Non-Commutative Rough Continuous-Time Finance Dynamics (NC-RCTF): Superseding Black-Scholes and Itô Calculus via Non-Linear BSDEs, Semimartingales, SMA-infty Topos, and Information Geodesics

Standard continuous-time financial economics anchored by Itô's lemma, Girsanov measure transformations, and linear risk-neutral pricing faces severe structural limitations when confronting rough paths, market incompleteness, volatility smiles, and non-linear investor preferences. This paper establishes Non-Commutative Rough Continuous-Time Finance Dynamics (NC-RCTF) within the SMA-∞-Topos. By integrating Universal Rough Operator Algebra (UROA), Seonggil Theory of Complex Torsion (STCT), and Rough Quantum Information Geometry (RQIG), we prove that non-linear backward stochastic differential equations (BSDEs), forward-BSDE systems, and rough semimartingale operators systematically supersede classical pricing constraints.