Geometric Thermodynamics as a First Principle:Complete Proof of Unified Physics

This paper presents the complete mathematical proof that Geometric Thermodynamics constitutes a genuine first principle of physics. Based on the energy-information relation $\mathcal{N} = \kappa E^2$ derived from black hole thermodynamics \cite{Bekenstein1973, Hawking1975}, we rigorously derive quantum mechanics, unify fundamental interactions, predict cosmological parameters, and describe condensed matter phenomena without free parameters. The theory provides five parameter-free predictions matching experimental data with remarkable accuracy: fine-structure constant $\alpha = 7.242\times 10^{-3}$ (0.8\% accuracy) \cite{Mohr2016}, dark energy density $\rho_\Lambda = 4.63\times 10^{-10}\,\text{J/m}^3$ (13\% accuracy) \cite{Planck2018}, neutrino mass $m_\nu \approx 0.1\,\text{eV}$ \cite{KATRIN2022}, Weinberg angle $\sin^2\theta_W = 0.225$ (2.6\% accuracy) \cite{Weinberg1967}, and galactic parameters $\lambda_{\text{eff}} = 4.2\,\text{kpc}$, $\Gamma = 0.12$ \cite{Famaey2012}. We establish Geometric Thermodynamics as the foundational principle unifying all of physics.