Classification of Transuranium Elements in Terms of `Winding' Numbers in the Bohr-Sommerfeld Model

We revisit the Bohr-Sommerfeld atomic model to explore hydrogen-like ions of Uranium $Z=92$, Oganesson $Z=118$, and hypothetical superheavy elements beyond. Although superseded by the Dirac equation and modern quantum electrodynamics, the semiclassical approach offers a historically and pedagogically valuable perspective. Using the Sommerfeld fine structure formula and computer algebra methods, we demonstrate the appearance of self-intersecting orbits in super strong Coulomb fields, beginning with Oganesson and hypothetical elements up to Zle137. These orbits can be classified by their `winding' numbers, providing a simple topological description of Coulomb field strength in this framework. Our results highlight a conceptual bridge between early quantum theory and modern superheavy element physics.