A Thermodynamic Derivation of the Cosmological Constant Λ and Resolution of the Vacuum Catastrophe from a New Quantum Scale

A thermodynamic and quantum derivation of the vacuum energy density u_Λ = Λ c^4/(8π G) is presented from first principles, resolving the long-standing vacuum catastrophe without recourse to Planck-scale physics. Using the Bekenstein–Hawking entropy and Gibbons–Hawking temperature of the de Sitter horizon, we apply E = T S to show that u_Λ arises naturally from a maximum-entropy bound of the universe. An independent derivation from zero-point energy follows by introducing a physically motivated cutoff at the Lambda scale L_Λ = (ℏ G/(Λ c^3))^(1/4), a new quantum–thermodynamic scale defined by G, ℏ, c, and Λ. The resulting Λ-units are unique: vacuum-matching to de Sitter horizon thermodynamics fixes the remaining affine freedom in the dimensional analysis. In this gauge, c and ℏ take unit value, while G and Λ appear symmetrically with their hierarchy encoded by the dimensionless gravitational fine-structure constant α_Λ ≡ c^3/(G ℏ Λ). This unifies thermodynamic and quantum perspectives, eliminating the 10^120-fold discrepancy in vacuum-energy predictions. We validate the framework across diverse domains—including the Casimir effect, boson and fermion gases, and electromagnetic radiation—each saturating at the same vacuum bound. The findings imply that the Planck system, complete within a purely mechanical framework, attains thermodynamic closure only when Λ is included, providing a unified description of the quantum and cosmological vacua.