The Cosmic Structure Diagnostic Tool (CSDT): A Geometry-Based Structural Diagnostic for Rotation-Curve Features and Lensing-Peak Alignment

We introduce the Cosmic Structure Diagnostic Tool (CSDT), a purely geometric and model-agnostic diagnostic for probing structural stability and leakage tendencies in galaxies and galaxy clusters using only observable tracer distributions. CSDT constructs a cumulative inverse-square geometry field from projected tracer positions and luminosity-weighted proxies. Two complementary modes are defined. In Dynamic Mode, derivative spike features in the cumulative geometry field align with observed galaxy rotation-curve features in accordance with Renzo’s rule. In Static Mode, geometry-field tension maps predict structural maxima that trace collisionless galaxy distributions in dissociative clusters and remain centrally aligned in relaxed systems. CSDT is formulated strictly as a diagnostic framework rather than a force law and provides a unified geometric method for comparing dynamical and lensing observables across diverse astrophysical systems.