PJ-RoPE: A Fourier-Jet-Affine Position Space for Relative Attention

We unify RoPE's Fourier phase, Jordan-RoPE's finite jets, and ALiBi's affine recency into a single learnable relative-position space, and study which regions of this space are selected by different tasks. PJ-RoPE is a Fourier-Jet-Affine formulation for relative attention, with an optional Poincare-type reading as the affine completion of a homogeneous Fourier-jet positional representation. Algebraically, the same primitives form a finite constant-coefficient difference module: simple roots of the lag-shift operator give Fourier/RoPE characters, repeated nonzero roots give Jordan/Fourier jets, and the repeated unit root gives ALiBi-like affine recency. The framework separates scalar PJ-bias kernels from exact PJ-rotary feature transforms, introduces adaptive sector diagnostics, and uses LC/rapidity coordinates to stabilize high-order jets. Controlled probes verify sector containment and selection; small language runs expose an affine/recency boundary; music-token streams provide the clearest case where LC/affine variants remain strong while carrying measurable high-order corrections; and LC diagnostics show a scale-stability gain coupled to phase-resolution loss.