This paper studies how Transformer models with Rotary Position Embeddings (RoPE) develop emergent, wavelet-like properties that compensate for the positional encoding's theoretical limitations.Through an analysis spanning model scales, architectures, and training checkpoints, we show that attention heads evolve to implement multi-resolution processing analogous to wavelet transforms.We demonstrate that this scale-invariant behavior is unique to RoPE, emerges through distinct evolutionary phases during training, and statistically adheres to the fundamental uncertainty principle.Our findings suggest that the effectiveness of modern Transformers stems from their remarkable ability to spontaneously develop optimal, multi-resolution decompositions to address inherent architectural constraints.

Attention sink (AS) is a consistent pattern in transformer attention maps where certain tokens (often special tokens or positional anchors) disproportionately attract attention from other tokens. We show that in transformers, AS is not an architectural artifact, but it is the manifestation of a fundamental geometric principle: the establishment of reference frames that anchor representational spaces. We analyze several architectures and identify three distinct reference frame types, centralized, distributed, and bidirectional, that correlate with the attention sink phenomenon. We show that they emerge during the earliest stages of training as optimal solutions to the problem of establishing stable coordinate systems in high-dimensional spaces. We show the influence of architecture components, particularly position encoding implementations, on the specific type of reference frame. This perspective transforms our understanding of transformer attention mechanisms and provides insights for both architecture design and the relationship with AS.