Why Eight Percent of Benford Sequences Never Converge

TL;DR

This paper investigates the convergence behavior of multi-digit correlations in Benford sequences across various bases, identifying a subset of bases with persistent correlations and providing theoretical and computational insights into their convergence properties.

Contribution

It introduces a classification of bases into convergent and persistent regimes using a resonance ratio and proves bounds on mutual information deviation, supported by extensive computational validation.

Findings

8.4% of bases show persistent correlations at N=10,000

Effective scaling exponent beta_eff = 1.72 +/- 0.19 for convergent bases

Persistence rate likely converges to 1/12 based on Gauss-Kuzmin distribution

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.