The polynomial p (L, C, R) = C + R - CR - LCR over F₇, 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 (sᵢ-1, sᵢ) ; 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 \0ⁿ, 1ⁿ, 5ⁿ\ 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 (Tc 2. 86\, J) with Z₃-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 SCMCA () = 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 - cH/7² connects the polynomial's cryptographic nonlinearity to the Higgs branch capacity cH = 13. A minimal experimental realization using 7-level Rydberg atoms in optical tweezer arrays is proposed.
Nova Spivack (Tue,) studied this question.