Symmetry Reveals the In-Context Classifier: Transformers Implement Mean-Shift Dynamics

Transformers can perform in-context classification from a few labeled examples, yet the inference-time algorithm remains opaque. We study multi-class linear classification in the hard no-margin regime and make the computation identifiable by enforcing feature-and labelpermutation equivariance at every layer. This yields highly structured weights and enables interpretability while maintaining functional equivalence. From these models we extract an explicit depth-indexed recursion-an end-to-end identified, emergent update rule inside a softmax transformer, to our knowledge the first of its kind. Attention matrices formed from mixed feature-label Gram structure drive coupled updates of training points, labels, and the test probe. The resulting dynamics implement a geometry-driven algorithmic motif, which can provably amplify class separation and yields robust expected class alignment.