We derive eight Standard Model relationships from the Feigenbaum constants α = 2.50291 and δ = 4.66920 at cascade Level 2 (post-EWSB). Five results stand out. First, the Koide sum rule Q = 2/3 follows exactly and algebraically from the cascade bifurcation amplitude δ_K = √2, requiring no numerical fitting. Second, the lightest-quark mass ratio m_d/m_u = √δ is reproduced to 0.056% — a Level 0 connection involving only the parameter-space stretching constant. Third, the charm-to-muon mass ratio m_c/m_μ = T⁴ (0.521%), where T = δ/α is the Transition Constant. Since Paper 44 established T² ≈ α₃/α₂ (strong-to-weak coupling ratio, 0.348%), this yields m_c/m_μ ≈ (α₃/α₂)² — a gauge-fermion bridge. Fourth, the top quark mass satisfies m_t = v/√2 (0.778%), confirming the top Yukawa as the Level 2 ground state with y_t ≈ 1. Fifth, and most precisely, the Higgs boson mass satisfies m_H = m_t × α/T² = (v/√2) × α³/δ² = 125.215 GeV (0.028%), the cleanest numerical result in this series. The Higgs self-coupling λ = (α/T²)²/4 (0.055%) follows directly. The generation ladder m_t/m_c = (αδ)² (0.638%) and m_b/m_μ = (αδ)^{3/2} (0.913%) complete the picture. Together with Paper 44, the full Standard Model gauge-fermion-Higgs structure — three coupling ratios, the Weinberg angle, the fine structure constant, the dominant fermion mass ratios, and the Higgs mass — emerges from two constants. Paper 45 of the Resonance Theory series.