A Concrete UV Completion and Markovian Validation for Coherential Vacuum Backreaction RCD, RCD Hilltop Dark Energy Series — Paper IV

Paper III of this series identified vacuum backreaction from a Caldeira- Leggett bath as the minimal mechanism stabilizing the RCD hilltop, but left the ef- fective coefficients (α₀, γ₀, Γ, α) as phenomenological parameters. Here we spec- ify a concrete UV toy model — N scalar fields with Ohmic spectral density J(ω) = ηωe−ω/Λ_UV coupled to the coherence field through C-dependent masses — and compute all effective coefficients from the three microscopic parameters (η, λ, Λ_UV). The static Coleman-Weinberg correction is α₀ = ηλΛ²_UV/(4π), the vacuum produc- tion rate is γ₀ = ηλΛ³_UV/(πε), and the feedback coupling is α = λε/2. The Marko- vian approximation is validated: both the dissipation and noise kernels decay on a timescale τ_c ~ 1/Λ_UV ≪ H⁻¹, with non-Markovian corrections of order (H/Λ_UV)² — negligible by many orders of magnitude for any sub-Planckian bath (§4.3). The weak-backreaction condition εS_eq ≪ v/4 reduces to ηλΛ³_UV/(πΓ_eff) ≪ v/4, with the suppression factor ε canceling identically. This requires the effective coupling ηλ to be small: ηλ ≪ Γ_eff v/(4Λ³_UV), which localizes the cosmological hierarchy to the bath coupling sector. A phase diagram in (ηλ, Λ_UV) space shows that dynami- cally dominated stabilization (Λ > α₀) is the only viable regime for Λ_UV ≫ H. The bath relaxation rate Γ remains partially phenomenological, requiring mode-mode in- teractions absent from the free Hamiltonian. A linearized mode analysis (Appendix A) confirms a block-diagonal structure with two damped coherence oscillation modes and one purely dissipative bath mode, preparing the ground for Paper V.