V42: Gevrey Regularization and Modular Rigidity of Wasserstein Gradient Flows

This paper investigates the profound interaction between the spectral regularization of the heat flow, the stability of Evolution Variational Inequalities (EVI), and the emergence of infinitesimal Hilbertian structures in non-smooth metric measure spaces satisfying the RCD (K, ) condition. We introduce a robust spectral regularization mechanism based on Gevrey-type operators, V_ = ( (-) ^s/2), demonstrating its efficacy in controlling the capacity of singular sets and ensuring stability under stochastic dynamics. The core contribution is the Global Rigidity Theorem, which establishes a strict equivalence between analytic optimality (saturation of the EVI inequality along Wasserstein gradient flows), geometric decomposition (saturation of the Bochner inequality and modular splitting of the tangent module), and noncommutative spectral reconstruction (recovery of the metric via a Connes-type distance and Dirac operators). Furthermore, we prove that this entire variational, geometric, and spectral architecture is stable under measured Gromov-Hausdorff and -convergence. This framework acts as a definitive selection principle for hidden Euclidean structures within metric measure spaces.