The S-Matrix from Foam: LSZ Reduction, Unitarity, and Compton Scattering

We derive the LSZ reduction formula and the S-matrix from the Unified Foam Field Theory. The vacuum is the foam at equilibrium density ρ₀. Asymptotic single-particle states are topological defects (T₂g torsion loops for electrons, D-mode excitations for photons) propagating freely in flat foam at t → ±∞. The time-ordering operator follows from Axiom Zero irreversibility — displacement events have absolute time stamps. The photon reduction formula uses the derived Maxwell equation □A_μ = μ₀J_μ. The electron reduction uses the T₂g torsion Dirac equation. Iterating both gives the standard LSZ formula ⟨f|S|i⟩ = Πⱼ i (pⱼ²−mⱼ²) × G̃ (p₁,. . . , pₙ). Unitarity S†S = 1 follows from Axiom Zero void-pair conservation: B+V=D ensures total foam displacement is conserved through every scattering event. The optical theorem follows from unitarity. The S-matrix of the foam IS the S-matrix of QED in the IR limit — with foam-derived α = 1/137. 036 (0. 21 ppb) and foam-derived mₑ = 0. 51096 MeV (0. 007%). Concrete verification: Compton scattering (Klein-Nishina formula), Thomson cross section σT = 6. 6523 × 10⁻²⁹ m² against observed 6. 6524 × 10⁻²⁹ m² (0. 002%). Zero free parameters.