# Invariant Atoms: Sparse Coordinates of Local Semantic Geometry in Language Model Representations

arXiv: https://arxiv.org/abs/2609.36451 · HTML: https://mlsanity.com/paper/2609.36451/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 17 references that are in the map.
- First posted on arXiv 29 Sep 2026.

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

- [Sparse Autoencoders Find Highly Interpretable Features in Language Models](https://mlsanity.com/paper/2309.08600/) (2023)
- [Inference-Time Intervention: Eliciting Truthful Answers from a Language Model](https://mlsanity.com/paper/2306.03341/) (2023)
- [The Linear Representation Hypothesis and the Geometry of Large Language Models](https://mlsanity.com/paper/2311.03658/) (2023)
- [Scaling and evaluating sparse autoencoders](https://mlsanity.com/paper/2406.04093/) (2024)
- [Null It Out: Guarding Protected Attributes by Iterative Nullspace Projection](https://mlsanity.com/paper/2004.07667/) (2020)
- [LEACE: Perfect linear concept erasure in closed form](https://mlsanity.com/paper/2306.03819/) (2023)
- [Scaling Monosemanticity: Extracting Interpretable Features from Claude 3 Sonnet](https://mlsanity.com/paper/2605.29358/) (2026)
- [The Geometry of Categorical and Hierarchical Concepts in Large Language Models](https://mlsanity.com/paper/2406.01506/) (2024)
- [Linear Adversarial Concept Erasure](https://mlsanity.com/paper/2201.12091/) (2022)
- [Sparse Autoencoders, Again?](https://mlsanity.com/paper/2506.04859/) (2025)
- [Invariant Features in Language Models: Geometric Characterization and Model Attribution](https://mlsanity.com/paper/2605.06458/) (2026)

## Abstract (arXiv)

Large language models often preserve meaning despite substantial changes in wording, style, and syntax, while small semantic edits can systematically alter their hidden representations. This suggests that semantic variation may be organized along recurring local directions. We propose the Invariant Atom Hypothesis: local semantic motion admits preferred sparse coordinates along directions that remain stable under meaning-preserving transformations. We learn a shared semantic frame and sparse coordinates that reconstruct semantic displacements while suppressing nuisance variation, with anchor-dependent diagonal modulation adjusting atom strengths without sample-specific rotations. Empirically, the atoms exhibit strong semantic--nuisance separation, sparse reconstruction, reproducible directions, and causal effects on model predictions. The learned geometry generalizes to unseen semantic neighborhoods and nuisance families, while local reweighting improves semantic selectivity and preserves a consistent global-to-local structure. Atom signatures also remain stable under model modification. These findings support reusable invariant directions as a sparse coordinate system for local semantic geometry in language models.