Abstract:We study integer-valued multiplicative dynamics driven by i.i.d. prime multipliers and connect their macroscopic statistics to universal codelengths. We introduce the Multiplicative Turing Ensemble (MTE) and show how it arises naturally -- though not uniquely -- from ensembles of probabilistic Turing machines. Our modeling principle is variational: taking Elias' Omega codelength as an energy and imposing maximum entropy constraints yields a canonical Gibbs prior on integers and, by restriction, on primes. Under mild tail assumptions, this prior induces exponential tails for log-multipliers (up to slowly varying corrections), which in turn generate Pareto-type tails for additive gaps, with the survival exponent shifted by summation over primes. We also prove time-average laws for the Omega codelength along MTE trajectories. Empirically, Debian, PyPI, and CRAN package-size histograms have fitted Omega slopes well below the pure-Omega value $\log 2$, indicating heavier-than-pure-Omega tails within this energy scale. Taken together, the theory--data comparison suggests a qualitative split: machine-adapted regimes (Gibbs-aligned, finite first moment) exhibit clean averaging behavior, whereas human-generated complexity appears to sit beyond this regime, with tails heavy enough to produce an unbounded first moment, and therefore no averaging of the same kind.
From: Alexander Kolpakov [view email]
[v1]
Sun, 5 Oct 2025 12:04:50 UTC (19 KB)
[v2]
Thu, 16 Oct 2025 22:19:43 UTC (168 KB)
[v3]
Wed, 22 Jul 2026 21:03:07 UTC (663 KB)
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