用确定性结构解释数字首位分布，突破传统概率模型局限。
Structural Foundations for Leading Digit Laws: Beyond Probabilistic Mixtures
- 基于平移不变函数方程，推导出显式仿射加周期解
- 解释了素数、递推序列等确定性数据中的首位数字规律
- 适用于实证与数学数据，尤其擅长捕捉非对数分布的异常
本文提出一种现代确定性框架，用于研究数值数据中首位有效数字的分布。不同于传统的概率或混合模型，我们证明首位数字频率由数据生成过程的算术、算法和结构性质决定。核心是平移不变函数方程，其通解为显式的仿射加周期形式。该结构化表述能解释实证与数学数据中多样化的数字分布，包括明显偏离对数或尺度不变特征的情形。系统分析了有限与无限数据集中的数字分布，涵盖素数、递推关系等确定性序列，并揭示了块状结构与分形特征的出现。文章批判性考察了概率模型，给出明确例证与反例，讨论了局限性与开放问题。整体上，本工作建立了数字现象的统一数学基础，为应用与理论场景中的数字模式建模与分析提供通用工具集。
原文摘要 · Abstract (English)
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.
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