This paper introduces a new geometric and structural formulation of four-dimensional Yang–Mills theory based on the heat-flow regularization of gauge fields and the spectral properties of the Faddeev–Popov operator. We construct a gauge-invariant structural functional ( \Phi[A] ) that encodes both ultraviolet regularity and proximity to the Gribov horizon. The functional is defined through a heat-kernel smoothed curvature density and is shown to vanish precisely on the boundary of the first Gribov region ( \Omega ). We demonstrate that ( \Phi[A] ) is quantitatively comparable to the lowest eigenvalue of the Faddeev–Popov operator near the Gribov horizon, establishing a rigorous link between geometric regularization and spectral degeneracy. Based on this, we derive a Fisher–Hardy-type singular potential associated with ( \Phi[A] ), which acts as a repulsive barrier preventing gauge configurations from approaching the Gribov boundary. The paper develops the analytical properties of this structural regularization, including gradient estimates, stability bounds, and its behavior under gauge transformations and heat flow. These results provide the mathematical foundation for the subsequent analysis of the mass gap in the companion work, where the ultraviolet limit of this framework is studied.