Thermodynamic Geometrodynamics of the Swirl Condensate: A Unified Master Equation Formulation for Hydrogenic Orbitals in Swirl-String Theory

This research report presents an exhaustive derivation of the thermodynamic structure underlying Swirl-String Theory (SST), positing that the quantum mechanical behavior of electron orbitals arises not from probabilistic wavefunctions, but from the deterministic thermodynamics of a frictionless, incompressible "swirl condensate." By integrating the Abe-Okuyama quantum-thermodynamic isomorphism with the hydrodynamic axioms of SST, we derive a "Master Formula" for the thermodynamic equilibrium of hydrogenic orbitals. We demonstrate that the Bohr radius represents a surface of vanishing "swirl temperature" (geometric strain), and that excited states correspond to discrete adiabatic acoustic resonances of the vortex filament. The report further details the partition functions, entropy contributions (Kelvin-mode vs. Topological), and the Golden-ratio-governed mass hierarchies that emerge from this framework. We conclude that the Schrödinger equation is fundamentally an equation of state for a superfluid vacuum, where energy quantization results from the topological conservation of circulation and the mechanical stability of vortex cores. This document serves as a comprehensive foundational text, synthesizing the hydrodynamic ontology with thermodynamic principles to offer a parameter-free derivation of atomic structure.