Non-Commutative Rough Trustworthy & Interpretable AI Dynamics (NC-RTAIA): Unifying Tropical Geometry, Equivariant Group Actions, Persistent Homology, and Certified Robustness via UROA, STCT, and RQIG

Transitioning artificial intelligence from empirical scale-driven correlation learning to provably trustworthy, interpretable, and safe architectures requires transcending classical i.i.d. statistical learning theory. This paper establishes Non-Commutative Rough Trustworthy & Interpretable AI Dynamics (NC-RTAIA) 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 group-equivariant structure preservation, tropical neural networkdecision geometry, persistent homology regularization, and certified Lipschitz robustness as topological manifold stabilization and information geodesic optimization operators. Specifically, we elevate symmetry-preserving feature representations via the UROA Roughness Operator Γ_R, regulate multi-objective optimization trade-offs and adversarial shear stress via the STCT Dynamic Torsion Tensor TSTCT, and route tropical hyperplane arrangements and persistence diagram Wasserstein distances along minimal-entropy RQIG geodesics. A self-contained Python numerical simulation suite is provided to empirically validate simultaneous risk minimization, topological stabilization, and certified robustness amplification.