We would first like to thank all three reviewers for their thorough, constructive and considered reviews. Appendix A, our model is a nonequilibrium variant of Derrida's Random Energy Model. We will update the final manuscript to describe this analogy more explicitly. As such, this is still a matter of active research. Conditions claimed in L181-184: We will amend the manuscript to indicate that the equation directly preceding eqn.

We introduce the attention-indexed model (AIM), a theoretical framework for analyzing learning in deep attention layers. Inspired by multi-index models, AIM captures how token-level outputs emerge from layered bilinear interactions over high-dimensional embeddings. Unlike prior tractable attention models, AIM allows full-width key and query matrices, aligning more closely with practical transformers. Using tools from statistical mechanics and random matrix theory, we derive closed-form predictions for Bayes-optimal generalization error and identify sharp phase transitions as a function of sample complexity, model width, and sequence length. We propose a matching approximate message passing algorithm and show that gradient descent can reach optimal performance. AIM offers a solvable playground for understanding learning in modern attention architectures.