The Hydrogen Spectrum as Torus Knot Resonance: Derivation of Atomic Energy Levels from Phase Geometry without the Schrodinger Equation

We derive the hydrogen atom energy spectrum E_n = -13.6/n^2 eV from a purely geometric framework in which the electron is a phase oscillator on a torus. Stable atomic states correspond to (n,1) torus knots: closed orbits where the radial phase completes exactly n cycles per angular orbit. The frequency hierarchy omega_n = omega_1/n^3 follows from Kepler's law in phase coordinates. The phase velocity v_n = v_1/n and energy E_n proportional to v_n^2 = 1/n^2 follow algebraically. The energy scale E_1 = (1/2) alpha^2 m_e c^2 = 13.6 eV is set by the fine structure constant alpha = 1/137, interpreted as the electromagnetic phase coupling strength. The Balmer series wavelengths match measurements to better than 0.06%. No wave functions, no Coulomb potential, and no Schrodinger equation are used. Atomic quantization emerges from the requirement that phase orbits close on a torus.