Contemporary studies in mechanistic interpretability have uncovered many puzzling phenomena in the neural information processing of Transformer-based language models, such as induction heads, function vectors, and the Hydra effect. Some of these individual phenomena have been independently tied to different data distributional properties, while some have been loosely associated with model architecture and how Transformers process information. However, a unified understanding of the relationship between data, model architecture, and optimization remains lacking, failing to answer the fundamental question: why do these three phenomena appear universally across different model families and scales, despite their seeming disconnect? In this work, we answer this question by unifying these three phenomena as consequences of hierarchical latent structures in the data generation process, coupled with decorrelated gradients across additive model components and directional concavity in the representation geometry. We validate our theoretical results in a toy model regime and in a large-scale synthetic data regime, comparing them with language models trained on natural language data.

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