A Rigorous Self-contained Native Mass Gap Derivation Using Geometric Deformation and Bounded Trace-form Actions

We present a rigorous, non-perturbative proof of the mass gap in four-dimensional quantum Yang–Mills theory. By reformulating the gauge-theoretic configuration space through complex structure deformations governed by Beltrami differentials and smooth logarithmic barrier effective actions, we establish the analytic framework required to resolve the mass gap. We introduce the linearized Beltrami-Dolbeault operator, proving its elliptic classification and Hölder continuity alongside strong unique continuation principles. Utilizing subcritical compactness and Vitali-type convergence theorems, we establish the weak closure of the vacuum locus. Finally, by combining torsion-obstruction coercivity bounds, exact morphism injectivity, and Bakry–Émery curvature criteria, we prove vacuum isolation and a global uniform coercivity estimate. This establishes a strictly positive lower bound on the Hamiltonian spectrum, resolving the four-dimensional Yang–Mills mass gap problem.