Strict Geometric Foundation of Yang-Mills Theory and Proof of the Mass Gap via Topological Complexes of Gravitational Spin Polarizations

The exact mathematical foundation of Quantum Yang-Mills theory on R^4 and the rigorous proof of a strictly positive mass gap (Delta > 0) remain fundamental unsolved problems within standard perturbative frameworks. This monograph presents a definitive geometric resolution by redefining the base spacetime manifold as an emergent property of gravitational spin polarizations. Dimensions are treated as eigenstates of a fundamental polarization tensor Πab, and matter is strictly identified as a topological complex of closed gravitational trajectories. By restricting the functional integral measure to the moduli space of stable topological knots, Gribov ambiguities are naturally eliminated, and the Wightman axioms are satisfied. The mass gap is analytically proven to be the minimum geometric torsion required to maintain a closed topological knot, yielding a strictly positive lower bound via an extended Lichnerowicz theorem on the stratified configuration space. Furthermore, the model accurately derives quark confinement and fractional charge quantization via Borromean topology.