CountScan: Simultaneous Lower Bounds on Watermarked Tokens over All Text Intervals

Text produced partly by a watermarked language model and partly by humans raises a question that global detection does not answer: how many watermarked tokens does a given paragraph contain, when the paragraph is chosen after looking at the text? We give finite-sample lower bounds on the number of watermarked tokens, meaning positions whose context and token reproduce an output of the keyed generator, that hold simultaneously for every interval of the document and every count. Our method, CountScan, runs one hedged e-process from each start position, combines them with Ville's inequality and a union over starts, and counts by trimming the strongest tokens; the price of simultaneity is logarithmic in the document length. Closing over all subsets instead loses power linearly in the length of the surrounding human text. Validity needs no entropy model and extends to editing that does not use the key, provided the calibrator is monotone; an editor who can query the detector breaks it. For power, we derive the limit of the certified proportion, its finite-length expansion with an explicit threshold bias, and an upper bound for a broad class of per-token calibrators. On a pre-registered holdout with two open models sampled at temperature one, the bound certifies on average about three fifths of each 1000-token watermarked document at level 0.05; at temperature 0.7 the generations have low entropy and little can be certified. In a pre-registered comparison on the same documents, CountScan covered more of the watermarked tokens than a recent paragraph test that controls the false discovery rate, and neither method made a false report; editing a fifth of the generated tokens removed most of the signal for both.