[
  "Even perfectly group-calibrated scores, including true class-conditional probabilities, provably violate predictive parity after thresholding, per Definitions 2.1 and 2.3 of sufficiency and predictive parity (Section 2, Definitions 2.1, 2.3).",
  "Theorem 3.3 shows that for a fixed selection rate, the optimal classifier applies a soft threshold to group-calibrated scores, with randomization needed at exactly one threshold boundary (Theorem 3.3).",
  "Theorem 3.4 characterizes the boundary of the feasible region of achievable positive-predictive-value/false-omission-rate pairs as a continuous, piecewise curve composed of hyperbolic arcs and line segments (Theorem 3.4).",
  "Algorithm 1 traces the intersection of group-specific feasible PPV/FOR regions to construct the optimal classifier satisfying sufficiency exactly (Algorithm 1).",
  "A COMPAS case study demonstrates that optimal fair classifiers under sufficiency generally require group-specific decision thresholds rather than a single universal threshold (Section 7)."
]