On the Derivation of the Observed Cosmological Constant from a Self-Consistency Requirement Between the Bekenstein Entropy Bound and de Sitter Thermodynamics

We investigate a sharply defined infrared consistency condition linking the Bekensteinentropy bound to de Sitter horizon thermodynamics. Consider a vacuum-dominated causalsphere of radius R, energy EΛ(R), and entropy S(R). The Bekenstein bound assigns theentropy ceilingS ≤ 2πkBERℏc ,while de Sitter thermodynamics assigns to the same horizon the Gibbons–Hawking entropySdS(R) = kBA4ℓ2P= πkBc3R2Gℏ .If one demands that the maximal late-time vacuum system be self-consistent by saturatingboth descriptions at the same radius, and if the interior energy is purely vacuum energyEΛ(R) = uΛ V with uΛ = Λc4/(8πG), then the unique consistency condition isΛR2 = 3.Equivalently, the horizon energy must satisfyEΛ(R) = c4R2G ,which is the Schwarzschild-horizon energy relation evaluated at the de Sitter radius.We then evaluate the relation using the observed late-time ΛCDM asymptotic horizonscale inferred from Planck 2018 cosmological parameters. With H0 = 67.36 km s−1 Mpc−1and ΩΛ = 0.6847, one obtainsR∗ = cH0√ΩΛ≃ 1.660 × 1026 m, Λcons = 3R2∗≃ 1.089 × 10−52 m−2,1in agreement with the observed cosmological constant scale. The paper argues that thisresult is best interpreted as an infrared thermodynamic self-consistency condition ratherthan as a microscopic resolution of the vacuum catastrophe. It explains why the observedpositive cosmological constant has precisely the de Sitter value compatible with horizonentropy saturation, but it does not explain why quantum vacuum contributions renormalizeto that small value in the first place.