Best-of-$N$ jailbreaking (BoN) bypasses safeguards of aligned models by drawing $N$ independent augmentations of an unsafe prompt and sampling $M$ completions of each. Previous works have shown that the attack success rate (ASR) seems to follow a power-law in $N$, which we challenge. The exponent drifts with $N$, with an exponential crossover which is a finite-size artifact of the adversarial dataset. Little work has been done to explore the entire two-budget ($N, M$) attack surface as well as its dependence on the generation temperature $T$. We introduce a simple barrier model where each prompt has a baseline safety level and each augmentation a random thermally activated barrier. Then four numbers, each backed by an interpretable safety mechanism, determine the entire ($N, M$) attack surface. They extrapolate predictions from $N \leq 100$ to $N = 10^4$, collapse five distinct models on the same scaling function and predict ASR at different temperatures from the one they were fitted at.

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