The three-line derivation of the hydrogen spectrum Three panels. First, a vibrating electron pumping isotropic ripples at the Compton wavelength. Second, the moving electron Doppler-compressing those ripples ahead of it to form the de Broglie pilot wave. Third, the pilot wave wrapping a closed orbit so that only whole numbers of wavelengths survive, fixing the energy levels. From heartbeat to orbit — the three-line derivation A vibrating particle in a wave-bearing medium quantizes itself. No probability postulate is added by hand. ① The engine λc Electron breathes at ωc λc = h ∕ mec ≈ 2.43 pm ② The lens v compressed → λB stretched Doppler bunches the wavefronts λB = λc · (c ∕ v) = 137 λc at v = αc ③ The lock p⁺ n = 4 shown Whole wavelengths self-reinforce 2πr = n λB ⟹ En = −13.6 ∕ n² eV Each step is ordinary fluid dynamics; quantization is the geometric consequence of closing the loop.