Core-Halo Complexity Maximizers and the Statistics of Zeta Zero Spacings: A Speculative Computational Bridge

Abstract

This essay proposes a speculative bridge between the variational theory of statistical complexity maximizers developed by Sharma and the distribution of spacings between consecutive zeros of the Riemann zeta function.

Source Paper: Complexity Maximization

Sharma (2026) develops a unified variational framework for maximizing generalized statistical complexity. The complexity functional combines Shannon entropy with Rényi-based disequilibrium. The central result is the "two-root theorem": every stationary solution of the variational problem possesses exactly two probability levels, creating a universal core–halo structure. The global maximum is attained by the smallest admissible core.

Target Domain: Zeta Zero Spacings

The Riemann Hypothesis predicts that non-trivial zeros of the zeta function lie on the critical line Re(s) = 1/2. The GUE conjecture posits that the normalized spacings between these zeros follow the statistics of eigenvalues from the Gaussian Unitary Ensemble from random matrix theory.

Proposed Analogy

We speculate that the core–halo structure of complexity maximizers might correspond to the two-scale behavior (short-range repulsion vs. long-range randomness) of zero spacings. We propose treating the empirical histogram of zero spacings as a discrete probability distribution and computing its Sharma complexity. This is rated as a suggestive metaphor; the analogy is weak because the source yields piecewise-constant two-level distributions while the GUE limit is smooth, and because the source framework lacks the Euler product and functional equation central to zeta function theory.

Proposed Experiment

The essay outlines (but does not execute) a Wolfram Language experiment to bin the normalized spacings of zeta zeros, compute the generalized complexity, and compare it to the theoretical two-level maximum. If the zero spacings exhibit near-maximal complexity, this would support the metaphor; if they exhibit minimal complexity (approaching Poisson), the metaphor fails.

This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.