textGrain watermarking is harder to detect: can sum-tests still do it optimally?

Swapnaneel Bhattacharyya, Soham Bonnerjee

Watermarking embeds an imperceptible signal in text from large language models that is statistically detectable with a secret key. The recent textGrain watermark (Li et. al. (2026b)) calibrates this signal against the model's sampling entropy through an explicit entropy budget, combining the stochasticity of Red-Green and the unbiasedness of Gumbel-max watermarks, and its accompanying report proposes a detector that sums pivotal statistics. We study when, and how efficiently, textGrain can be detected after human edits, modeled as a sparse mixture in which a fraction $\varepsilon_n\asymp n^{-p}$ of positions retain the watermark and token uncertainty decays as $n^{-q}$. First, entropy calibration comes at a cost: the minimax detection boundary of textGrain is $p+q=1/2$, so its detectable region is strictly smaller than that of Gumbel-max and maximal-coupling watermarks, whose boundary is $2p+q=1$. This cost stems from textGrain's construction rather than from entropy calibration itself: an unbiased scheme with the same entropy loss remains detectable where textGrain is not. Second, perhaps surprisingly, this loss makes the sum detector optimal at the boundary when the number of…