Computational Universality and the Standard Model: Rule 110, Z5 Rings, and Mod-7 Structure in the Universal Generative Principle

The Standard Model's generation orbit uniquely determines Rule 110-the only elementary cellular automaton rule with a complete, published proof of computational universality Cook2004,Cook2009-as the sole binary CA rule consistent with SM winding structure and vacuum transparency. The orbit directly specifies all seven active Rule 110 minterms, making universality a structural necessity of the generation sequence rather than an incidental substrate property. Rule 110 is independently forced by the UGP canonical substrate Spivack2025_UGPDynamicsUniversality, so the rule is determined from both above (substrate) and below (orbit constraints). The central result (CUP-4, Lean 4, zero sorry): the SM generation orbit algebraically fixes all 8 bits of Rule 110, selecting exactly one vacuum-transparent rule among all 256 (Lean-verified false-positive rate p_raw = 1.36\%). CUP-8/9 establish odd parity for all generation-3 particles and a closed Z_5 ring on the five SM families. CUP-11/12 identify a mod-7 CA f_MDL whose MDL-minimal completion has 76-bit description length, is Lean-uniquely determined, and is computationally universal; generation-1 is a Garden of Eden under f_MDL, certifying first-generation stability as arithmetic unreachability. A Cook-independent algebraic certificate (zero sorry) derives universality via p(L,C,R)=C+R-CR-LCR over F_7, which reduces to NAND when C=1, bypassing the cyclic tag system entirely. Three-generation uniqueness, colour confinement, and charge quantisation are machine-certified at . Hadron predictions include -' mixing _P = -13.08^ 3.74^ and topological susceptibility _top^1/4 = 166.5\,MeV. The complete formalization has zero bare sorry; three named physics bridge axioms remain, of which the Cook bridge is independently bypassable via this algebraic route.