Dark Energy Equation of State in the Four-Sector Cosmology (FSC): Numerical Convergence to ΛCDM and Constraints from DESI 2025

AbstractWe systematically investigate the dark energy equation of state w(z) within theFour-Sector Cosmology (FSC), a framework in which the observed cosmic composition—baryonic matter (5%), dark matter (27%), and dark energy (68%)—emerges dynam-ically from inter-sector density flows governed by a coupled ordinary differentialequation (ODE) system. Employing five distinct numerical models that vary thetransition-point condition and the definition of ρΛ, we demonstrate that constantinter-sector transition rate ̃νII inevitably produces wa ∈ [−0.06, 0] in the CPL pa-rameterization w(z) = w0 + waz/(1 + z). We prove analytically that this result isa structural consequence of the ODE architecture: the logarithmic curvature of ̃ρIIwith respect to ln a is non-positive, forcing wa ≤0. Comparison with the DESI 2025measurement w0 = −0.838, wa = −0.620 reveals that while the sign of wa is cor-rectly reproduced, the magnitude differs by one to two orders. We interpret this asevidence that FSC with constant ̃νII provides a structural explanation for the near-ΛCDM behavior of the present universe, and that a cosmological time-dependence ̃νII(τ) is required to fully reproduce the DESI signal. This paper serves as Paper Gin the FSC series and constitutes the first systematic numerical verification of FSCdark energy dynamics.