Why Three Generations? Twin Prime Structure and the Termination at D = 3

We explain why the Standard Model contains exactly three generations of fermions. The answer emerges from twin prime structure at factorial resonances: the pairs (5,7), (11,13), and (17,19) bracket the multiples 6, 12, 18 = kD! for k = 1, 2, 3 where D! = 6. The value D = 3 is uniquely selected by the Borwein integrals where D(D+1) = 2×D!. The sequence terminates at k = 3 because 24 ± 1 = (23, 25) fails to be a twin pair — the number 25 = 5² is composite, where 5 is the icosahedral vertex degree. This provides a number-theoretic derivation of the three-generation structure. The twin products 35 + 143 + 323 = 501 equal the icosahedral pairwise count C(E,2) + C(V,2), connecting generation structure to spatial geometry. The exchange term 77 = 7 × 11 governs inter-generation mixing. The Higgs mass m_H = 501/4 = 125.25 GeV emerges as the total twin structure divided by spacetime dimensions.