The Structural Radiante: A Pentadimensional Instrument for Visualization and Analysis of Structural Trajectories in Complex Systems

The Structural Radiante is the visualization and formal analysis layer of the Kappa Method (Ohio, 2026a). The Kappa Method detects structural instability in time series via five topological observables: the Ohio Number Oh(t), structural memory Phi(t), dynamic rigidity eta(t), structural diversity Xi(t), and state-phase deficit DEF(t). The Radiante transforms these observables into a dimensional navigation instrument that answers the questions the Kappa raises but does not directly resolve: what regime is the system in? How fast is it moving toward rupture? What is the energetic cost of reorganization? Where does it go next? This paper presents: (i) the geometry of the pentadimensional space K(t) and the three archetypal regimes — Nagare (JP: nagare, flow), Utsuroi (JP: utsuroi, transition), and Katashi (JP: katashi, crystallization); (ii) the gravitational sub-observable Psi(t), which quantifies the cost of structural reorganization as gravitational action; (iii) the domain-agnostic exchange schema radiante.v1; and (iv) empirical validations across four domains: the 2008 Global Financial Crisis via Kappa-FIN v3 (DOI: 10.5281/zenodo.18917558), the polarized political blogosphere (Adamic & Glance, 2005), educational dropout (OULAD dataset), and LLM hallucination detection via Kappa-LLM (DOI: 10.5281/zenodo.18883790). The GFC 2008 analysis demonstrates that the Katashi onset detected in November 2007 — ten months before the Lehman Brothers bankruptcy — constitutes the first retrospective validation case of the framework. Section 7 presents the multi-entity network extension that constitutes the architectural gateway for Kappa-GEO, an ongoing geopolitical application.