GTE Unification: A Bidirectional Correspondence between Rule 110 Orbit Arithmetic and the Standard Model Electroweak Sector

We present a bidirectional correspondence between four mathematical structures: the Generative Triple Evolution (GTE) arithmetic over Z_7, the MDL-minimal cellular automaton , its continuous φmdl 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; φmdl 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\% from PDG; corrected to 384729/1664000 ≈ 0.23121 with Wolfenstein λ^3/(2c_H) radiative correction, 0.24σ from PDG, CMCA (three-layer Chiral Minkowski CA, P41~)), (GUT) = 3/8 (exact SU(5) prediction), and (Wolfenstein) = 9/40 = 0.225 (0.000\% from PDG). In the reverse direction ( → ), the observed value ≈ 0.231 determines = 3 via the arithmetic pivot + = 2^, and the SM colour structure identifies the 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 φmdl 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) = π^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\% to 0.31 percentage points. QCD predictions from = include asymptotic-freedom~ coefficient b_0 = 7 (machine-certified in Lean~4, zero sorry), strong CP angle θ_ 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 χ_ top^1/4 = 166.5\,MeV (PROVISIONAL), θ_P = -13.08^ ± 3.74^ in the PDG range (computationally verified), and σ_4D = 440.6\,MeV (computationally verified). Two structurally independent Lean-certified derivations of α_ EM^-1 = 137 converge to 0.107\% agreement. Deeper SM consequences---the baryon-to-photon ratio ≈ 6.54 × 10^-10 (7.2\% from Planck~2018), a parameter-free neutrino mass scale, and a structural 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.