Non-Commutative Rough Humanoid & Robotics Dynamics (NC-RHRD) Subsuming Physical Data Scarcity and Energy-Mass-Thermal Coupling via UROA, STCT, and RQIG

Humanoid and robotics learning data scarcity, along with energy-mass-thermal bottlenecks, are conventionally treated as isolated engineering hurdles in control and material science. This paper supersedes independent views by establishing Non-Commutative Rough Humanoid & Robotics Dynamics (NC-RHRD) within the SMA-∞ Topos. By integrating Universal Rough Operator Algebra (UROA), the Seonggil Theory of Composite Torsion (STCT), and Rough Quantum Information Geometry (RQIG), we strictly prove that interaction data scarcity and mass-power feedback loops are macroscopic topological manifestations of manifold sparsity and torsional shear stress. Furthermore, synthetic data generation and elastic energy recovery actuators are formally derived as geodesic optimization sub-loops protecting the physical-digital manifold.