To make a universal Turing machine out of an LLM only requires a loop and the ability to make a model that will look up a 2x3 matrix of operations based on context and output operations to the context on the basis of them (the smallest Turing machine has 2 states and 3 symbols or the inverse).

So, yes, you can.

Once you have a (2,3) Turing machine, you can from that build a model that models any larger Turing machine - it's just a question of allowing it enough computation and enough layers.

It is not guaranteed that any specific architecture can do it efficiently, but that is entirely besides the point.

LLMs cannot loop (unless you have a counterexample?), and I'm not even sure they can do a lookup in a table with 100% reliability. They also have finite context, while a Turing machine can have infinite state.

Are you saying that LLMs are Turing complete or did I misunderstand it?