Is someone who solely contributes negative feedback to a paper considered an author?

To moderate Yemon Choi's comment: yes. Your first assertion is that A and B are collaborating, which means they should be co-authors, unless one actively backs out.

For a mathematical project, it's easy to have lots of ideas but not enough time to pursue them all to see which (if any) work. If someone can shoot down ideas and tell you they definitely (or with high likelihood) won't work, this can help put you on the right track. So in your situation I would say B was instrumental in finding a correct solution.

(In a somewhat different abstract scenario where B dismisses some approaches to a problem that A suggests, and B does not otherwise actively work on the project, it may depends on the situation and they should have a discussion about whether B is a co-author or not. And some people will have different opinions about the same situation, e.g. RS versus A in the RSA example.)

In general in a mathematical collaboration, if there's one key idea it's unlikely that both collaborators arrive at it together. Maybe through discussion they enhance each other's understanding of the problem, and then one will have the key idea and the other will encourage/validate it. That doesn't mean was the other person was unnecessary, even if you can't pinpoint parts of the final paper as being "their contribution."

(And if collaborations became competitive to the point of dropping co-authors just because the didn't see the final solution first, who would want to collaborate?)

Pointing out errors is certainly a contribution to the paper, especially if done repeatedly. Indeed, it's possible that B's contribution exceeds A's. If A is getting lots of information of the form "this won't work" and "that won't work", the possibilities for something that does work can get narrowed down to the point where it's rather easy to find a proof (especially if the "won't work" information comes with an indication of why things won't work).

Rather than talking about two people collaborating, think about what happens when you write a solo paper. If your experience is like mine, it may well happen that discovering that early attempts won't work and understanding why they won't work is a bigger part of the job than finding the proof that finally does work.

Pointing flaws isn't something negative in any way. Does the paper/theory worth anything if it has several flaws ? In my opinion he is doing an excellent job. Find every possible flaw in a paper, and fixing it, is what makes it consistent and reliable.

Get an example for something else external to mathematics, like the engineering process of a car. Those who test the car for safety, aren't doing a purely negative contribution, is a step that is mandatory, that is to check for any flaws. The car resulting from a process where no tests were made, will be a choice for you ? Will be accepted by the market ? The same goes for the paper, if B didn't work finding flaws, someone else will, it happens all the time, even with more than one person focusing on finding flaws, happens to someone else, external to the group, to find something.