A candidate prediction is included or not before the score is computed. Suppose it will match some annotation with probability $p$ and, when it does, with IoU $j$. Whether it matches or not, the denominator rises by exactly $½$: a match turns a false negative into a true positive ($n + 1$, $fn − 1$, net $+½$), a miss adds a false positive ($+½$). The expected score after inclusion is therefore the exact quotient $pqWithCandidate$.

The main statement is the inclusion rule: including the candidate raises the expected Panoptic Quality if and only if its expected contribution $p · j$ exceeds half the current score. The remaining statements are the two directions in the form in which a decision procedure uses them, and the same rule solved for the match probability.