__Yang-Mills

Through the enforcement of a global Bakry-Émery curvature-dimension condition, we derive an associated Logarithmic Sobolev Inequality, proving the existence of a strict spectral mass gap. Furthermore, we demonstrate that the geometric boundaries of the effective action induce complex conjugate pole structures in coloured sectors, which—when evaluated through Osterwalder-Schrader reflection positivity and Krein space quotienting—provide a rigorous, first-principles proof of colour confinement. This framework satisfies all Osterwalder-Schrader axioms, offering a fully consistent bridge between Euclidean path integral formulations and physical Lorentzian quantum field theory.