Paper FBW-D: Dynamical Origin of beta = 2|p|/9 and Unification of Three Descriptions of eta_b in Four-Sector Cosmology

We resolve all three open questions (OQ-1, OQ-2, OQ-3) left by Paper FBW-C in the Four-Sector Cosmology (FSC) framework. Theorem A: the Z4 two-phase decay has dynamical eigenvalues lambda_I = Gen(Z4)/Sub(Z4) = 2/3 and lambda_II = Gen(Z2)/Sub(Z2) = 1/2, with CPT-symmetric product Lambda_CPT = |Z4|*lambda_I*lambda_II = 4/3. Theorem B (resolving OQ-1): combining the Dedekind eta-function T-transform eta(tau+1) = exp(i*pi/12)*eta(tau) with the Z4 orbital structure gives |p| = 6*ln(R)/pi and beta = Lambda_CPT * ln(R)/pi = 4*ln(R)/(3*pi) = 2|p|/9, where the denominator 9 = |Z4|+|Sub(Z4)|+|Gen(Z4)| = 4+3+2 and numerator 2 = |Gen(Z4)| are dynamically fixed by the group structure of Z4. Theorem 4.1 (resolving OQ-2): the FSC partition function Z_FSC(tau) is invariant under the full SL(2,Z) action (modular weight k=0), with beta/|p| = 2/9 as the internal fixed ratio. Theorem C (resolving OQ-3): the three descriptions of eta_b are unified by eta_b^QI/eta_b^TH = R = Omega_DM/Omega_b and eta_b^TH = eta_b^GT with no free parameters. Section 6 decomposes the 2.3% deviation into five components (eps_eta + eps_R + eps_beta + eps_ft + eps_delta = 2.28%), yielding eta_b^precise = 6.17e-10 (Delta = 1.1%). Observational predictions: w_a = +0.003 (Euclid), mu = 3.3e-4 (PIXIE), DeltaH0 = 2.69 km/s/Mpc. All results from two inputs only: R = 5.4 and |Z4| = 4.