Non-Commutative Rough Pseudo-Riemannian Manifold Dynamics (NC-RPRMD): Superseding Riemannian and Lorentzian Geometries via UROA Low-Regularity Metrics, STCT Torsion Bounces, and Information Geodesic Unitarity

Classical Riemannian manifolds enforce positive-definite metrics that prohibit relativistic time-like causal structures, while standard smooth pseudo-Riemannian geometries break down at low-regularity shockwaves and unmediated Hawking-Penrose curvature singularities. Furthermore, contemporary alternatives such as synthetic Lorentzian geometry and Connes' non-commutative geometry remain dynamically disconnected from Einstein's field equations. This paper establishes Non-Commutative Rough Pseudo-Riemannian Manifold Dynamics (NC-RPRMD) 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 arbitrary-signature low-regularity rough metrics (L^∞_loc ∩ C0,1), torsion-regulated Planck bounces, and spectral-geodesic syntheses systematically supersede classical geometric and singularity limitations. An executable Python simulation verifying cosmological scale-factor bounce and energy density bounding is provided.