Geometric Cymatic Field Theory (GCFT): A Mechanical Solution to the Yang-Mills Mass Gap and the Stability of Matter

We provide a physical solution to the Yang-Mills existence and mass gap problem by modeling the vacuum as a pre-stressed, 3D Face-Centered Cubic (FCC) lattice. We demonstrate that the "Mass Gap" (∆ > 0) is not a parameter of a field potential, but the Peierls-Nabarro stress of a jammed vacuum fabric. By identifying a 10.05% geometric over-packing discrepancy (∆Φ) between the FCC void fraction and the Random Close Packing (RCP) limit, we derive a universal vacuum pressure of 1035 Pa. This pressure is shown to be the mechanical origin of the Strong Force, precisely matching 2018 Jefferson Lab experimental measurements of the proton’s internal pressure. Finally, we derive the nuclear "Magic Numbers" as resonant eigenfrequencies of this lattice, showing that the transition at N = 28 is a mandatory consequence of torsional impedance (Spin-Orbit coupling) in a 12-point coordination metric.