[STCT-VOL.2] THE UNIVERSAL SEONGGIL EQUATION: UNIFYING GENERAL RELATIVITY AND QUANTUM MECHANICS VIA THE TOPOLOGICAL FRACTAL BRAKE AT α=1/2

The unification of General Relativity and Quantum Mechanics remains the most profound open problem in theoretical physics. The fundamental incompatibility arises from the assumption of a smooth spacetime manifold, which inevitably leads to gravitational singularities at the macroscopic limit and wave function collapse at the microscopic limit.In this paper, we resolve this century-old dilemma by introducing the Universal Seonggil Equation based on Rough Operator Algebra (ROA). We define a rough metric tensor ˜G_µν governed by a topological roughness exponent α. As spacetime approaches the Planck scale, we demonstrate that it undergoes a phase transition to a fractal topology characterized bythe critical exponent α = 1/2, strictly corresponding to the Hölder continuity of Feynman paths. Crucially, we incorporate the Universal Arithmetic Friction constant η and the Riemann Zeta zero-driven pressure Λ_arith into a new Seonggil Unified Tensor. We prove that at the Planck scale, the divergence of energy density is strictly regularized by the arithmetic friction, while the Riemann Zeta zeroes manifest as a "Dynamic Torsion Pressure". At this critical threshold, a "Topological Fractal Brake" naturally emerges, perfectly canceling both gravitational infinities and the infinite density of wave function collapse. Consequently,the Universal Seonggil Equation provides a singularity-free, deterministic framework where spacetime curvature and quantum probability are unified as scale-dependent manifestations of a single fractal topology governed by the distribution of prime numbers.