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 m_e = 0.51096 MeV (0.007%). Concrete verification: Compton scattering (Klein-Nishina formula), Thomson cross section σ_T = 6.6523 × 10⁻²⁹ m² against observed 6.6524 × 10⁻²⁹ m² (0.002%). Zero free parameters.