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

DR-RoPE replaces RoPE's geometric rotation base with angles derived from the digital-root doubling sequence (1→2→4→8→7→5), the multiplicative group of units mod 9, S=⟨2⟩=(ℤ/9ℤ)ˣ. Revision 2: the non-aliasing claim is restated precisely — the digital-root sequence supplies the frequency-spectrum shape (period-6, 3-coprime invariant), while position-wise non-repetition is supplied by the scale_factor term; pure cyclic angles would make the encoding exactly periodic. Algebraic structure is stated as S=⟨2⟩=(ℤ/9ℤ)ˣ, with 2 as a primitive root mod 9 (order φ(9)=6). A modulus-convention note distinguishes mod-9 routing from the 64-state operators' base-10 manifest-frequency convention.