Prime-Star Localization and Second-Order Spectra of Finite Divisibility Cover Graphs

Abstract

Let G_X be the graph on the positive integers n <= X in which n is joined to np whenever p is prime and np <= X. Let A_X be its adjacency matrix, let lambda_1(A_X) be its spectral radius, and let pi(X) count the primes up to X. Then lambda_1(A_X)^2 = pi(X) + 4 sum_{p <= X} 1/(p-1) + o(1). More generally, fix a finite set S of primes and retain only integers not divisible by a prime in S. Let A_{S,X} be the resulting adjacency matrix, let pi_S(x) count primes up to x that do not belong to S, and let a_1 < a_2 < ... list the retained positive integers. For every fixed j, lambda_j(A_{S,X})^2 = pi_S(X/a_j) + 4 sum_{p <= X, p not in S} 1/(p-1) + c(a_j) + o_{S,j}(1), where c(a) is an explicit divisor correction. The proof splits the edges at the square-root prime threshold. The large-prime edges form disjoint stars, while the remaining adjacency has norm of order X^(1/4)/sqrt(log X) but zero first-order matrix element on each star eigenvector. This produces an explicit finite prime-sum correction with error O_{S,j}(1/log X), extreme-eigenvector localization, and centre-shell adjacency evolution. The prime number theorem and Mertens estimates identify the finite sum with 4 log log X plus an explicit constant. The same constant layer and localization hold uniformly for retained centres a <= (log X)^(1/2-epsilon). AI disclosure: Generative AI, including OpenAI Codex, assisted with mathematical exploration, drafting, finite diagnostics, and the Lean 4 formalization workflow. The author selected the theorem architecture, checked the mathematics, and accepts responsibility for the manuscript.

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