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Why Eight Percent of Benford Sequences Never Converge

James M. Hyman

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.

Why Eight Percent of Benford Sequences Never Converge

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.
Paper Structure (46 sections, 17 theorems, 42 equations, 5 figures, 8 tables)

This paper contains 46 sections, 17 theorems, 42 equations, 5 figures, 8 tables.

Table of Contents

  1. Introduction
  2. The discovery
  3. Why this matters
  4. What Benford's Law does and does not guarantee
  5. Main contributions
  6. Connections to prior work
  7. Theoretical Framework
  8. Notation and definitions
  9. Information-theoretic preliminaries
  10. Digit representation via equidistribution
  11. Continued fractions and Diophantine approximation
  12. Cylinder sets and joint distributions
  13. Discrepancy and convergence
  14. Key technical lemma: CMI and discrepancy
  15. Main Results
  16. ...and 31 more sections

Key Result

Lemma 2.3

Let $b > 1$ and $\alpha = \log_{10}(b)$. For the sequence $a_n = b^n$, the significand is and the $k$-th significant digit is determined by the interval in $[0,1)$ containing $\{n\alpha\}$.

Figures (5)

  • Figure 1: Classification phase diagram in the $(Q^*, N)$ plane, where $Q^* = \max_k(a_{k+1} q_k)$ is the resonance parameter (Definition \ref{['def:resonance']}). Three regions emerge: convergent (green, $Q^* < N/\log N$, power-law decay), persistent (red, $Q^* > N\log N$, constant CMI), and transitional (yellow). Boundaries follow Theorem \ref{['thm:dichotomy']} and Observation \ref{['obs:transitional']}. For any practical $N \leq 10^5$, bases in the upper-left region never show convergent behavior.
  • Figure 2: Conditional mutual information $I(D_1; D_3 \mid D_2)$ versus sample size $N$ on logarithmic axes. Convergent sequences ($2^n$, $3^n$, $5^n$) show linear decay corresponding to power laws $\beta \approx 1.7$--$1.8$. The $7^n$ sequence (red diamonds) shows no decay, remaining near 0.68 bits across all sample sizes. Dashed lines show power-law fits.
  • Figure 3: Bimodal distribution of decay exponents across 996 integer bases. A dominant peak at $\beta \approx 1.72$ (774 convergent bases, 77.7%) and a secondary concentration below $\beta = 0.3$ (84 persistent bases, 8.4%). The near-absence of bases with $\beta \in [0.5, 1.3]$ supports the convergent-persistent dichotomy. Extended computation to $N = 200{,}000$ resolves most transitional cases (Table \ref{['tab:survey_summary']}).
  • Figure 4: Survey classification overview. (a) Distribution across categories at $N = 10{,}000$: convergent 77.7%, persistent 8.4%, transitional 13.9%; extended to $N = 200{,}000$: 92.4% and 5.3%. (b) Anomaly strength versus maximum partial quotient, confirming the mechanistic role of exceptional rational approximations.
  • Figure 5: The resonance parameter $Q^*$ separates convergent from persistent bases. Cumulative distribution of $Q^*$ across 996 bases. The empirical threshold $Q^*_{\mathrm{emp}} \approx 145{,}000$ achieves $>98\%$ classification accuracy: bases above it are overwhelmingly persistent (96.5%), while bases below are convergent or transitional.

Theorems & Definitions (41)

  • Definition 2.1: Significant digits
  • Definition 2.2: Benford distribution
  • Lemma 2.3: Digit-fractional part correspondence
  • proof
  • Theorem 2.4: Weyl, 1916
  • Theorem 2.5: Best approximation property
  • Definition 2.6: Irrationality type
  • Remark 2.7
  • Definition 2.8: Cylinder sets
  • Lemma 2.9: Cylinder set measures
  • ...and 31 more