Geometric Thermodynamics II:Scalar-Tensor Realization and Natural Emergence of Dark Energy

This paper presents the cosmological realization of geometric thermodynamics, a unified framework based on the black hole constant $\kB = 4\pi G/(\hbar c^5)$. Building upon the fundamental energy-information relation $\Nstruct = \kB E^2$ established in Paper I, we construct a minimally coupled scalar-tensor theory that naturally incorporates geometric thermodynamics. The theory features a Planck-scale symmetry-breaking phase transition, described by a quartic potential with an exceptionally small symmetry-breaking parameter $\epsilon \sim 10^{-122}$. We provide complete variational derivations of the field equations and demonstrate that the residual vacuum energy density after symmetry breaking is $\rho_\Lambda = \epsilon V_0 \sim 10^{-123} \EP^4$, in precise agreement with the observed dark energy density. The geometric temperature formula $T = 1/(2k_B\kB E)$ applied to the de Sitter horizon yields $T_{\text{dS}} \sim 10^{-29}$ K, consistent with the Gibbons-Hawking temperature of de Sitter space. Furthermore, we show how local scalar field perturbations generate effective dark matter behavior on galactic scales, with universal parameters $\lambdaeff \approx 4.2$ kpc and $\Gammapar \approx 0.12$ that will be tested in Paper III. All mathematical derivations are presented in full detail, establishing a self-consistent cosmological framework that naturally explains dark energy without fine-tuning or additional dark components.