Benford's law describes the distribution of leading digits in many naturally occurring datasets: 1 appears as the first digit about 30% of the time, 2 about 17%, down to 9 at about 4.6%. For sequences of powers (2^n, 3^n, ...), the leading digits follow Benford's distribution in the limit. But “in the limit” hides a question: how quickly do the multi-digit correlations between successive terms decay? If the first digit of 2^n is correlated with the first digit of 2^(n+1), do those correlations eventually vanish?
For about 8% of integer bases studied (bases 2 through 1000), they do not. The correlations persist indefinitely — or at least far beyond 10^6 terms. A resonance ratio derived from the continued fraction expansion of log(base) classifies which bases converge and which persist. The persistent bases are those where the continued fraction has specific structural features that prevent the multi-digit correlations from washing out. The asymptotic fraction of persistent bases is conjectured to approach 1/12, grounded in the Gauss-Kuzmin distribution — the same distribution that governs the statistics of continued fraction coefficients.
The connection to information theory is through conditional mutual information: the researchers measure how much knowing the current digit tells you about the next digit, conditioning on depth in the sequence. For convergent bases, this mutual information decays to zero. For persistent bases, it plateaus at a nonzero value. The digits remember.
The through-claim is about the boundary between ergodic and non-ergodic behavior hiding in elementary number theory. The same continued fraction that determines how well log(base) is approximated by rationals also determines whether the digit correlations in the Benford sequence ergodically mix. The number-theoretic obstruction (persistent resonance) manifests as an information-theoretic observable (nonzero mutual information). 1/12 of the number line, measured by Gauss-Kuzmin weight, is permanently non-mixing.
(arXiv:2603.18243)