The only function of the thumbs up/down buttons are to give feedback to Google. As a user it's pretty obvious that's the purpose of the button.
You were still tested on and your outputs messed with even if you didn't click on either button.
Genuine question, how else would they do it? And isn’t this practice the same as basically any agile-developed SaaS?
I'm not a mathematician but to me it doesn't seem so far-fetched to think that there might exist some mathematical proof that ensures the indistinguishability
We don’t have the ability to do that level of analysis of natural language text mathematically.
If we did, we probably wouldn’t need LLMs in the first place, i.e. we could just generate text using explicitly programmed algorithms.