Dark Geometry: The Origin of Dark Matter via Exotic Smoothness and Roughness Renormalization

Dark Matter, the invisible mass constituting 27% of the universe, remains one of the greatest unsolved mysteries in physics. Traditional particle physics approaches have failed to detect dark matter particles. In this paper, we propose a geometric solution based on Rough Operator Algebra (ROA) and the unique topological properties of 4-dimensional spacetime. We prove that as the universe undergoes Roughness Renormalization (α → 1), the spacetime manifold does not converge to a single smooth structure but undergoes a "4D Bifurcation," generating infinite Exotic Smoothness structures. We posit that Dark Matter is not a new particle, but ordinary matter residing in an "Exotic Layer" of spacetime. We derive the Decoupling Theorem, showing that while gravity (topological) permeates these layers, electromagnetic interaction (differential) is mathematically blocked due to the mismatch of differential operators. This explains why Dark Matter is gravitationally attractive yet electromagnetically invisible. This work extends Sunggil's Grand Unified Mathematics to the cosmological domain.