Paper FBW-C: First-Principles Derivation of the Sector-IV Decay Rate, Scaling Exponent, and Baryon-to-Photon Ratio in the Four-Sector Cosmological Framework

We derive three central quantities of the Four-Sector Cosmology (FSC) framework from first principles, using no free parameters beyond the single dimensionless ratio R = Omega_DM/Omega_b = 5.4 and the Z4 symmetry structure. Theorem FBW-C.1 establishes lambda_IV = R/t_0 = 1.240e-17 s^-1 (0.80% agreement with Paper D). The Master Equation |p| = 6 ln R/pi = 3.221 (0.007% accuracy) is derived from the T-transformation of the Dedekind eta-function in the FSC partition function (CQ-IX), yielding the scaling exponent beta = 4 ln R/(3 pi) = 0.7158 (0.03% accuracy). The transition time t_switch = beta * tau_IV = 1.829 Gyr defines the baryon-generation window (t_Page, t_switch) = (1.771, 1.829) Gyr -- a 58 Myr interval determined entirely by R, t_0, and the Z4 group structure. Theorem X (first derivation): N_IV(t_switch) = N_IV,0 * R⁻ᵇᵉᵗᵃ, proved via lambda_IV * t_switch = beta ln R = ln R^beta. The baryon-to-photon ratio eta_b = 6.24e-10 (2.3% agreement with BBN+CMB value 6.10e-10) follows with no free parameters. OQ-1 partial answer: the coefficient 2/9 in beta = 2|p|/9 has group-theoretic origin 9 = |Z4| + |Sub(Z4)| + |Gen(Z4)| = 4+3+2. Complete derivation chain: ei pi+1=0 -> Z4 -> (R, t_0, delta) -> lambda_IV -> beta -> t_switch -> eta_b = 6.24e-10.