Benford's law from Turing ensembles and integer partitions

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 nonergodicity (ensemble vs. trajectory averages). We also perform numerical experiments that corroborate our conclusions.