Four Closing Theorems: Anomaly, Generations, Gravity, and the Continuum Limit

Paper #59 established the Central Theorem: the BCC lattice of truncated octahedra with action S = ψ†LTψ flows in the continuum limit to the Standard Model coupled to General Relativity, with all parameters from seven cell integers. The proof chain had four acknowledged gaps. This paper closes all four. Theorem 60. 1 (Chiral Anomaly): the modified Ginsparg-Wilson relation produces the correct ABJ anomaly coefficients 3, 2, 1 for SU (3) ×SU (2) ×U (1), determined entirely by the T₁u irrep dimension and the torsion eigenvalue spectrum. Theorem 60. 2 (Three Generations): the number of fermion generations equals dim (T₁u) = 3 in Oₕ; a fourth generation is geometrically forbidden by the fixed multiplicity theorem of Paper #57. Theorem 60. 3 (General Relativity): long-wavelength compression fluctuations of the Kelvin cell metric produce a massless spin-2 graviton from the symmetric traceless component of T₁u ⊗ T₁u; the Einstein-Hilbert action emerges at leading order with GN = ℏc/MP². Theorem 60. 4 (Lattice-to-Continuum Completeness): the Bloch expansion of S = ψ†LTψ reproduces the full SM+GR Lagrangian exactly, with all 14 face modes and the graviton collective mode accounted for and no missing or extra sectors. All four theorems are Tier 1. The Central Theorem is complete without qualification. Not peer reviewed.