Double Layer Stabilization of Spacetime Defects: \\ Geometric Origin of the Exotic Matter in Traversable Wormholes

The stability of Traversable Wormhole solutions traditionally requires the violation of the Null Energy Condition (NEC) by ``exotic matter.'' In this work, we demonstrate that such violation is an artifact of neglecting the distributional structure of singular spacetime boundaries. Utilizing the Gelfand-Leray Double Layer formalism, we perform a detailed, step-by-step derivation of the tangential pressure $S_{\theta\theta}$ at the throat of a Morris-Thorne wormhole. We prove that the required ``exotic'' stress is exactly provided by the jump in extrinsic curvature induced by the Double Layer. Furthermore, we define the effective potential $V_{eff}(R)$ of the throat dynamics and prove that the topological stiffness of the vacuum (governed by the Dilaton coupling $\xi \ge 1$) guarantees radial stability against perturbations. This identifies the ``exotic matter'' not as a bulk substance, but as the elastic response of the Amplitude Geometry to the violation of the fundamental tubular thickness.