Non-Commutative Rough Human-Environment Dynamics (NC-RRD): Unifying Endogenous Feedback, Non-Autonomous Pullback Attractors, Tipping Points, and Planetary Boundaries via UROA, STCT, and RQIG

Treating human socio-economic activity as an exogenous forcing in Earth system models is fundamentally insufficient for predicting coupled regime shifts, mitigating climate cascades, and maintaining planetary viability. This paper establishes Non-Commutative Rough Human-Environment Dynamics (NC-RRD) within the SMA-∞ Topos [1]. By integrating Foundations of Roughness Operator Algebra (ROA) [3], Universal Rough Operator Algebra (UROA) [4], Seonggil Tensor Calculus (STCT) [2], and Rough Quantum Information Geometry (RQIG), we formulate human society (H_t) and the natural environment (E_t) as a single endogenously coupled, non-autonomous stochastic dynamical manifold. Specifically, we elevate multiscale socio-environmental feedback via the UROA Roughness Operator ΓR, regulate bifurcation-, rate-, and noise-induced tipping stresses (B-, R-, and N-tipping) via the STCT Dynamic Torsion Tensor T_STCT, and route random pullback attractors A(t,ω) and Mean-Field Game (MFG) policy controls along minimal-entropy RQIG geodesics. Furthermore, we prove that persistent homology bottleneck distances of branched pullback attractors serve as rigorous topological early-warning precursors, enabling stochastic viability control that confines coupled trajectories inside the Planetary Safe Operating Space S. A self-contained Python numerical simulation suite empirically validates endogenous phase-space stabilization and boundary compliance.