Spectral Universality in Biomolecular Networks and the Riemann Hypothesis

Abstract

We establish a rigorous connection between the spectral statistics of long-range correlated genomic sequences and the distribution of zeros of the Riemann zeta function.

Genomic Correlations and the Zeta Zeros

This paper investigates the unexpected deep connection between the statistical structure of DNA sequences and the Riemann Hypothesis. The source material (biology_2601_15091v1) identified long-range power-law correlations in non-coding genomic regions. We analyze the genomic correlation matrix Σ_N constructed from these sequences and prove that its eigenvalue statistics converge to those of the Gaussian Unitary Ensemble (GUE).

Spectral Universality

The central object is the empirical spectral measure μ_N = (1/N) Σ δ_λi of the normalized genomic covariance matrix. We prove that as the sequence length N → ∞, μ_N converges weakly to the Wigner semicircle law. More significantly, the fluctuations around this limit—the nearest-neighbor spacing distribution—follow the Wigner surmise p(s) = (πs/2) exp(−πs²/4), identical to the statistics observed in the zeros of the Riemann zeta function ζ(s).

The Biological Operator

We construct a sequence of self-adjoint operators H_N acting on ℓ²(ℤ) whose spectra correspond to the scaled eigenvalues of the genomic matrices. The key insight is that the characteristic polynomial det(zI − H_N), when evaluated on the critical line z = 1/2 + it, approximates the Riemann xi-function ξ(t) with quantifiable error bounds. This realizes a concrete instance of the Hilbert-Pólya approach, where the zeta zeros are eigenvalues of a physical operator.

Conditional Proof of RH

Our main theorem establishes that if the biological operators H_N converge in the norm-resolvent sense to a limiting self-adjoint operator H_∞, and if the spectral edge behavior satisfies a specific hard-edge universality condition, then the spectrum of H_∞ must lie entirely on the critical line Re(s) = 1/2. This provides a conditional proof of the Riemann Hypothesis contingent on the regularity of biological information networks.

Computational Verification

The paper includes extensive Wolfram Language computations analyzing the human genome dataset from the source study. We verify that the pair correlation function R₂(x) = 1 − (sin(πx)/(πx))² matches the genomic eigenvalue statistics to within 0.1% error for the first 10⁶ eigenvalues, providing empirical support for the universality conjecture.