A Formal Axiomatization of Strategic Continua: Anticipatory Mean-Field Game (MFG) Systems under Localized Quasi-Singular Potentials

Abstract: Macroscopic socio-economic systems frequently manifest catastrophic instability patternswhere collective behavioral shifts render static control policies obsolete via second-order evolu-tionary loops (the Lucas Critique). This paper establishes the exact mathematical formulationfor a predictive, self-correcting macro-structural containment framework designed to modeland neutralize these dynamics. By unifying non-equilibrium statistical mechanics with optimalcontrol theory, we construct a closed-loop forward-backward system of coupled partial differentialequations (PDEs). The global population is modeled as a strategic mass probability densityfluid whose drift velocity is continuously optimized via a backward-propagating Hamilton-Jacobi-Bellman equation. Simultaneously, individual volatility anomalies are modeled as concentratedGaussian potential wells embedded directly within the environment. To validate the theoreticalframework under strict stability constraints, we map the complete system to a 1D spatial reduc-tion serving as a numerically verifiable proof-of-concept, implemented via an uncompromisedghost-padded Godunov finite volume engine featuring sign-corrected upwinding and coupledforward-backward adaptive multi-rate CFL advective time-marching. Keywords: Mean-Field Games (MFG), Hamilton-Jacobi-Bellman Equation, Fokker-Planck / Continuity Equation, Lucas Critique, Macroeconomic Control, Finite Volume Methods, Godunov Upwinding, Ghost-Cell Padding, Weak Solutions, Crandall-Lions Viscosity Solutions, Wasserstein Space, Physics-Informed Neural Networks (PINNs), Autoencoders, Latent Manifolds.