The Persistent Digit

Benford’s law predicts the leading-digit distribution of numbers drawn from many natural processes: 1 appears ~30% of the time, 9 about 5%. The convergence to this distribution is well-studied. What Hyman shows is that approximately 8% of integer bases never converge — and the structural cause is number-theoretic, not statistical.

The mechanism is continued fraction resonance. Each base b has a log₁₀(b) with a continued fraction expansion. Certain partial quotient patterns create persistent correlations in multi-digit sequences that resist the usual averaging. For these “persistent” bases, the convergence threshold exceeds 10⁶ samples at standard precision — effectively infinite for practical observation. The resonance ratio derived from the continued fraction classifies bases into convergent and persistent regimes.

The conjectured asymptotic persistence rate is 1/12, grounded in the Gauss-Kuzmin distribution of partial quotients. This is the distribution that governs how “typical” a continued fraction expansion is — and the persistent bases are exactly the atypical ones.

The structural finding: a universal statistical law (Benford) has base-dependent exceptions whose frequency is governed by a deeper universal distribution (Gauss-Kuzmin). The exceptions to one universality class are predicted by another universality class. The law’s failure is as lawful as the law itself.