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

Abstract

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 introducing a fitted late-time Λ parameter; the quantitative scaling is conditional on the stated RG-matching assumptions. 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 under that matching 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 regime-limited gravitational-slip target η = 5/4 in the quasi-static, linear, unscreened sub-Compton regime of metric 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.18728935); shares the R+γR² Lagrangian. CRM-V (Saturation Theorem, Concept DOI 10.5281/zenodo.19036188) provides the universal tanh normal form; CRM-VI (QG-CRM, Concept DOI 10.5281/zenodo.19352448) provides the UV completion programme. 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 1.11 (August 2026; corrective audit release) Provenance: this version documents the outcome of an internal proof/paper-mathematics audit (2026-07-16), which re-derived every load-bearing claim from the original manuscript and withdrew those that did not survive re-derivation. It is a transparency correction, not a new result. Mathematical core: EN/DE now prove only the dimensionally closed reduced functional and its unique Lambert-W minimiser; the original mode-sum expression is retained as dimensionally inhomogeneous motivation. Numerical correction: the conditional Hubble-scale ansatz evaluates to 3.8 × 10^-55 m^-2, about 290 below observation, not within a factor of three. Scalar correction: the Pöschl–Teller profile has an unstable hilltop; the tanh history is a phenomenological proxy and not a derived stable de-Sitter attractor. Perturbation correction: standard metric f(R) limits are slip 1/2 (or reciprocal 2) and lensing Σ ≃ 1/F; the former 5/4 and 9/8 forecasts are withdrawn. Screening guardrail: the environment thresholds are retained, but all passes remain source-blocked; neither tested |f_R0| value is by itself a Cassini proof. Open constructive gates (unresolved, not claimed here): explicit RG-matching coefficients, a stable scalar history, a source-bound local environment profile, and an official Planck/DESI likelihood run remain outstanding future work. This version corrects the record; it does not close these gates. EN/DE status: English and German PDFs were rebuilt synchronously; Unicode umlauts, cross-references, links, extracted text, and representative rendered pages were checked. Version 1.10 (June 2026) Proof/paper-math correction: stationarity is synchronized to B[1+ln(rho_*/rho_Pl)] + 2C rho_* = 0; the older shortcut through T_dS*(Lambda_UV^3/rho_*) is treated as historical only. RG/C2 guardrail: the raw mode sum is explicitly UV-dominated; the physical H_0^2 scaling remains conditional on RG matching, C2/Minkowski subtraction and UV renormalization. Screening correction: the convexity-screening no-go uses derivative boundary conditions at the density endpoints, not a divergent Phi(rho) condition at rho->0+. Claim calibration: RH/FST analogies are structural rather than evidential; Hu--Sawicki/chameleon screening is framed as a conditional consistency and thin-shell gate, not as a completed Cassini-safe proof. Strict-review hardening: DESI comparison is relabelled as a CPL proxy; eta = 5/4 is treated as an observable-window target; screening toy numerics are no longer presented as Cassini-safe proof. Official likelihood gate: v1.10 does not claim a new DESI inference; the next quantitative gate is reproduction or overlay against the published DESI DR2 chains, posterior-maximization products and likelihoods. EN/DE status: English and German are synchronized with the post-v1.9 guardrail state and retain verified Unicode umlauts. Version 1.9 (May 2026) Source-check correction: DESI DR2 dark-energy constraints updated from arXiv-only metadata to the published Physical Review D version (PRD 112, 083515; DOI 10.1103/tr6y-kpc6), with the BAO+CMB CPL value kept at wa = -1.75 +/- 0.58. Bibliography corrections: MOND/EUP, Finsler-Friedmann, Copeland-Liddle-Wands, RH, RFEP, Navier-Stokes, Yang-Mills and BSD entries corrected to verified author/title/journal/DOI or Concept-DOI metadata. Claim guardrails: non-peer-reviewed MOND comparator retained only as low-weight context; DE Starobinsky/Pöschl-Teller paragraph corrected so that the Pöschl-Teller scalar is not presented as the Starobinsky R^2 Einstein-frame potential. EN/DE synchronisation: German text now includes the Cohen-Kaplan-Nelson UV-IR cutoff citation and matches the English DESI and source-status framing. 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 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.