Yare 3-6-9 Rotary Positional Encoding: A Digital-Root Frequency Schedule for RoPE

Rotary Positional Encoding (RoPE; Su et al. 2021) uses a geometric frequency progression thetaᵢ = 10000^ (-2i/d) that, while effective, has no principled structure governing its period or extrapolation behavior. We propose Yare 3-6-9 RoPE, two frequency schedules tuned to digital-root structure: a vortex₃69 schedule alternating between two poles, and a digitalᵣootdoubling schedule walking the orbit of 1 under doubling modulo 9 — the cycle (1, 2, 4, 8, 7, 5). Both schedules produce closed-form period structure analyzable without empirical probing. The reference implementation is a non-destructive runtime patch over the HuggingFace RoPE invfreq buffer, preserving the standard contract and fully revert-able. Unit tests verify norm preservation, idempotency, and determinism. We outline a long-context benchmark suite (LongBench, needle-in-a-haystack at 8k/32k/128k, passkey retrieval) comparing both Yare schedules against vanilla RoPE, NTK-aware scaling, YaRN, and LongRoPE. Empirical results are deferred to v1. 1.