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

This paper develops a thermodynamic research framework for dark energy, with a proved reduced variational model and open physical connections.

For the dimensionally closed reduced free-energy density Phi_red = A + B*rho*ln(rho/rho_Pl) + C*rho^2, B,C > 0, existence, uniqueness, strict convexity, and an explicit Lambert-W minimiser are established. The original momentum-space expression is dimensionally inhomogeneous and does not determine the matching coefficients.

The scaling Lambda_eff ~ H0^2/(c^2 ln(M_Pl/H0)) remains an external RG-matching ansatz. Its value is about 3.8 x 10^-55 m^-2, roughly a factor of 290 below observation. The Poschl-Teller potential has an unstable maximum at phi = 0; a tanh history is a phenomenological proxy rather than a derived stable attractor.

Standard quasistatic metric f(R) gives an unscreened slip of 1/2 (or reciprocal 2, depending on convention) and a lensing combination Sigma ~= 1/F. Earlier forecasts of eta = 5/4, Sigma = 9/8, and a stable attractor are not supported; their derivations and counterarguments remain documented. The Hu-Sawicki environment ledger provides necessary curvature thresholds, while a source-bound local profile and Cassini compatibility remain unestablished.

The programme continues from this mathematical and diagnostic foundation. Explicit RG matching, gravitating-density identification and conservation, a stable scalar history, local screening, horizon/ensemble specification, and official Planck/DESI likelihood tests remain open. No additional physical evidence or validation is introduced in version 1.13.

Series position

This paper belongs to two programme lines:

• FST programme — Sub-component of the FST-IV Cosmology application slot. 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). CRM-V (Saturation Theorem, Concept DOI 10.5281/zenodo.19036188) and CRM-VI (QG-CRM, Concept DOI 10.5281/zenodo.19352448) remain companion references, not closed proof inputs for the open gates above.

Code & sources

GitHub: research-line/functional-stability-theory/tree/main/fst-cosmology/dark-energy. The public repository is a supplementary source path; the v1.13 EN/DE sources and PDFs describe the current state of the same ongoing research programme.

External scalar-tensor and interacting-quintessence models do not determine the identification of the reduced minimizer with a conserved gravitating density, the subtraction term A, or the RG matching. A quintom comparator includes a phantom field and is distinct from ghost-free interacting quintessence. The tanh proxy alone establishes no phantom-crossing mechanism. No new independent physical validation is claimed.

Changes in Version 1.13 (October 2026)

• Readable research architecture: Figure 1 now separates headings, status lines, equations, and explanatory text; overlapping labels and text overflow are removed in both languages.
• Current research state: the abstract, title-page version line, introduction, and evidence tables present the ongoing project directly. Version-to-version development is described here and in a dedicated manuscript note.
• Established results retained: the reduced variational theorem and Lambert-W minimiser are unchanged. Bibliography, numerical values, and evidentiary status retain the v1.12 state.
• Counterarguments and remaining work: the factor-290 matching deficit, unstable hilltop, and failure of earlier slip/lensing forecasts remain explicit. RG matching, stable dynamics, local screening, and official likelihood tests remain open.
• Evidence change: no new computation, counterexample, strengthened physical evidence, or model validation is claimed in this presentation update.

Changes in Version 1.12 (October 2026; inventory correction)

• Source identity: replaced a wrongly attributed local Pfeifer PDF with the verified arXiv:2504.08062v2 source; retained the misidentified historical artifact for traceability.
• Bibliographic corrections: corrected Vasak/Kirsch/Struckmeier author order, Efstratiou and Sahoo journal references and DOIs, and Antusch/Sahoo arXiv primary categories in both languages.
• Interpretation: separated interacting canonical quintessence from phantom-field quintom comparators; stated the independent gravitating-density, conservation/exchange and subtraction gates. No external result closes the factor-290 matching deficit.
• Transparency: synchronized English and German, corrected AI disclosure, preserved the reduced mathematical core and all open physical gates. No new computation or unpublished conserved-mapping claim is promoted.
• Files: separate English and German PDFs; historical versions and changelogs retained.

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 x 10^-55 m^-2, about 290 below observation, not within a factor of three.
• Scalar correction: the Poschl-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 Sigma ~= 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 w_a = -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/Poschl-Teller paragraph corrected so that the Poschl-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 were 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 Gamma-convergent phase separation.
• Major: DESI DR2 comparison: best-fit kappa=2.63, a_trans=0.326 yields 0.39 sigma compatibility.
• Major: Callan-Symanzik RG section with beta function and UV/IR fixed points.
• Major: Pattern B (Flow Selection) explicitly identified.

Version 1.3 (March 2026)

• Critical: Entropy term corrected from ln(rho) to rho*ln(rho/rho_Pl), restoring unique minimum.
• Critical: Lagrangian standardised to f(R) = R + gamma R^2.