The Standard Model's gauge group SU(3)×SU(2)×U(1) is chosen to match observations. Pentagon Physics derives it. The Galois involution √5 → −√5 acting on the 600-cell fixes exactly 24 vertices — the vertices of the 24-cell, whose symmetry group is D₄. D₄ is the unique group compatible with the Galois structure of ℚ(√5) and the binary icosahedral group 2I. The gauge group selection rule: sin²θ_W = φ⁻³ at μ ≈ 243 GeV (≈ Higgs VEV), g₁² = λ = 2φ/25, chirality from Galois ℤ/2ℤ. The three forces of nature are D₄'s three non-trivial representations.