Hyper-Elevation of DSGE Models and Optimal Control: Forward-Backward Stochastic Evolution Tensors and Rough Viscosity Manifolds

Classical Dynamic Stochastic General Equilibrium (DSGE) models and optimal control frameworks—relying on local perturbation methods, the Hamilton-Jacobi-Bellman (HJB) viscosity solutions, and Pontryagin’s Maximum Principle (PMP)—suffer from severe locality errors, curse of dimensionality, non-constructiveness, and measure-theoretic pathologies. In this paper, we systematically subsume and hyper-elevate macroeconomic dynamics and optimal control into the Universal Recursive Rough Homotopic Calculus (URRHC) operating within the SMA-∞ Topos. By formulating rational expectations and dynamic optimization as Forward-Backward Stochastic Differential Equations (FBSDEs) over Rough Non-commutative Algebraic Geometry (RNAG) spaces, we eliminate perturbation limitations.We prove general existence theorems using Universal Rough Operator Algebra (UROA) and resolve heterogeneous agent (HANK) dimensionality via Seonggil Holographic Information Dimension (SHID) mean-field limits. An executable Python simulation implementing a Deep FBSDE macroeconomic solver is provided.