L 2 M: Mutual Information Scaling Law for Long-Context Language Modeling

We present a universal theoretical framework for understanding *long-context language modeling* based on a *bipartite* mutual information scaling law that we rigorously verify in natural language. We demonstrate that bipartite mutual information captures multi-token interactions distinct from and scaling independently of conventional two-point mutual information, and show that this provides a more complete characterization of the dependencies needed for accurately modeling long sequences. Leveraging this scaling law, we formulate the **L**ong-context **L**anguage **M**odeling (**L**$^2$**M**) condition, which lower bounds the necessary scaling of a model's history state--the latent variables responsible for storing past information--for effective long-context modeling.