Geometric Thermodynamics: A Unified Framework from Black Holes to Quantum Condensates and the Mathematical Origin of Fundamental Constants

This paper presents a comprehensive theoretical framework of geometric thermodynamics. Starting from the black hole constant $\kappa_B = 4\pi G/(\hbar c^5)$, we establish a universal energy-information relation $\mathcal{N}_{\text{struct}} = \kappa E_{\text{irr}}^2$, successfully unifying the descriptions of black hole thermodynamics, cosmological dark energy, galaxy rotation curves, quantum condensed matter, and high-temperature plasma. Further, we discover precise mathematical relations between fundamental physical constants and the golden ratio $\phi = (\sqrt{5}-1)/2$: $\alpha = (2\pi/7)\phi^{10}$ and $M_P/m_e = \phi^{-36}$, with accuracy exceeding $10^{-10}$. These relations reveal that physical constants are not arbitrary but determined by pure mathematical constants. More importantly, we demonstrate that the golden ratio $\phi$ actually functions as a physical scale factor, with length being the fundamental building block of the universe - time, mass, and other physical quantities can all be derived from length. This framework not only unifies gravitation and quantum phenomena but also provides concrete evidence for the mathematical origin of the physical world.