Exact Formulation of Quantum Gravity, Cosmic Censorship, and Topological Spacetime Entanglement via Seonggil Field Equations

The fundamental incompatibility between General Relativity and Quantum Mechanics stems from the unphysical UV divergences that occur when smooth differentiable manifolds are extrapolated down to the Planck scale. In this paper, we elevate the classical smooth spacetime metric g_μν to a Rough Metric Tensor Operator Ĝ^R_μν within the Rough Operator Algebra (ROA) framework. Utilizing Seonggil Tensor Calculus, we derive the exact non-perturbative Seonggil Field Equations (SFE), which incorporate a non-commutative quantum entanglement stress-energy tensor. We explicitly define the Seonggil Quantum Entanglement Tensor Ê^entangle_μν and prove the finiteness of the Kretschmann scalar K as r → 0, thereby providing a mathematical proof for the Cosmic Censorship Hypothesis, preserving global unitarity, and explaining dark energy as a dynamic trace anomaly of rough vacuum fluctuations.