Rigidity and Stability of RCD(K,∞) Path Spaces under Singular Kato–Voigt Perturbations

We study the rigidity and stability properties of RCD(K,∞) path spaces constructed over metric measure spaces via projective limits. The analysis is performed within the framework of L^2-normed modules and Gigli’s differential calculus. We establish the closure of the Γ_2-operator on D2,2, identify the intrinsic Hessian as a Riesz representative, and prove stability under singular Kato–Voigt perturbations via Mosco convergence. The main rigidity result characterizes curvature shifts through the Evolution Variational Inequality (EVI) on Wasserstein space P_2(Ω), linking geometric curvature bounds with spectral perturbations of the associated Dirichlet form.