Arithmetic Derivation of the Electroweak Mixing Angle from Rule 110 Orbit Arithmetic

We derive the electroweak mixing angle = 3/13 ≈ 0.23077 from the combinatorial arithmetic of the Rule~110 cellular automaton orbit that the Standard Model generation sequence is forced to satisfy~. The derivation is a ten-step chain starting from = 3 (the Garden-of-Eden orbit depth, machine-certified in Lean~4 with zero ) through the palindrome decomposition of the 13~active f_ MDL neighborhoods. The chain is fully machine-certified in Lean~4, zero bare , zero new axioms, conditional on the import of one Lean-certified result from the UGP dynamics paper~. The predicted value = 3/13 = 0.23077 lies -0.195\% (15.0σ given PDG precision of 0.013\%) below the PDG 2022 experimental value 0.23122 ± 0.00003 at M_Z~. A companion GUT-scale formula (M_ GUT) = 3/8 reproduces the standard SU(5) prediction exactly and is also machine-certified in Lean~4. The derivation rests on two arithmetic inputs, both Lean-certified: = 3 (three SM generations, forced by the Garden of Eden orbit structure) and = 13 (the Higgs branch capacity, equal to + 2 with = 5 the Z_5 family-ring size). The physical identification of palindromic f_ MDL neighborhoods with U(1)_Y channels follows from a geometric Parity Restriction Theorem (zero new axioms) and a single Lean-certified bridge from the UGP dynamics framework. A complementary bare prediction = 3456/15101 ≈ 0.22886 (-1.02\%) is derived directly from Lean-certified gauge couplings in the GTE framework~ without any running. The two numbers are distinct predictions at different energy scales: the bare value is the CA-orbit–scale coupling and the 3/13 formula is the infrared value after Z_5 ring running shifts the denominator from 2^ = 8 to = 13. This constitutes, to our knowledge, the most precise derivation of the electroweak mixing angle from first-principle discrete mathematics.