Paper CQ-VI: Four-Sector Cosmology - Unified Complex FSC Metric on a Riemann Surface, Lindblad Operators from Quantum Field Theory, CMB Polarisation Transfer Map, Hubble Tension Resolution via Shell Oscillation, and Primordial Gravitational Wave Spec

Abstract We present Paper CQ-VI of the Four-Sector Cosmology (FSC) programme, resolving five high-priority open questions (OQ-CQ6-1, 3, 5, 9, 11) from Paper CQ-V and establishing five new theorems (Theorems E-I). Theorem E (OQ-CQ6-1): We construct the unified complex FSC metric as a holomorphic family on the FSC Riemann surface Sigma_FSC = C*/Z_4, whose Z_4 monodromy group implements the sector cycle I -> II -> III -> IV -> I. The geometric structure is governed by the Euler identity exp(i pi) + 1 = 0. Theorem F (OQ-CQ6-3): We derive the Lindblad collapse operators L_i = sqrt(lambda) |vac_i><0bar| + |B><1bar| in FRW spacetime and derive the CMB polarisation power spectrum prediction C_ell^EE / C_ell^BB -> 1 for ell <= ell_*, with ell_* = pi/(H_0 T_osc) approximately 80, where T_osc approximately 7.96 Gyr is the FSC shell oscillation period. This is qualitatively distinct from the standard LambdaCDM prediction C_ell^BB << C_ell^EE. Theorem H (OQ-CQ6-9): The oscillating Israel thin shell Sigma_{I/IV} with period T_osc approximately 7.96 Gyr contributes Delta H_0^(6) = 2.71 +/- 0.30 km/s/Mpc to the Hubble tension, giving the total FSC prediction Delta H_0^FSC = 2.69 + 2.71 = 5.40 +/- 0.30 km/s/Mpc, consistent with the observed 5.6 +/- 0.4 km/s/Mpc at 0.5 sigma, with zero new free parameters. Theorem I (OQ-CQ6-11): The FSC saddle-point approximation in the complex time plane yields the primordial gravitational wave spectrum n_t^FSC approximately -0.064, r^FSC approximately 0.012, n_t/(-r/8) approximately 43, both within the LiteBIRD detection window and providing a decisive test against slow-roll inflation via the violated consistency relation n_t ≠ -r/8. All five results emerge from the single timescale hierarchy t_Page : tau_IV : T_osc = ln 2 : 1 : 5.13, determined by the single FSC parameter lambda = 1.25 x 10^{-17} s^{-1}. Paper CQ-VI establishes FSC as a geometrically unified framework in which the cosmic composition 5:27:68, the Hubble tension, the CMB polarisation structure, and the primordial gravitational wave background all derive from the Euler identity exp(i pi) + 1 = 0. Eleven open questions for Paper CQ-VII are registered.