We do have proofs that hallucination will always be a problem. We have proofs that the "reasoning" for models that "think" are actually regurgitation of human explanations written out. When asked to do truly novel things, the models fail. When asked to do high-precision things, the models fail. When asked to do high-accuracy things, the models fail.
LLMs don't understand. They are search engines. We are experience engines, and philosophically, we don't have a way to tokenize experience, we can only tokenize its description. So while LLMs can juggle descriptions all day long, these algorithms do so disconnected from the underlying experiences required for understanding.