Applications of Geometric Thermodynamics: A Unified Theory of Fundamental Interactions Based on the Black Hole Constant

This paper presents a comprehensive unification of the four fundamental interactions (gravitational, electromagnetic, weak, and strong) within the framework of geometric thermodynamics based on the black hole constant $\kappa_B \equiv 4\pi G/(\hbar c^5)$. The theory originates from a single fundamental principle: the energy-information relation $\mathcal{N} = \kappa_B E^2$, which establishes a profound connection between energy and structural information. We demonstrate mathematically that all interaction strengths can be expressed in a unified mathematical form $g_i^2(E) = \kappa_B E^2 \gamma_i(E) \approx (E/E_P)^{\beta_i}$, with all couplings converging to unity at the Planck scale: $g_i(E_P)=1$. This framework yields five parameter-free predictions with remarkable accuracy: (1) the fine-structure constant $\alpha = 7.242 \times 10^{-3}$ (0.8\% accuracy); (2) the dark energy density $\rho_\Lambda = 4.63 \times 10^{-10}\,\text{J/m}^3$ (13\% accuracy); (3) the neutrino mass scale $m_\nu \approx 0.1\,\text{eV}$; (4) the Weinberg angle $\sin^2\theta_W = 0.225$ (2.6\% accuracy); (5) universal galactic parameters $\lambda_{\text{eff}} = 4.2\,\text{kpc}$, $\Gamma = 0.12$. The theory also provides geometric explanations for dark matter phenomena and quantum ordered states. Geometric thermodynamics thus offers a mathematically complete, empirically validated framework that unifies physics from black holes to elementary particles.