We derive the structural form of the Standard Model gauge kinetic term and Higgs potential from the Recognition Composition Law (RCL), a single functional equation whose unique solution is J(x) = ½(x + x⁻¹) − 1. The forcing chain RCL → J → φ → 8-tick → D = 3 → Q₃ produces the cube graph whose hyperoctahedral automorphism group B₃ decomposes into three layers with ranks (3, 2, 1). We construct explicit Gell-Mann and Pauli generators on the fundamental representation spaces, prove Hermiticity, tracelessness, and the su(2) and su(3) commutation relations. Matrix-valued gauge potentials on the ℤ³ lattice yield a nonabelian field strength F_μν = ∂_μA_ν − ∂_νA_μ + i[A_μ, A_ν] whose Wilson action Tr(FF†)/2 is nonnegative, vanishes on the vacuum, and reduces to −¼Tr(F²_μν) for Hermitian F. The J curvature at the vacuum forces the Higgs quartic λ = 1/2, the Mexican-hat potential shape, and spontaneous symmetry breaking with a VEV at rung 55 on the φ-ladder. Yukawa couplings inherit the φ-scaling of the mass law. Every theorem is compiled in Lean 4 with zero sorry and zero axioms beyond standard mathematics. The work converts three of seven structural gaps in the SM Lagrangian emergence program from open to closed.