Presenting a unified framework in which a single variational functional K = K_ent + K_rec + K_bdry, defined on the space of quantum density operators, recovers thermodynamics, the Born rule, and the Einstein field equations as limiting cases of one principle. The functional contains two parameters (alpha_0 ~ 10^{-78}, epsilon ~ 10^{-45}), fixed by matching to the Planck scale and the cosmological constant. From these, we derive 18 quantitative predictions with zero adjustable constants, including CKM matrix elements (to 2.5%), PMNS mixing angles, the baryon-to-photon ratio (eta = 5.5e-10, observed 6.1e-10), and a resolution of the strong CP problem without an axion. We test the framework at galactic scales against 117 SPARC galaxies with compressed dark matter halo fits from Li et al. (2022). The K-functional predicts that the inner dark matter density slope gamma is controlled by Phi_total/sigma^2_NFW. We find gamma = 0.834 log10(Phi/sigma^2) + 0.299 with Pearson r = 0.896 (p = 2.6e-42). The coefficient 0.834 is reproduced to 1.5% by Blumenthal adiabatic contraction, with K-entropy conservation providing the information-theoretic basis for the adiabatic invariant. The cusp-core problem is resolved: inner slopes are set by the K_bdry/K_ent balance, with a transition at M_200 ~ 2e10 solar masses.