Context: As I understand, in geometric complexity theory, the existence of obstructions serves as a proof-certificate, so to speak, for the nonexistence of an efficient computational circuit for the explicit hard function in the lower bound problem under consideration. Now there are some other assumptions for obstructions that they must be short, easy to verify and easy to construct.
Question: My question is that say I have a problem that I conjecture to be solvable in polynomial time. Then how can I show that there exist no obstruction for this problem, i.e. if no obstructions exist then the problem can be computed efficiently and it is indeed in polynomial time.
Approach: I think, and I may be wrong in this assertion, that showing no obstructions exist can be equivalent to standard reduction of NP problems to other problems whose complexity is yet unknown, in the proof that they themselves are in NP. So then in that case one can, if possible, show that obstructions exist as one tries to reduce an NP problem to the problem under consideration, that way, the reduction is intractable. Also what role does postselection play in all of this? Is it possible to simply postselect on the nonexistence of obstructions? Thanks and pardon the lack of precise statements in my approach and questions.
Just an another example, consider a problem X that we know to be in P. Now let's say we didn't know about that problem being solvable in polynomial time, then is it possible, that one can make the following assertion:
Since no obstructions exist in the computation of X we can say that it is in the class P
From there on, the problem is the easy (computationally) discovery of those obstructions, if even one exists, would show that X is not in polynomial time. However going the other way, i.e. finding that no obstructions exist is a difficult task.