We present a bidirectional correspondence between four mathematical structures: the Generative Triple Evolution (GTE) arithmetic over Z_7, the MDL-minimal Z_7 cellular automaton , its continuous field limit, and the Standard Model (SM). The electroweak correspondence is summarised in six arrows and packaged as a single 7-conjunct machine-certified theorem, ugp_r110_sm_joint_unification, in Lean 4 (zero sorry, zero new axioms). The discrete Rule 110 layer is the binary shadow of under parity projection; is the continuum limit as lattice resolution M . In the forward direction ( ), the single integer = 3 (the Garden-of-Eden orbit depth of , machine-certified) generates via the arithmetic cascade = 2^ - = 5, = 2^+1 - = 13 a complete set of Standard Model electroweak parameters: (EW) = 3/13 0.23077 (-0.195 384729/1664000 0.23121 with Wolfenstein ^3/(2c_H) radiative correction, 0.24 from PDG, CMCA (three-layer Chiral Minkowski CA, P41 )), (GUT) = 3/8 (exact SU(5) prediction), and (Wolfenstein) = 9/40 = 0.225 (0.000 In the reverse direction ( ), the observed value 0.231 determines = 3 via the arithmetic pivot + = 2^, and the SM colour structure identifies the Z_7 winding sectors via the double Mersenne endpoint theorem. The correspondence extends to spacetime and QCD. The MDL minimality principle that selects also selects the Einstein-Hilbert action via Lovelock's uniqueness theorem (computationally confirmed, with numerical Ricci-curvature support). The causal-graph spectral dimension d_s = 4 is machine-certified in Lean 4 in the thermodynamic limit (zero sorry). The Algebraic Lifting Theorem and Algebraic Descent Theorem connect beable-level, discrete, and continuous results: every -algebraic property of holds for at every resolution M 1; every PSC-admissible beable with positive [D]-weight lifts to a physical observable at Compton scale. The lattice correction (M) = ^2/(3M^2) quantifies the sole geometric deviation from exact Lorentz invariance; at M=7 this predicts 6.71 matching the measured AFCA (Asynchronous Fractal Cellular Automata) SR error of 6.4 QCD predictions from = Z_7 include asymptotic-freedom coefficient b_0 = 7 (machine-certified in Lean 4, zero sorry), strong CP angle QCD = 0 (three independent proofs, machine-certified), Casimir invariants C_F = 4/3, C_A = 3, T_F = 1/2 (machine-certified), and a zero-PDG-input hadronic chain yielding top^1/4 = 166.5 MeV (PROVISIONAL), _P = -13.08^ 3.74^ in the PDG range (computationally verified), and _4D = 440.6 MeV (computationally verified). Two structurally independent Lean-certified derivations of EM^-1 = 137 converge to 0.107 Deeper SM consequences—the baryon-to-photon ratio 6.54 10^-10 (7.2 dark-sector classification—follow from the same integer inputs. Exactly three generations are uniquely predicted (machine-certified in Lean 4, zero sorry); charge neutrality and anomaly cancellation are machine-certified bundle theorems (zero sorry). Zero free parameters are used anywhere in these derivations.