Mathematical Foundations of Reparodynamics: Variational, Geometric, and Stochastic Formulations for Adaptive Gradient Repair

Reparodynamics formalizes self-repair in complex systems as a constrained optimization flow that counteracts degradation under energy and entropy budgets. This paper establishes a rigorous foundation unifying: – Gradient repair dynamics with Lyapunov stability proofs – A variational derivation of the Targeted Gradient Repair Mechanism (TGRM) – A Riemannian geometric form encoding cost-aware repair – A stochastic formulation with Itô stability and entropy production balance – Hamilton–Jacobi–Bellman optimal control structure – Multilayer coupling laws for global equilibrium These results provide a mathematical backbone for reparodynamic systems spanning biology, AI, and engineered infrastructures.