# Finding Belief Geometries with Sparse Autoencoders

arXiv: https://arxiv.org/abs/2604.02685 · HTML: https://mlsanity.com/paper/2604.02685/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Sparse autoencoder training and evaluation, one of 13 directions in Mechanistic interpretability.
- Builds on 11 of its 24 references that are in the map.
- First posted on arXiv 3 Apr 2026.

## Placed in

- Mechanistic interpretability / [Sparse autoencoder training and evaluation](https://mlsanity.com/mechinterp/sparse-autoencoder-training-and-evaluation/)

## Builds on (most cited first, 11 of 11)

- [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)
- [Transformers learn in-context by gradient descent](https://mlsanity.com/paper/2212.07677/) (2022)
- [What learning algorithm is in-context learning? Investigations with linear models](https://mlsanity.com/paper/2211.15661/) (2022)
- [Language Models Represent Space and Time](https://mlsanity.com/paper/2310.02207/) (2023)
- [Transformers represent belief state geometry in their residual stream](https://mlsanity.com/paper/2405.15943/) (2024)
- [Language Models Use Trigonometry to Do Addition](https://mlsanity.com/paper/2502.00873/) (2025)
- [When Models Manipulate Manifolds: The Geometry of a Counting Task](https://mlsanity.com/paper/2601.04480/) (2026)
- [Constrained belief updates explain geometric structures in transformer representations](https://mlsanity.com/paper/2502.01954/) (2025)
- [Finding Archetypal Spaces Using Neural Networks](https://mlsanity.com/paper/1901.09078/) (2019)
- [Neural networks leverage nominally quantum and post-quantum representations](https://mlsanity.com/paper/2507.07432/) (2025)

## Built on by (most cited first, 2 of 2)

- [Sparse Autoencoders Encode Both Concepts and Functions: The Downstream Geometry of Feature Effects](https://mlsanity.com/paper/2607.24645/) (2026)
- [Small transformers track Bayesian evidence for latent common causes via a context-invariant mechanism](https://mlsanity.com/paper/2609.35161/) (2026)

## Abstract (arXiv)

Understanding the geometric structure of internal representations is a central goal of mechanistic interpretability. Prior work has shown that transformers trained on sequences generated by hidden Markov models encode probabilistic belief states as simplex-shaped geometries in their residual stream, with vertices corresponding to latent generative states. Whether large language models trained on naturalistic text develop analogous geometric representations remains an open question. We introduce a pipeline for discovering candidate simplex-structured subspaces in transformer representations, combining sparse autoencoders (SAEs), k-subspace clustering of SAE features, and simplex fitting using AANet. We validate the pipeline on a transformer trained on a multipartite hidden Markov model with known belief-state geometry. Applied to Gemma-2-9B, we identify 13 priority clusters exhibiting candidate simplex geometry (K ≥ 3). A key challenge is distinguishing genuine belief-state encoding from tiling artifacts: latents can span a simplex-shaped subspace without the mixture coordinates carrying predictive signal beyond any individual feature. We therefore adopt barycentric prediction as our primary discriminating test. Among the 13 priority clusters, 3 exhibit a highly significant advantage on near-vertex samples (Wilcoxon p < 10-14) and 4 on simplex-interior samples. Together 5 distinct real clusters pass at least one split, while no null cluster passes either. One cluster, 768596, additionally achieves the highest causal steering score in the dataset. This is the only case where passive prediction and active intervention converge. We present these findings as preliminary evidence that genuine belief-like geometry exists in Gemma-2-9B's representation space, and identify the structured evaluation that would be required to confirm this interpretation.