Unified Geometric Mechanics: A Resolution to Yang-Mills, Mass Gap, and Fundamental Particle Masses.

This paper provides a formal resolution to the Yang-Mills existence and mass gap prob-lem through the framework of Unified Geometric Mechanics (UGM). By substituting thecontinuous manifold with a Discrete Rhombic Dodecahedral (RD) lattice, we eliminate ul-traviolet singularities at the Planck scale. The SU (3) color symmetry is shown to emergefrom the torsional shear mechanics of a 31 trefoil knot defect. We provide an analytical proofof the mass gap ∆ > 0 using the discrete Hamiltonian of the Dual Tetrahedral Oscillator(DTO). Furthermore, we derive exact geometric formulas for the masses of the electron,proton, and neutron, incorporating the vibrational energy of the electron’s core stars (δstar )and the vacuum’s thermal elasticity (δ2K ).