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

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. If B_1 is the prime Meissel-Mertens constant and kappa = B_1 + sum_p 1/(p(p-1)), then lambda_1(A_X)^2 = pi(X) + 4 log log X + 4 kappa + 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 log log X + 4 kappa_S + c(a_j) + o_{S,j}(1), where kappa_S and c(a) are explicit prime and divisor corrections. 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 same constant layer and localization hold uniformly for retained centres a <= (log X)^(1/2-epsilon).