Analytical Derivation of the Critical Angular Momentum and VEV Stability Displacement in the Four-Sector Complex Cosmological Model

Abstract We present an analytical framework for deriving the critical angular momen-tum ℓc and the vacuum expectation value (VEV) stability displacement ε in theFour-Sector Complex (FSC) cosmological model introduced in Papers D and F. Byanalyzing the effective potential of the complex dark energy scalar field Φ = reiθunder the Mexican-hat potential with centrifugal correction, we derive the exactcondition for the critical orbit rc = (1 + ε)v, where v is the VEV. The displace-ment ε ≈ 0.0303 is shown to arise self-consistently from the asymmetry parameterδ = 0.00195 and the cosmological phase rotation timescale τeff = 5.69 ×10−3 (inln a units), yielding the master relationε = δ2r4c/32λv6τ2eff,which is self-consistent to 7% precision. Furthermore, we identify a “quintic reso-nance” between the field rotation timescale and the Sector-IV decay rate:Λ−1IV ≈5 ×2π ×τcosmo (2.3% precision),suggesting a deeper topological structure to be explored in Paper .