We present a novel resolution to the Yang-Mills mass gap problem and cosmological singularities by reformulating the physical action through the Projective Monge Method. By treating spacetime as a tubular neighborhood of finite thickness $\lambda$, we derive a modified Einstein-Hilbert action governed by a 'Geometric Amplitude' $\Psi_g \propto \text{sgn}(\sqrt{g})\sqrt{|g|}$. This formalism introduces a rigorous 'Hard Wall' boundary condition at singular loci, providing a microscopic geometric origin for the Gribov horizon. We demonstrate two major physical consequences: (i) the emergence of a repulsive pressure that replaces the Big Bang with a 'Cosmological Bounce', and (ii) the enforcement of total information reflection at black hole horizons ($|\mathcal{S}|^2 = 1$), acting as a modified Cauchy surface that preserves quantum unitarity. Furthermore, we provide a formal proof that the tubular geometry induces a strictly positive mass gap in the energy spectrum.