For all practical purposes, calling it a LUT is somewhat too reductive to be useful here I think. But we can try: leaving aside LLMs for a second; with this LUT reasoning model you're using, would you be able to prove the existence of just a computer?

What state is lacking? There is a result which requires computation to be output. The model is the state. The computation must be performed for each input to produce a given output. What are you even objecting to?