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 label-permutation equivariance at every layer. This enables interpretability while maintaining functional equivalence and yields highly structured weights. From these models we extract an explicit depth-indexed recursion---an end-to-end identified, emergent update rule inside a standard 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 label-aware mean-shift, which provably amplifies class separation and yields robust expected class alignment.