The paper's authors' reply to the comment is really bad. E.g., it says this: > With respect to the assumption, we have clearly elaborated on why a mathematical

The paper's authors' reply to the comment is really bad. E.g., it says this: > With respect to the assumption, we have clearly elaborated on why a mathematical assumption is essential for proving computational hardness results in Appendix A of our paper as follows: In fact, the Turing machine itself is an assumption about machines (i.e., mechanized finite formal systems) that can be realized in the physical universe. In essence, the Turing machine represents a fundamental physical assumption, and Turing’s findings on uncomputability signify the physical limits of mechanically solving all instances of a problem. But Turing's assumption was necessary precisely because it was not mathematical. He had to cross the gap between physics (where computation takes place) and mathematics (which he wanted to use to discuss computation) with an assumption about physics that allowed him to construct a mathematical definition of computation. The authors of the paper, on the other hand, make an assumption within mathematics that could simply be mathematically wrong. It's like "proving" Golbach's conjecture by assuming that if an even number were ever not the sum of two primes, then that number must also be divisible by 17 and proceeding from there. Their "proof" is essentially: we assume that if a tractable algorithm (a mathematical construct) for SAT existed, it would be done in a certain way [1], but that way isn't possible ergo a tractable algorithm for SAT cannot exist. Physics is based on (mathematically) unprovable assumptions. In mathematics, though, "assumptions" are conditionals. The paper should have said, we make a mathematical conjecture that, if true, would lead to our conclusion. You can't assume a mathematical conjecture and then call the entailment (from that conjecture) of X a proof of X. It's calling a proof of `A ⇒ B` a proof of `B`. I agree that a paper that aims to present a mathematical proof that is based on a redefinition of what a mathematical proof means should have led to the paper's outright rejection. [1]: The paper says: "We only need to assume that this task is finished by dividing the original problem into subproblems" (... of the same kind)