I know $P$ vs $NP$ is an open problem in Computer Science. However the people concerned (at least according to Wikipedia) believe $P \neq NP$. I have two questions:$(1.)$ Given that it is possible to prove lower bounds for problems. Does there exist any problem(s) for which the solution can be verified in polynomial time but no known algorithm can solve it in polynomial time which has a proven lower bound?  $(2.)$ Is this proven lower bound polynomial or exponential?
Lower bounds and $P$ vs $NP$
algorithms;time complexity;asymptotics