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 $\Leff$ without fine-tuning. Starting from a free-energy function $\Phi(\rhov)$ defined over trial vacuum energy densities and regularised between an ultraviolet cutoff $\LUV$ and an infrared cutoff set by the Hubble radius $H_0^{-1}$, we argue that the unique minimum of $\Phi$ yields a residual vacuum energy density scaling as $\Leff \sim H_0^{2}/\ln(M_{\mathrm{Pl}}/H_0)$. The resulting numerical estimate lies within an order of magnitude of the Planck 2018 value. Embedding the variational bound in a scalar-tensor Lagrangian ($R + \gamma R^{2}$ 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 prediction for the gravitational slip parameter $\eta = 5/4$ on sub-Compton scales, a result that holds universally across all metric $f(R)$ theories including the Hu–Sawicki model used for chameleon screening (see Remark ). The framework naturally addresses the coincidence problem by linking the vacuum energy to the current expansion rate. Limitations (honest scope statement). The scaling $\Leff \sim H_0^2/\ln(M_\mathrm{Pl}/H_0)$ follows from the variational principle conditional on a renormalisation-group matching that contributes a second logarithm $\ln(M_\mathrm{Pl}/H_0)$ from integrating $\beta_\Lambda^\mathrm{QQG}$ between $H_0$ and $M_\mathrm{Pl}$. This RG-matching is not derived from first principles in the present paper. A structured reduction onto two explicit open sub-conditions ((B1) existence of a perturbatively accessible QQG renormalisation-group trajectory ending in a Hu–Sawicki-type effective $f(R)$; (B2) the $\beta_\Lambda^\mathrm{QQG}$-integral evaluating to $\ln(M_\mathrm{Pl}/H_0)$ with controlled $O(1)$ prefactor) is documented in the companion proof note \path{BRIDGE_TO_HU_SAWICKI.md}. We classify this as a Cosmology-Pattern-A reduction (sub-reduction with unsolved root): the present paper structures the cosmological constant problem rather than closing it. Changelog: Series-Hub Verlinkung: This paper is now formally part of the FST Programme via Related Identifier isPartOf=10.5281/zenodo.20130499. Title page footnote added pointing to the FST Hub paper. Affiliation simplified from 'Bernau im Schwarzwald' to 'Bernau' for consistency across FST companions. Date updated to May 2026.