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

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.

Vehicles Swarm Intelligence: Cooperation in both Longitudinal and Lateral Dimensions

Longitudinal-only platooning methods are facing great challenges on running mobility, since they may be impeded by slow-moving vehicles from time to time. To address this issue, this paper proposes a vehicles swarming method coupled both longitudinal and lateral cooperation. The proposed method bears the following contributions: i) enhancing driving mobility by swarming like a bee colony; ii) ensuring the success rate of overtaking; iii) cruising as a string of platoon to preserve sustainability. Evaluations indicate that the proposed method is capable of maneuvering a vehicle swarm to overtake slow-moving vehicles safely and successfully. The proposed method is confirmed to improve running mobility by 12.04%. Swarming safety is ensured by a safe following distance. The proposed method's influence on traffic is limited within five upstream vehicles.