An Analysis of Residual-Stream Geometry Across Transformer Depth

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.

From Direction to Magnitude: How Multimodal Instruction-Tuning Reorganizes the Geometric Encoding of Identity-Specifying Prompts in Transformer Hidden States

We investigate whether identity-specifying system prompts produce statistically distinguishable geometric fingerprints in the hidden-state trajectories of four open-weight transformer language models spanning four post-training regimes: no training (Gemma-4-E4B base), multimodal RLHF (Gemma-4-E4B-it), RL distillation (DeepSeek-R1-Distill-Qwen-7B), and SFT (Qwen2.5-7B-Instruct). Three prompt conditions (an identity-specifying axis prompt, a length-matched generic-assistant prompt, and a 26-token vanilla baseline) are compared via five geometric metrics, principally the 1-Wasserstein distance between edge-wise distributions of Ollivier-Ricci curvature on k-NN trajectory graphs. Claims rest on trajectory-level permutation tests with multiple geometric controls (teacher-forced content controls, temporal-chain vs k-NN topology, ABT-projected k-NN, angular vs Euclidean graph construction, B=5000 permutations on borderline statistics). The central finding is a qualitative reorganization of identity encoding across the instruction-tuning boundary: in the base model the fingerprint is direction-coded (separation 0.034, p=0.002 under angular k-NN); in the multimodal instruction-tuned model it migrates into the magnitude (angular separation collapses to p=0.439 while Euclidean survives at p=0.042, and the mean norm of the first generated state inverts its length-ordering, being lowest for the identity prompt). This direction-to-magnitude reorganization is specific to the multimodal instruction-tuning regime, absent under RL distillation and SFT. A teacher-forced control attributes ~30% of the free-running cosine signal to prompt-driven effects. We position W_1 on edge-wise Ollivier-Ricci distributions on k-NN trajectory graphs as a methodological contribution of independent interest.

What Does Debiasing Really Remove? A Geometric Study of PCA-Based Gender Debiasing in Word Embeddings

Debiasing methods based on principal component analysis (PCA) are broadly used to reduce gender bias in word embeddings used in LLMs, yet it remains unclear what aspects of bias they actually remove and how destructive this process is. These methods are based on the understanding that bias resides in a low-dimensional subspace, with the assumption that most of it can be captured by a few principal components. In this work, we conduct a systematic geometric analysis of PCA-based gender debiasing and investigate what is actually removed from the embedding space. Our experiments across multiple embeddings show that direct gender bias is primarily concentrated in the first principal component, supporting the low-rank bias hypothesis. However, associative bias measured by WEAT does not align with these principal directions and is instead spread across multiple embedding dimensions. Furthermore, as expected, we demonstrate that removing an increasing number of principal components leads to a consistent degradation of the embedding geometry, affecting semantic structure and vector relationships. These results reveal that PCA-based debiasing operates as a trade-off: while it effectively reduces certain forms of direct bias, it fails to eliminate distributed associations and introduces geometric distortion. Moreover, there is no universal optimal level of debiasing, as the balance between bias reduction and semantic preservation depends on the chosen metric and embedding. Overall, our findings suggest that bias in word embeddings is not purely low-rank and that simple subspace removal methods may be insufficient for comprehensive debiasing.

A geometric relation of the error introduced by sampling a language model's output distribution to its internal state

GPT-style language models are sensitive to single-token changes at generation points where the predicted probability distribution is spread across multiple tokens. Viewing this sensitivity as a geometric property, we derive an $\mathfrak{so}(n)$-valued 1-form that depends only on the geometry of the token embeddings. Despite this purely geometric origin, we show that its curvature is semantically meaningful: On chess reasoning tasks, the curvature couples to the world model of an off-the-shelf instruction-tuned model, with transformations clustering by board region and respecting piece importance. Our findings suggest that token space geometry directly reflects how models internally represent problems.