Thermodynamic Rule of the Cosmological Constant: Λ = 2π/S and Its Theoretical Prediction for the Dark Energy Attractor Ω_Λ = 2/3

The observed value of the cosmological constant Λ differs from its quantum field theory expectation by approximately 120 orders of magnitude, constituting one of the most severe numerical puzzles in fundamental physics. Based on the established thermodynamic properties of the cosmic event horizon (temperature T = 1/(2πR_h), entropy S = πR_h²), this paper rigorously applies the first law of thermodynamics while incorporating matter conservation. We demonstrate that thermodynamic self-consistency necessitates a parameter-free relation between Λ and the event horizon entropy S: Λ = 2π/S. In the asymptotic de Sitter limit, this relation precisely predicts the dark energy density parameter Ω_Λ = 2/3 ≈ 0.6667, which is in remarkable agreement with the current best observational value Ω_Λ = 0.6847 ± 0.0073 (a relative deviation of 2.7%). The theory further naturally yields a stable attractor state valid within the current accelerated expansion phase: Ω_m = 1/3, Ω_Λ = 2/3, indicating the universe is approaching this stage-specific thermodynamic equilibrium state. This provides a natural explanation for the low-redshift cosmic coincidence problem. For the first time, this work determines from first principles the key proportionality constant c = √(2/3) ≈ 0.8165 in the holographic dark energy model, elevating a previously phenomenological adjustable parameter to a fundamental constant determined by basic principles. Numerical solutions of the dynamical equations constrained by Λ = 2π/S show that the matter density parameter Ω_m(z) asymptotically converges to 1/3 at low redshifts, while the dark energy equation of state w(z) exhibits a tiny deviation from −1 (e.g., w_0 ≈ −0.998). These clear predictions provide testable targets for next-generation cosmological observations.