Abstract We establish that the vacuum expectation value (VEV) amplitude βV = 0.5177of the Sector-V complex scalar field is the single master parameter of the Four-Sector Cosmological (FSC) model, governing the complete energy structure of theuniverse from the phase transition epoch to the present day.Building on Papers P and Q (7; 8), we prove Theorem R-0: the present-dayCPT-conjugate sector density satisfiesΩIV (t0) = 1 + εO2 [βV (1 −βV )]2 = 1 + εO2 ΩII (1 −βV )2,to 0.053% accuracy, where εO = 0.01206 is the Paper O asymptotic parameter (6).The effective dilution coefficientg = ΩIV (t0)Ω(0)IV= 1 + εO2 βV (1 −βV ) = 0.12640is numerically independent of βV (coefficient of variation 0.0000%) once εO is heldfixed, confirming that βV alone drives the CPT-sector evolution.Theorem R-1 then gives the FSC Hubble constant asH∗0 = Hobs0√1 + g βV (1 −βV ) = 68.46 km s−1 Mpc−1,within 0.07% of the independent FSC prediction 68.41 km s−1 Mpc−1.Together with the prior results of Papers P and Q, these theorems complete thecausal chain:V (Φ) μ/√λ−−−→ βV −→ {ΩII, Ω(0)IV , zV , Tc, H∗0 , g, C}.The residual 0.053% in Theorem R-0 and the Euler-number connections in Theo-rems Q-2 and Q-4 are formulated as Open Problems for Paper S.