Simple Lie Algebras with 3-Smooth Representation Dimensions

A positive integer is 3-smooth if its only prime factors are 2 and 3. I classify the simple compact Lie algebras for which both the fundamental and adjoint representation dimensions are 3-smooth. The answer is exactly {A₁, A₂, B₄, C₄}, corresponding to su(2), su(3), so(9), and sp(8). The proof uses Størmer's 1897 theorem on consecutive smooth numbers for the A-series, Zsygmondy's theorem for the B and C-series, modular arithmetic for the D-series, and direct computation for the five exceptional algebras. As an application, I show that the four Standard Model anomaly cancellation conditions force a hypercharge ratio structure 1 : 4 : −2 : −3 : −6 whose coefficients are all 3-smooth. Combined with observed electric charges, every hypercharge value is itself 3-smooth. I also prove that the only simple Lie algebras with prime-power adjoint dimension are su(2) and su(3), selecting the Standard Model gauge group by a single arithmetic condition on the adjoint alone.