Square-Root Collision Thresholds and Zero Statistics: A Gaussian Field Analogy for the Riemann Hypothesis

Abstract

This essay speculates that the phase transition observed in Linjun Li's Gaussian random-field model—where the total correlation of selected minima vanishes below a square-root budget and persists above it—mirrors the transition between Poisson and random-matrix statistics in the spacings of Riemann zeta zeros.

Overview

This essay examines a speculative bridge between the stochastic geometry of Gaussian random fields and the distribution of zeros of the Riemann zeta function. The source paper by Linjun Li (arXiv:2607.17522v1) analyzes a one-step selection process where the K smallest values of a correlated score field are selected, and a "total correlation" cost is assigned to the selected set. The paper establishes a sharp phase transition: when the budget K grows slower than the square root of the system size N, the dependence cost vanishes, but at the square-root scale, it persists with positive probability.

We propose that this "square-root collision threshold" structurally resembles the transition between Poisson (independent) and GUE (repulsive) statistics observed in the spacings of Riemann zeta zeros when sampled at different densities. In this analogy, the indices [N] correspond to an interval on the critical line, the score field corresponds to the logarithmic height |log ζ|, and the total correlation corresponds to the pairwise logarithmic energy of the Coulomb gas model from random matrix theory.

The analogy is classified as a suggestive metaphor. While both systems exhibit a phase transition in the "energy" of selected points at a critical density, the source model relies on short-range AR(1) correlations and Gaussianity, whereas the zeta function exhibits long-range logarithmic correlations and non-Gaussian tail behavior. The essay details these failure modes and proposes computational tests on the Riemann zeta function to assess whether extreme values exhibit a similar dependence threshold.

This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.