Theorem IV (Extension to Yang-Mills) - Derivation of σ

We derive the vacuum elasticity coefficient of pure SU(3) Yang-Mills theory within the Topological Density Functional Theory (T-DFT) framework. Starting from the Yang-Mills action in background-field gauge with Faddeev-Popov ghosts, we compute the one-loop effective action, apply the Slavnov-Taylor identity to establish transversality, and invoke the holographic boundary reduction of Theorem I to restrict the spectral average to the boundary three-sphere S3. The reality of the adjoint representation provides an exact factor of 1/2. The resulting vacuum elasticity coefficient is: where CA = 3 is the quadratic Casimir of SU(3) in the adjoint representation. Combined with the topological solid angle Ω3 = 2π2 (Theorem I) and the confinement fraction fadj⊗adj = 1/64 (Theorem II, Corollary II-B), the full Yang-Mills geometric index is: The primary testable prediction of the T-DFT framework is the lightest scalar glueball mass: in agreement with Lattice QCD at the 0.35% level. The relationship of this result to the Clay Millennium Problem on the Yang-Mills mass gap is discussed, and three open mathematical problems are identified.