Rigorous Multiscale Analysis and Topological Control of Stochastic Reaction-Diffusion SPDEs on Deformable Manifolds

This manuscript establishes a strict functional-analytic framework for the topological control ofadvective-diffusive processes on stochastically forced, time-dependent Riemannian manifolds. Weresolve heuristic multiscale obstructions by executing two-scale convergence formally within thecotangent bundle of a reference Gelfand triple, stabilized via a pullback diffeomorphism. We rigorously prove that the homogenized macroscopic operator satisfies strict coercivity conditions in thesense of Krylov-Rozovskii. To control the topological morphology of the infinite-dimensional state,we introduce a Sobolev-regularized controller mapped through a sublevel persistence landscape. Byestablishing the Fréchet sub-differentiability of this energy functional, we formulate the controller asa maximal monotone operator utilizing the Clarke generalized gradient. We invoke the Kuratowskiand Ryll-Nardzewski measurable selection theorem to mathematically guarantee the existence of astrongly measurable drift, intrinsically proving strict control coercivity. Global well-posedness andexponential energy dissipativity are derived via the variational stochastic Itô formula for movingdomains. Finally, we formulate an unconditionally strictly stable Semi-Implicit Euler-Maruyama(IMEX) operator-splitting scheme for GPU architectures.