The definitions behind the math of LLMs.

This isn't true.

A neural network can do new things outside the training data. Random initialization sees to that.

Whether those things are useful is a different question, but it turns out that yes, some of those things are useful. It also turns out that in a large-enough LLM if you give it some kind of idea of what you see as useful then yes, it can pick out the newly useful things.