THEOREM IV - EXTENSION TO YANG-MILLS: DERIVATION OF σQCD INCLUDING FADDEEV-POPOV GHOST CONTRIBUTIONS AND THE SLAVNOV-TAYLOR IDENTITY

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: σQCD = (CA αs) / 2 = 3αs / 2 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: ΓYM = 3π2αs / 64 The primary testable prediction of the T-DFT framework is the lightest scalar glueball mass: Mglueball = 8ΛQCD ≈ 1704 MeV 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 formal propositions are established within the T-DFT framework: RG-invariance of fadj⊗adj = 1/64 (Proposition IV-B), autoconsistency of the mass formula with the Dyson-Schwinger equations (Proposition IV-A), and Osterwalder-Schrader reflection positivity of the T-DFT restricted measure dμf (Proposition IV-C).