> ...generate answers near the center of existing thought.

This is right in the Wikipedia's article on universal approximation theorem [1].

[1] https://en.wikipedia.org/wiki/Universal_approximation_theore...

"n the field of machine learning, the universal approximation theorems (UATs) state that neural networks with a certain structure can, in principle, approximate any continuous function to any desired degree of accuracy. These theorems provide a mathematical justification for using neural networks, assuring researchers that a sufficiently large or deep network can model the complex, non-linear relationships often found in real-world data."

And then: "Notice also that the neural network is only required to approximate within a compact set K {\displaystyle K}. The proof does not describe how the function would be extrapolated outside of the region."

NNs, LLMs included, are interpolators, not extrapolators.

And the region NN approximates within can be quite complex and not easily defined as "X:R^N drawn from N(c,s)^N" as SolidGoldMagiKarp [2] clearly shows.

[2] https://github.com/NiluK/SolidGoldMagikarp