The Geometry of Singular Distributions in Gauge Theories: A Gelfand-Leray Layer Approach

This paper investigates the analytic structure of the functional integration measure in non-Abelian gauge theories, establishing a rigorous correspondence between the Gelfand-Shilov distribution formalism and the Faddeev-Popov quantization procedure. We demonstrate that the Faddeev-Popov functional determinant naturally emerges from the geometric definition of the Dirac delta distribution concentrated on singular hypersurfaces via Leray forms. Crucially, we provide the \textbf{first rigorous variational derivation} of the ``double layer'' distribution $\delta^{(1)}$ directly from the second variation of the action, proving that it acts as a reflective boundary condition (Hard Wall) necessary for the stability of the vacuum. This induces a spectral gap in the Hamiltonian via the Poincaré-Friedrichs inequality. Finally, we extend this formalism to the configuration space of General Relativity, deriving a Generalized Topological Stress-Energy Tensor that stabilizes spacetime singularities through both surface tension and vacuum stiffness.