Dynamics of the Transformer Residual Stream: Coupling Spectral Geometry to Network Topology

Recent methods in language model interpretability employ techniques such as sparse autoencoders to decompose residual stream contributions into linear, semantically meaningful features. Such methods are commonly interpreted as identifying features that persist in the residual stream and that subsequent layers build upon. We challenge this view by identifying the Transformer Layer Correction Mechanism (TLCM), wherein adjacent transformer layers systematically counteract portions of each other's contributions. TLCM appears in 5 out of 7 major open-source model families and activates across nearly all tokens in diverse texts. We show that TLCM emerges during pretraining, operates most strongly on contextually dependent tokens, and adaptively calibrates its correction strength based on the preceding layer's output. Using the layer Jacobian, we further show that TLCM selectively corrects specific subspaces while reinforcing others, which we interpret through a ``propose-and-reject'' framework in which layers propose candidate features and subsequent layers selectively remove inappropriate ones. This dynamic suggests that the residual stream at any layer contains transient proposals alongside persistent features, helping explain why SAE feature descriptions often have low specificity, why effective model steering requires extreme feature amplification, and why transcoders hold a theoretical advantage over SAEs.

Training loss and throughput can hide distinct internal representation in language-model training. To examine these hidden mechanics, we use spectral measurements as practical and operational diagnostics. Using a controlled family of decoder-only models adapted from the modded NanoGPT codebase, we introduce an empirical protocol based on activation covariance and per-sample gradient SVD spectra. This dual-view reveals three empirical findings and one mechanistic explanation. First, batch size acts as a latent determinant of representation geometry: runs that reach equal loss settle into systematically distinct activation spectra. Second, the activation covariance tail measured early in training reliably forecasts downstream token efficiency. Third, movement of the activation spectrum head (leading modes), together with gradient spectra, characterizes underlying learning-dynamics changes, separating learning-side architectural improvements from primarily execution-side gains. These predictive and diagnostic signals persist across the 12-, 36-, and 48-layer model tiers. Finally, a mechanistic model proves the main observations and explains how activation covariance spectra correlate with task-aligned feature learning.

We investigate the geometry of predictive information across the layers of large language models (LLMs). We repurpose representation lenses-learned affine maps trained to predict the next token from intermediate residual streams-as geometric diagnostic tools. Rather than asking what the model predicts at each layer, we ask where predictive information resides and how it evolves across depth. We define at each layer a predictive readout subspace as the dominant k-dimensional singular subspace of such a map on the d-dimensional residual stream (where k is a resolution parameter), and track its trajectory on the Grassmann manifold as a similarity profile across layers. The profile is well described by unimodal distributions exhibiting a rise, near-plateau, and descent; varying k from 1% to 50% of d traces a Pareto frontier between visibility and energy retention, yet the same structure emerges at all scales. Across eight models from two families (Qwen2.5 and OLMo2, 1B-32B), we identify three geometric phases. Updates are approximately orthogonal to the residual stream throughout; what distinguishes the phases is their effect on the effective rank, which expands, stabilizes, and concentrates. In the first, Seeding Multiplexing, feed-forward memories and attention layers seed a candidate set in superposition in family-specific proportions, with the final token rising as leading candidate from 20% to 35% of positions across this phase. In the second, Hoisting Overriding, updates override existing subspaces to concentrate the candidate distribution without expanding the rank. In the third, Focal Convergence, high-energy low-rank updates write the winner into a form aligned with the unembedding direction. Phases 1 and 3 grow slowly with model depth, while Phase 2 expands linearly. The additional capacity of deeper LLMs is largely absorbed by candidate disambiguation.