The Fermion Level: Standard Model Mass Hierarchy from Cascade Geometry

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 δK = √2, requiring no numerical fitting. Second, the lightest-quark mass ratio md/mᵤ = √δ is reproduced to 0. 056% — a Level 0 connection involving only the parameter-space stretching constant. Third, the charm-to-muon mass ratio mc/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 mc/m_μ ≈ (α₃/α₂) ² — a gauge-fermion bridge. Fourth, the top quark mass satisfies mₜ = v/√2 (0. 778%), confirming the top Yukawa as the Level 2 ground state with yₜ ≈ 1. Fifth, and most precisely, the Higgs boson mass satisfies mH = mₜ × 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 mₜ/mc = (αδ) ² (0. 638%) and mb/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.