I've not been able to find any way to distinguish "mere surface statistics" from the deeper, richer, and more meaningful kind of something that it is meant to b

I've not been able to find any way to distinguish "mere surface statistics" from the deeper, richer, and more meaningful kind of something that it is meant to be contrasted with, except that "surface statistics" are un-compressed. For example, surface statistics might be the set of output measurements generated by a compact process, such as the positions of planets over time; knowing the laws of gravity means we can generate gigabytes of these statistics correctly and easily, which will accurately match future observations. But then going the other way, from statistics to a causal model, is just an inverse problem -- just like, say, going from a set of noisy magnetic field measurements at the boundary of a container to a pattern of electric current flow inside a volume, or going from planet positions to orbit shapes and periods to an inverse square law of gravity. Generating a compressed inverse model from surface statistics is exactly the sort of thing that deep learning has proven to be very good at. And by now we've seen no shortage of evidence that LLMs and other deep networks contain stateful world models, which is exactly what you'd expect, because for all their parameters, they aren't nearly big enough to contain an infinitesimal fraction of the statistics they were trained on. So I think it's overly dismissive to regard LLMs as mere surface statistics.