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.

翻译：我们提出了两种互补的生成机制，用以解释本福特定律（首位数字定律）在何种条件下及为何会出现。首先，概率图灵机（PTM）系综导出了编码长度的几何分布规律。在给定停机长度约束条件下最大化其熵值，即可推导出本福特统计特性。该模型揭示了关于停机概率的相变现象。其次，通过受约束的分割模型（爱因斯坦固体组合模型）在粗粒度熵率约束下，得到了与最大熵解相同的对数分布特征，从而阐明了非遍历性（系综平均与轨迹平均的差异）所起的作用。我们还通过数值实验验证了所得结论。