# A geometric relation of the error introduced by sampling a language model's output distribution to its internal state

arXiv: https://arxiv.org/abs/2605.04899 · HTML: https://mlsanity.com/paper/2605.04899/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 25 references that are in the map.
- First posted on arXiv 6 May 2026.

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

- [The Curious Case of Neural Text Degeneration](https://mlsanity.com/paper/1904.09751/) (2019)
- [Root Mean Square Layer Normalization](https://mlsanity.com/paper/1910.07467/) (2019)
- [Fantastically Ordered Prompts and Where to Find Them: Overcoming Few-Shot Prompt Order Sensitivity](https://mlsanity.com/paper/2104.08786/) (2021)
- [Semantic Uncertainty: Linguistic Invariances for Uncertainty Estimation in Natural Language Generation](https://mlsanity.com/paper/2302.09664/) (2023)
- [Eliciting Latent Predictions from Transformers with the Tuned Lens](https://mlsanity.com/paper/2303.08112/) (2023)
- [Emergent World Representations: Exploring a Sequence Model Trained on a Synthetic Task](https://mlsanity.com/paper/2210.13382/) (2022)
- [Emergent Linear Representations in World Models of Self-Supervised Sequence Models](https://mlsanity.com/paper/2309.00941/) (2023)
- [Language Models Represent Space and Time](https://mlsanity.com/paper/2310.02207/) (2023)
- [How Language Model Hallucinations Can Snowball](https://mlsanity.com/paper/2305.13534/) (2023)
- [Chain-of-Thought Reasoning Without Prompting](https://mlsanity.com/paper/2402.10200/) (2024)
- [Chess as a Testbed for Language Model State Tracking](https://mlsanity.com/paper/2102.13249/) (2021)
- [Emergent World Models and Latent Variable Estimation in Chess-Playing Language Models](https://mlsanity.com/paper/2403.15498/) (2024)
- [What's the Magic Word? A Control Theory of LLM Prompting](https://mlsanity.com/paper/2310.04444/) (2023)
- [Linear Latent World Models in Simple Transformers: A Case Study on Othello-GPT](https://mlsanity.com/paper/2310.07582/) (2023)
- [The Curved Spacetime of Transformer Architectures](https://mlsanity.com/paper/2511.03060/) (2025)

## Abstract (arXiv)

GPT-style language models are sensitive to single-token changes at generation points where the predicted probability distribution is spread across multiple tokens. Viewing this sensitivity as a geometric property, we derive an mathfrakso(n)-valued 1-form that depends only on the geometry of the token embeddings. Despite this purely geometric origin, we show that its curvature is semantically meaningful: On chess reasoning tasks, the curvature couples to the world model of an off-the-shelf instruction-tuned model, with transformations clustering by board region and respecting piece importance. Our findings suggest that token space geometry directly reflects how models internally represent problems.