PRISM: Deriving a White-Box Transformer as a Signal-Noise Decomposition Operator via Maximum Coding Rate Reduction

Deep learning models, particularly Transformers, are often criticized as "black boxes" and lack interpretability. We propose Prism, a white-box attention-based architecture derived from the principles of Maximizing Coding Rate Reduction (MCR2). By modeling the attention mechanism as a gradient ascent process on a distinct signal-noise manifold, we introduce a specific irrational frequency separation (π-RoPE) to enforce incoherence between signal (semantic) and noise (syntactic) subspaces. We show empirical evidence that these geometric inductive biases can induce unsupervised functional disentanglement alone. Prism spontaneously specializes its attention heads into spectrally distinct regimes: low-frequency heads capturing long-range causal dependencies (signal) and high-frequency heads handling local syntactic constraints and structural artifacts. To provide a theoretical grounding for these spectral phenomena, we draw an analogy between attention mechanism and a Hamiltonian dynamical system and identify that the standard geometric progression of Rotary Positional Embeddings (RoPE) induces dense resonance networks (Arnold Tongues), leading to feature rank collapse. Empirical validation on 124M-parameter models trained on OpenWebText demonstrates that Prism spontaneously isolates the Attention Sink pathology and maintains isentropic information flow across layers. Further, we suggest a physics-informed plug-and-play intervention KAM-RoPE for large language models (LLMs). Our results suggest that interpretability and performance can be unified through principled geometric construction, offering a theoretically grounded alternative to heuristic architectural modifications