Thermodynamics of the Swirl Condensate: Temperature, Heat, and the Geometric Deformation of Equilibrium Boundaries

In Swirl--String Theory (SST), a bound electronic state is represented by a closed swirl filament whose geometry contains two intrinsic length scales: the core radius \( r_c \), which characterises the topological filament itself, and the Bohr radius \( a_0 \), which sets the equilibrium size of the Coulomb envelope. Although previous SST thermodynamic formulations were expressed solely in terms of \( r_c \), a physically complete treatment of atomic systems requires both scales. Using the canonical SST relation we show that \( a_0 \) is not an independent constant but an emergent, dynamically coupled outer boundary of the ground--state hydrogen torus. This leads naturally to a two--scale swelling theory in which thermodynamic deformation involves simultaneous variations of the core and orbital radii, showing that orbital and core strains coincide to first order because \(a_0\) inherits its variation from \(r_c\). The resulting enthalpy \( H_{(R,r)} \) produces an effective stiffness \[ K = K_{RR} a_0^2 + 2K_{Rr} a_0 r_c + K_{rr} r_c^2, \] yielding a low--temperature equation of state with both thermodynamic and chronometric behaviour dominated by the coupling between the two geometric scales. This framework unifies swelling dynamics, heat capacity, and swirl--clock time dilation in a single analytic structure, providing a consistent thermodynamic description of hydrogen and establishing a foundation for SST--based models of atomic stability and excitation.