Spectral Geometry of the Primes

TL;DR

This paper constructs a family of operators on prime numbers to explore an emergent arithmetic geometry, revealing spectral properties that indicate a non-Euclidean, coherence-limited structure intrinsic to the distribution of primes.

Contribution

It introduces a novel spectral framework based on prime number divergences, connecting spectral geometry with number theory and revealing intrinsic constraints on prime distributions.

Findings

Eigenvalues grow sublinearly, indicating spectral compression.

Spectral dimension calculated as 1/2, showing non-classical diffusion.

Heat trace scales as t^{-1/4}, reflecting non-Euclidean geometry.

Abstract

We construct a family of self-adjoint operators on the prime numbers whose entries depend on pairwise arithmetic divergences, replacing geometric distance with number-theoretic dissimilarity. The resulting spectra encode how coherence propagates through the prime sequence and define an emergent arithmetic geometry. From these spectra we extract observables such as the heat trace, entropy, and eigenvalue growth, which reveal persistent spectral compression: eigenvalues grow sublinearly, entropy scales slowly, and the inferred dimension remains strictly below one. This rigidity appears across logarithmic, entropic, and fractal-type kernels, reflecting intrinsic arithmetic constraints. Analytically, we show that for the unnormalized Laplacian, the continuum limit of its squared Hamiltonian corresponds to the one-dimensional bi-Laplacian, whose heat trace follows a short-time scaling proportional to $t^{-1/4}$. Under the spectral dimension convention $d_s=-2\,d\log\Theta/d\log t$, this result produces $d_s = 1/2$ directly from first principles, without fitting or external hypotheses. This value signifies maximal spectral compression and the absence of classical diffusion, indicating that arithmetic sparsity enforces a coherence-limited, non-Euclidean geometry linking spectral and number-theoretic structure.