# Cross-Trajectory Chimera Interventions Reveal Dissociable Roles of Weight Magnitude and Direction in Grokking

arXiv: https://arxiv.org/abs/2607.06628 · HTML: https://mlsanity.com/paper/2607.06628/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 12 references that are in the map.
- First posted on arXiv 7 Jul 2026.

## Builds on (most cited first, 9 of 9)

- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [Linear Mode Connectivity and the Lottery Ticket Hypothesis](https://mlsanity.com/paper/1912.05671/) (2019)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [Git Re-Basin: Merging Models modulo Permutation Symmetries](https://mlsanity.com/paper/2209.04836/) (2022)
- [Deep Ensembles: A Loss Landscape Perspective](https://mlsanity.com/paper/1912.02757/) (2019)
- [The Role of Permutation Invariance in Linear Mode Connectivity of Neural Networks](https://mlsanity.com/paper/2110.06296/) (2021)
- [Towards Understanding Grokking: An Effective Theory of Representation Learning](https://mlsanity.com/paper/2205.10343/) (2022)
- [Explaining grokking through circuit efficiency](https://mlsanity.com/paper/2309.02390/) (2023)
- [A Basin-Selection Perspective on Grokking via Singular Learning Theory](https://mlsanity.com/paper/2603.01192/) (2026)

## Abstract (arXiv)

Which properties of a partially trained network are causally portable to a different, independently trained network? Single-trajectory interventions show necessity within one run, not portability across runs. We introduce cross-trajectory chimera interventions: given two runs from different seeds, we split each weight vector into a norm and a unit direction, recombine one run's norm with the other's direction, and continue training. On two modular-arithmetic tasks that grok, the components dissociate. Direction carries a transferable, donor-specific circuit identity: implanting a donor's direction at the recipient's norm drives the run to the donor's circuit in 40/40 cases, while an angle-matched random control yields no shift. The transfer is threshold-like, and its location is predicted by the recipient's norm, separating perfectly by norm class over all 20 pairs (joint permutation probability 1.9e-4). Norm carries only a modest, distributed delay effect and no identity signal. An adaptive bisection procedure localizes the threshold to +/-1/64. Direction indexes which solution a trajectory approaches; norm governs how susceptible that identity is to being overwritten.