What question did this study set out to answer?

The aim is to introduce Digital-Root RoPE, enhancing position encoding in sequence representation.

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June 3, 2026Open Access

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

Key Points

The aim is to introduce Digital-Root RoPE, enhancing position encoding in sequence representation.
Replaces geometric rotation base with a digital-root-doubling sequence.
Implements a periodic structure with a cycle of 6.
Applies the cycle to LoRA training using a non-destructive runtime patch.
Establishes a unique period-6 structure avoiding symmetry degeneracies.
Improves mapping of token positions to Tesla invariant classes.
Integrates seamlessly with HuggingFace's existing RoPE framework.

Abstract

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 invfreq buffer.

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