FRC 840.101: The Phase--Attention Boundary: Recall Smearing and the Limits of Continuous State Compression (Version 2.0)

This paper formulates the Phase–Attention Boundary: a structural separation between continuous phase-state architectures and discrete attention-based architectures. Within the Fractal Resonance Cognition (FRC) program, the Large Lambda-Tensor Model (LLTM) was developed as a continuous recurrent phase-coupled architecture inspired by Kuramoto dynamics and low-rank coherence fields. Controlled comparisons against Transformer baselines revealed a fundamental limitation: continuous state compression blends historical information into a finite evolving state, producing recall smearing. We prove a formal Recall Smearing Theorem: under γ-contractive recurrence, mutual information about a past token decays exponentially with distance at rate γ2(T-τ). This bound is derived from the Data Processing Inequality and applies universally to fixed-state recurrent systems, including structured state-space models such as S4, Mamba, RWKV, Griffin, and xLSTM. We show that data-dependent selectivity (as in Mamba) can reduce the rate of smearing but cannot eliminate it; only explicit key–value addressability achieves zero-smearing recall. The conclusion is not that phase-state models fail. Rather, FRC 840.101 establishes a domain-separation principle: continuous phase-state architectures are naturally suited to analog, noisy, dynamical, sensorimotor, biological, market, audio, and field-regime systems, whereas attention architectures are structurally superior for language, code, symbolic logic, and exact recall. This paper defines the mathematical boundary with formal proof, presents controlled empirical evidence from parameter-matched LLTM experiments on Tiny Shakespeare, and positions hybrid architectures as the next step.