Structural Origin of Cosmological Density Ratios: Discrete L1 cosmological realizations and a Lorentz-type square-root closure

This work investigates a discrete structural origin of the cosmological density relation \[\Omega_m^2 = 3\,\Omega_\Lambda\,\Omega_b.\] The coefficient \(3\) is not introduced as an observational fit parameter. It is inherited from the pre-observational L1 construction through the fixed update \(q=2\) and \(q+1=3\). For a positive structural ratio \(T\), the cosmological density hierarchy is defined by \[\Omega_\Lambda:\Omega_b:\Omega_m=1:\frac{1}{3T^2}:\frac{1}{T},\] which implies the quadratic relation exactly. Using the primitive-divisor structure inherited from the companion foundational construction, the first pre-exception window \(n=1,2,3,4\) generates eight discrete candidate cosmological realizations from the two L1 ratio maps \(R(n)\) and \(S(n)\). For each candidate, the dimensionless quantity \[Z_c(T)=\frac{\Omega_{dm}+\Omega_m}{\Omega_\Lambda}\] is evaluated. Requiring the Lorentz/light-cone square-root factor \(\sqrt{Z_c(T)}\) to close rationally reduces the eight candidates directly to \[R(q)=R(2)=\frac{13}{6},\qquadS(q^2)=S(4)=\frac{31}{24}.\] For the \(R(q)\) survivor, \[(\Omega_\Lambda,\Omega_{dm},\Omega_b,\Omega_m)=\frac{1}{247}(169,66,12,78),\] and the corresponding square-root structure gives \[\sqrt{Z_c(R(q))}=\frac{12}{13}.\] It also admits a real timelike Lorentz factorization with \[\cosh\eta=\frac{13}{12},\qquad\sinh\eta=\frac{5}{12},\] and light-cone eigenvalues \(3/2\) and \(2/3\). The main structural result is therefore a pre-observational reduction \(8\to2\). No observational density value is used in this selection. Planck 2018 and DESI DR2 data are used only as downstream comparison diagnostics. The proposed relation is a structural constraint among standard cosmological density parameters. It does not by itself specify the microscopic nature of dark matter or dark energy, nor does it replace standard FLRW expansion dynamics.