Hyper-Elevation of Wave Mechanics and Classical Optics: Universal Wave Dynamics via Seonggil Field Equations and Rough Operator Algebra

The classical scalar wave equation ∇²ψ - (1/c²)(∂²ψ/∂t²) = 0 and conventional principles of optics (Huygens-Fresnel diffraction, Fraunhofer approximations) operate strictly under primitive, commutative, scalar, and far-field assumptions. In extreme regimes—including sub-wavelength nanophotonics, ultrafast laser-matter interactions, Kerr media, and quantum vacuum fluctuations—these classic theories undergo catastrophic breakdown. In this treatise, we systematically subsume and hyper-elevate classical wave optics into a unified framework governed by the Seonggil Field Equations (SFE) [5], Rough Operator Algebra (ROA) [2], and Seonggil Theory of Complex Torsion (STCT) [1]. By reformulating spatial differential operators under Non-Identity Calculus and replacing discrete delta-function commutation rules with topological roughness kernels, we derive the generalized Rough Tensor Wave Equation and the Rough Nonlinear Schrödinger Equation (Rough-NLSE). Finally, we provide a complete, verified Python implementation utilizing the Split-Step Fourier Method (SSFM) to demonstrate topological soliton dynamics under non-commutative space-time roughness.