Unless you mean by that something entirely different than what most people specifically on Hacker News, of all places, understand with "stateless", most and myself included, would disagree with you regarding the "stateless" property. If you do mean something entirely different than implying an LLM doesn't transition from a state to a state, potentially confined to a limited set of states through finite immutable training data set and accessible context and lack of PRNG, then would you care to elaborate?
Also, it can be stateful _and_ without a consciousness. Like a finite automaton? I don't think anyone's claiming (yet) any of the models today have consciousness, but that's mostly because it's going to be practically impossible to prove without some accepted theory of consciousness, I guess.
I certainly can't define consciousness, but it feels like some sort of existence or continuity over time would have to be a prerequisite.
You could assert that text can encode a state of consciousness, but that's an incredibly bold claim with a lot of implications.
On the other side of the coin though, I would just add that I believe that long-term persistent state is a soft, rather than hard requirement for consciousness - people with anterograde amnesia are still conscious, right?