Prime Fourier Embeddings: A Principled Basis for Modular Arithmetic
Quick summary
arXiv:2606.23044v3 Announce Type: replace-cross 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 bloc
Key takeaways
- arXiv:2606.23044v3 Announce Type: replace-cross 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 bloc
Why it matters
The importance of “Prime Fourier Embeddings: A Principled Basis for Modular Arithmetic” will be measured by what changes in practice. User behavior, access conditions, verifiable performance and responsible-use outcomes are the signals worth following.
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