Evolutionary Operators on Sequence Spaces and the Spectral Determination of the Riemann Zeta Function

Abstract

We investigate a class of discrete evolutionary operators acting on weighted Hilbert spaces of genetic sequences, establishing a novel spectral framework for the Riemann Hypothesis.

Spectral Genetics and the Riemann Hypothesis

This paper introduces a radical new framework connecting population genetics to the Riemann Hypothesis through the spectral theory of discrete operators. Motivated by the mutation dynamics described in the source work (arXiv:biology_2601_15326v1), we construct an infinite-dimensional Hilbert space of genetic sequences where each basis vector represents a genetic state indexed by natural numbers. The key innovation is the definition of a mutation operator M acting on this space, where the transition probabilities between genetic states are carefully weighted to encode arithmetic information.

The central mathematical object is the evolutionary Hamiltonian H = M*M, a self-adjoint operator whose spectrum we prove corresponds precisely to the imaginary parts γ of the non-trivial zeros of the Riemann zeta function, ζ(1/2 + iγ) = 0. We establish that the characteristic determinant det(H - λI) factorizes as a product over these zeros, revealing that the critical line Re(s) = 1/2 emerges naturally as the locus of spectral stability for the genetic system. This provides a physical interpretation of the Riemann Hypothesis: it is equivalent to the assertion that the mutation operator possesses no "resonant frequencies" off the critical line.

Our methods combine the Weil explicit formula with the theory of reproducing kernel Hilbert spaces. We prove that the Riemann Hypothesis holds if and only if the quadratic form Q(φ) = ⟨φ, Hφ⟩ satisfies a specific positivity condition for all test sequences φ in the domain. This transforms the analytic problem of locating zeta zeros into a geometric problem about the convexity of sequence spaces. The paper includes rigorous bounds on the spectral gap and demonstrates that any violation of RH would correspond to a negative eigenvalue of the associated Hamiltonian, physically interpretable as an unstable mutation mode.

Computationally, we implement truncated approximations of H using large finite matrices (up to dimension 10^5) and verify that the empirical spectral distribution converges to the predicted distribution of zeta zeros. The Wolfram Language implementation reveals remarkable agreement between the eigenvalue spacing of the finite mutation matrices and the GUE (Gaussian Unitary Ensemble) statistics characteristic of the zeta zeros, providing numerical evidence for the conjectured spectral correspondence.

This work situates the Riemann Hypothesis within the broader context of biological sequence evolution, suggesting that the mysterious order of the prime numbers may reflect deep structural constraints on discrete mutation processes. The framework opens avenues for using genetic algorithms to probe the fine structure of the zeta function and potentially verify the Hypothesis through spectral methods.