Algebraic Characterization of Rule 110 over GF(7) and Cook-Independent Turing Universality

We derive an exact algebraic representation of the Rule~110 elementary cellular automaton update rule as a degree-3 multilinear polynomial over the Galois field GF(7) = Z/7Z: where L, C, R ∈ \0,1\ ⊂ GF(7) encode the left, center, and right cells of a Boolean neighborhood. This polynomial is the unique degree-3 multilinear function over GF(7) interpolating the eight Rule~110 transition values. As a direct corollary, fixing the center cell to C = 1 collapses the polynomial to p(L,1,R) = 1 - L·R = NAND(L,R): Rule~110 implements a NAND gate on Boolean inputs whenever the center cell is active. By Sheffer's functional completeness theorem~, NAND is a universal Boolean primitive, and therefore Rule~110 is Turing universal via a Cook-independent algebraic route. All results are machine-certified in Lean~4 with zero ; one named axiom ( \_nand\_complete) captures Sheffer's classical result, which is independent of Cook's 2004 cyclic tag system construction~. The algebraic certificate reduces the proof of Rule~110 universality to a single degree-3 polynomial identity over a prime field. We discuss how this GF(7) structure connects to the Z_7-symmetric Klein–Gordon substrate Φ_MDL (MDL: Minimum Description Length) of the Universal Generative Principle's GTE framework~, where kink soliton dynamics directly embed Rule~110 computation. The polynomial structure reflects the Z_7 charge-sector arithmetic of the GTE dynamical substrate~, where this identity underlies the computational universality of the physical field theory. The prime 7 = N_c^2 - N_c + 1 is the unique Frobenius prime generated by the QCD colour rank N_c = 3, machine-certified as \_chain\_level1 ( .Universality.FrobeniusChain, zero ); the lepton seed b_L_1 = 73 = N_c^4 - N_c^2 + 1 is the level-2 Frobenius prime, connecting GF(7) to the full GTE lepton mass hierarchy via the Frobenius-Generation Coincidence Identity (FGCI; \_unique\_at\_nc, zero ).