Wave-free EBK quantization from proper-time holonomy: the 1/n^2 hydrogenic Coulomb ladder

De Broglie’s “phase harmony” suggests that quantization can be understood as a requirement of global phase consistency after completing the fundamental cycles of a bound orbit. We show that the Einstein–Brillouin–Keller (EBK) quantization rules for planar Coulomb orbits can arise without introducing a matter wave, from a long-lived localized internal clock whose phase advances in proper time at a fixed rest-frame angular frequency. In the slow–fast regime, special-relativistic time dilation makes the clock accumulate a small phase lag relative to laboratory time. After subtracting the trivial rapid ticking at the carrier frequency, the remaining drift defines two independent loop phases on the Kepler torus: one around the azimuthal cycle and one around the radial cycle (the torus generators). To leading order these generator phases are proportional to the corresponding Kepler actions, with a single action scale set by the reduced mass, the speed of light, and the clock frequency. Requiring the clock factor to return to itself after each generator loop—its holonomy, meaning the accumulated loop phase—yields closure conditions identical in form to EBK quantization, including the universal turning-point and Coulomb-origin phase slips known as the Maslov and Langer shifts. Because the Coulomb energy depends only on the principal action given by the sum of the radial and azimuthal actions, these closure conditions reproduce the familiar hydrogenic inverse-square bound-state ladder with the expected reduced-mass scaling. We also show how closure can carry an energetic bias in a frequency-local sealed-clock regime: if electric-type dipole radiation is cancelled exactly at the carrier frequency but nearby detuned sidebands radiate, then a holonomy mismatch shifts spectral weight into detuned lines and increases near-carrier leakage quadratically near closure. The identification of the action scale with Planck’s constant is treated as an external calibration.