You're confusing yourself w/ fancy words like "proof space". The LLM is not doing any kind of traversal in any meaningful sense of the word b/c the "proof" is often just grammatically coherent gibberish whereas an actual traversal in an actual space of proofs would never land on incorrect proofs.
My reading of their comment is that a proof space is a concept where a human guesses that a proof of some form q exists, and the AI searches a space S(q) where most points may be not valid proofs, but if there is a valid proof, it will hopefully be found.
So it is not a space of proofs in the sense that everything in a vector space is a vector. More like a space of sequences of statements, which have some particular pattern, and one of which might be a proof.
So it's not a proof space then. It's some computable graph where the edges are defined by standard autoregressive LLM single step execution & some of the vertices can be interpreted by theorem provers like Lean, Agda, Isabelle/HOL, Rocq, etc. That's still not any kind of space of proofs. Actually specifying the real logic of what is going on is much less confusing & does not lead readers astray w/ vague terms like proof spaces.