# Learned Structure Shows Up Earlier in the Task's Own Basis: A Controlled Look at a Synchronisation Precursor in Grokking

arXiv: https://arxiv.org/abs/2606.12966 · HTML: https://mlsanity.com/paper/2606.12966/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 22 references that are in the map.
- First posted on arXiv 11 Jun 2026.

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

- [Locating and Editing Factual Associations in GPT](https://mlsanity.com/paper/2202.05262/) (2022)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [In-context Learning and Induction Heads](https://mlsanity.com/paper/2209.11895/) (2022)
- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [Towards Automated Circuit Discovery for Mechanistic Interpretability](https://mlsanity.com/paper/2304.14997/) (2023)
- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [Causal Abstractions of Neural Networks](https://mlsanity.com/paper/2106.02997/) (2021)
- [Towards Understanding Grokking: An Effective Theory of Representation Learning](https://mlsanity.com/paper/2205.10343/) (2022)
- [Hidden Progress in Deep Learning: SGD Learns Parities Near the Computational Limit](https://mlsanity.com/paper/2207.08799/) (2022)
- [The Clock and the Pizza: Two Stories in Mechanistic Explanation of Neural Networks](https://mlsanity.com/paper/2306.17844/) (2023)
- [Omnigrok: Grokking Beyond Algorithmic Data](https://mlsanity.com/paper/2210.01117/) (2022)
- [A Toy Model of Universality: Reverse Engineering How Networks Learn Group Operations](https://mlsanity.com/paper/2302.03025/) (2023)
- [Explaining grokking through circuit efficiency](https://mlsanity.com/paper/2309.02390/) (2023)
- [Grokking as the Transition from Lazy to Rich Training Dynamics](https://mlsanity.com/paper/2310.06110/) (2023)
- [The Slingshot Mechanism: An Empirical Study of Adaptive Optimizers and the Grokking Phenomenon](https://mlsanity.com/paper/2206.04817/) (2022)
- [Grokking modular arithmetic](https://mlsanity.com/paper/2301.02679/) (2023)
- [A Tale of Two Circuits: Grokking as Competition of Sparse and Dense Subnetworks](https://mlsanity.com/paper/2303.11873/) (2023)
- [Unifying Grokking and Double Descent](https://mlsanity.com/paper/2303.06173/) (2023)
- [Grokfast: Accelerated Grokking by Amplifying Slow Gradients](https://mlsanity.com/paper/2405.20233/) (2024)
- [Grokking Group Multiplication with Cosets](https://mlsanity.com/paper/2312.06581/) (2023)

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

- [Structure-Specific Representational Priors Causally Control the Grokking Delay](https://mlsanity.com/paper/2607.04333/) (2026)
- [At-Grok Is Not Converged:A Measurement-Validity Audit for Grokking Representation Metrics](https://mlsanity.com/paper/2607.06639/) (2026)

## Abstract (arXiv)

Analyses of grokking usually read hidden activations in one basis, most often the Fourier basis modulo p. We ask how much the answer to "has the network organised yet?" depends on that choice. The measure FSDrm gen averages each unit's activation over inputs with the same answer and counts how many units share the same strongest irreducible representation. For modular multiplication we read one answer-averaged profile in three orders: discrete-logarithm order, in which the characters of the multiplicative group become Fourier modes; natural order; and random orders. Across 60 networks at p∈59,97,113, the discrete-log reading reached a fixed threshold before both other readings in 60 of 60 networks (59 of 60 when it had to hold 500 steps). Because untrained networks already score higher in discrete-log order, we also required an equal rise above each reading's untrained level; the discrete-log reading was still first in at least 16 of 20 networks at every prime. At p=59 and p=97 this happened during the memorisation plateau in 35 of 39 networks; at p=113 in 12 of 20. Memorisation preceded grokking in all 179 runs that grokked, by more than any synchronisation signal on average, so a lead over grokking is not by itself evidence of a detected circuit. Synchronisation time is only weakly related to grokking time (pooled ρ=0.18). On six finite groups, read with the same measure in every basis, the irreducible-representation reading came before random element orders on all six; with equal-rise thresholds this held in every case on four groups, in 7 of 10 on mathbbZ5×mathbbZ5 and in none on D16, where about half the profile power lies in one-dimensional representations the measure leaves out. We also correct a conjugate-pair artefact that roughly halves the measure on abelian groups. This version withdraws the causal and predictive claims of v1.