Prime Fourier Embeddings: A Principled Basis for Modular Arithmetic
For machine learning practitioners, PFE provides a principled embedding that eliminates the need for neural networks to discover algebraic structure in modular arithmetic tasks.
Prime Fourier Embeddings (PFE) encode integers as prime-indexed (cos, sin) pairs, enabling modular arithmetic by selecting relevant prime channels. Empirical results show specialization ratios exceeding 500x between task-relevant and irrelevant channels, with perfect in-distribution test accuracy across all square-free composite moduli tested.
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.