Structural Origin of Cosmological Density Ratios: $\Omega_m^2 = 3\Omega_\Lambda \Omega_b$

This work investigates a simple algebraic relation among cosmological density parameters $$\Omega_m^2 \approx 3\,\Omega_\Lambda\,\Omega_b$$ suggested by representative late-time cosmological parameter estimates reported by Planck and DESI analyses. We present a minimal structural framework in which the density sector is organized into three phases: an unconstrained component ($\Omega_\Lambda$), a semi-constrained component ($\Omega_{dm}$), and a fully constrained component ($\Omega_b$), with $\Omega_m = \Omega_b + \Omega_{dm}$. The hierarchy is parameterized by a single structural ratio $$R = \frac{26}{12} = \frac{13}{6},$$ which leads to $$\Omega_\Lambda : \Omega_m : \Omega_{dm} : \Omega_b=3R^2 : 3R : (3R-1) : 1.$$ After normalization the framework predicts $$\Omega_\Lambda = \frac{169}{247}, \quad\Omega_m = \frac{78}{247}, \quad\Omega_{dm} = \frac{66}{247}, \quad\Omega_b = \frac{12}{247}.$$ These values satisfy the quadratic closure relation $$\Omega_m^2 = 3\,\Omega_\Lambda\,\Omega_b$$ and correspond to the closure diagnostic $$Q = \sqrt{\frac{\Omega_m^2}{3\,\Omega_\Lambda\,\Omega_b}} = 1.$$ Using representative parameter combinations from Planck 2015, Planck 2018, and DESI 2024 cosmological analyses, the inferred values cluster near this predicted closure relation. The results suggest that the cosmological density sector may exhibit a simple structural organization at the level of parameter ratios.