Source Clarifications

Proposition 2: pairwise rate and finite-sample bound

• One-sided support: if the score gap is nonnegative and has zero-gap probability $p_0>0$, the rate is $-\log p_0$ and the error probability is $p_0^N$. If every positive-mass gap is strictly positive, errors vanish for every positive $N$; this is represented by extended rate $+\infty$.

Proposition 4: outcome rate and $M^2$ bound

• One-sided pairs: a zero-gap pair contributes rate $-\log p_0$; an eventually error-free pair contributes extended rate $+\infty$ and cannot lower the minimum. This includes score gaps with no negative realization.