Prime Fourier Embeddings: A Principled Basis for Modular Arithmetic
arXiv:2606.23044
The paper proposes Prime Fourier Embeddings, a sinusoidal encoding of integers based on prime indices that makes modular arithmetic operations explicit, and shows theoretically that equivariant linear maps must be block‑diagonal per prime, with empirical validation on square‑free composite moduli.
Abstract
Numbers have algebraic structure that standard neural embeddings often fail to expose. We introduce Prime Fourier Embeddings (PFE), which encode integers as prime-indexed (cos, sin) pairs derived from the harmonic analysis of Q, providing a pre-structured representation in which modular arithmetic reduces to selecting the relevant prime channel rather than discovering algebraic structure from scratch. We prove that any linear map equivariant with respect to the product group action on PFE must be block-diagonal with one independent block per prime -- a consequence of Schur's lemma applied to the resulting character decomposition. For square-free composite moduli, the Chinese Remainder Theorem predicts which prime channels are task-relevant. Both predictions are confirmed empirically: ablation studies show specialization ratios exceeding 500x between task-relevant and task-irrelevant channels, with perfect in-distribution test accuracy across all square-free composite moduli tested.