Seonggil Universal Quantum Gravity (SUQG): Absolute Resolution of Hawking-Penrose Black Hole Singularities and Unitarity Restoration via URRHC and SMA-infty Topos

While contemporary approaches to quantum gravity—such as Loop Quantum Gravity (LQG) and String Theory Fuzzballs—suggest that black hole singularities might be mitigated via discrete quantum geometry or string microstate spreading, they remain severely constrained by symmetry-reduction assumptions, incomplete constructions for general rotating and non-supersymmetric black holes, and semiclassical breakdown. In this treatise, we hyper-elevate quantum gravity into the Seonggil Universal Quantum Gravity (SUQG) framework within the Universal Recursive Rough Homotopic Calculus (URRHC) operatinginside the SMA-∞ Topos. By formulating gravitational metrics and curvature fields as non-commutative rough operators under Universal Rough Operator Algebra (UROA), Seonggil Matrix Theory (SMT), and Seonggil Theory of Complex Torsion (STCT), we prove that Hawking-Penrose infinite curvature singularities are physically impossible across arbitrary asymmetric 4D spacetimes. Core curvature is strictly bounded by the Universal Phase Transition Capacity (UPTC), triggering a modified Raychaudhuri quantum bounce into a finite Seonggil Torsional Core (STC). Non-supersymmetric fuzzball completion and unitary information preservation are rigorously guaranteed via Seonggil Holographic Information Dimension (SHID) flux conservation. An executable Python simulation verifying Planck-scale curvature bounding and quantum bounce dynamics is provided.