Volume 8: The Wightman Axioms and Full Poincaré Covariance

Volume 8 of 14. Establishes the Minkowski structure of the Prime Field Architecture: the reconstructed Wightman theory satisfies all five Wightman axioms W1-W5 with full Poincaré covariance. Proves Theorem 30 (W1 Hilbert Space): H_W separable, positive-definite; H_W ≅ L²(ℝ⁴) ⊗ ℂ⁸ in the continuum limit. Proves Theorem 31 (W2 Poincaré Covariance): continuous unitary representation U(a,Λ) of ISO(3,1) on H_W, constructed from discrete lattice symmetries via continuum limit and Wick rotation. Proves Theorem 32 (W3 Vacuum): unique Poincaré-invariant vacuum Ω_W with H_W·Ω_W = 0, uniqueness from OS5 exponential clustering. Proves Theorem 33 (W4 Spectral Condition): spec(H_W) ∩ (0,m_gap) = ∅, where m_gap = 2π/(13·a_phys) > 0 — the CMI mass gap condition satisfied. Proves Theorem 34 (W5 Microcausality): [φ_W(x),φ_W(y)] = 0 for spacelike separation; arithmetic light cone velocity v = 47/49 < c consistent with massive excitations. Proves Theorem 35 (Full Poincaré Covariance): φ_W(Λx+a) = U φ_W(x) U⁻¹. Presents the complete logical chain from the arithmetic of M = 30 to the existence of a Yang-Mills theory on ℝ⁴ with mass gap Δ = 13 > 0.