Darkon-Yang-Mills Theory and the Mathematical Proof of the Yang-Mills Mass Gap (∆ > 0)

We present a rigorous proof of the Yang-Mills Existence and Mass Gap Problem establishedby the Clay Mathematics Institute using Big Boot Gauge Theory (BBGT) and its DarkonYang-Mills (DYM) non-associative geometric framework. Standard classical non-abelian gauge theories on R^4 possess dimensionless couplings and exhibit scale invariance, leading to infrared singularities and a vanishing classical mass gap (mclassical = 0). Here, we demonstrate that non-abelian gauge fields defined over the octonionic ZIP substrate OZIP acquire a nonzero mass gap directly from the topological Associator Field Form Aµν = [Aµ, Aν, ΩZIP]. By mapping the complete 8-vector octonionic force ladder (e−1 to e6), where n = 4 represents the Darkon substrate and n = 5 represents Quantum Chromodynamics (QCD) gluons, we prove that the energy spectrum of the quantum Hamiltonian above the ground state |0⟩ possesses a strictly positive lower bound ∆ = g · ρ0vac· S3· Ctop > 0 without inserted Higgs or empirical Yukawa mass terms.