Geometric Closure of the Hydrogen Rydberg Series from a Discrete Shell Ladder

This preprint examines what structural information the hydrogen principal series determines independently of dynamical explanation. It presents a constrained geometric correspondence in which spectral modes are treated on discrete shells under explicit tangential phase-closure assumptions. With the Rydberg constant R∞ used as an empirical calibration anchor, the manuscript derives the wavelength-radius relation λ = 4π²rₙ and the radius ladder rₙ = n²/(4π²R∞), from which the standard Rydberg transition formula follows algebraically for principal-series transitions. Comparisons with NIST reference wavelengths for the Lyman, Balmer, and Paschen families show agreement at the principal term-value level within the stated scope. The work is shared as a public preprint and has not been peer reviewed.