That doesn't seem true to me at all. Let's say you fit y=c+bx+ax^2 on the domain -10,10 with 1000 data points uniformly distributed along x and with no more tha

That doesn't seem true to me at all. Let's say you fit y=c+bx+ax^2 on the domain -10,10 with 1000 data points uniformly distributed along x and with no more than 1% noise in observed y. Your model will be pretty damn good and absolutely will be able to generate "synthetic a priori" y outputs for any given x within the domain. Now let's say you didn't know the true function and had to use a neural network instead. You would probably still get a great result in the sense of generating "new" outputs that are not observed in the training data, as long as they are within or reasonably close to the original domain. LLMs are that. With enough data and enough parameters and the right inductive bias and the right RLHF procedure etc, they are getting increasingly good at estimating a conditional next token distribution given the context. If by "synthetic" you mean that an LLM can never generate a truly new idea that was not in it's training data, then that becomes the question of what the "domain" of the data really is. I'm not convinced that LLMs are strictly limited to ideas that they have "learned" in their data. Before LLMs, I don't think people realized just how much pattern and structure there was in human thought, and how exposed it was through text. Given the advances of the last couple of years, I'm starting to come around to the idea that text contains enough instances of reasoning and thinking that these models might develop some kind of ability to do something like reasoning and thinking simply because they would have to in order to continue decreasing validation loss. I want to be clear that I am not at all an AI maximalist, and the fact that these things are built largely on copyright infringement continues to disgust me, as do the growing economic and environmental externalities and other problems surrounding their use and abuse. But I don't think it does any good to pretend these things are dumber than they are, or to assume that the next AI winter is right around the corner.