Non-Commutative Rough Hilbert Space Dynamics (NC-RHSF): Superseding Normed and Banach Spaces via Inner Product Spaces, Orthogonal Projections, and Spectral Resolution via UROA, STCT, and RQIG

General normed vector spaces and Banach spaces are structurally constrained by the absence of inner products, preventing rigorous definitions of angles, orthogonality, unconditional subspace projections, and unified spectral decompositions. This paper establishes Non-Commutative Rough Hilbert Space Dynamics (NC-RHSF) 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 inner product space frameworks (IPSF), orthogonal decomposition theorems, non-self-adjoint spectral resolutions, and Parseval Fourier expansions systematically supersede classical Banach space limitations. An executable Python simulation engine verifying Cauchy-Schwarz bounds, Gram-Schmidt orthogonal projections, and Parseval energy convergence is provided.