equal PPV and equal FDR across groups; (2) **equal false positive rates** — $\text{FPR}_0 = \text{FPR}_1$; and (3) **equal false negative rates** — $\text{FNR}_0 = \text{FNR}_1$. Conditions 2 and 3 together constitute equalized odds. The theorem proves that calibration and equalized odds are mathema