# Discovering Cross-Language Reasoning Invariance in LLMs with Geometry-Invariant Sparse Autoencoders

arXiv: https://arxiv.org/abs/2608.23809 · HTML: https://mlsanity.com/paper/2608.23809/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Reasoning benchmarks and evaluation, one of 17 directions in LLM reasoning.
- Builds on 21 of its 27 references that are in the map.
- First posted on arXiv 24 Aug 2026.

## Placed in

- LLM reasoning / [Reasoning benchmarks and evaluation](https://mlsanity.com/lr/reasoning-benchmarks-and-evaluation/)
- Mechanistic interpretability / [Steering and interpreting LLM reasoning](https://mlsanity.com/mechinterp/steering-and-interpreting-llm-reasoning/)

## Builds on (most cited first, 21 of 21)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [The Curious Case of Neural Text Degeneration](https://mlsanity.com/paper/1904.09751/) (2019)
- [Locating and Editing Factual Associations in GPT](https://mlsanity.com/paper/2202.05262/) (2022)
- [Phi-3 Technical Report: A Highly Capable Language Model Locally on Your Phone](https://mlsanity.com/paper/2404.14219/) (2024)
- [Gemma 3 Technical Report](https://mlsanity.com/paper/2503.19786/) (2025)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [Sparse Autoencoders Find Highly Interpretable Features in Language Models](https://mlsanity.com/paper/2309.08600/) (2023)
- [Towards Automated Circuit Discovery for Mechanistic Interpretability](https://mlsanity.com/paper/2304.14997/) (2023)
- [Scaling and evaluating sparse autoencoders](https://mlsanity.com/paper/2406.04093/) (2024)
- [Language Models are Multilingual Chain-of-Thought Reasoners](https://mlsanity.com/paper/2210.03057/) (2022)
- [Do Llamas Work in English? On the Latent Language of Multilingual Transformers](https://mlsanity.com/paper/2402.10588/) (2024)
- [Language-Specific Neurons: The Key to Multilingual Capabilities in Large Language Models](https://mlsanity.com/paper/2402.16438/) (2024)
- [Reliability of CKA as a Similarity Measure in Deep Learning](https://mlsanity.com/paper/2210.16156/) (2022)
- [On the Multilingual Ability of Decoder-based Pre-trained Language Models: Finding and Controlling Language-Specific Neurons](https://mlsanity.com/paper/2404.02431/) (2024)
- [Large Language Models Share Representations of Latent Grammatical Concepts Across Typologically Diverse Languages](https://mlsanity.com/paper/2501.06346/) (2025)
- [Beneath the Surface of Consistency: Exploring Cross-lingual Knowledge Representation Sharing in LLMs](https://mlsanity.com/paper/2408.10646/) (2024)
- [Large Language Models Are Cross-Lingual Knowledge-Free Reasoners](https://mlsanity.com/paper/2406.16655/) (2024)
- [Unveiling Language-Specific Features in Large Language Models via Sparse Autoencoders](https://mlsanity.com/paper/2505.05111/) (2025)
- [The Transfer Neurons Hypothesis: An Underlying Mechanism for Language Latent Space Transitions in Multilingual LLMs](https://mlsanity.com/paper/2509.17030/) (2025)

## Abstract (arXiv)

Multilingual language models can solve the same mathematical problem in different languages, but it remains unclear whether they rely on shared features or on language-specific computations that only produce similar outputs. We study this question in five models from four families using the Multilingual Grade School Math (MGSM) dataset, with problems solved in English, German, French, Spanish, Russian, and Chinese, retaining problems with valid reasoning traces in all six languages and replaying those traces through the model to record representations at multiple layers. For each model, we first use Centered Kernel Alignment (CKA) to identify layers with cross-language alignment. At each selected layer, we train two sparse autoencoders (SAE): a baseline reconstruction-only model and a contrastive variant introduced in this work, the Geometry-Invariant SAE (GI-SAE). GI-SAE supplements the reconstruction loss with an Information Noise-Contrastive Estimation (InfoNCE) loss that trains the encoder to produce similar activations for traces of the same problem, regardless of language or token position. We then test whether the resulting shared features are functionally interchangeable by swapping their values between languages during the model's forward pass and measuring the resulting change in output, quantified by Kullback-Leibler (KL) divergence per feature. Although GI-SAE yields higher CKA and Jaccard similarity at nearly every layer, higher geometric similarity does not consistently imply greater functional interchangeability. We find that cross-language feature sharing is model- and architecture-dependent in this sample and appears at different depths in different models. GI-SAE primarily amplifies cross-language structure already present: the pattern is model-specific, with strengthening in Qwen, no functional benefit in Gemma, and mixed layer-dependent effects in Llama and Phi.