Quite literally! http://theconversation.com/millennium-prize-p-vs-np-4246

NP means that a solution can be verified to be valid. Like the reviewer saying “yep, this is a car all right”. But verifying whether a solution is optimal is not generally even in NP.

You can use the decision version of an optimization problem to answer the optimization problem.

For example when you're looking for the fastest route, you're not directly asking "is this the fastest route?" You're asking "Is there a route shorter than length x?" It's a yes or no question, that's easy to verify given a certificate. If the answer is "yes" then the certificate is a route produced that is shorter than length x.

Assuming the existence of an algorithm that can reliably produce a verifiable "yes" with a certificate or "no", you ask "Is there a route shorter than length x?" and if the answer is yes (which you can easily verify), you keep reducing the length of x in the question until the answer to the question is not yes (no certificate can be produced). The certificate of the limiting "yes" is the optimal solution.

That technique isn't as powerful as sometimes believed. In your example, you've assumed the uncomputable thing, an algorithm which reliably produces a solution to a problem for which one exists. Consider, "Is there a proof in fewer than x lines of the Riemann hypothesis?"

I don’t think that a candidate solution to your problem statement can be deterministically verified in polynomial time.

Any candidate solution produced would be subjective to the interpreter who is verifying it as to whether it satisfies the requirements of a proof (maybe they want more definitions, less shorthand notation, maybe in french, references in APA style with full names, etc, all of which would effect the length of the proof).

So whereas a candidate solution to your question is difficult to verify, it is not in this P vs NP discussion.

Verifying a solution is harder than producing one. Identifying a flaw in a solution is easier than either.

Not sure what you are referring to -- at least in the P/NP problem complexity classification verifying the solution is easy, coming up is P/NP-hard, i.e hard.

So if I gave you a flawless solution to a sudoku problem, would you tell me that it was harder for you to verify it than it was for the solution to be produced?

I had to do exactly this in a practical coding interview. Fun was had.

I think that really depends on the problem space. If I had to hire a sales team to increase revenue, then I hire them and check the revenue numbers (with basic accounting for other secondary variables), its pretty straight forward to validate.