SORT-CX: A Projection-Based Structural Framework for Complex Systems Operator Geometry, Non-Local Kernels, Drift Diagnostics, and Emergent Stability

Complex systems across physics, biology, ecology, technology, and society exhibit emergent structures that cannot be reduced to microscopic rules or simple dynamical laws. While standard approaches rely on differential equations, agent-based simulations, or data-driven models, many emergent phenomena are fundamentally structural in nature, characterized by stability islands, collective modes, scale-dependent correlations, and critical transitions. In this work, we introduce SORT-CX, the complex-systems application layer of the Supra-Omega Resonance Theory (SORT). SORT-CX applies a projection-based operator framework—comprising idempotent resonance operators, a global consistency projector, and a non-local projection kernel—to the structural analysis of complex systems. Emergence is formulated as a projective process rather than being defined purely as a temporal evolution. Structural stability corresponds to idempotent fixed points under operator projection, while structural change is diagnosed through drift metrics defined in resonance space. The framework enables a principled classification of complex systems into operator-dominated, kernel-dominated, and drift-dominated regimes, independent of specific dynamical equations and dataset-specific modeling assumptions. We develop a series of representative use cases, including network mode analysis, stability landscapes of complex fields, pattern formation, critical transitions, and multilayer coupled systems. SORT-CX positions projection-based structural analysis as a unifying perspective for emergent phenomena.