Non-Commutative Rough Higher-Order Topological Dynamics (NC-R3DI): Unifying Simplicial Complexes, Hodge Laplacians, Topological Localization, and Non-Autonomous Tipping via UROA, STCT, and RQIG

Complex systems across neuroscience, Earth climate dynamics, and multi-agent AI architectures are fundamentally governed by multi-body (higher-order) interactions and global topological invariants that defy reduction to pairwise graphs. This paper establishes Non Commutative Rough Higher-Order Topological Dynamics (NC-R3DI) 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 prove that simplicial Hodge Laplacians L_k, topological Dirac operators D_topo, homological state localization (ker L_k), and non-autonomous tipping points (B-, R-, and N-tipping) are macro-manifestations of non-commutative boundary operator fluctuations and information geodesic routing. Specifically, we elevate multi-body boundary operators via the UROA Roughness Operator Γ_R, regulate homological memory trapping and metastable shear stress via the STCT Dynamic Torsion Tensor T_STCT, and formulate Fokker–Planck tipping escape statistics along minimal-entropy RQIG geodesics. A self-contained Python numerical simulation suite is provided to empirically validate harmonic mode localization and tipping point resilience.