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) but 3.3σ below the flat-ΛCDM DESI DR2+CMB combination (ΩΛ = 0.6973 ± 0.0036): the falsifiability criterion set in earlier versions of this work is formally met by that dataset combination, while the 2.3σ internal BAO–CMB tension and the 2.8–4.2σ preference of the same data for an evolving equation of state leave the verdict open. Against the DESI DR2 BAO distances alone the relation remains an acceptable absolute fit (χ² = 14.1 for 12 d.o.f., p = 0.30; Δχ² = 3.8 from the best fit). Three quantitative analyses are added in v6. (1) Beyond ΛCDM: defining α_eff ≡ ρ_DE(t₀) G t₀², the published w₀wₐCDM best fits give α_eff ≈ 0.061–0.073 — between −18% and −2% of 2/27, with the Pantheon+ combination closest (−0.7σ) — so the relation does not survive generically outside ΛCDM but defines a specific locus in (w₀, wₐ, Ωm). (2) A trials-factor estimate: within narrow low-complexity classes of candidate coefficients the probability of a chance match as good as the one observed is ≈ 0.11, rising to ≈ 0.8–0.9 in broader classes. (3) The relation holds within the Planck 1σ band only inside a ±1.2% window of cosmic age (−180/+146 Myr). An erratum corrects the effective equation of state quoted in v5 for the dynamical promotion of the ansatz (w₀ ≈ −0.30, not −1.70). The coefficient α = 2/27 admits the decomposition 2/3³ but remains an empirical input, not derived from first principles. The archive includes the LaTeX source, figures, and a self-contained Python script reproducing every number from the official DESI DR2 BAO likelihood.