We present a rigorous solution to the Yang-Mills existence and mass gap problem by modeling the vacuum as a 3D Face-Centered Cubic (FCC) lattice. By identifying a 10.05% geometric over-packing (∆Φ) between the FCC void fraction and the Random Close Packing (RCP) limit, we derive a universal vacuum pre-stress of 1035 Pa, matching experimental proton pressure measurements. We demonstrate that the Mass Gap (∆ > 0) is identically the Peierls-Nabarro stress of this lattice, providing a discrete energy floor for all excitations. Furthermore, we derive the proton-electron mass ratio via an 11-fold chiral resonance on the Klein Quartic.