The Central Theorem: From Foam to the Standard Model

We prove the central theorem of the Unified Foam Field Theory: the continuum limit of the BCC lattice action S = Σ ψ†LTψ is the Standard Model coupled to General Relativity, with all parameters determined by seven cell integers. The proof assembles results from Papers #48–58 into a single logical chain with five links: (1) the gauge kinetic terms emerge from plaquette sums on the 24-triangle, 42-plaquette face graph; (2) the Dirac equation emerges from the T₁u Wilson fermion mechanism with mass gap √17; (3) the Yukawa couplings emerge from the torsion cross-block T₂₁ = 2U connecting the T₁u bands through the A₂u Higgs sector; (4) spontaneous symmetry breaking is forced by the A₂u torsion eigenvalue −1; (5) the RG flow is unique because SU (3) and SU (2) are asymptotically free, the Oh → O (3) lattice artefacts are irrelevant in 4D, and the 14-dimensional face space admits no additional sectors. Each link is either a theorem or a consequence of established lattice field theory. The Symanzik matching at O (a²) has been computed explicitly: the corrections scale as (E/MP) ² ~ 10⁻³⁵ at the electroweak scale and are negligible.