The paper proposes a new definition for copyright infringement by AI: an output infringes if it couldn’t be generated without a specific work in the training data. This counterfactual criterion addresses AI’s ability to imitate style without copying content. Using mathematical models (closure operators), the authors characterize permissible generation sets and prove key properties: monotonicity (larger corpora expand permissible sets) and stability (combining permissible outputs yields only permissible outputs). The main theorem reveals an asymptotic dichotomy: with light-tailed innovation (gradual, normal distribution-like), the permissible ratio approaches one almost surely as corpora grow, meaning infringement vanishes. 

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