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 exact transfer matrix acts on pair states (s_i-1, s_i); its Perron eigenvalue at = 1 is _1 = 1.4846, the correlation length grows from 2.9 sites at = 1 to 15.0 at = 2 and diverges as , and the local zero-energy windows number 43 = _6(7) (consistent with the Cyclotomic Point Count Theorem). The global zero-energy rings are exactly the three uniform configurations \0^n, 1^n, 5^n\ at every ring length n 3 - an exact threefold ground-space degeneracy, machine-certified in Lean 4 - so the ground space realizes the ternary repetition code of length n and distance n. In 2D, Metropolis Monte Carlo simulations on L L tori (L = 4-16) reveal a phase transition at _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) 7. A secondary feature at _c,2 1.7 exhibits strong hysteresis consistent with domain-wall locking. The zero-temperature support of the 1D transfer matrix is the deterministic pair-state digraph of the CMCA vacuum sector, and the thermodynamic entropy rate S_CMCA() = _1() interpolates from 7 = 1.946 nats at infinite temperature through 0.395 nats per site per update at = 1 to 0 at zero temperature, where only the three rigid ground states survive. 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.