[Lerc]

That still seems unfalsifiable. If it fails one instance the claim is that the failure is representative of things outside the training set. If it succeeds the claim is that it is in the training set. Without a definitive way to say something is not in the training set (a likely impossible task) the measure of success or failure is the only indicator of the purported reason reason for the success or failure.

Given models can get things wrong even when the training data contains the answer, failure cannot show absence.

[voidhorse]

I do think there are cases which, in controlled environments, there is some degree of knowledge as to what is in the training set. I also don't thin it's as impossible as you assume.

If you really wanted to ensure this with certainty just use the natural numbers to parameterize an aspect of a general problem. Assume there are N foo problems in the training set, then there is always a case N+1 parameter not in the training set, and you can use this as an indicative case. Go ahead and generate an insane number of these and eventually the probability that the Mth instance is not in the set is effectively 1.

Edit: Of course, it would not be perfect certainty, but it is probabilistically effectively certain. The number of problem instances in the set is necessarily finite, so if you go large enough you get what you need. Sure, you wouldn't be able to say there is a specific problem instance not in the set, but the aggregate results would evidence whether or no the LLm deals with all cases or (on assumption) just known ones.

[Lerc]

Well there are models that can sum two many-digit numbers. They certainly have not been trained on every pair of integers up to that level. That either makes the claim they can't do things that they haven't seen trivially false, or the criteria for counting something as being in the training data includes a degree of inference.

What happens when someone makes a claim that they have gotten a model to do something not in the training data and another person claims it must be encoded in the training data in some form. It seems like an impasse.