Mostly ordered scores for Cases 1..12 are 0.90,0.80,0.70,0.70,0.60,0.40,0.80,0.50,0.40,0.30,0.20,0.10. Reversed uses 1 − score with the same actual labels; All tied uses 0.50. Scores and threshold are compared on integer 0.05 ticks. Every dataset has six actual positives and six negatives, so no zero-denominator ROC case is introduced.

ROC uses actual-class denominators. It does not plot precision, accuracy or predicted-positive proportions. Sweep score groups in descending order; lowering the inclusive threshold can increase or leave unchanged each rate. Tie groups jump together, so a straight area segment does not make every intermediate point a separate hard-threshold outcome.

AUC is the trapezoidal area and equals (positive-over-negative pair wins+half ties)/36. Here each of the 36 pairs chooses one positive and one negative uniformly from this finite sample. Mostly ordered has 29 wins and 2 ties, Reversed 5 wins and 2 ties, All tied 0 wins and 36 ties. AUC 0.5 need not mean random score generation. AUC does not select an optimal threshold, measure calibrated probabilities, encode error costs or guarantee future performance. No fitting, ROC uncertainty, class-prevalence editing or precision–recall curve is implemented.