# From Symmetry to Invariance: Learning Galois Equivalent Representations in Finite Fields

arXiv: https://arxiv.org/abs/2608.22513 · HTML: https://mlsanity.com/paper/2608.22513/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 12 references that are in the map.
- First posted on arXiv 23 Aug 2026.

## Builds on (most cited first, 12 of 12)

- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [Towards Understanding Grokking: An Effective Theory of Representation Learning](https://mlsanity.com/paper/2205.10343/) (2022)
- [Grokking as the Transition from Lazy to Rich Training Dynamics](https://mlsanity.com/paper/2310.06110/) (2023)
- [Grokking modular arithmetic](https://mlsanity.com/paper/2301.02679/) (2023)
- [Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokking](https://mlsanity.com/paper/2311.18817/) (2023)
- [Grokking as a First Order Phase Transition in Two Layer Networks](https://mlsanity.com/paper/2310.03789/) (2023)
- [Emergence in non-neural models: grokking modular arithmetic via average gradient outer product](https://mlsanity.com/paper/2407.20199/) (2024)
- [To grok or not to grok: Disentangling generalization and memorization on corrupted algorithmic datasets](https://mlsanity.com/paper/2310.13061/) (2023)
- [Grokking Group Multiplication with Cosets](https://mlsanity.com/paper/2312.06581/) (2023)
- [Can Neural Networks Learn Small Algebraic Worlds? An Investigation Into the Group-theoretic Structures Learned By Narrow Models Trained To Predict Group Operations](https://mlsanity.com/paper/2601.21150/) (2026)
- [Grokking Finite-Dimensional Algebra](https://mlsanity.com/paper/2602.19533/) (2026)

## Built on by (most cited first, 1 of 1)

- [Representation Redundancy and Structural Complexity in Finite-Field Inversion](https://mlsanity.com/paper/2609.04583/) (2026)

## Abstract (arXiv)

Neural networks can learn algebraic operations from finite examples, but it remains unclear whether this ability transfers across mathematically equivalent representations of the same operation. We study this question through multiplication in finite fields under changes of basis. The Galois action organizes basis representations into orbits, and bases in the same orbit induce the same coordinate multiplication map. This structure allows us to separate learning multiplication from transferring it to basis representations that are not used for training. We examine several ways of providing or recovering the relevant orbit structure, including invariant labels, basis matrices, orbit recognition, and algebraic decomposition. Our main approach trains a model to predict the Galois action between basis representations. Repeated applications of the learned transformation are then used to construct a canonical representative for each orbit, which supports multiplication on held-out bases through exact canonical matching. This provides a concrete mechanism for converting a learned algebraic symmetry into an invariant representation that can be used for transfer.