Spectral Geometry of the Primes

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 $d_s=-2\,d\logΘ/d\log t$, 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.

Figures (4)

• Figure 1: Coherence spectral profile $d_s(t)$ derived from the heat trace $\Theta(t)$ for various divergence models. All prime-based Hamiltonians ($\delta_{ij}^{(1)}$–$\delta_{ij}^{(4)}$) display sharp suppression and scale-dependent decay, whereas the GUE-based model shows a slower decline and higher plateau values, corresponding to weaker spectral compression. All curves deviate from classical Weyl scaling, illustrating persistent non-Euclidean spectral characteristics.
• Figure 2: Coherence spectral profile for $\boldsymbol{\delta_{ij}^{(3)}}$. The numerical $d_s(t)$ (solid) exhibits the characteristic PCP morphology—gradual activation, a single peak, and rapid decay—with no constant-dimension plateau; $d_s(t)$ returns toward $0$ for large $t$. A four-parameter $t/\tau$ model, $d_s(t)=A\,\!\left(\frac{t}{\tau}\right)^{\alpha} e^{-\left(t/\tau\right)^{\beta}}$, provides good qualitative agreement but does not capture all fine-scale features. A representative fit yields $(A,\alpha,\beta,\tau)=(10,\,3.0858,\,1.2644,\,2.14418)$ with $R^2_{\log}=0.4451$.
• Figure 3: Normalized comparison of spectral dimension profiles. All curves are scaled by the maximum of the prime--kernel $d_s(t)$ so that the prime peak equals 1. The arithmetic prime coherence kernel (blue) shows the characteristic sharp rise–peak–decay morphology. The 1D bi-Laplacian control (orange) is smoother and more plateau-like, while the GUE control (brown dashed) is flatter still. Although all three models share the same asymptotic spectral dimension ($d_s = 1/2$ for the bi-Laplacian order–4 operator), their intermediate-scale shapes differ qualitatively. The prime model’s compressed morphology thus remains distinctive of arithmetic structure.
• Figure 4: Coherence kernel $K_{ij} = \exp(-\delta_{ij}/\delta_0)$ for the first $N=20$ primes under two divergence functions: (a) logarithmic $\delta^{(2)}_{ij} = (\log(p_i/p_j))^2$ and (b) entropic $\delta^{(3)}_{ij} = \log((p_i + p_j)/(2\sqrt{p_i p_j}))$. Brightness indicates coherence strength between prime pairs. In both cases, coherence decays rapidly with arithmetic separation, while the entropic divergence produces slightly broader coupling.

Theorems & Definitions (12)

• Remark 2.1
• Conjecture 6.1: Prime–Zeta Spectral Correspondence
• Theorem 7.1: Tauberian dimension law
• proof
• Remark 7.2
• Theorem 7.3: Continuum limit yields $d_s=\tfrac{1}{2}$
• proof
• Remark 7.4
• Proposition 7.5: Uniform sub-Euclidean envelope for $H=L^2$
• proof