Wightman Quantum Field Theory From Osterwalderschrader Reconstruction in the Topological Density Functional Theory Framework

We prove that the Topological Density Functional Theory (T-DFT) framework defines a Wightman quantum field theory in four-dimensional Minkowski space-time. The argument proceeds in two stages. Stage I: we verify that the projected Euclidean Schwinger functions of T-DFT satisfy all five Osterwalder-Schrader axioms (OS1–OS5). The key ingredients are exponential decay of the gauge-projected two-point function at rate M = 8ΛQCD > 0, Euclidean E(4) covariance of the Reynolds projector, collapse of the indefinite Krein space onto the colour-singlet sector (reflection positivity), bosonic symmetry of the physical spectrum, and the cluster bound inherited from the positive mass gap. Stage II: the Osterwalder-Schrader Reconstruction Theorem is invoked to construct the unique Wightman quintuple (HMink, U, |Ω⟩, D, φ), and each Wightman axiom W1–W5 is verified by explicit correspondence with the matching OS axiom. The spectral condition W3 includes the mass gap inf spec(P2)|H⊥ = M2 = 64Λ2QCD ≈ (1704 MeV)2 > 0. A machine-checked formalisation of the logical skeleton of the proof is provided as a Lean 4 supplementary file.