We propose a geometric unification framework where gauge interactions and gravitation correspond to distinct strata of a single singular distribution on the space of connections. Building upon the Projective Monge Method, we demonstrate that the ``simple layer'' distribution describes propagating gauge fields, while the ``double layer'' at the tubular boundary induces mass and inertia. We derive the \textbf{Modified Einstein Field Equations} from the ``Amplitude Gravity'' action, stabilized by a dynamical Dilaton field. Finally, we propose a laboratory experiment using counter-propagating Petawatt Laguerre-Gaussian lasers to nucleate a ``micro-horizon.'' We predict a detectable Topological Vacuum Birefringence that is strictly distinguishable from Standard Model backgrounds by its linear dependence on the Optical Angular Momentum (OAM) charge $\ell$.