Maybe, I don't know, not be based on a statistical model?

Come on. If you are actually entertaining the idea that LLMs can possibly be intelligent, you don't know how they work.

But to take your silly question seriously for a minute, maybe I might consider LLMs to be capable of intelligence if they were able to learn, if they were able to solve problems that they weren't explicitly trained for. For example, have an LLM read a bunch of books about the strategy of Go, then actually apply that knowledge to beat an experienced Go player who was deliberately playing unconventional, poor strategies like opening in the center. Since pretty much nobody opens their Go game in the center (the corners are far superior), the LLM's training data is NOT going to have a lot of Go openings where one player plays mostly in the center. At which point you'll see that the LLM isn't actually intelligent, because an intelligent being would have understood the concepts in the book that you should mostly play in the corners at first in order to build territory with the smallest number of moves. But when faced with unconventional moves that aren't found anywhere on the Internet, the LLM would just crash and burn.

That would be a good test of intelligence. Learning by reading books, and then being able to apply that knowledge to new situations where you can't just regurgitate the training material.

Have you seen the now-years-old transcripts of "ancient" LLMs inventing new languages with grammar and syntax structures completely different from our own?