Title:An Analysis of Residual-Stream Geometry Across Transformer Depth

Abstract:We propose a transition-centred geometric analysis of transformer residual streams. Relative displacement measures how \emph{far} representations move between consecutive layers, and orthogonal Procrustes analysis separates each transition into a rigid rotation and a non-rigid residual. Across six instruction-tuned models, on code generation and cross-lingual translation, these measurements reveal reproducible depth regularities. Relative displacement is strongly layer-dependent; typically larger early and late, with a quieter middle third; and nearly invariant across conditions within each model. Rotation magnitude is nearly constant across depth, while Procrustes residual and angle concentration remain depth-modulated, with residual peaking at the final transition. During generation, non-English targets show larger final-layer displacement and residual than English targets. We present these as descriptive geometric regularities, not as measures of computational effort or causal explanations. The contribution is a measurement framework for residual-stream transitions and evidence that, in the settings studied here, depth curves are model-dependent and largely condition-stable.

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