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 \[Ω_m^2 = 3\,Ω_Λ\,Ω_b.\] 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 \[Ω_Λ:Ω_b:Ω_m=1:1/3T^2:1/T,\] 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 \[Z_c(T)={Ωdm+Ω_m}{Ω_Λ}\] is evaluated. Requiring the Lorentz/light-cone square-root factor \(√Z_c(T)\) to close rationally reduces the eight candidates directly to \[R(q)=R(2)=13/6,(q^2)=S(4)=31/24.\] For the \(R(q)\) survivor, \[(Ω_Λ,Ωdm,Ω_b,Ω_m)=1/247(169,66,12,78),\] and the corresponding square-root structure gives \[√Z_c(R(q))=12/13.\] It also admits a real timelike Lorentz factorization with \[η=13/12,η=5/12,\] and light-cone eigenvalues \(3/2\) and \(2/3\). The main structural result is therefore a pre-observational reduction \(8→2\). 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.