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This paper introduces a bounded spectral framework to analyze rotary phase alignment and semantic continuity in transformer language models, moving beyond traditional vector geometry. By focusing on ordered hidden-state sequences for spectral decomposition, the authors derive the Rotary Position Embedding (RoPE) attention score and establish a local stability lemma that limits score degradation through bounded phase displacement. The findings highlight a critical distinction between representational continuity and execution-boundary governance, providing a new lens for understanding the dynamics of transformer architectures.
A novel spectral framework reveals that while internal coherence maintains task relevance, it does not guarantee successful execution transitions in transformer models.
Transformer language models are usually analyzed through vector geometry, yet ordered context and rotary position encoding introduce explicit phase structure into query-key interactions. This paper develops a bounded spectral framework for examining rotary phase alignment, hidden-state continuity, and semantic drift without treating language models as literal physical wave systems. It first identifies ordered hidden-state sequences, rather than vocabulary indices, as valid domains for spectral decomposition. It then derives the Rotary Position Embedding (RoPE) attention score as a sum of magnitude-weighted cosine terms and proves a local stability lemma: uniformly bounded phase displacement limits degradation of the corresponding pre-softmax score. To extend phase analysis beyond native RoPE coordinates, the paper defines complex modal coordinates over fixed orthonormal direction pairs and introduces a weighted coherence functional for hidden-state trajectories. These constructions support a strict distinction between representational continuity and execution-boundary admissibility. Internal coherence may describe preservation of task-relevant relations, but it cannot authorize a consequential transition. Positioned against existing geometric, spectral, phase-modulation, representation-analysis, and mechanistic-interpretability accounts, the framework contributes a theoretical and methodological program for determining when spectral structure explains continuity and when governance must remain an external predicate over execution.
