Assuming P provably != NP, banks and online retailers win (certain forms of encryption even theoretically can't be broken in P time), NSA supercomputer contractors might lose (certain forms of encryption even theoretically can't be broken in P time). UPS and FedEx likely lose (traveling salesman problem is NP-complete). Any other business that hinges on solving hard problems might lose, though counterintuitively, some might win--firms that do a really good job at approximate solutions to NP problems (is ITA an example?) are extremely talented at doing something really hard, whereas if P = NP, it would be easier for any old firm to develop a perfect solution.

But considering the fact that we've all been operating under the assumption that P != NP, the losers don't lose much (if anything) and the winners don't win much (if anything). If P = NP was proven, it's possible but not guaranteed that bank robbers and whatever software firms can pivot fast enough to exploit P = NP get huge windfalls, internet retailers would be ruined, banks who didn't shut off their data links fast enough would be ruined, etc. This worst case scenario hinges on a proof that took the form of a proof by counterexample, which happened to neatly solve an NP-complete problem in efficient P time. In better case scenarios, where other proof forms were used or the P-time solution was a horrific factor like O(n^100,000), online retailers would still lose a little if only based on media hype about the discovery scaring grandmothers away.

There sure is: http://www.claymath.org/millennium/P_vs_NP/

Ah, yes. No, I meant for the outside investor with early news of the purported proof who is willing to bet that it holds up.

For example, if there is some crypto company whose business is premised on hedging that a "P == NP" proof is just around the corner - short them. Alternatively, maybe buy the firm we think of as RSA. That kind of thing.

This paper hasn't yet got a lot of press attention and I'm only about 1/4 joking when I say I'm curious as to what effect it will have on various stocks if it isn't quickly debunked.

HP will get some prestige out of this, and not much else will happen short term. Almost everyone was already assuming P != NP.

And that doesn't even scratch the security implications. Cryptography based on complexity theory would become an instant relic. The only cryptography I know of that could stand up to a P vs NP solution is quantum cryptography. Of course, there's the distinct possibility the NSA and other government sponsored institutions are already aware of a solution.