$ and $P^T = P$. \item $\Tr(\widetilde{A}_N P) = 0$, i.e., $\e^T \widetilde{A}_N \e = 0$. \item $\widetilde{A}_N = A_N - c_N P$ is an algebraic identity of finite matrices, not an asymptotic approximation. \end{enumerate} \end{lemma} \begin{proof} (i) $(P^2)_{mn} = \frac{1}{N^2}\sum_k 1 \cdot 1 = \frac{1}{N} = P_{mn}$. (ii) $\e^T\widetilde{A}_N\e = S_1 - c_N \cdot N = S_1 - S_1 = 0$. (iii) Tautological. \end{proof} \begin{remark} The identity $\e^T \widetilde{A}_N \e = 0$ does \textbf{not} imply $\widetilde{A}_N \e = 0$. The vector $\widetilde{A}_N \e$ has $m$-th entry $\phi_N(m) - S_1/N$, where $\phi_N(m) = \#\{n \leq N : \gcd(m,n) = 1\}$. This is generically nonzero. \end{remark} \begin{definition}[Transfer matrix and weight vector] For a prime $p$, define the $2 \times 2$ transfer matrix and weight vector: \begin{equation} M_p = \begin{pmatrix} p-1 & p-1 \\ 1 & 0 \end{pmatrix}, \qquad \bw_p = \begin{pmatrix} p-1 \\ 1 \end{pmatrix}, \qquad \one = \begin{pmatrix} 1 \\ 1 \end{pmatrix}. \label{eq:transfer} \end{equation} The two states represent residue classes modulo $p$: state~$1$ for nonzero residues (weight $p-1$) and state~$0$ for zero residue (weight~$1$). \end{definition} \begin{definition}[Cycle and path local factors] \label{def:cycle-path} For a prime $p$ and integer $k \geq 0$: \begin{align} C_k(p) &= \Tr(M_p^k), \label{eq:Ck}\\ P_k(p) &= \one^T M_p^k \bw_p. \label{eq:Pk} \end{align} \end{definition} \begin{definition}[Eigenvalues and discriminant] The eigenvalues of $M_p$ are \begin{equation} \lambda_\pm(p) = \frac{(p-1) \pm \sqrt{(p-1)(p+3)}}{2}. \label{eq:eigenvalues} \end{equation} Define the discriminant $\Delta(p) = \sqrt{(p-1)(p+3)}$. Then $\lambda_+(p) > 0 > \lambda_-(p)$ for all primes $p \geq 2$, and $\lambda_+(p)\lambda_-(p) = -(p-1)$, $\lambda_+(p)+\lambda_-(p)=p-1$. \end{definition} %===================================================================== \section{Path Euler Products} \label{sec:paths} %===================================================================== The cycle moments $m_k = \lim_{N\to\infty} N^{-k}\Tr(A_N^k)$ were computed in \cite{paper1}. We now establish the analogous result for \emph{path densities}. \begin{definition}[Path densities] \label{def:sigma} For $k \geq 0$, define \begin{equation} \sigma_k = \lim_{N\to\infty} \frac{S_k}{N^{k+1}} = \lim_{N\to\infty} \frac{\e^T A_N^k \e}{N^{k+1}}, \label{eq:sigma} \end{equation} with $\sigma_0 = 1$. \end{definition} \begin{theorem}[Path Euler product] \label{thm:path-euler} For each fixed $k \geq 1$, the limit $\sigma_k$ exists and factors as \begin{equation} \sigma_k = \prod_p \frac{P_k(p)}{p^{k+1}}, \label{eq:path-euler} \end{equation} where the product runs over all primes $p$. \end{theorem} \begin{proof} The entry $(\e^T A_N^k \e)$ counts all walks of length $k$ in the coprimality graph with free (unmatched) endpoints: \[ S_k = \sum_{\substack{a_0,a_1,\ldots,a_k=1}}^N \prod_{i=0}^{k-1} [\gcd(a_i,a_{i+1})=1]. \] Apply M\"obius inversion to each coprimality constraint: $[\gcd(a_i,a_{i+1})=1] = \sum_{d_i | \gcd(a_i,a_{i+1})} \mu(d_i)$. Exchange summation order and restrict to squarefree $d_i$. By the Chinese Remainder Theorem, for fixed squarefree $d_1,\ldots,d_k$, the number of solutions $(a_0,\ldots,a_k) \in [N]^{k+1}$ factors over primes: each prime $p$ dividing at least one $d_i$ contributes a local constraint count, while primes dividing no $d_i$ contribute the unconstrained count $p^{k+1}$. The local count at prime $p$ for walks with free endpoints and $k$ edges, where the state (divisible or not divisible by $p$) at each vertex is tracked by the transfer matrix $M_p$, is exactly $\one^T M_p^k \bw_p = P_k(p)$. Here $\bw_p$ encodes the initial state weights and $\one$ the final state summation. Dominated convergence applies: each local factor satisfies $P_k(p)/p^{k+1} = 1 + O(p^{-2})$, so the product converges absolutely, and the tail $\prod_{p > P} P_k(p)/p^{k+1}$ is bounded by $\exp(C k / (P \log P))$ for a universal constant $C$. \end{proof} \begin{proposition}[Explicit path local factors] \label{prop:path-explicit} The path local factors for small $k$ are: \begin{align} P_0(p) &= p, \label{eq:P0}\\ P_1(p) &= (p-1)(p+1) = p^2-1, \label{eq:P1}\\ P_2(p) &= (p-1)(p^2+p-1), \label{eq:P2}\\ P_3(p) &= p(p-1)^2(p+2), \label{eq:P3}\\ P_4(p) &= (p-1)^2(p^3+2p^2-p-1), \label{eq:P4}\\ P_5(p) &= (p-1)^3(p+1)(p^2+2p-1). \label{eq:P5} \end{align} \end{proposition} \begin{proof} Direct computation: $P_k(p) = \one^T M_p^k \bw_p$ by matrix multiplication for each $k$. \end{proof} \begin{corollary} \label{cor:sigma1} $\sigma_1 = 1/\zeta(2) = 6/\pi^2$. \end{corollary} \begin{proof} $P_1(p)/p^2 = (p^2-1)/p^2 = 1 - 1/p^2$, so $\sigma_1 = \prod_p(1-1/p^2) = 1/\zeta(2)$. \end{proof} \begin{corollary} \label{cor:sigma-m} $\sigma_1 = m_2$ and $\sigma_k = m_{k+1} \cdot R_k$ where $R_k = \prod_p P_k(p)/(p \cdot C_k(p))$ is the path-to-cycle ratio. In general $R_k \neq 1$. \