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 Λeff without fine-tuning. Starting from a free-energy functional Φ[ρ] defined over trial vacuum energy densities and regularised between an ultraviolet cutoff ΛUV and an infrared cutoff set by the Hubble radius H₀⁻¹, the unique minimum of Φ yields a residual vacuum energy density scaling as Λeff ~ H₀²/ln(MPl/H₀). The resulting numerical estimate lies within an order of magnitude of the Planck 2018 value. Embedding the variational bound in a scalar-tensor Lagrangian (R + γR² plus a Pöschl–Teller scalar field), we derive a tanh saturation dynamics for the dark-energy density parameter, an effective equation of state in the thawing regime, and a falsifiable universal gravitational slip prediction η = 5/4 across f(R) theories. A convexity-screening incompatibility theorem (no-go) shows that linear R+γR² cannot produce Solar System screening; a Hu–Sawicki-type nonlinear f(R) is required in the IR — a Cosmology-Pattern-A reduction (the bridge structures the open problem rather than closing it). Series position This paper belongs to two programme lines: FST programme — Sub-component of the FST-IV Cosmology application slot (the dark-energy chapter of the cosmological scale). Programmatic umbrella: FST — A Programmatic Hub (Concept-DOI 10.5281/zenodo.20130499). CRM series — Follow-up paper to CRM I–IV (10.5281/zenodo.18728936); shares the R+γR² Lagrangian. CRM-V (Saturation Theorem) provides the universal tanh normal form; CRM-VI (QG-CRM, in preparation) provides the UV completion. Code & sources GitHub: research-line/functional-stability-theory, sub-path fst-cosmology/dark-energy/. Scripts for the MCMC fit (Hu–Sawicki) and the w(z) comparison against DESI DR2 are available there. Changelog Version 1.8 (May 2026) Description rewrite: raw LaTeX macros (\Leff, \LUV, \rho, \gamma, \eta, \path) eliminated; replaced by Unicode Greek letters and HTML markup. Visual layout cleaned up. Dual-programme positioning: Series Position section added — FST-IV Cosmology sub-component AND CRM follow-up paper. Related Identifiers extended: isPartOf relations now include both FST Hub and CRM I-IV; CRM-V and CRM-VI added as references. GitHub source link added. No substantive content changes since v1.7. Version 1.7 (May 2026) Series-Hub linkage: isPartOf Related Identifier pointing to FST — A Programmatic Hub. Title-page footnote added. Affiliation simplified from "Bernau im Schwarzwald" to "Bernau" for consistency across FST companions. Version 1.6 (March 2026) Recompiled with latest cross-references; no substantive content changes. Version 1.5 (March 2026) Critical: Fixed \thanks placement. 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. 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. Changelog: Description rewrite: LaTeX macro residues (Leff, LUV etc.) eliminated and replaced by Unicode/HTML. Dual-programme positioning (FST-IV Cosmology sub-component AND CRM follow-up). Related Identifiers extended. GitHub source link added.