Paper R: The Universal Master Parameter β_V — Single Origin of Dark Matter, CPT-Conjugate Sector, CMB Dispersion, and Hubble Constant in Four-Sector Cosmology

                                                      Abstract

     We establish that the vacuum expectation value (VEV) amplitude βV = 0.5177
of the Sector-V complex scalar field is the single master parameter of the Four-
Sector Cosmological (FSC) model, governing the complete energy structure of the
universe from the phase transition epoch to the present day.
Building on Papers P and Q (7; 8), we prove Theorem R-0: the present-day
CPT-conjugate sector density satisfies
ΩIV (t0) = 1 + εO
2 [βV (1 −βV )]2 = 1 + εO
2 ΩII (1 −βV )2,
to 0.053% accuracy, where εO = 0.01206 is the Paper O asymptotic parameter (6).
The effective dilution coefficient
g = ΩIV (t0)
Ω(0)
IV
= 1 + εO
2 βV (1 −βV ) = 0.12640
is numerically independent of βV (coefficient of variation 0.0000%) once εO is held
fixed, confirming that βV alone drives the CPT-sector evolution.
Theorem R-1 then gives the FSC Hubble constant as
H∗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 the
causal 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.