Phase-Hybrid NSI: A Phase-Coherence Metric for Unifying Discrete and Continuous Representations

The representation of information often involves a dynamic interplay between discrete (symbolic, quantized) and continuous (signal, field-based) forms. Existing complexity metrics typically assume fixed representational modes or employ hard switching mechanisms, failing to capture smooth transitions or inherently hybrid states. We introduce Phase-Hybrid NSI (NSI_hyb), a novel informational metric designed to unify these representational modalities through a phase-driven interference mechanism. NSI_hyb leverages phase fields extracted from both continuous (e.g., φ_cont via FFT) and discrete (e.g., φ_disc via token rank) encodings of an object. A hybrid interference phase field (φ_hyb) is constructed, and its Kuramoto coherence (κ_φ) is computed. This coherence factor κ_φ, reflecting the degree of phase alignment between the representations, adaptively weights the contribution of standard continuous (K_cont) and discrete (K_disc) complexity proxies (e.g., DCT entropy, LZMA compression size) in the final metric: NSI_hyb = κ_φ·K_cont + (1-κ_φ)·K_disc. Unlike previous approaches relying on comparison or minimization, NSI_hyb provides a smooth, continuous interpolation driven by internal phase dynamics, mirroring physical interference phenomena. We demonstrate its efficacy on synthetic time-series and text data, showing that κ_φ serves as an interpretable measure of structural consistency and that NSI_hyb accurately tracks representational shifts, identifying phase equilibrium points. NSI_hyb offers a fundamentally new mechanism for modeling self-organizing representations and provides a foundation for applications in phase-aware adaptive coding, dynamic neural routing, multimodal analysis, and cognitive modeling.