Great post. Here are a few things pointers if you want more information about your points:
B) The situation is actually worse than exhausted. We know (have proven) that the regular methods simply won't work (check up "the reletaivzation barrier" and "natural proofs" and "algebrization")
C) We actually can construct problems which take at least a certain amount of time to solve (though admittedly many of them are kind of unnatural). Search up "time hierarchy theorem". This is probably one of the theorems you mention. However, you're right, as far as natural problems go, it appears to be extreme difficult to prove most natural problems takes at least linear time (https://mathoverflow.net/questions/4953/super-linear-time-co...). Note the log*... factor grows extremely slowly.
In fact, it is not known whether SAT requires more than linear time. Since SAT is NP-complete, we expect it to have no polynomial time algorithm... and yet here we are wondering whether it requires even more than linear time. If you are interested in this sort of thing, maybe look up fine-grained complexity. Also note that to solve P != NP, we will have to atleast shown P != PSPACE, since PSPACE contains NP. However, even this seemingly easier problem has had no progress and seems like it would require a huge breakthrough.
From this it seems that the question itself might have been hopelessly naive
Is there any evidence that P might be equal to NP? This seems different from other famous conjectures like Fermat's Last Theorem or Riemann's Theorem where what's hard is finding a counterexample that disproves the theory.
I think RJ Lipton (blog author) and Knuth believe it could be true. However, most complexity theorists don't (I think Fortnow does a survey of theorists on this every few years). Knuth says something like "well all it would take is a single algorithm for any of these hundreds of problems". A fine take for the godfather of algorithms. At the same time, there are many people who have in some sense tried to find such an algorithm, even non-explicitly (e.g. factoring would be in P if P = NP). Moreover, there is a lot of complexity theoretic work which makes it hard to believe it to be true (I think Scott Aaronson's blog has some presentable information about this)
>>B) The situation is actually worse than exhausted. We know (have proven) that the regular methods simply won't work (check up "the reletaivzation barrier" and "natural proofs" and "algebrization")
I've heard of those, even tried to read the proof. Still, my understanding is those claims are ultimately "informal proofs" despite having formal step so the situation isn't entirely closed-up but still very bad.
Maybe this will help with understand the revitalization barrier. Most proof of separations of complexity classes relativize, in that not only do they show
A strict subset B
they also show
A^O strict subset B^O
for ANY oracle O. Most of these proofs don't actually explicitly state this is true, but they are readily to be extended this way (e.g. the diagonlizing proofs of the time hierarchy theorem)
However, we know (Baker, Gill, Solovay) that there exists and oracle O1 where
P^O1 subset NP^O1
as well as an oracle O2 such that
P^O2 not subset NP^O2
Therefore, we know we a proof of P != NP must not relativize or else that would contradict the aforementioned result.
B) The situation is actually worse than exhausted. We know (have proven) that the regular methods simply won't work (check up "the reletaivzation barrier" and "natural proofs" and "algebrization")
C) We actually can construct problems which take at least a certain amount of time to solve (though admittedly many of them are kind of unnatural). Search up "time hierarchy theorem". This is probably one of the theorems you mention. However, you're right, as far as natural problems go, it appears to be extreme difficult to prove most natural problems takes at least linear time (https://mathoverflow.net/questions/4953/super-linear-time-co...). Note the log*... factor grows extremely slowly.
In fact, it is not known whether SAT requires more than linear time. Since SAT is NP-complete, we expect it to have no polynomial time algorithm... and yet here we are wondering whether it requires even more than linear time. If you are interested in this sort of thing, maybe look up fine-grained complexity. Also note that to solve P != NP, we will have to atleast shown P != PSPACE, since PSPACE contains NP. However, even this seemingly easier problem has had no progress and seems like it would require a huge breakthrough.