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Benford's law from Turing ensembles and integer partitions.

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We present two models explaining Benford's first-digit law. These generative mechanisms reveal how data naturally follows a logarithmic distribution under specific constraints, offering insights into data patterns.

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Area of Science:

  • Data Science
  • Statistical Analysis
  • Computational Mathematics

Background:

  • Benford's Law describes the common occurrence of first digits in numerical datasets.
  • Existing explanations for Benford's Law are diverse and sometimes lack unifying principles.

Purpose of the Study:

  • To develop novel generative mechanisms explaining the emergence of Benford's first-digit law.
  • To elucidate the conditions and reasons behind the prevalence of Benford's Law in data.

Main Methods:

  • Developed a probabilistic Turing machine (PTM) ensemble model.
  • Utilized constrained partition (Einstein-solid combinatorics) modeling.
  • Performed numerical experiments to validate theoretical findings.

Main Results:

  • The PTM ensemble model, under entropy maximization with a halting length constraint, yields Benford statistics.
  • A phase transition in Benford statistics was observed concerning halt probability.
  • The constrained partition model reproduces the logarithmic profile, clarifying nonergodicity's role.

Conclusions:

  • Two complementary mechanisms provide a comprehensive explanation for Benford's Law.
  • The findings highlight the interplay between entropy, constraints, and data distribution.
  • The study offers a robust theoretical framework for understanding first-digit phenomena.