Wightman Quantum Field Theory from Osterwalder–Schrader Reconstruction in the Holographic Vacuum Elasticity Framework

We prove that the Holographic Vacuum Elasticity (HVE) framework defines a Wightman quantum field theory in four-dimensional Minkowski space-time via the Osterwalder–Schrader (OS) reconstruction pathway. The argument proceeds in two stages. In Stage I we verify that the gauge-projected Euclidean Schwinger functions {Sn}n≥0 of HVE satisfy all five OS axioms (OS1–OS5): exponential decay of the Reynolds-projected two-point function at rate M = 8ΛQCD > 0 guarantees temperedness (OS1); commutativity of the Reynolds projector P̂G with the Euclidean group E(4) yields covariance (OS2); collapse of the indefinite Krein space onto the colour-singlet sector establishes reflection positivity (OS3); bosonic quantum numbers of the physical glueball field give permutation symmetry (OS4); and the positive mass gap implies exponential cluster decay (OS5). In Stage II the OS Reconstruction Theorem is applied to construct the unique Wightman quintuple (HMink, U, |Ω⟩, D, φ), and each Wightman axiom W1–W5 is verified by explicit correspondence with its OS counterpart. The spectral condition (W3) yields the mass gap inf spec(P2)|H⊥ = M2 = 64Λ2QCD ≈ (1704 MeV)2 > 0. The governing equation of the HVE framework is the Vacuum Suppression Law (VSL), Oobs(x) = Oideal · exp(−χ · σ0G · W(x) · Ω3 · fG), with universal parameters σ0G = α/2 (Atiyah–Singer index), Ω3 = 2π2 (volume of S3, holographic boundary for D = 4 bulk), fG = 1/64 (Schur–Reynolds fraction, SU(3) adjoint), W(x) = 1 (flat vacuum), and chirality χ = +1 (suppression). Under the HVE-PARv4 anti-fit protocol the mass gap prediction MVSL ≈ 1702 MeV is verified with 0.46% agreement against lattice QCD (1710 ± 50 MeV) with zero free parameters. This constitutes a Class I (post-diction, structurally independent) confirmation. A machine-checked Lean 4 formalisation of the logical skeleton of the proof is provided as supplementary material (Companion O2_HVE.lean), achieving zero sorry invocations for all algebraic assertions.