Copyright law focuses on whether a new work is "substantially similar" to an existing one, but generative AI can closely imitate style without copying content, a capability now central to ongoing litigation. We argue that existing definitions of infringement are ill-suited to this setting and propose a new criterion: a generative AI output infringes on an existing work if it could not have been generated without that work in its training corpus. To operationalize this definition, we model generative systems as closure operators mapping a corpus of existing works to an output of new works. AI generated outputs are \emph{permissible} if they do not infringe on any existing work according to our criterion. Our results characterize structural properties of permissible generation and reveal a sharp asymptotic dichotomy: when the process of organic creations is light-tailed, dependence on individual works eventually vanishes, so that regulation imposes no limits on AI generation; with heavy-tailed creations, regulation can be persistently constraining.

翻译：版权法关注新作品是否与现有作品“实质性相似”，但生成式人工智能能够在不复制内容的情况下高度模仿风格，这一能力已成为当前诉讼的核心焦点。我们认为现行侵权定义不适用于此情境，并提出新标准：若生成式人工智能输出在其训练语料库中未包含特定作品时无法生成，则该输出构成对现有作品的侵权。为实现该定义的可操作性，我们将生成系统建模为闭包算子，将现有作品语料库映射至新作品输出。根据我们的标准，若人工智能生成输出未侵犯任何现有作品，则视为**可允许**输出。我们的研究结果刻画了可允许生成的结构特性，并揭示了一种尖锐的渐近二分现象：当有机创作过程呈轻尾分布时，对个体作品的依赖性最终消失，监管对人工智能生成不构成限制；而在重尾创作分布下，监管可能产生持续性约束。