Structural Lower Bounds on w0 and the Minimal Topology Principle in FSC Four-Sector Cosmology Paper I: Extended Dark-Energy Definition, Topology Perturbations, and DESI 2025 Tension

Abstract We investigate structural lower bounds on the dark-energy equation-of-state pa-rameter w0 within the Four-Sector Cosmology (FSC) framework, extending theresults of Paper H (w0 ≥−0.64 for the baseline serial-ring topology). Two indepen-dent modifications are examined: Candidate B, an extended dark-energy definitionρDE = ̃ρII + δ ̃ρIII with a free mixing parameter δ ≥0; and Candidate D, a directSector-I → Sector-III coupling with rate γ ≥ 0. Numerical optimisation (modelsM9 and M10) establishes that the combined lower bound is w0 ≥ −0.73 ±0.01,only marginally below the Paper H bound despite substantially enlarged parameterspaces. Critically, both β →0 (Sector-IV decay, first observed in Paper H / M8) andγ → 0 (direct I→III coupling, new in M10) exhibit spontaneous zero-convergenceduring unconstrained optimisation, indicating that the minimal serial-ring topologyI →II →III →IV →I is the unique thermodynamically preferred ground state ofFSC. We formalise this observation as the Minimal Topology Principle (The-orem 5.5). The persistent gap ∆w0 = |−0.838|−|−0.73| ≈ 0.10 relative to DESI2025 is shown to be of structural, not parametric, origin, and we identify CandidateC (complex density field Φ = |Φ|eiθ) as the minimal extension capable of resolvingthis tension. A falsifiable FSC prediction, wFSC0 ≥ −0.73, is stated for verificationby DESI 2027.