Counterexamples to a best-of-n win-rate bound at every budget

For every n≥2 and δ>0, there is a finite positive reference distribution whose largest atom is at most δ and whose best-of-n policy violates the proposed divergence–win-rate curve. A four-response best-of-two example gives exact probabilities and a strict logarithmic certificate. The proof uses finite interval bounds and gives a quantitative positive gap. It does not determine the sharp replacement curve.