# Geometry of Ordinal Representations in Language Models

arXiv: https://arxiv.org/abs/2607.04167 · HTML: https://mlsanity.com/paper/2607.04167/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 26 references that are in the map.
- First posted on arXiv 5 Jul 2026.

## Builds on (most cited first, 20 of 20)

- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Gemma 2: Improving Open Language Models at a Practical Size](https://mlsanity.com/paper/2408.00118/) (2024)
- [Sparse Autoencoders Find Highly Interpretable Features in Language Models](https://mlsanity.com/paper/2309.08600/) (2023)
- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [The Linear Representation Hypothesis and the Geometry of Large Language Models](https://mlsanity.com/paper/2311.03658/) (2023)
- [The Geometry of Truth: Emergent Linear Structure in Large Language Model Representations of True/False Datasets](https://mlsanity.com/paper/2310.06824/) (2023)
- [Eliciting Latent Predictions from Transformers with the Tuned Lens](https://mlsanity.com/paper/2303.08112/) (2023)
- [Gemma Scope: Open Sparse Autoencoders Everywhere All At Once on Gemma 2](https://mlsanity.com/paper/2408.05147/) (2024)
- [Towards Best Practices of Activation Patching in Language Models: Metrics and Methods](https://mlsanity.com/paper/2309.16042/) (2023)
- [Language Models Represent Space and Time](https://mlsanity.com/paper/2310.02207/) (2023)
- [Not All Language Model Features Are One-Dimensionally Linear](https://mlsanity.com/paper/2405.14860/) (2024)
- [Scaling Monosemanticity: Extracting Interpretable Features from Claude 3 Sonnet](https://mlsanity.com/paper/2605.29358/) (2026)
- [Is This the Subspace You Are Looking for? An Interpretability Illusion for Subspace Activation Patching](https://mlsanity.com/paper/2311.17030/) (2023)
- [Transformers represent belief state geometry in their residual stream](https://mlsanity.com/paper/2405.15943/) (2024)
- [The Origins of Representation Manifolds in Large Language Models](https://mlsanity.com/paper/2505.18235/) (2025)
- [When Models Manipulate Manifolds: The Geometry of a Counting Task](https://mlsanity.com/paper/2601.04480/) (2026)
- [Priors in Time: Missing Inductive Biases for Language Model Interpretability](https://mlsanity.com/paper/2511.01836/) (2025)
- [Understanding sparse autoencoder scaling in the presence of feature manifolds](https://mlsanity.com/paper/2509.02565/) (2025)
- [Mechanistic Indicators of Understanding in Large Language Models](https://mlsanity.com/paper/2507.08017/) (2025)

## Abstract (arXiv)

Recent work showed that language models represent character counts on curved 1D manifolds, with attention heads performing geometric transformations to enable computation. We test whether this generalizes across four ordinal tasks (bracket depth, indentation, table position, numeric magnitude) in Gemma-2-2B, Gemma-2-9B, and Qwen3-4B. We find that 1D manifolds with place-cell feature tiling emerge for tasks where the ordinal variable is locally computable from token identity, while tasks requiring cross-position integration or semantic extraction produce higher-dimensional or incoherent representations. Geometric computation is architecture-dependent: Qwen3-4B shows substantially stronger twisting than Gemma models for indentation, and its twisters preserve ordinal order, unlike its numeric twisters. Activation patching confirms that the identified manifold subspaces concentrate task-relevant information, with manifold-direction ablation causing dramatically larger probe accuracy drops than random-direction controls.