# The Geometric Anatomy of Capability Acquisition in Transformers

arXiv: https://arxiv.org/abs/2602.15997 · HTML: https://mlsanity.com/paper/2602.15997/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Circuits and mechanistic analysis, one of 13 directions in Mechanistic interpretability.
- Builds on 13 of its 17 references that are in the map.
- First posted on arXiv 17 Feb 2026.

## Placed in

- Mechanistic interpretability / [Circuits and mechanistic analysis](https://mlsanity.com/mechinterp/circuits-and-mechanistic-analysis/)

## Builds on (most cited first, 13 of 13)

- [Emergent Abilities of Large Language Models](https://mlsanity.com/paper/2206.07682/) (2022)
- [Pythia: A Suite for Analyzing Large Language Models Across Training and Scaling](https://mlsanity.com/paper/2304.01373/) (2023)
- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [Are Emergent Abilities of Large Language Models a Mirage?](https://mlsanity.com/paper/2304.15004/) (2023)
- [An Investigation into Neural Net Optimization via Hessian Eigenvalue Density](https://mlsanity.com/paper/1901.10159/) (2019)
- [Empirical Analysis of the Hessian of Over-Parametrized Neural Networks](https://mlsanity.com/paper/1706.04454/) (2017)
- [RankMe: Assessing the downstream performance of pretrained self-supervised representations by their rank](https://mlsanity.com/paper/2210.02885/) (2022)
- [The Local Learning Coefficient: A Singularity-Aware Complexity Measure](https://mlsanity.com/paper/2308.12108/) (2023)
- [Tracing the Representation Geometry of Language Models from Pretraining to Post-training](https://mlsanity.com/paper/2509.23024/) (2025)
- [Neural Networks Learn Statistics of Increasing Complexity](https://mlsanity.com/paper/2402.04362/) (2024)
- [What can linear interpolation of neural network loss landscapes tell us?](https://mlsanity.com/paper/2106.16004/) (2021)
- [The Full Spectrum of Deepnet Hessians at Scale: Dynamics with SGD Training and Sample Size](https://mlsanity.com/paper/1811.07062/) (2018)
- [From Condensation to Rank Collapse: A Two-Stage Analysis of Transformer Training Dynamics](https://mlsanity.com/paper/2510.06954/) (2025)

## Built on by (most cited first, 1 of 1)

- [From Syntax to Semantics: Unveiling the Emergence of Chirality in SMILES Translation Models](https://mlsanity.com/paper/2605.09949/) (2026)

## Abstract (arXiv)

Neural networks gain capabilities during training, but the internal changes that precede capability acquisition are not well understood. In particular, the relationship between geometric change and behavioral change, and the effect of task difficulty and model scale on that relationship, is unclear. We track geometric measures and linear probes across six transformer sizes (405K--151M parameters), eight algorithmic tasks (144 task×level×model combinations), and three Pythia language models (160M--2.8B). Across all settings, representations first collapse to a low-dimensional state, then recover, and only then does behavioral performance improve. Linear probes show that the model's hidden states already contain task-relevant information before the model can act on it. The collapse floor is task-specific, the collapse propagates top-down through the network, and of the geometric measures tested, only rankme reliably precedes capability acquisition for hard tasks. Whether this precursor is detectable depends on task difficulty relative to model capacity. For hard tasks, there is a clear gap: geometry changes first, behavior follows. For easy tasks, the model learns so quickly that both happen simultaneously and no precursor is detectable. On Pythia-2.8B, a logical deduction task that is genuinely hard for the model shows a precursor gap of ~49K training steps, while easy benchmarks show none. This suggests that geometric patterns observed in small proxy models can persist at larger scale when the task remains difficult relative to model capacity.