Project T Volume XLVIII : The OS Bridge Osterwalder-Schrader Reconstruction on W(Ã₂)

This volume establishes the Osterwalder-Schrader (OS) bridge between the lattice construction of Volumes XLVI–XLVII and continuum quantum field theory. The transfer matrix T = exp(−L) on the W(Ã₂) Cayley lattice has all strictly positive eigenvalues, since L ≥ 0 implies exp(−λᵢ) > 0. This yields OS reflection positivity (OS2) automatically, without any additional assumptions. The OS reconstruction theorem then provides: a physical Hilbert space ℋ, a positive self-adjoint Hamiltonian H ≥ 0 with unique vacuum Ω, and Wightman distributions satisfying all axioms. The mass gap Δ(H) = 1 follows directly from the spectral gap of L (Volume XLVI). The connected two-point function decays as exp(−t) with effective mass converging to 1.0000, confirming exponential clustering. The construction extends to 4D Euclidean spacetime (Γ⁴(W(Ã₂))) with the same gap Δ = 1, satisfying the dimensional requirement of the Clay Millennium Problem. The topological protection from C(3) = 0 prevents any deconfinement phase transition at any temperature, ensuring the mass gap persists universally. Zero free parameters throughout.