Paper Q: The Sector-V Phase Transition and the Cosmological Master Redshift — Exact Determination of z_V = 3697.56 from the FSC Distribution Theorem, and the Structural Constant C = 1.038

                                                         Abstract
     We establish the cosmological master redshift zV = 3697.56 of the Four-Sector
Cosmological (FSC) model as the unique epoch at which the radiation density
parameter Ωr equals the Sector-V vacuum expectation value (VEV) amplitude
βV = 0.5177, and simultaneously ΩM = 1 −βV . This result, Theorem Q-0, holds to
machine precision (< 10−14) and constitutes the cosmological realisation of the alge-
braic identity Theorem P-1 of Paper P (7). We further derive a closed-form expres-
sion for the Sector-V critical temperature: kBTc = kBT0 ·(ΩM/Ω(0)r ) ·βV /(1 −βV ) =
0.8687 eV (Theorem Q-3), corresponding to Tc = 10080 K at zV = 3697.56. A
structural constant C = 1.038 emerges independently in two contexts: (i) the βV
precision formula of Paper P (Theorem P-2 improved), and (ii) the redshift hierar-
chy zV −zeq = C ×4(βV −1
2 ) ×zeq (Theorem Q-1 candidate). The near-coincidence
kBTc ≈ e · ΩM [eV] (error 0.81%, Theorem Q-2 candidate) is traced to the ap-
proximate identity kBT0/Ω(0)r ≈ e · (1 − βV )/βV (Theorem Q-4 candidate, error
0.79%), suggesting a deep connection between the present CMB temperature and
FSC parameters. The theoretical origin of C is stated as Open Problem Q-C for
Paper R.