Geometric Thermodynamics I:} \\ \textbf{The Black Hole Constant and the Universal Energy-Information Relation

This paper establishes the theoretical foundation of \textbf{geometric thermodynamics}, a new framework that unifies gravitational and quantum phenomena through a single fundamental constant: the \textbf{black hole constant} $\kB = 4\pi G/(\hbar c^5)$. We demonstrate that the relation $\Nstruct = \kB E^2$, exact for Schwarzschild black holes, can be elevated to a universal principle connecting energy to structural information. By decomposing total energy into irreducible (structural) and dynamic components, we introduce a bipartite information measure: $\Nstruct = \kB E_{\irr}^2$ for structural information, and $\Ndyn = \alpha (E_{\dyn}/E_{\irr}) \ln \Nstruct$ for dynamic information, where $E_{\dyn}$ represents extractable energy (rotational, electromagnetic, etc.). From this, a fundamental temperature law $T = 1/(2k_B\kB E)$ emerges naturally. We provide complete derivations for Schwarzschild, Kerr, and Reissner-Nordstr\"om black holes, establishing $\kB$ as the fundamental conversion rate between energy and spacetime information. The framework is shown to be fully consistent with black hole thermodynamics and provides a unified description from static to dynamic black holes. All derivations are presented in full mathematical detail, forming a self-contained theoretical basis for subsequent applications in cosmology and condensed matter physics.