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arXiv · 2502.16314

Benford's Law from Turing Ensembles and Integer Partitions

Alexander Kolpakov·Aidan Rocke
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Abstract

We develop two complementary generative mechanisms that explain when and why Benford's first-digit law arises. First, a probabilistic Turing machine (PTM) ensemble induces a geometric law for codelength. Maximizing its entropy under a constraint on halting length yields Benford statistics. This model shows a phase transition with respect to the halt probability. Second, a constrained partition model (Einstein-solid combinatorics) recovers the same logarithmic profile as the maximum-entropy solution under a coarse-grained entropy-rate constraint, clarifying the role of non-ergodicity (ensemble vs. trajectory averages). We also perform numerical experiments that corroborate our conclusions.

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Alexander Kolpakov, Aidan Rocke. 2025-02-22. Benford's Law from Turing Ensembles and Integer Partitions. https://doi.org/10.1103/xjlr-sg7r

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