The Spin-7 Lattice Model: Phase Transitions and Statistical Mechanics of the GTE Polynomial

The polynomial p(L,C,R) = C + R - CR - LCR over F_7, derived as the unique MDL-minimal substrate encoding the Standard Model from first principles, defines a natural statistical mechanics system: a 7-state nearest-neighbor spin model with three-body interactions. We analyze this spin-7 model in one and two spatial dimensions. In 1D, the transfer matrix has dominant eigenvalues (10.42, 6.73, 0, 0, 0, 0, -0.67) at beta = 1, a spectral gap Delta ≈ 3.69, a correlation length xi ≈ 7.77 sites near beta ≈ 2, and a ground-state degeneracy of 43 = Phi_6(7) local configurations (consistent with the Cyclotomic Point Count Theorem). In 2D, Metropolis Monte Carlo simulations on L x L tori (L = 4--16) reveal a phase transition at beta_c ≈ 0.35 (T_c ≈ 2.86 J) with Z_3-like symmetry breaking among three degenerate global ground states: the uniform configurations {0, 1, 5}, algebraically characterized as the roots of p(x,x,x) = -x(x-1)(x-5) (mod 7). A secondary feature at beta_{c,2} ≈ 1.7 exhibits strong hysteresis consistent with domain-wall locking. The 1D transfer matrix coincides exactly with a single CMCA spacetime evolution step, yielding a thermodynamic entropy rate S_CMCA = log(10.42) = 2.343 nats per update at beta = 1. The three ground-state sectors {0, 1, 5} map to the color charge sectors of the Standard Model, and the NL identity NL(p) = 1 - c_H/7^2 connects the polynomial's cryptographic nonlinearity to the Higgs branch capacity c_H = 13. A minimal experimental realization using 7-level Rydberg atoms in optical tweezer arrays is proposed.