Predicted broad-line ratios as a function of ionization parameter, log U, from our CLOUDY grid. The curves illustrate the sensitivity of these diagnostics to the hardness of the incident continuum, which becomes softer for larger black-hole masses, and to the gas density. All calculations adopt the same BLR gas parameters as in Figure 6 (metallicity 0.1 Z_⊙, density log(n_H/cm⁻³) = 10, and total hydrogen column log(N_H/cm⁻²) = 23). Top panel: He IIλ4686/Hβ. The solid, dotted, and dashed curves correspond to models with log(M_BH/M_⊙) = 7.5 (fiducial), 7.8, and 8.2, respectively. For comparison, the dashed magenta line and shaded magenta band show the value reported by Übler et al. (2023) for the z = 5.55 AGN GS_3073 (which has an estimated black-hole mass of log(M_BH/M_⊙) = 8.2 ± 0.4), while the dashed red line indicates the upper limit from the stack of LBDs (Zucchi et al. 2026) (downward arrows). Bottom panel: He IIλ4686/He Iλ5876. The solid, dotted, and dashed curves correspond to models with (n_H, M_BH) = (10¹⁰ cm⁻³, 10^7.5 M_⊙),(10¹⁰ cm⁻³, 10⁸ M_⊙), and (10¹¹ cm⁻³, 10^7.5 M_⊙), respectively.