Top panel: solid lines indicate metal-free (Z = 0) Rosseland mean opacities κ_R(T) for different densities ρ. A representative κ_R(T) profile for solar metallicity opacity with dust for ρ = 10⁻¹² g cm⁻³ is shown as a green dashed line for comparison. Lower panel: effective temperature calculated by Equation (4) for different densities. The shaded region represents solutions with T_eff > T that are no longer consistent with the optically thick assumption, implying a stellar-UV rather than thermalized emission. For Z = 0 opacities, there is a unique and universal transition at T_eff ≈ T ≈ 4000–5000 K (gray circle), regardless of density.