A closed-form empirical relation for the dark energy density parameter

We report an empirical observation: the dimensional ansatz ρΛ = α/(G t₀²), with the single empirical input α = 2/27, yields a closed-form expression ΩΛ = tanh²(2√π/3) = 0.6855 when combined self-consistently with the Friedmann equations in flat ΛCDM. This value lies within 0.11σ of the Planck 2018 CMB+lensing result (ΩΛ = 0.6847 ± 0.0073) and within 0.6σ of the Planck+BAO constraint (ΩΛ = 0.6889 ± 0.0056). The coefficient admits the decomposition α = 2/3³, relating the numerator to a small integer and the denominator to the cube of the spatial dimensionality; among forms p/qⁿ with p, q, n small positive integers, this is the unique one compatible with the Planck 1σ band, but the restriction to this class of forms is chosen a posteriori and is not an argument of physical uniqueness. The ansatz further implies Λ = 16π/(27 t₀²), with Newton's constant G absent from the expression; this is a structurally automatic consequence of any ansatz of the form ρΛ ∝ 1/G and is not an independent geometric result. Within the holographic dark energy framework, the coefficient corresponds to d = 4√π/9 ≈ 0.788. The relation is strictly falsifiable: upcoming surveys with δ(ΩΛ) ~ 0.002–0.003 will determine whether this is a physically meaningful relation or a numerical coincidence. We stress that α = 2/27 is an empirical input, not derived from first principles.