reviews: 
Comments: The main manifest of the paper is that LLM such as GPT are inherently incapable in the problem of producing rigorous proofs, build on chain of reasoning, unless such a proof has been exactly observed in the training set. It is also mentioned that this fundamental impossibility is due to the attention mechanism, and/or due to the finite-temperature prediction of the architecture, meaning that the next output token is not always the most likely token. 

The proof is based on ideas from first-order logic, which I am not familiar with, but if I understand correctly, goes something like this: Since the proof requires a chain of reasoning comprised of multiple connections, and since each such connection there is a probability that it will not be sampled correctly, then overall the probability of the LLM to output the correct proof decays exponentially with the length.
Strengths: 1) The question whether LLMs can produce rigorous proofs or not is fascinating, and at the center of scientific interest. 
2) The logic-based model is potentially an interesting model to address such questions.
Weaknesses: 1) There is an empirical evidence that LLMs have some capability in providing rigorous proofs, even if currently a weak one. The paper does not seem to address this evidence too much (except the Trinh et al 2024 paper on line 229), and explain why its reasoning is not applicable in those cases. Much more detailed comparison is required. 

2) Section 2.1 presents the first order-logic model, and the main theorems. I am not familiar with first-order logic (as said), but this model appear to be general, and not tightly connected to neither any specific machine learning algorithm, nor LLMs or GPT. Theorem 1 appears colloquial – what does it rigorously mean “necessitates veracity of inference”? How is “literal learning” is defined? To establish the statement targeted by the authors, I think that a much more detailed model is required. Afterwards, there are examples of inference problems the are not solvable by LLMs. These are just mentioned without any justification. 

2) Experiments are lacking: Section 2.2 pretenses a couple of “question and answer” examples, with a few models, and it is claim that “Calude was the most pleasant and sensible”. This part is not scientific. These are just simple experimentation withe models and general impression. 

3) Section 2.3: It is impossible for me to understand this example from the displayed code.

4) The concluding paragraph following – “There is a widespread belief that because the training set” – reads more like a personal opinion rather than a solid conclusion from the paper. 

5) Proofs: These are difficult to follow because there is no methodological structure of accurate definitions of terms, follows by a clear definition of the probabilistic model, before the proofs. For a quick read, there is no technical challenge in the proofs, and these are just a description in mathematical terms of the ideas that explained in the paper in words (except for the use of Borell-Cantelli lemmas to prove 0-1 laws)

6) While it is occasionally mentioned that the attention mechanism of GPT is an inherent cause for its impossibility to provide novel rigorous proofs, this is not actually discussed in the paper.
Recommendation: 1
Reviewer confidence: 4
Soundness/correctness: 1
Clarity: 1