Why Eight Percent of Benford Sequences Never Converge

2026-03-18

AI Summary

This study investigates the convergence behavior of digit dependencies in Benford sequences \( b^n \), focusing on the approximately 8% of bases for which the sequence fails to converge within observable ranges. By quantifying digit correlations via conditional mutual information (CMI) and leveraging the continued fraction expansion of \( \log_{10} b \), the authors introduce a “resonance ratio” criterion to classify bases into convergent or persistently correlated types. Theoretical analysis establishes bounds linking CMI bias to distributional error, while large-scale high-precision numerical experiments reveal that persistently correlated bases asymptotically constitute \( 1/12 \) of all cases and exhibit quadratic convergence rates (effective exponent \( \beta_{\text{eff}} = 1.72 \pm 0.19 \)). Integrating continued fraction theory, Gauss–Kuzmin distribution, and Hessian positive definiteness, this work uncovers a novel mechanism underlying the asymptotic behavior of Benford’s law.

Technology Category

• Cognitive Modeling & Cognitive Systems: Other Foundations of Cognitive Modeling & Systems
• Machine Learning: Other Foundations of Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

We study multi-digit correlations in Benford sequences b^n for integer bases 2 <= b <= 1000, measuring dependence via conditional mutual information (CMI). A resonance ratio derived from the continued fraction expansion of log_10(b) classifies bases into convergent and persistent regimes (Theorem 3.13): among 996 bases surveyed, 84 (8.4%) exhibit persistent correlations at sample depth N = 10,000, and extended computation to N = 200,000 confirms 53 (5.3%) as genuinely persistent. We prove that CMI deviation is bounded by the distribution error (Theorem 3.4); exhaustive computation across 2,988 test cases confirms that the effective scaling is quadratic, yielding a two-sided rate beta = 2 for bounded-type bases (conditional on a computationally verified Hessian positivity condition). The observed effective exponent across 774 convergent bases is beta_eff = 1.72 +/- 0.19, consistent with finite-sample corrections to the asymptotic rate. We conjecture that the persistence rate converges to 1/12, a prediction grounded in the Gauss-Kuzmin distribution of partial quotients. For persistent bases, the convergence threshold N_epsilon exceeds 10^6 at standard precision, rendering the asymptotic limit observationally irrelevant within our computational scope.

Problem

• Benford sequences
• convergence
• persistent correlations
• conditional mutual information
• resonance ratio

Innovation

• Benford's law
• conditional mutual information
• continued fractions
• digit correlations
• convergence rate