Benford's law --- the logarithmic distribution of leading digits --- is a convergence theorem: for most bases b, the sequence b^n converges to Benford's distribution as n grows. Hyman shows that 8% of integer bases between 2 and 1,000 never get there. Among 996 bases tested, 84 exhibit persistent multi-digit correlations at sample depth 10,000, with 53 confirmed persistent at depth 200,000. The mechanism is arithmetic. A resonance ratio derived from the continued fraction expansion of log_10(b) separates convergent bases from persistent ones. When the continued fraction has large partial quotients --- meaning log_10(b) is well-approximated by rationals --- the resulting near-commensurability creates correlations between successive digits that do not decay on any computationally accessible timescale. Convergence thresholds for persistent bases exceed 10^6, making the asymptotic limit observationally irrelevant.
The structural point is that Benford's law is not a single phenomenon but two: a convergent regime and a persistent regime, separated by a number-theoretic boundary. The Gauss-Kuzmin distribution over partial quotients predicts that the fraction of persistent bases converges to 1/12 --- close to the observed 8.4% --- grounding the empirical finding in the measure theory of continued fractions. The effective convergence exponent across the 774 convergent bases is beta_eff = 1.72, which is subquadratic. The quadratic scaling of conditional mutual information deviation, proved rigorously and confirmed across 2,988 test cases, means the convergence rate is itself a predictable function of the resonance ratio. The law has fine structure.
Asymptotic theorems guarantee behavior at infinity. Finite systems live in the transient. When the transient is long enough to exceed any practical observation window, the asymptotic result becomes a mathematical truth with no empirical content --- a theorem that is correct and irrelevant simultaneously. The resonance ratio identifies exactly which bases live in this gap. The general principle: for any convergence theorem, map the region where convergence is too slow to matter, because that region is where the system actually operates.
(arXiv:2603.18243)