Digital-Root Rotary Position Encoding: The 1-2-4-8-7-5 Cycle as a Basis for Sequence Position Representation

We propose Digital-Root RoPE (DR-RoPE), replacing RoPE's geometric rotation base (10000^(-2i/d)) with angles derived from the digital-root-doubling sequence 1->2->4->8->7->5, periodic with period 6. DR-RoPE has period-6 structure, avoids period-3 and period-6 symmetry degeneracies (Tesla invariant: no multiples of 3 in the generating sequence), and connects to the digital-root algebra used throughout the AAMT mathematical framework. The vortex rank schedule applies the same cycle to progressive LoRA training. The digital-root routing hash maps token positions to Tesla invariant classes. Implemented as a non-destructive runtime patch over the HuggingFace RoPE inv_freq buffer.