Structural Foundations for Leading Digit Laws: Beyond Probabilistic Mixtures
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916906041081856 |
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| author | Berman, Vladimir |
| author_facet | Berman, Vladimir |
| contents | This article presents a modern deterministic framework for the study of leading significant digit distributions in numerical data. Rather than relying on traditional probabilistic or mixture-based explanations, we demonstrate that the observed frequencies of leading digits are determined by the underlying arithmetic, algorithmic, and structural properties of the data-generating process. Our approach centers on a shift-invariant functional equation, whose general solution is given by explicit affine-plus-periodic formulas. This structural formulation explains the diversity of digit distributions encountered in both empirical and mathematical datasets, including cases with pronounced deviations from logarithmic or scale-invariant profiles.
We systematically analyze digit distributions in finite and infinite datasets, address deterministic sequences such as prime numbers and recurrence relations, and highlight the emergence of block-structured and fractal features. The article provides critical examination of probabilistic models, explicit examples and counterexamples, and discusses limitations and open problems for further research. Overall, this work establishes a unified mathematical foundation for digital phenomena and offers a versatile toolset for modeling and analyzing digit patterns in applied and theoretical contexts. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_13237 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structural Foundations for Leading Digit Laws: Beyond Probabilistic Mixtures Berman, Vladimir Machine Learning Statistics Theory Methodology This article presents a modern deterministic framework for the study of leading significant digit distributions in numerical data. Rather than relying on traditional probabilistic or mixture-based explanations, we demonstrate that the observed frequencies of leading digits are determined by the underlying arithmetic, algorithmic, and structural properties of the data-generating process. Our approach centers on a shift-invariant functional equation, whose general solution is given by explicit affine-plus-periodic formulas. This structural formulation explains the diversity of digit distributions encountered in both empirical and mathematical datasets, including cases with pronounced deviations from logarithmic or scale-invariant profiles. We systematically analyze digit distributions in finite and infinite datasets, address deterministic sequences such as prime numbers and recurrence relations, and highlight the emergence of block-structured and fractal features. The article provides critical examination of probabilistic models, explicit examples and counterexamples, and discusses limitations and open problems for further research. Overall, this work establishes a unified mathematical foundation for digital phenomena and offers a versatile toolset for modeling and analyzing digit patterns in applied and theoretical contexts. |
| title | Structural Foundations for Leading Digit Laws: Beyond Probabilistic Mixtures |
| topic | Machine Learning Statistics Theory Methodology |
| url | https://arxiv.org/abs/2508.13237 |