The Crystal Vacuum and the Monster: Completing Monstrous Moonshine from the 24-Cell

We complete the six-step Monster chain established in Paper 4 by proving Step 5: the vertex operator algebra of the 24-cell crystal vacuum is isomorphic, after the Turyn-Golay construction, to the Leech lattice VOA, whose Z2_Golay orbifold is the Frenkel-Lepowsky-Meurman moonshine module V♮. The Monster group M = Aut(V♮) is therefore the symmetry group of crystal quantum gravity. The proof delivers five new exact results. First, the j-function constant decomposes as 744 = n_v + |S_6| = 24 + 720, where n_v = 24 is the crystal vertex count (untwisted sector) and |S_6| = 720 = N_c × |E_8| is the Golay orbifold twisted-sector contribution. This is the first physical decomposition of the constant 744 since its discovery in 1979. Second, this decomposition implies M_F5 = 1 + |S_6|/n_v = 1 + 30 = 31: the fifth Mersenne prime equals one plus the ratio of twisted to untwisted ground states. A new exact identity connecting number theory to VOA physics. Third, the crystal partition function satisfies Z_crystal(τ) = Θ_Leech(τ)/η(τ)^24 = j(τ) − 720, with constant term n_v = 24, confirming the identification V_crystal ≅ V_Leech modulo zero modes. Fourth, the number 720 = |S_6| which appeared in Paper 3 as the count of solar Wilson loops entering the Leech theta function Θ_Leech = E4³ − 720Δ is identified as the twisted-sector contribution to V♮. The solar loops of the crystal association scheme are the twisted sector of the moonshine module. In Part B we establish the wave equation of primes: the chain 24-cell → E8 → Θ_E8 = E4(τ) → j(τ) → ζ(s)ζ(s−3) → all primes, showing that the prime distribution is encoded in the crystal spectrum. In Part C the McKay-Thompson series for the two Monster involution classes are computed from crystal eigenmodes, with coefficients C(24,2) = 276 (class 2B) and 2^h(F4) + C(24,2) = 4372 (class 2A) both exact in crystal numbers. All results are verified symbolically. The six-step Monster chain is complete.