# Rotary Positional Embeddings as Phase Modulation: Theoretical Bounds on the RoPE Base for Long-Context Transformers

arXiv: https://arxiv.org/abs/2602.10959 · HTML: https://mlsanity.com/paper/2602.10959/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 34 references that are in the map.
- First posted on arXiv 11 Feb 2026.

## Builds on (most cited first, 22 of 22)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [Lost in the Middle: How Language Models Use Long Contexts](https://mlsanity.com/paper/2307.03172/) (2023)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [RULER: What's the Real Context Size of Your Long-Context Language Models?](https://mlsanity.com/paper/2404.06654/) (2024)
- [Mixed Precision Training](https://mlsanity.com/paper/1710.03740/) (2017)
- [DeepSeek-V2: A Strong, Economical, and Efficient Mixture-of-Experts Language Model](https://mlsanity.com/paper/2405.04434/) (2024)
- [Self-Attention with Relative Position Representations](https://mlsanity.com/paper/1803.02155/) (2018)
- [Train Short, Test Long: Attention with Linear Biases Enables Input Length Extrapolation](https://mlsanity.com/paper/2108.12409/) (2021)
- [On Layer Normalization in the Transformer Architecture](https://mlsanity.com/paper/2002.04745/) (2020)
- [Extending Context Window of Large Language Models via Positional Interpolation](https://mlsanity.com/paper/2306.15595/) (2023)
- [YaRN: Efficient Context Window Extension of Large Language Models](https://mlsanity.com/paper/2309.00071/) (2023)
- [The Impact of Positional Encoding on Length Generalization in Transformers](https://mlsanity.com/paper/2305.19466/) (2023)
- [LongRoPE: Extending LLM Context Window Beyond 2 Million Tokens](https://mlsanity.com/paper/2402.13753/) (2024)
- [On the Relationship between Self-Attention and Convolutional Layers](https://mlsanity.com/paper/1911.03584/) (2019)
- [Scaling Laws for Transfer](https://mlsanity.com/paper/2102.01293/) (2021)
- [Exponential expressivity in deep neural networks through transient chaos](https://mlsanity.com/paper/1606.05340/) (2016)
- [Understanding the Difficulty of Training Transformers](https://mlsanity.com/paper/2004.08249/) (2020)
- [Base of RoPE Bounds Context Length](https://mlsanity.com/paper/2405.14591/) (2024)
- [Understanding the RoPE Extensions of Long-Context LLMs: An Attention Perspective](https://mlsanity.com/paper/2406.13282/) (2024)

## Built on by (most cited first, 7 of 7)

- [RoPE Distinguishes Neither Positions Nor Tokens in Long Contexts, Provably](https://mlsanity.com/paper/2605.15514/) (2026)
- [Phase Structure in Rotary Attention: A Spectral Framework for Semantic Continuity and Execution-Boundary Governance](https://mlsanity.com/paper/2607.25507/) (2026)
- [Three-Phase Transformer](https://mlsanity.com/paper/2604.14430/) (2026)
- [Self-Attention Dynamics with Rotary Position Embeddings: Twisted States and Explicit Consensus Rates on the Sphere](https://mlsanity.com/paper/2607.24502/) (2026)
- [Transcript-Managed Transformers: Monotone Multi-Agent Collapse and Universality with Two Pop-Enabled Transcripts](https://mlsanity.com/paper/2607.29496/) (2026)
- [Shifting Mechanisms: How Positional Encoding Choice Shapes In-Context Retrieval](https://mlsanity.com/paper/2609.38530/) (2026)
- [RoPE at the End of Its Rope? Theory, Diagnosis, and Mitigation of Long-Context Failures](https://mlsanity.com/paper/2609.39929/) (2026)

## Abstract (arXiv)

Rotary positional embeddings (RoPE) are widely used in large language models to encode token positions through multiplicative rotations, yet their behavior at long context lengths remains poorly characterized. In this work, we reinterpret RoPE as phase modulation applied to a bank of complex oscillators, enabling analysis through classical signal processing theory. Under this formulation, we derive principled lower bounds on the RoPE base parameter that are necessary to preserve positional coherence over a target context length. These include a fundamental aliasing bound, analogous to a Nyquist limit, and a DC-component stability bound that constrains phase drift in low-frequency positional modes. We further extend this analysis to deep transformers, showing that repeated rotary modulation across layers compounds angular misalignment, tightening the base requirement as depth increases. Complementing these results, we derive a precision-dependent upper bound on the RoPE base arising from finite floating-point resolution. Beyond this limit, incremental phase updates become numerically indistinguishable, leading to positional erasure even in the absence of aliasing. Together, the lower and upper bounds define a precision- and depth-dependent feasibility region a Goldilocks zone for long-context transformers. We validate the framework through a comprehensive case study of state-of-the-art models, including LLaMA, Mistral, and DeepSeek variants, showing that observed successes, failures, and community retrofits align closely with the predicted bounds. Notably, models that violate the stability bound exhibit attention collapse and long-range degradation, while attempts to scale beyond one million tokens encounter a hard precision wall independent of architecture or training.