After skimming its interesting that the majority of the proofs in your list claim P equals NP. I would have guessed it would be more common for proofs to claim the opposite because P != NP makes sense intuitively. That being said it would certainly be more exciting if P did equal NP.
A simple but wrong proof of "P = NP" is easier to write in some ways, since you "just" need to provide a single algorithm for one NP-hard problem, and show that it runs in polynomial time. It looks like many or most of the proof attempts in that list take this form.
A plausible proof of "P != NP" won't be quite as simple to express, since it needs to prove that all such algorithms do not run in polynomial time.
> A plausible proof of "P != NP" won't be quite as simple to express, since it needs to prove that all such algorithms do not run in polynomial time.
That sounds hard but, If for any NP-Complete problem there exists no P solution then for all NP problems there is no P solution. So this proof sounds like it has the right shape.
Well yes! Two things: You have a successful algorithm that runs in P time that solves an NP Hard problem and 2) you can map other NP Hard problems to your problem. Without the second factor, it is only a demonstration of a "range" in the computational realm in question, where p = np or whatever the declaration. Being able to show that your pizza slice is actually an ocean of pizza, and also show that any other shape of pizza slice can be appropriately transformed into the shape you have means p = np. a "solution." Or perhaps better put, it is a funnel through which complex computational patterns can either be reduced or simplified or elegantly correlated, approaching absolutely perfect parallelization of operations. This is just one way to look at it, but essentially intractability is an interesting term to consider.
Please forgive me if my liberal use of the language is an offense
Your 2 is the easy part: "NP-hard" is exactly the set of problems to which any NP problems can be reduced in polynomial time, and there are many known existing examples, both within NP (aka NP-complete) and outside it.
No, some NP-hard problems are outside NP (i.e. harder than NP), while some are inside. The wikipedia article has a good explanation and a good diagram: https://en.wikipedia.org/wiki/NP-hardness
Thanks many for your kind explanation. It makes more sense now. I forgot that important detail that mapping is pretty easy as most involvements reduce to and from 3-SAT
The funny thing is that all these "solutions" are implications but not proof, and one contrary demonstration would actually render them all irrelevant as far as I understand the theory of computational p v np time complexity
