The Totient Mixing Basis and Complete Spectral Resolution of Centered Coprimality Matrices (Paper 4)

\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}} \newcommand{\At}{\widetilde{A}} \title{The Totient Mixing Basis and Complete Spectral\\Resolution of Centered Coprimality Matrices} \author{Christopher Michael Pompetzki} \date{February 2026} \begin{document} \maketitle \begin{abstract} We construct an explicit basis that captures the entire $O(N)$ spectrum of the centered coprimality matrix $\At_N$. For each squarefree $d \geq 2$, we define two vectors: the centered divisibility indicator $f_d(m) = [d \mid m] - 1/d$ and the totient-weighted indicator $h_d(m) = [d \mid m]\,\varphi(m)/m$ (centered), where $\varphi$ is the Euler totient function. We prove that neither the $f$-subspace nor the $h$-subspace alone captures the extremal eigenvalues (both suffer $\approx 35\%$ error on $\alpha_{\min}$), but their union captures the full $O(N)$ spectrum to within numerical precision ($< 0.01\%$). The operator action formula from \cite{paper3} explains this: $\At_N$ maps $f_d$ into the $h_d$ direction, so every $O(N)$ eigenvector must have nonzero components in both subspaces. We compute the $2 \times 2$ block structure of the kernel restricted to $\text{span}\{f_p, h_p\}$ for each prime $p$, verify the cross-coupling entry $K_{fh}$ against the operator action formula (agreement $< 0.5\%$ for all primes), and exhibit a convergence hierarchy showing how each layer of totient mixing systematically closes the spectral gap. The structural parallel with the MUB problem for composite dimensions is made precise: both problems are governed by factorization obstructions on squarefree integers where the Euler totient mediates inter-factor coupling. All results are finite linear algebra computations in ZFC without the Axiom of Choice. \end{abstract} %===================================================================== \section{Introduction} \label{sec:intro} %===================================================================== In \cite{paper3} we proved that the $O(N)$ eigenvalues of the centered coprimality matrix $\At_N$ are determined by a generalized eigenvalue problem on the indicator subspace $\{f_d : d \geq 2,\, d\;\text{squarefree}\}$, where $f_d(m) = [d \mid m] - 1/d$. We showed that the Rayleigh quotient for each prime $p$ converges to $-1/\bigl((p+1)\zeta(2)\bigr)$, and identified a multiplicative kernel operator $\mathcal{K}$ whose generalized eigenvalues determine the full arithmetic spectrum. However, \cite{paper3} also reported a fundamental gap: the restricted eigenvalue problem on the indicator subspace $V_f = \text{span}\{f_d\}$ misses the extremal eigenvalues by approximately $35\%$. The restricted $\alpha_{\min}$ stabilizes at $-0.224$ regardless of how many squarefree indices $d$ are included, while the true value is $-0.346$. The same computation at $d_{\max} = 300$ (182 squarefree indices) showed identical $35\%$ error. The operator action formula (Theorem~5.1 of \cite{paper3}) explains the source of the gap: $\At_N$ maps $f_p$ not back to $f_p$, but into the direction $[p \mid n]\,\varphi(n)/n$, a fundamentally different function. This means the $O(N)$ eigenvectors cannot lie in the indicator subspace alone; they require a second component carrying the totient structure. This paper constructs that component and proves that the resulting enlarged basis resolves the entire $O(N)$ spectrum. %===================================================================== \section{The Enlarged Basis} \label{sec:basis} %===================================================================== \begin{definition}[Totient-weighted indicators] For squarefree $d \geq 2$, define \begin{equation}\label{eq:hd} h_d(m) = [d \mid m]\,\frac{\varphi(m)}{m} \end{equation} and its centered version $\overline{h}_d(m) = h_d(m) - \frac{1}{N}\sum_{j=1}^N h_d(j)$. \end{definition} \begin{remark} The function $\varphi(m)/m = \prod_{p \mid m}(1 - 1/p)$ is multiplicative: it depends only on the set of prime divisors of $m$, not their multiplicities. For $m$ with $d \mid m$, the constraint $d \mid m$ forces all prime factors of $d$ into $m$, so $\varphi(m)/m = \prod_{p \mid d}(1 - 1/p) \cdot \prod_{p \mid m,\, p \nmid d}(1 - 1/p)$. This multiplicative factorization is why $h_d$ carries spectral information that $f_d$ does not. \end{remark} \begin{definition}[The enlarged basis] For a cutoff $D$, define \begin{equation}\label{eq:V} \mathcal{V}_D = \text{span}\{f_d,\; \overline{h}_d : d \leq D,\; d\;\text{squarefree},\; d \geq 2\}. \end{equation} The dimension of $\mathcal{V}_D$ is at most $2 |\{d \leq D : d\;\text{squarefree}\}|$. \end{definition} %===================================================================== \section{The Spanning Theorem} \label{sec:spanning} %===================================================================== \begin{theorem}[Main result: spectral completeness] \label{thm:spanning} The enlarged basis $\mathcal{V}_D$ captures the full $O(N)$ spectrum of $\At_N$ as $D$ and $N$ grow. Precisely: for any $\varepsilon > 0$, there exists $D_0(\varepsilon)$ such that for all $D \geq D_0$ and all sufficiently large $N$, every eigenvalue $\lambda$ of $\At_N$ with $|\lambda| > \varepsilon N$ satisfies \begin{equation} \min_{\alpha \in \text{spec}(\At_N|_{\mathcal{V}_D})} |\lambda - \alpha| \leq \varepsilon N. \end{equation} \end{theorem} Before giving the proof, we establish two key ingredients. \begin{lemma}[Operator action in the enlarged basis] \label{lem:action_enlarged} For squarefree $d \geq 2$ and any $n \in \{1, \ldots, N\}$: \begin{equation}\label{eq:action_d} (\At_N f_d)(n) = \begin{cases} -\dfrac{N}{d}\,\dfrac{\varphi(n)}{n} + O(\sqrt{N} \log N) & \text{if } d \mid n, \\[6pt] O(\sqrt{N} \log N) & \text{if } d \nmid n. \end{cases} \end{equation} That is, $\At_N f_d \approx -(N/d)\,h_d$ on the support of $d$. \end{lemma} \begin{proof} Extends the proof of Theorem~5.1 in \cite{paper3} from primes to general squarefree $d$ by multiplicativity. If $d \mid n$, then $\gcd(m,n) \geq d > 1$ for all $m$ with $d \mid m$, so the coprimality constraint $[\gcd(m,n) = 1]$ kills the indicator part and leaves only the $-1/d$ tail of $f_d$, which sums to $-(1/d)\varphi_N(n) = -(N/d)\varphi(n)/n + O(2^{\omega(n)})$. If $d \nmid n$, then $\gcd(d,n) < d$, and the sum exhibits cancellation between the $[d \mid m]$ and $1/d$ components by multiplicativity of the M\"obius function, leaving $O(\sqrt{N}\log N)$ by standard bounds on character-like sums. \end{proof} \begin{lemma}[Image lies in $\mathcal{V}_D$] \label{lem:image} For any $v \in \mathcal{V}_D$, the image $\At_N v$ lies within $O(\sqrt{N}\log N)$ of the subspace $\mathcal{V}_D$ in the $\ell^2$ sense: \begin{equation} \|(I - \Pi_D)\,\At_N\, v\| \leq C_D\, \sqrt{N}\log N\, \|v\|, \end{equation} where $\Pi_D$ is the orthogonal projection onto $\mathcal{V}_D$ and $C_D$ depends only on $D$. \end{lemma} \begin{proof} By linearity, it suffices to check for $v = f_d$ and $v = \overline{h}_d$. Lemma~\ref{lem:action_enlarged} shows $\At_N f_d \approx -(N/d)\,h_d \in \mathcal{V}_D$ up to $O(\sqrt{N}\log N)$. For $v = \overline{h}_d$: the M\"obius decomposition gives $\At_N \overline{h}_d = \sum_e \mu(e)\,(u_e^T \overline{h}_d)\,u_e$, and each $u_e$ projects onto the indicator subspace. The dominant terms (those with $e$ sharing prime factors with $d$) produce linear combinations of $f_e$ and $h_e$ for $e \leq D$ when $d \leq D$, with the tail contributing $O(\sqrt{N}\log N)$ by the absolute convergence of $\sum_e \mu(e)/e^2$. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:spanning}] Let $\lambda$ be an eigenvalue of $\At_N$ with $|\lambda| > \varepsilon N$, and let $v$ be the corresponding eigenvector with $\|v\| = 1$. Write $v = v_\parallel + v_\perp$ where $v_\parallel = \Pi_D v$ and $v_\perp = (I - \Pi_D)v$. Then $\lambda v = \At_N v$, so \begin{align*} \lambda v_\perp &= (I - \Pi_D)\,\At_N\,(v_\parallel + v_\perp) \\ &= (I - \Pi_D)\,\At_N\, v_\parallel + (I - \Pi_D)\,\At_N\, v_\perp. \end{align*} By Lemma~\ref{lem:image}, the first term has norm $\leq C_D \sqrt{N}\log N\,\|v_\parallel\|$. By the Hilbert--Schmidt bound from \cite{paper3} Theorem~6.1, $(I - \Pi_D)\,\At_N\,(I - \Pi_D)$ has operator norm $o(N)$ as $D \to \infty$. Choose $D$ large enough that this operator norm is $\leq (\varepsilon/2)N$. Then $|\lambda|\,\|v_\perp\| \leq C_D\sqrt{N}\log N + (\varepsilon/2)N\,\|v_\perp\|$, giving $\|v_\perp\| \leq C_D\sqrt{N}\log N / (|\lambda| - \varepsilon N/2) = O(\log N/\sqrt{N}) \to 0$ for $|\lambda| > \varepsilon N$. Thus $v$ lies asymptotically in $\mathcal{V}_D$: $\|v_\parallel\| \to 1$. The restriction of $\At_N$ to $\mathcal{V}_D$ therefore has an eigenvalue within $O(\sqrt{N}\log N)$ of $\lambda$, which is $o(N)$. \end{proof} %===================================================================== \section{Ablation: Neither Subspace Alone Suffices} \label{sec:ablation} %==