Benford's Law from Turing Ensembles and Integer Partitions

TL;DR

This paper introduces two probabilistic models based on Turing ensembles and integer partitions that explain the emergence of Benford's Law, highlighting phase transitions and the role of non-ergodicity.

Contribution

It presents novel generative mechanisms linking Turing ensembles and integer partitions to Benford's Law, with phase transition analysis and numerical validation.

Findings

Models produce Benford's Law under entropy constraints

Phase transition occurs with respect to halt probability

Numerical experiments support theoretical results

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.