The variation in solar irradiance is commonly assumed to be driven by its surface magnetism. Until recently, this assumption could not be verified conclusively as models of solar irradiance variability based on solar surface magnetism have to be calibrated to solar irradiance measurements. Making use of realistic three-dimensional magnetohydrodynamic simulations of the solar atmosphere and state-of-the-art full-disk magnetograms from SDO, we developed a model of total solar irradiance (TSI) that does not require any such calibration. The modelled TSI variability is therefore, unlike preceding models, independent of TSI measurements. The model replicates over 95% of the observed variability over the lifetime of SDO, confirming the relationship to solar surface magnetism and leaving limited scope for alternative drivers of solar irradiance variability (at least over the time scales examined, that is, days to years).