Four-Sector Cosmology (FSC): Dual Equation-of-State Structure of the Complex Scalar Field and Observational Constraints from DESI BAO 2025 Paper J

Abstract We present Paper J of the Four-Sector Cosmology (fsc) series, establishing thedual equation-of-state (eos) structure of the fsc complex scalar field Φ and itsobservational confrontation with the DESI BAO Year-1 2025 results.The fsc framework assigns the four cosmic density components — baryonicmatter ( ̃ΩI ≈ 0.050), dark matter ( ̃ΩII ≈ 0.268), dark energy ( ̃ΩIII ≈ 0.683), andthe CPT-conjugate sector ( ̃ΩIV ) — to four branches of complex time, unified by asingle Mexican-hat complex scalar potential.Within Policy B ( ̃ρII ≡ |Φ|2), we demonstrate that the scalar field admits twophysically distinct eos representations : (i) the fluid-dynamical eos wB derived fromthe logarithmic field velocity, and (ii) the microscopic kinetic/potential eos wKE/PEderived from the ratio KE/PE. These two representations bracket the DESI centralvalue:wB = −1.000 < wdesi0 = −0.838 < wKE/PE = −0.778. (1)A comprehensive eight-dimensional parameter optimisation (Phases 3a through3c, over 105 evaluations) establishes that wKE/PE = −0.778 is a structural upperlimit of Policy B: no extension of the parameter space (τ0 ≤ 12, n ≤ 20, λ ≤ 2,Φ0 ≤4v) overcomes this bound. This constitutes Theorem J.5.The bracketing relation (1) is a falsifiable prediction : future Euclid or DESI DR2measurements of w0 will determine which fsc eos representation is observationallyprimary.