Actually: I think you meant what I'm saying in the earlier version of this comment. But yes: if "P==NP is undecidable" holds then for any provably-polynomial sa

Actually: I think you meant what I'm saying in the earlier version of this comment. But yes: if "P==NP is undecidable" holds then for any provably-polynomial sat-"solver" the following possibilities hold: - you'd be able to find a proof it was an invalid "solver" (eg: the algorithm that assigns 'YES' to each of the N variables is provably polynomial and provably incorrect) - you wouldn't be able to find a proof it was incorrect (in a strong sense: this isn't 'because you aren't clever enough' but b/c you've stumbled upon one of the 'true' polynomial-time algorithms, ergo there can't be a valid proof it's an invalid solver) I'd guess you couldn't prove that a given polynomial-time algorithm couldn't be shown to be "provably known to be incapable of invalidation"; such a proof => doesn't exist a counterexample (in this case a SAT instance it doesn't get the right answer for), b/c such a counterexample is a refutation => proof the algorithm is correct => contradiction of "P==NP undecidable", but I won't remove the 'I'd guess' from that unless I had a better understanding of the field, as this kind of reasoning is the kind of thing where the technicalities are the important things, and I'm not up to speed on them. Thus in the "P==NP undecidable" universe if you found a creepy algorithm -- in your example provably-polynomial but not known to be correct or incorrect, but "empirically" correct over a very extended period of time -- you'd know not to waste time trying to prove it correct (b/c such a proof would be obviously impossible) but you'd see every attempt at proving it invalid also fail, like you said. Edited for clarity. Edited again for correction.