Title: Multiplicative Turing Ensembles, Pareto's Law, and Creativity Show Chinese title

Title: 乘法图灵集合，帕累托定律和创造力

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 tails for additive gaps. We also prove time-average laws for the Omega codelength along MTE trajectories. Empirically, on Debian and PyPI package size datasets, a scaled Omega prior achieves the lowest KL divergence against codelength histograms. 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. Show Chinese abstract

Abstract: 我们研究由独立同分布素数乘数驱动的整数值乘法动力学，并将其宏观统计特性与通用码长联系起来。我们引入了乘法图灵系综（MTE），并展示它如何自然地——尽管不唯一地——从概率图灵机的系综中出现。我们的建模原则是变分的：以埃利斯的Omega码长作为能量，并施加最大熵约束，可以得到整数上的规范吉布斯先验，并通过限制得到素数上的规范吉布斯先验。在温和的尾部假设下，这种先验会对数乘数产生指数尾部（加上缓慢变化的修正项），从而生成加性间隔的帕累托尾部。我们还证明了沿MTE轨迹的Omega码长的时间平均定律。经验上，在Debian和PyPI包大小数据集上，缩放后的Omega先验相对于码长直方图实现了最低的KL散度。综合来看，理论与数据的比较表明存在定性区分：机器适应的区域（吉布斯对齐、一阶矩有限）表现出清晰的平均行为，而人类生成的复杂性似乎超出这一区域，其尾部足够重以至于产生无界的首阶矩，因此不存在相同类型的整体平均。