Dark Energy as Residual Vacuum Free Energy: A Thermodynamic Bound on the Cosmological Constant

The cosmological constant problem remains one of the deepest puzzles in theoretical physics: naive quantum field theory estimates of the vacuum energy density exceed the observed value by roughly 120 orders of magnitude. We propose a thermodynamic variational principle that constrains the effective cosmological constant Λ_eff without fine-tuning. Starting from a free-energy functional Φ[ρ] defined over trial vacuum energy densities and regularised between an ultraviolet cutoff and an infrared cutoff set by the Hubble radius H₀, we show that the unique minimum yields a residual vacuum energy density scaling as Λ_eff ~ H₀². The framework naturally addresses the coincidence problem by linking the vacuum energy to the current expansion rate. CHANGELOG Changes in Version 1.6 (March 2026) No substantive changes. Recompiled with latest cross-references. Changes in Version 1.5 (March 2026) Critical: Fixed \thanks placement Changes in Version 1.4 (March 2026) Major: Screening recast as Γ-convergent phase separation Major: DESI DR2 comparison: best-fit κ=2.63, a_trans=0.326 yields 0.39σ compatibility Major: Callan–Symanzik RG section with β function and UV/IR fixed points Major: Pattern B (Flow Selection) explicitly identified Changes in Version 1.3 (March 2026) Critical: Entropy term corrected from ln(ρ) to ρ·ln(ρ/ρ_Pl), restoring unique minimum Critical: Lagrangian standardised to f(R) = R + γR²