Comparative Thermodynamics of Topological Defects and Quantum Ensembles:\\ Structural Divergence between Swirl--String Theory and the Abe--Okuyama Framework

This article presents a comparative thermodynamic analysis of two conceptually distinct frameworks for microscopic physics: the information-theoretic thermodynamics of Abe and Okuyama (AO), and the hydrodynamic thermodynamics of Swirl--String Theory (SST). AO operates entirely within standard quantum mechanics, defining temperature and entropy from the Shannon information of pure-state expansion coefficients in a fixed Hilbert space. SST, by contrast, posits a physical, incompressible swirl condensate and identifies particles with knotted vortex filaments; thermodynamic variables are reinterpreted as geometric and topological properties of these defects. We formulate and compare the definitions of temperature, heat, work, and entropy, and derive the respective equations of state. SST predicts a quadratic energy law $E }^{2}$ and a linear low-temperature heat capacity $C_V }$, in contrast to the gapped/exponential or equipartition-type behavior of the AO particle-in-a-box model. We further discuss the Golden Layer hypothesis in SST, which endows the vacuum enthalpy with discrete scale invariance, and contrast it with the smooth Shannon entropy structure in AO. The comparison highlights an ontological schism: AO treats thermodynamics as an emergent information calculus over quantum amplitudes, whereas SST treats it as the elastic mechanics of a continuous fluid substrate. We identify falsifiable differences in low-temperature heat capacity, stability mechanisms, and possible temperature dependence of decay rates, and outline experimental and analogue-gravity tests that could discriminate between the two pictures.