Disentangling Direction and Magnitude in Transformer Representations: A Double Dissociation Through L2-Matched Perturbation Analysis

Transformer hidden states encode information as high-dimensional vectors, yet whether direction (orientation in representational space) and magnitude (vector norm) serve distinct functional roles remains unclear. Studying Pythia-family models, we discover a striking cross-over dissociation: angular perturbations cause up to 42.9 more damage to language modeling loss, while magnitude perturbations cause disproportionately more damage to syntactic processing (20.4% vs.1.6% accuracy drop on subject-verb agreement).This finding is enabled by L2-matched perturbation analysis, a methodology ensuring that an gular and magnitude perturbations achieve identical Euclidean displacements. Causal intervention reveals that angular damage flows substantially through the attention pathways (28.4% loss recovery via attention repair), while magnitude damage flows partly through the LayerNorm pathways(29.9% recovery via LayerNorm repair). These patterns replicate across scales within the Pythia architecture family. These findings provide evidence that direction and magnitude support partially distinct computational roles in LayerNorm based architectures. The direction preferentially affects attentional routing, while magnitude modulates processing intensity for fine-grained syntactic judgments. We find different patterns in RMSNorm-based architectures, suggesting that the dissociation depends on architectural choices. Our results refine the linear representation hypothesis and have implications for model editing and interpretability research