The Prime Field Architecture of the Arithmetic Modulation Field: Yang-Mills Existence and Mass Gap on R⁴ — Volume 5: The Hilbert Space and the Vacuum Sector

Volume 5 of 14. Establishes the quantum structure of the Prime Field Architecture: the Hilbert space H_PFA, the vacuum sector V, and the arithmetic mass gap Δ = |V| = 13 > 0. Proves Theorem 12 (H_PFA separable, complete, positive-definite): H_PFA = L²(ℤ_{≥0} × P1) ≅ l²(ℕ) ⊗ ℂ⁸, satisfying Wightman Axiom W1. Proves Theorem 13 (Purity of the Vacuum): every element of V = {1,7,11,13,17,19,23,29,31,37,41,43,47} is prime (or the Cornerstone n=1); zero P1 composites exist in [1,48]. Proves Theorem 14 (The Arithmetic Mass Gap): Δ = |V| = 13 > 0 — an exact theorem about counting integers, independent of any parameter. Proves Theorem 15 (First Excitation): n₀ = 49 = 7² is the first P1 composite. Proves Theorem 16 (Vacuum Uniqueness): |Ω₀⟩ = (1/√8) Σ_{r∈P1} |0,r⟩ is the unique SU(3)-invariant minimum-energy state. Establishes the PFA Light Cone (the vacuum sector V as the arithmetic light cone), the energy gap Δ_E = 1 > 0, and the completeness relation (PFAT Decomposition = resolution of identity on H_PFA). Answers Wolfgang Pauli's 1954 question "What is the mass?" with the arithmetic theorem Δ = 13.