RTLI Dual-Inertia: A Variance-Geometry Diagnostic for Structural Integration

This technical note introduces a variance-geometry diagnostic framework for analyzing structural integration across complex systems. Building on eigenvalue analysis of covariance matrices, the framework distinguishes between integrated variance (physical inertia, m_p) and distributed variance (relational inertia, m_r), and operationalizes their ratio through the eigenvalue metric \lambda_{\text{eff}}. An empirically observed transition corridor near \lambda_{\text{eff}} \approx 1.7 is identified as a region where variance integration commonly occurs. The framework is extended with an additional diagnostic quantity, effective viscosity (\eta_{\text{eff}}), defined as the ratio of relational inertia to the gradient of integrated inertia with respect to a system-specific ordering variable. This dimensionless measure characterizes resistance to variance integration and complements existing diagnostics based on slope and scatter. Applied across multiple astrophysical datasets—including galaxy scaling relations and exoplanet populations—the combined diagnostics (\lambda_{\text{eff}}, N, \eta_{\text{eff}}) reveal structural differences that are not captured by traditional scatter-based analyses alone. In particular, systems with similar tightness can exhibit markedly different integration resistance, while some populations display signatures of partial or reversed integration. The framework is presented as an operational, falsifiable diagnostic tool rather than a new physical theory. Explicit null tests, limitations, and domain-specific caveats are provided. The results suggest that variance-geometry diagnostics may offer a useful lens for comparing structural organization across disparate empirical systems.