Heavy-tailed bandit learners must tell a poor arm from an arm whose favorable rewards have not yet appeared. We characterize this obstruction through a scale of one-sided visibility conditions on the optimal reward law: the mean hidden in a rare event is at most a constant multiple of a retained central moment. A quantile-trimmed UCB index that is given the visibility constants attains arm-specific moment rates at every exponent, with no lower bound on the exponent. Conversely, every strategy with a fixed exploration budget has polynomial regret on each law that violates the corresponding condition. For the variance benchmark we determine the order of both budget constants in the visibility level. The frontier depends on the largest exponent the benchmark promises, and the resulting classes are strictly nested. The same index has logarithmic expected regret when rewards are integrable, with a coefficient that can be exponential in the inverse gap.

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