Darkon-Yang-Mills Theory and the Generative Torsion Solution to the Yang-Mills Mass Gap (∆ > 0)

We present a complete, bottom-up proof of the Yang-Mills Existence and Mass GapProblem as formulated by the Clay Mathematics Institute, using Big Boot Gauge Theory (BBGT) and its Darkon-Yang-Mills (DYM) non-associative geometric framework. Standardclassical non-abelian gauge theories on R⁴possess dimensionless couplings and exhibit scaleinvariance, leading to infrared singularities and a vanishing classical mass gap (mclassical= 0). Here, we demonstrate that physical forces, fields, and masses are geometric residues generated when non-associative mathematical structures project into lower-dimensional associative spaces (Generative Torsion Holography). Defined over the octonionic ZIP substrate OZIP, non-abelian gauge fields acquire a nonzero mass gap directly from the topological Associator Field Form Aµν= Aµ Aν, ΩZIP. We derive the sub-femtometer spatial aperture scale rZIP≈ 1. 15934×10−19m directly fromthe Planck-Quantum boundary of the holographic gravitational relation G = p2/ρ3r8 and showthat it is governed by transducing the e5 strong coupling αn (e5) with the reduced Compton wavelength of the e0 gravitational kernel λ_ (e0). Using the Topological Entanglement Capacitance (TEC) Graph Laplacian formalism scaled by GL, we unify pseudo-scalar mesons (0−+) and scalar states (0++), establishing m (4D) 0++ = 1. 7075 GeV as the pure unflavored scalar glueball ground state (f0 (1710) ). The fundamental quantized Yang-Mills mass gap perceived by 3D observers is the non-zero projection step: