Algebraic Characterization of Rule 110 over GF(7) and Finite NAND Functional Completeness

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 in {0,1} subset 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 theorem (1913), NAND generates all Boolean functions, establishing Rule 110 as a universal Boolean primitive (finite functional completeness). This is a finite Boolean result: it does not imply Turing universality (which requires unbounded computation), and is independent of Cook's 2004 proof that Rule 110 is Turing universal via cyclic tag system simulation. Both results are true and complementary: Cook's route establishes Turing universality; this paper establishes finite NAND functional completeness via an algebraic certificate over GF(7). All results are machine-certified in Lean 4 with zero sorry; one named axiom (nand_complete) captures Sheffer's classical result. We discuss how this GF(7) structure connects to the Z_7-symmetric Klein-Gordon substrate Phi_MDL of the Universal Generative Principle's GTE framework, where kink soliton dynamics directly embed Rule 110 computation. 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 sorry); 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 sorry).