AI 中文总结
该研究针对齐普夫定律的等终性问题，在统一观测设计下比较四种生成过程，通过分析序列结构差异与拟合指数分布等，提出区分机制需依赖候选模型存在差异的观测。
AI 中文摘要
类齐普夫的频次-秩标度现象出现在语言、城市规模、生物数据及动物交流中。由于多个生成过程可产生相同的边缘模式，仅指数的机制内涵有限。我们将这种歧义表述为等终性问题，并在统一观测设计下比较四种构造：独立同分布有限齐普夫过程、持久马尔可夫链、典型样本空间约简（SSR）具有相同的平稳边缘分布p_j=(jH_V)⁻¹，而潜在尺度混合则通过聚合产生相似边缘分布。前三种构造在不改变总体秩分布的前提下，分离序列结构的差异。马尔可夫依赖改变拟合指数的有限样本分布；在中等持久度下，块调整有效样本量可紧密重现这种偏移。超额滞后1阶互信息可区分可交换过程与序列过程，转移方向可区分可逆持久度与SSR内置的定向收缩。对潜在尺度的条件分析揭示了聚合路径，而拟合窗口、字母表及序列边界分析则表明哪些结论依赖于观测设计。本文借鉴动物交流学习的最新研究来构建前瞻性测试，而非对模型进行经验验证。因此，匹配标度律是一种兼容性条件；区分机制需要候选模型存在差异的观测。
英文摘要
Scaling laws summarize complex systems through low-dimensional regularities, but the same marginal law can arise from different stochastic dynamics. We examine this ambiguity for Zipf rank--frequency scaling. An i.i.d. finite-Zipf process, a persistent Markov chain, and canonical sample-space reduction (SSR) are constructed to have exactly the same stationary marginal, $p_j=(jH_V)^{-1}$, while a latent-scale mixture produces a similar marginal through aggregation. The first three models therefore hold the population rank distribution fixed while changing temporal organization. Markov dependence shifts the finite-sample distribution of fitted exponents; at moderate persistence, a block-adjusted effective sample size reproduces most of this shift. Excess lag-1 mutual information detects serial dependence relative to a shuffle null, whereas transition direction separates reversible Markov persistence from the directional contraction of SSR. Conditioning on latent scale reveals the aggregation mechanism. Fit-window, alphabet, and sequence-boundary analyses quantify sensitivity to the observation design. These examples separate the population marginal, the finite-sample behavior of a fitted exponent, and temporal structure. Matching a scaling law is therefore a compatibility condition, not a mechanism identifier: discrimination requires observables on which candidate processes make different predictions.
