Sufficient Representations and the GUE Conjecture: A Semi-Supervised Analogy for Zero Statistics

Abstract

We explore a structural analogy between the semi-supervised generative model RepG, which uses sufficient representations to factorize conditional distributions, and the statistical distribution of zeros of the Riemann zeta function and L-functions.

Overview

This essay mines the machine learning paper Semi-Supervised Conditional Generative Learning through Stochastic Interpolation and Sufficient Representations (arXiv:2607.16725v1) for patterns relevant to the Riemann Hypothesis. The source paper proposes RepG, a two-stage generative model that decomposes the task of sampling from a high-dimensional conditional distribution P_X|Y into a low-dimensional latent stage Y → R and a reconstruction stage R → X. The key structural property is the conditional independence X ⊥⊥ Y | R, which renders R a sufficient representation. The authors prove that the Kullback-Leibler divergence of the generative model decomposes into stage-wise errors plus a structural bias equal to the conditional mutual information I(X; Y | R).

The Analogy

We map this structure onto the statistics of zeta zeros. In random matrix theory, the GUE conjecture posits that the fine-scale spacing statistics of high zeros are universal—they depend only on the symmetry type (unitary, orthogonal, or symplectic) and not on the specific arithmetic of the L-function. This suggests a Markov chain: Arithmetic → Symmetry Type → Zero Statistics. The symmetry type acts as a sufficient representation: once it is fixed, the specific coefficients of the L-function provide no additional information about the spacing distribution in the bulk.

The conditional mutual information I(Zero Statistics; Arithmetic | Symmetry Type) thus measures the deviation from universality. The source paper’s error bound suggests that learning zero distributions is statistically efficient when one first classifies the symmetry type (low-dimensional, few labeled examples) and then learns the universal spacing distribution (high-dimensional, many unlabeled examples pooled across families).

Assessment

We rate this a formal analogy: the mathematical structures (conditional independence, mutual information decomposition) have identical shapes, but the axioms differ significantly. The source paper assumes Euclidean spaces and deterministic representations, whereas zero statistics form point processes with asymptotic (not exact) universality. The essay proposes computational experiments to test whether the conditional mutual information indeed vanishes for high zeros, but notes critical failure modes regarding the lack of Euler product structure and the asymptotic nature of the correspondence.

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