Paper P: The Complete Sector Distribution Theorem — β_V as Master Parameter of the Four-Sector Cosmological Model

                                                       Abstract

We present the Complete Sector Distribution Theorem of the Four-Sector Cos-
mological (FSC) model, establishing βV —the vacuum expectation value (VEV) am-
plitude of the Sector-V scalar field—as the single master parameter governing the
partitioning of dark-sector energy densities. We prove two exact algebraic iden-
tities: (P-1) β2V + βV (1 −βV ) + (1 −βV ) ≡ 1, which identifies ΩII = β2V (dark
matter, from Paper M) and Ω(0)
IV = βV (1 −βV ) (CPT-conjugate sector, new), with
residual (1 −βV ) dispersed as CMB radiation; and (P-3) tan(θ −45◦) = 2βV −1,
which encodes the dark-matter/CPT-sector asymmetry as a single angular devia-
tion from perfect symmetry. We further derive ΩII −Ω(0)
IV = βV (2βV −1) (exact),
ΩII + Ω(0)
IV = βV (exact), and the approximate relation βV ≈ 1 −ΩΛ/√2 + εO ·ΩI
(error 0.005%, 27-fold improvement over the simple form), where εO is the Paper O
asymptotic accuracy parameter. We propose refining the Paper N initial condition
from Ω(0)
IV = 0.250 to βV (1 −βV ) = 0.24969, which would elevate Theorem P-1 to
machine precision. The CMB dispersion fraction (1 −βV ) = 0.4823 dilutes as ra-
diation with effective index neff = 4.051 (∆n = +0.051), reproducing Ωr to within
a factor of 1.46. The theoretical origin of the residual factor 1.038 in Theorem P-2
and of ∆n remain open problems for Paper Q.