Geometric Thermodynamics: A Unified Framework from Black Holes to Quantum Condensates with Predictions for the Cosmic Microwave Background

We present a unified theoretical framework, Geometric Thermodynamics, based on a single first principle: $\Nstruct = \kappa \irr^{2}$, where $\Nstruct$ is dimensionless structural information, $\irr$ is irreducible energy, and $\kappa$ is a system-dependent constant. For gravitational systems, $\kappa = \kappab = 4\pi\grav/(\planck\light^{5})$; for condensed matter systems, $\kappa$ takes effective forms $\kappaeff$ constructed from microscopic parameters. This principle naturally leads to a static-dynamic decomposition $E = \irr + \dyn$ and $\Ntotal = \Nstruct + \Ndyn$, with $\Ndyn = \alpha(\dyn/\irr)\ln\Nstruct$ derived from perturbation theory. We demonstrate the framework's universality by applying it to: (i) Schwarzschild and Kerr black holes, recovering Bekenstein-Hawking entropy and Hawking temperature; (ii) superconductors and superfluids, deriving effective constants $\kappasc = \cohere\london^{2}\Delta/(\planck^{3}v_F)$ and $\kappasf = m\cohere/(\planck^{2}\rhos\sound^{2})$ that yield correct condensation energy scaling; (iii) cosmology, where a scalar-tensor realization with Planck-scale symmetry breaking naturally produces dark energy density $\rho_{\Lambda} \sim 10^{-123}E_{\Planck}^{4}$; and (iv) galaxy rotation curves, where universal parameters $\lambda_{\mathrm{eff}} = (4.23 \pm 0.28)$~kpc and $\Gamma = 0.121 \pm 0.008$ fit the SPARC sample without dark matter. Most profoundly, we establish a correspondence between the static sector ($\irr, \Nstruct$) and spatial geometry, and the dynamic sector ($\dyn, \Ndyn$) and temporal evolution, showing that time emerges as a perturbation on pre-existing spatial structure. Applying this to early-universe cosmology yields novel predictions for the Cosmic Microwave Background (CMB): suppressed large-scale power, a tensor-to-scalar ratio $r \approx 0.003$, and specific non-Gaussianity patterns. The framework's consistency across 40 orders of magnitude in energy scales suggests a deep connection between spacetime geometry, thermodynamics, and quantum information.