An arXiv preprint (2610.11833) examines how two-layer neural networks learn modular addition. Past work has shown that trained networks often develop Fourier-structured representations that support exact generalization. The new paper focuses on the process behind that outcome, proposing a "probability-signature dynamics" framework to describe how gradient-based learning shapes these circuits.
The abstract indicates that prior research identified the Fourier circuits themselves, but not the mechanism by which training produces them. The new framework appears intended to fill that gap by modeling the dynamics of learning, rather than only describing the final representation.
Because only the abstract was available for this digest, the full derivation is not summarized here. No conflicting sources were provided, so the claims are presented as reported by the preprint.