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.

翻译：我们研究整数底数2 ≤ b ≤ 1000的本福特序列b^n中的多位数字相关性，通过条件互信息（CMI）度量依赖关系。基于log_10(b)连分数展开导出的共振比将底数分为收敛型与持久型两类（定理3.13）：在调查的996个底数中，84个（8.4%）在样本深度N = 10,000时呈现持久相关性，扩展计算至N = 200,000后确认其中53个（5.3%）为真正持久型。我们证明CMI偏差受分布误差界约束（定理3.4）；对2,988个测试案例的穷举计算证实有效缩放为二次型，对有界型底数给出双边速率β = 2（基于经计算验证的海森矩阵正定性条件）。774个收敛型底数的观测有效指数为β_eff = 1.72 ± 0.19，与渐近速率的有限样本修正一致。我们推测持久率收敛于1/12，该预测基于部分商的高斯-库兹明分布。对于持久型底数，收敛阈值N_ε在标准精度下超过10^6，使得渐近极限在我们的计算范围内观测上不可及。