Kappa: A Method for Informational Regime Detection via Geometry and Dynamics

Technical report presenting the complete theoretical framework of the Kappa Method — a domain-agnostic system for detecting structural regime transitions in complex informational systems via five topology-aware observables (Oh, Φ, η, Ξ, DEF) derived from persistent homology and Forman-Ricci curvature of time-evolving dependency networks. The document covers: (I) Fundamentals of flow and structural stability; (II) Scientific lineage from Navier-Stokes to persistent homology; (III) The paradox of collective predictability; (IV) The five structural observables and regime classification (Nagare, Utsuroi, Katashi); (V) Complete mathematical formalism including Ledoit-Wolf shrinkage, edge weights, H₁ persistent homology, Ohio Number Oh(t), and structural memory Φ(t) with γ ∈ [0.98, 0.99]; (VI) Empirical validation summary across financial markets (GFC 2008: ~60-day lead time), LLMs (AUC 94.2%), and education (dropout 3 weeks ahead); (VII) Regime-based decision framework. Proposes the Law of Ohio (Organized Hallucination In Optimization): when information-processing systems lose access to external validation, they optimize internal coherence at the expense of reality correspondence — detectable via topological invariants before collapse. Part of the Kappa Method ecosystem. Related work: Kappa-FIN (doi:10.5281/zenodo.18883585).