Paper CQ-V: Resolution of Open Questions in FSC Horizon Thermodynamics - Israel Thin Shell, Quasar Peak Timing, Continuous Approximation, and Transfer Map

We resolve five open questions (OQ-CQ4-1 through OQ-CQ4-5) posed in Paper CQ-IV of the Four-Sector Cosmology (FSC) series. (OQ-CQ4-1) The FSC static metric for Sector IV is derived from w_IV = -1/3 via two consistent descriptions: a real-time power-law a_IV ∝ t4/9 and a complex-time relation a_IV = i t_R. (OQ-CQ4-2) Applying the Israel junction conditions to the Sector-I/IV boundary ΣI/IV, we derive a dust-like thin shell with surface energy density σ = c²λ/(4πG), zero pressure, and oscillation period T_osc ≈ 7.96 Gyr. A universal relation σ/T_H = c²k_B/(2Gℏ), independent of the FSC decay rate λ, is established as Theorem A. The shell entropy satisfies S_shell = A/(2G) = 2 S_BH. (OQ-CQ4-3) The Δt ≈ 0.5–1.5 Gyr offset between the FSC Page time t_Page = (ln 2)/λ ≈ 1.76 Gyr and the observed quasar peak at z ≈ 2–3 is explained analytically: the entropy-rate centroid is t_center = 1/λ = τ_IV ≈ 2.54 Gyr, giving Δt = (1−ln 2)/λ ≈ 0.778 Gyr and a predicted quasar peak at z_QSO^FSC ≈ 2.4. (OQ-CQ4-4) The continuous approximation |R|(t) = 4f(t) is rigorously justified as the exact expectation value of a four-sector Lindblad decay process. The systematic error vanishes exactly due to the linearity of the entropy function S = n_WH log 2. (OQ-CQ4-5) The transfer map T = |E⟩⟨0̄| + |B⟩⟨1̄| is explicitly constructed, satisfying all four required conditions including isometry, code-word correspondence, symmetry intertwining T Π_code T† = Π_pol, and flat Schmidt spectrum preservation. Prediction P6 (C_ℓ^EE/C_ℓ^BB → 1) is thereby elevated from hypothesis to theorem. Open questions for Paper CQ-VI are enumerated.