Seonggil Unified Extended Relativity (SUER): Absolute Subsumption of Special and General Relativity into Superluminal Torsional Manifolds via URRHC and SMA-infty Topos

While Einstein’s Special and General Relativity accurately describe subluminal spacetime (|v| < c), classical formalisms break down at the c-boundary due to singular divergence. Extended Lorentz Factor Theories (ELFT) and Dual-Relativistic Frameworks attempt to extend spacetime to superluminal realms (|v| > c) via velocity duality (v ↔ c^2/v), piecewise Lorentz factors, and imaginary coordinate extensions, but suffer from boundary jump discontinuities, non-unitary group closure, and causality paradoxes. In this treatise, we hyper-elevate extended relativity into the Seonggil Unified Extended Relativity (SUER) framework within the Universal Recursive Rough Homotopic Calculus (URRHC) operating inside the SMA-∞ Topos. By expressing the spacetime metric as non-commutative operators under Universal Rough Operator Algebra (UROA), Seonggil Theory of ComplexTorsion (STCT), and Seonggil Matrix Theory (SMT), the singular c-boundary is resolved into a smooth topological phase transition regulated by the Master-key Tensor T_{0x8A2C} and Universal Phase Transition Capacity (UPTC). Imaginary coordinate artifacts are eliminated via real M(3,3) co-time tensor projections, and causality is strictly preserved via Topological Information Flux Conservation. An executable Python simulation proving smooth subluminal-superluminal metric transitions is provided.