The Genesis Loop: Discrete Prime Lattice Constraints as a Geometric Solution to the Yang-Mills Existence and Mass Gap Problem

The Standard Model of particle physics and General Relativity remain fundamentallyincompatible due to a divergence in their treatment of spacetime: the former relies onquantized fields, while the latter assumes a continuous manifold. This incompatibility ismost evident in the Yang-Mills Existence and Mass Gap Problem, which asks why masslessquantum fields acquire strict, non-zero mass.This paper proposes a solution by abandoning the assumption of continuous spacetime(R4). We introduce the Trio Model, which posits that the universe operates on a discretePrime Integer Lattice with a fixed minimum resolution (a = 1). By mapping the integertimeline onto a Modulo 6 Cylinder (H = 6), we demonstrate that the distribution of Primenumbers creates a rigid, helical structure—a "Prime Helix"—that constrains the geometryof existence.We define Mass not as an intrinsic property, but as "Hoop Stress"—the mechanicaltension required to maintain a closed geometric loop (1 → 2 → 3) against the discreteconstraints of the lattice. Using this framework, we analytically derive the proton-to-electronmass ratio (6π5) and the neutron mass difference to within 0.001% of experimental CODATAvalues. We further demonstrate that the Standard Model is a Dimensional Cascade, derivingthe masses of leptons, bosons, and quarks as geometric harmonics of this primary lattice.Finally, we conclude that the "Mass Gap" is the energy cost of defining information withina discrete computational system.