Why Eight Percent of Benford Sequences Never Converge
This addresses a theoretical problem in number theory and statistics regarding the convergence behavior of Benford sequences, with incremental computational verification of existing mathematical frameworks.
The paper investigates multi-digit correlations in Benford sequences for integer bases 2 to 1000, finding that 8.4% of bases exhibit persistent correlations at a sample depth of 10,000, with 5.3% confirmed as genuinely persistent at 200,000 samples. It proves that conditional mutual information deviation is bounded by distribution error and observes an effective scaling exponent of 1.72 ± 0.19 for convergent bases.
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.