arXiv:2601.14013v3 Announce Type: replace Abstract: Allowing for adversarial contamination and heavy tails, we study testing whether the mean of a high-dimensional random vector equals zero. Because standard max-tests based on sample averages are highly non-robust, we propose a max-test based on quantile-winsorized observations. The test controls asymptotic size under adversarial contamination and only requires $m>2$ moments, while allowing dimension to grow exponentially with sample size. We fully characterize its asymptotic power function. Comparing with the standard max-test, for which we also derive a power characterization as a benchmark, we show that robustness is obtained for free at first order: under the stronger conditions that make the standard max-test valid, our robust test has identical asymptotic power. We show that the decay condition on the contamination-rate imposed is necessary for this no-first-order-loss property. Finally, we also study the role of bootstrap critical values, showing that their use never decreases power, can strictly improve asymptotic power in extremely correlated designs, but often has no first-order asymptotic effect.