Witness Density and the Cosmological Constant: A Unified Equation from Quantum Error Correction

We propose a unified equation linking dark energy density to the thermodynamic cost of information preservation: ρ_Λ = λ(a) n_w (k_B T ln 2), where n_w is the witness number density (the number of independent degrees of freedom that protect the logical qubit), T is the cosmic microwave background temperature, and λ(a) is a running coupling. We show that this equation is dimensionally consistent, and we use observational data to constrain n_w^0 ≈ 2.3 × 10^13 m^-3 today. We derive the running coupling λ(a) ∝ a^4 from the code structure via the renormalization group, with beta function β(λ) = 4λ, and independently from the constancy of ρ_Λ and the holographic bound. We show that the inferred witness density is remarkably close to the number density of ultralight dark matter particles with mass m ∼ 0.1 meV, suggesting a possible physical identification. We also show that the cosmological constant problem is partially resolved, with the smallness of ρ_Λ decomposed into a density factor n_w/n_w^max ∼ 10^-30 and a temperature factor T_0/T_Pl ∼ 10^-32. We discuss consistency with the [[4,1,2]] Four-Sector Cosmology (FSC) code of Katakura (2026). This work is a physics-only extension of CQ-III; no theological or metaphysical assumptions are used.