On the Recall Scaling Laws in Mamba: A Theoretical and Mechanistic Study via Hashing

Associative Recall (AR) is the cognitive ability to learn and retrieve links between items in memory. In NLP, AR is used as a benchmark for evaluating the in-context memory capacity of architectures such as Mamba, and has been found to strongly correlate with language modeling performance. This paper explores AR from the perspective of mechanistic interpretability, aiming to reverse-engineer the exact internal algorithm used by Mamba to perform recall. Our key insight is that Mamba performs recall by implicitly learning linear hash functions, and we identify the low-level circuit that enables this behavior. Building on these findings and inspired by theoretical tools in similarity-preserving hashing, such as the Johnson–Lindenstrauss (JL) lemma, we develop a theoretical framework for analyzing AR, which we term recall scaling laws. For example, given a model’s state, context length, and embedding dimensions, our theory predicts a lower bound on the largest vocabulary size that can be perfectly recalled in the AR task. Empirical results show that this bound is tight and predictive, offering insights into how AR capacity scales with vocabulary size, state size, embedding size, and model architecture.