The duty cycle in our model, f_duty(z) = 0.0004(1 + z)³ (black solid line), is shown together with a gray shaded region indicating the adopted uncertainty in the duty-cycle normalization arising from observational calibration and extrapolation beyond z ≲ 6. Colored points indicate measurements for galaxies of different stellar masses (as specified in the legend), with 1σ confidence intervals. These observational duty cycles are inferred by multiplying the AGN fraction by the mean accretion rate of the full galaxy population, following the methodology of J. Aird et al. (2017) and W. Ren et al. (2025). Colored curves show the duty cycles measured directly from our simulations with different feedback prescriptions (legend labels).