I think you are a bit confused about proof and logic, it is possible to prove absence of a property X if the presence of X has a necessary condition Y. Should Y fail, X cannot be present. This is one of the basic tools for proving theorems in math, for example if I can prove that the derivative of a function f is never zero, I can indeed prove that f has no localized minimum or maximum. More formally, given sets A and B we want to prove that their intersection is empty by proving that the complement of A is a superset of B.

Or in more laymans terms (I don't mean to be condescending I have no clue how well you know this stuff and want to cover my bases) If the streets are wet after it has rained, the fact that they are dry is proof of the absence of rain.

So there absolutely is a sufficient condition to prove the absence of sentience, the "ability to experience feelings and sensations". Or rather, there is a necessary condition to having sentience, since at the bare minimum it requires an organism to be capable of collecting sensory inputs, processing them and reacting to them. That is, it needs a nervous system. A lack of such facilities therefore means a lack of sentience.

A tree has no capacity to "feel" what's happening to it's branches. Lopping one of will have the rest of the tree completely unaffected, only the cut will "experience" the change and heal. An animal on the other hand can anticipate (even if erroneously) damage and react to it, by processing input with it's nervous system. A clam will close, a cat might jump, a human might scream etc.

This argument is presented in the link that sparked that comment chain just to be clear.

Now this isn't a formal proof of course. Such a proof would require the careful dissection and experimentation of plants to show that indeed no nervous system or similarly functioning facility is present. I think this is an inappropriate amount of work to ask of someone when arguing on the internet, which is why I would consider the above argument proof enough. Most people I would think can see clearly that salad is not sentient but a pig is. To pin this on the nervous system should also make sense to most and if someone is intrigued by such a topic usually they ask follow-up questions and don't combatively dismiss everything that's said (this argument is wrong therefore the claim is wrong is also some-type of fallacy).

The reason I didn't spell all of this out is because I had enough of these "that's not a proper proof/argument" counterpoints, lazily shifting the burden for even the most trivial arguments on to vegans, which is why I made the point that it is a proof, just not a formal one. That there are different layers of formalization and depending on context being lax can be more appropriate than a formalized argument.

What then followed was a continued stream of claims with 0 arguments provided and a prompt dismissal of the conversation when pushed for even one.