Prime-Star Localization and Second-Order Spectra of Finite Divisibility Cover Graphs
A mix of human-written and AI-generated textHuman understanding: all partsmath.CO — Combinatoricsmath.NT — Number Theory
Contributed by Idris Ali Shaik ↗
Version 1 / Oct 20, 2026 / CC BY 4.0
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 the positive integers divisible by none of the primes in S. Let a_1 < a_2 < ... list these integers, and let pi_S count the primes outside S. For every fixed j, the j-th largest positive eigenvalue satisfies
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) = sum_{q divides a, q prime} (q - 5)/(q - 1). The proof splits the edges according to whether their prime label exceeds sqrt(X). 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. An explicit first-exit resolvent calculation gives a degree-ratio correction with remainder O_{S,j}(1/log X) and localization of the extreme eigenvectors. The prime number theorem and Mertens estimates then evaluate that correction as 4 log log X plus an explicit constant, yielding the displayed o(1) formula. The same constant-layer formula and localization hold uniformly for retained centres a <= (log X)^(1/2-epsilon), with finite-sum error O_{S,epsilon}(a^2/log X). The paired positive and negative modes also yield uniform centre--shell adjacency evolution. The explicit finite correction determines its phase with uniform o(1) error whenever |t| = o(sqrt(X log X)/a^(5/2)). The uniform finite-prime-sum law, its largest-eigenvalue specialization, and the explicit-frequency centre--shell dynamics theorem are formalized in Lean 4 and registered in the Palomar Registry as PALOMAR-2026-08-29-000017, version 2.
Provenance statement
During preparation, the author used ChatGPT and OpenAI Codex for proof exploration, manuscript editing, diagnostic code, and Lean 4 formalization. The author reviewed and edited all outputs and takes responsibility for the content of the manuscript.
Formalizations
Tools used
- OpenAI
- ChatGPT
- OpenAI
- Codex
- Lean
- Lean
References
No bibliography entries.
Version history
- v1Submitted by Idris Ali ShaikInitial depositCurrentOct 18, 2026