Structural Constraints on the Dark-Energy Equation-of-State Parameters in the Four-Sector FSC Cosmological Model: Analytic Proof of the Lower Bound on w₀ and Incompatibility with DESI 2025

Abstract We investigate the equation-of-state (EoS) parameters (w0,wa) of dark energy withinthe Four-Sector FSC (Four-Sector Cosmological) framework, a cyclic cosmologicalmodel in which the observable universe (Sector I), dark energy (Sector II), dark mat-ter (Sector III), and a CPT-conjugate sector (Sector IV) exchange energy throughtime-dependent transition rates ̃νk(τ). Using an extended suite of numerical ex-periments (Models M6 through M8) combined with analytic estimation, we demon-strate that the FSC four-sector ordinary differential equation (ODE) system imposesa structural lower bound w0 ≥ wmin0 ≈ −0.64,independent of the functional form of ̃νk(τ), provided that the observed densityfractions (Ωb,ΩDM,ΩΛ) = (0.049, 0.268, 0.683) are satisfied. This bound is incom-patible with the DESI 2025 best-fit value w0 = −0.838, establishing that the current FSC ODE topology cannot reproduce the DESI result. We further show that thespontaneous convergence β →0 observed in Model M8 is a numerical manifestationof the underlying CPT symmetry of the FSC framework. Three minimal structuralmodifications capable of resolving this tension are proposed, providing a roadmaptoward a next-generation FSC model (Paper I).