Coulomb, Rydberg and Phase Closure: A Minimal Carrier-Flux Reconstruction of the Circular Hydrogen Spectrum

We give a minimal semiclassical reconstruction of the circular hydrogen spectrumby separating two roles that the usual Bohr construction combines. The exteriorCoulomb field is represented as a conserved isotropic carrier momentum flux on theexterior domain, fixed by a single Coulomb normalisation. A charge on a circularorbit laterally intercepts this flux as it sweeps; the intercepted exposure per radianequals the kinetic scale of the orbit, and the virial theorem makes the Rydbergbinding scale one half of this counted flux exposure. The energy scale is thus fixed,and the corresponding flux-exposure count is continuous in the orbital radius,before any integer is introduced. The integer is supplied separately by phase: aPlanck–Einstein matter clock of the electron, with Lorentz covariance, gives thede Broglie phase, and closed phase return around the orbit selects the admissibleorbits. Combining this phase closure with circular Coulomb balance yields theBohr–Rydberg radii, velocities, energies, and angular momenta, so that Bohr–Sommerfeld quantisation is recovered as a closed-phase condition. The distinctivefeature is that the Rydberg scale is counted from the flux before any quantisationis imposed. As an internal consistency check, the same carrier-counting factorsapplied to a moving source yield the standard retarded Li´enard–Wiechert exteriorpotential structure. The spectral construction uses only α, me, ℏ, and c, with noadjustable parameter, and is falsifiable through explicit failure modes; it does notderive full quantum mechanics or quantum electrodynamics, and is offered as aself-contained reconstruction of the circular Coulomb sector.