The Proton-to-Electron Mass Ratio from Cascade Geometry: A First-Principles Derivation of the QCD–Electroweak Scale Interface

The proton-to-electron mass ratio mp/me = 1836.15267343 (CODATA 2018) has no accepted derivation from first principles. The Wyler approximation 6π⁵ ≈ 1836.118 [0.002%] has been known for fifty years but resists theoretical explanation. This paper derives the cascade tree-level prediction from the two Feigenbaum constants α = 2.502907875 and δ = 4.669201609: mp/me = 4α⁵δ = 4α²/(T·α_em) = 1834.539 [error 0.088%], where T = δ/α = 1.865510775 is the cascade transit ratio and α_em = 1/136.576 is the cascade-derived fine structure constant (Paper 44). Five equivalent forms are presented, the most compact being (mp/me) × α_em = 4α²/T = 13.432 [0.088%], which connects the mass ratio, the electromagnetic coupling, and the cascade geometry in a single equation with no free parameters. The residual 0.088% = 0.12 × α_em is consistent with known QCD electromagnetic corrections to the proton mass. This result completes the cascade derivation of the Standard Model mass hierarchy from the electron (0.511 MeV) through the proton (938 MeV), top quark (174 GeV), and Higgs boson (125 GeV) — twelve orders of magnitude from two numbers. Paper 49 of the Resonance Theory series.