Based on the SynthID-Text paper
https://www.nature.com/articles/s41586-024-08025-4 I agree that the LLM's learned distribution isn't modified, but I don't think it's correct to say that the sampling process is not modified. Also I just read the paper today so I could be misinterpreting things.
As described in the paper, you're right that it doesn't affect the main sampling technique, but what they do is they sample the distribution for 2^m samples, and then use Tournament sampling to choose the tokens among those 2^m samples, and the watermark key changes the scoring of the tournament options, using the watermark key as an input to the random generator that generates the scoring functions.
Then, to calculate the watermark, they take the text, and compute the mean g-values of the text, and a higher score means that it's more likely that it was sampled using the provided selection of tournament watermarking functions.
let's say you had some top P words: mango, banana, pineapple, guava, and you sampled 8 times, and got each one twice in the following order:
1. mango
2. banana
3. pineapple
4. guava
5. mango
6. banana
7. pineapple
8. guava
without tournament sampling, you'd truly see any of those come through. But in tournament sampling, you take those 8 options, create m scoring functions based on the pseudorandom generator, and score the 'tournament' by sampling the biased distribution you create from the g values. That does change the sampling from based purely on the LLM and entropy, but i mean, if the watermark key is also generated from some entropy, it's probably representative of the original sampling options as expected?
this is a very fascinating topic! I do still stand by my point that anthropic is the only one who can tell if something is watermarked or not and feeling icky, but the paper has mostly quelled my concern on impacting the intelligence part.