Revisiting Anisotropy in Language Transformers: The Geometry of Learning Dynamics

Understanding how transformer representations evolve across layers, not merely what they encode, remains an open problem in mechanistic interpretability. We recast the transformer forward pass as a discrete population trajectory through a high-dimensional representation manifold, drawing on geometric tools from computational neuroscience. Rather than probing for pre-specified features, we characterize trajectory geometry using five metrics computed directly in the ambient space: trajectory length, curvature, a semantic convergence index, layerwise cosine similarity, and representational stability. Across three model families (GPT-2, TinyLlama, Qwen2.5) and five controlled prompt families, we report four findings. First, semantically related prompts converge significantly in middle-to-late layers (peak CI 0.41--0.58, p<0.001, Mann-Whitney U), consistent with attractor-like dynamics. Second, reasoning tasks produce trajectories of greater curvature than lexical variations (0.71--0.83 rad vs. 0.27--0.31 rad), suggesting curvature encodes computational complexity. Third, ambiguous tokens exhibit trajectory bifurcation with up to 5.6x representational separation by the final layer, absent in unambiguous controls. Fourth, layerwise cosine similarity reveals a universal three-phase structure: encoding, elaboration, and output preparation, consistent across all three architectures. All four effects vanish under shuffled-layer and random-embedding controls. We release a fully open-source, model-agnostic pipeline and argue that trajectory geometry constitutes a principled, probe-free lens for mechanistic interpretability.

Pretraining and alignment update the same transformer weights, and we ask in which directions each stage changes them. The residual stream gives every matrix a shared coordinate system, in which a weight change can be compared with the principal subspace of the activations the matrix reads and with the prediction subspace of the unembedding. In this frame, alignment updates to the query and key matrices are enriched in the principal input subspace, while its updates to the output matrices are close to random relative to the prediction subspace, and the same signature appears in late pretraining and under every objective we apply to one checkpoint, so it is a property of the training stage rather than of the objective. The gradient of a linear map is a sum of outer products $δ_t a_t^\top$, and the residual stream enters this product through the activation $a_t$ for matrices that read from it and through the upstream gradient $δ_t$ for matrices that write into it. Read gradients therefore inherit the spectrum of the activations, which is the same under every objective, while write gradients inherit the geometry of the upstream gradients, which point into the prediction subspace only early in pretraining, when the output matrices acquire their prediction geometry. The raw read gradient follows the activation spectrum, the optimiser attenuates the concentration it inherits, and cancellation between tokens sets the scale of the update but not its direction. A write direction derived from the unembedding still changes predictions after alignment, and adaptation restricted to any single group of matrices remains effective, so the signature describes where training places its updates, not which matrices can carry them.

Current transformer language models are trained with uniform computational budgets across all layers, implicitly assuming layer homogeneity. We challenge this assumption through empirical analysis of SmolLM2-135M, a 30-layer, 135M-parameter causal language model, using five diagnostic metrics: weight predictability (R2), ablation degradation, recovery speed, weight manipulation robustness, and structural analysis. We find profound anatomical heterogeneity: (1) Layer weights follow strong mathematical regularity (R2 = 0.91) with a universal oscillatory delta pattern (correlation ~= -0.50), yet predicted weights cause catastrophic failure due to nonlinear error accumulation. (2) Layer importance spans a 10^7 range, from a critical core (L8-11, up to +63,419% PPL degradation) to anti-layers (L14, L17) whose removal improves performance. (3) Recovery speed correlates with layer importance, indicating differential training requirements. (4) Only weight scaling (alpha = 0.9) preserves model quality among five tested manipulation strategies. (5) Growth Transformer Training, allocating budget by layer importance, achieves ~54% cost reduction. A proof-of-concept experiment confirms this: 4.7x lower validation loss than uniform training at identical parameter count, while being 13% faster.