# A Geometric Perspective on Next-Token Prediction in Large Language Models: Three Emerging Phases

arXiv: https://arxiv.org/abs/2605.09011 · HTML: https://mlsanity.com/paper/2605.09011/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 21 references that are in the map.
- First posted on arXiv 9 May 2026.

## Builds on (most cited first, 15 of 15)

- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Locating and Editing Factual Associations in GPT](https://mlsanity.com/paper/2202.05262/) (2022)
- [Decoupled Weight Decay Regularization](https://mlsanity.com/paper/1711.05101/) (2017)
- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [Transformer Feed-Forward Layers Are Key-Value Memories](https://mlsanity.com/paper/2012.14913/) (2020)
- [The Linear Representation Hypothesis and the Geometry of Large Language Models](https://mlsanity.com/paper/2311.03658/) (2023)
- [Intrinsic Dimensionality Explains the Effectiveness of Language Model Fine-Tuning](https://mlsanity.com/paper/2012.13255/) (2020)
- [Eliciting Latent Predictions from Transformers with the Tuned Lens](https://mlsanity.com/paper/2303.08112/) (2023)
- [2 OLMo 2 Furious](https://mlsanity.com/paper/2501.00656/) (2024)
- [Emergent Linear Representations in World Models of Self-Supervised Sequence Models](https://mlsanity.com/paper/2309.00941/) (2023)
- [Patchscopes: A Unifying Framework for Inspecting Hidden Representations of Language Models](https://mlsanity.com/paper/2401.06102/) (2024)
- [Jump to Conclusions: Short-Cutting Transformers With Linear Transformations](https://mlsanity.com/paper/2303.09435/) (2023)
- [Transformers represent belief state geometry in their residual stream](https://mlsanity.com/paper/2405.15943/) (2024)
- [When Models Manipulate Manifolds: The Geometry of a Counting Task](https://mlsanity.com/paper/2601.04480/) (2026)
- [A multiscale analysis of mean-field transformers in the moderate interaction regime](https://mlsanity.com/paper/2509.25040/) (2025)

## Built on by (most cited first, 3 of 3)

- [Off-Axis, On Purpose: Where a Transformer Computes Concepts and Why it Does So](https://mlsanity.com/paper/2608.10251/) (2026)
- [The Depth Flow of Token Representations Is Nonlinear and Does Not Descend Its Own Density](https://mlsanity.com/paper/2608.29706/) (2026)
- [The Geometry of Inference in Transformer Residual Streams](https://mlsanity.com/paper/2609.37824/) (2026)

## Abstract (arXiv)

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.