Beyond Entropy and Bits: NSI v2.0 Formalism for Coherence-Gated Structural Information

Classic information measures like Shannon entropy and Kolmogorov complexity face inherent limitations when applied to complex, structured data prevalent in physical, biological, and cognitive systems. They often fail to distinguish meaningful organization from random noise or adequately capture the role of internal coherence. This paper introduces NSI v2.0 (Nonlinear Structural Information), a fundamentally revised theoretical framework and operational metric designed to overcome these limitations. NSI v2.0 is built upon a rigorous axiomatic foundation using functional analysis. It uniquely defines informational complexity through the interplay of three core components: Structural Intensity (Iform), quantifying gradient energy or form; Coherence (κ), measuring phase or directional alignment; and Residual Entropy (H), capturing remaining unpredictability. These are integrated into a novel composite metric, INSI, reformulated as a normalized ratio κ*I_rel / (κ*I_rel + (1-κ)*H_rel + ε). This revised formulation posits that information manifests primarily through structure made coherent, rendering information negligible as coherence vanishes (κ → 0). This new version explicitly addresses critical issues of previous approaches, including: Contextual Normalization: Replacing unstable global min/max scaling with robust intra-contextual relative metrics (I_rel, H_rel) for improved stability and generalizability. A Priori Decorrelation: Introducing analytical methods (e.g., geometric mean decomposition Z = sqrt(I*H)) to separate collinearity between Iform and H without statistical training, ensuring component independence. Computable Complexity Proxy: Defining a pseudo-complexity measure K* based on the coding length of NSI features (-log2 p(I, κ, H)) as a practical bridge to algorithmic complexity. Tensor Generalization: Extending the framework naturally to multichannel and multimodal data. We present the complete axiomatic system (NSI v2.0), operational formulas for metrics, the context-aware normalization scheme, the decorrelation technique, the K* construction, and the tensor extension. Preliminary validation on synthetic and semi-realistic data confirms the theoretical predictions, including limit behaviors and high discriminative power (AUC ≈ 1.0) for classifying order vs. chaos and detecting anomalies. NSI v2.0 offers a robust, interpretable, and computationally tractable foundation for analyzing and modeling information in structure-rich, dynamic systems across diverse scientific and technological domains.