Geometric Thermodynamics IV:Superconductivity and Superfluidity as Quantum Ordered States of Vanishing Dynamic Information

This paper establishes a profound correspondence between geometric thermodynamics and quantum ordered states in condensed matter physics, specifically superconductivity and superfluidity. Building upon the fundamental framework developed in Paper I, where the black hole constant $\kB = 4\pi G/(\hbar c^5)$ connects energy to structural information via $\Nstruct = \kB E^2$, we demonstrate that analogous relations govern quantum condensates through effective constants $\kappa_{\text{eff}}$. For superconductors, we derive $\kappa_{\text{eff}}^{\text{(SC)}} = \frac{16\pi^2 \xi}{c^2 \hbar v_F H_c^2}$ from macroscopic electrodynamics, where $\xi$ is the coherence length, $H_c$ the thermodynamic critical field, and $v_F$ the Fermi velocity. For superfluid $^4$He, we obtain $\kappa_{\text{eff}}^{\text{(SF)}} = \frac{m_4 \xi}{\hbar^2 \rho_s c_s^2}$ from the Gross-Pitaevskii equation, with $\xi \approx \SI{1}{\angstrom}$ the healing length, $\rho_s$ the superfluid mass density, and $c_s = \SI{238}{\meter\per\second}$ the sound velocity. In both cases, the condensation energy $E_{\text{cond}}$ serves as the irreducible energy $E_{\irr}$, yielding structural information $\Nstruct = \kappa_{\eff} E_{\text{cond}}^2$. The ideal ordered states (Meissner state for superconductors, ground state for superfluids) correspond to vanishing dynamic information ($\Ndyn = 0$) due to $E_{\dyn} = 0$. Dynamic excitations (vortices, currents) inject finite $\Ndyn$ according to $\Ndyn = \alpha (E_{\dyn}/E_{\irr}) \ln \Nstruct$. We derive testable predictions: critical current density $j_c \propto H_c/(\lambda_L \sqrt{\ln \Nstruct})$ for superconductors and vortex nucleation condition $\Omega_c \propto (\hbar/m_4 R^2) \ln \Nstruct$ for superfluids. Experimental data from NbTi, YBCO, and liquid helium validate these predictions within 10--15\% accuracy after consistent unit treatment. This work establishes geometric thermodynamics as a universal framework bridging gravitational and quantum ordered phenomena, revealing deep connections between spacetime geometry and quantum coherence.