This isn’t my area of expertise, but I think I have a sketch for a very simple weak proof:
The conjecture states that V runtime and π length are polynomial in C size, but leaves the constant open. Therefore a counterexample would have to be an infinite family of circuits satisfying P(C), with their corresponding π growing faster than polynomial. To prove the existence of such a counterexample, you would need a proof that each member of the family satisfies P(C). But that proof has finite length, and can be used as the π for any member of the family with minor modification. Therefore there can never be a proven counterexample.
That’s an interesting point! I think it only applies to constructive proofs, though: you could imagine disproving the counterexample by showing that for every V, there is some circuit that satisfies P(C) but that V doesn’t flag, without exhibiting a particular such circuit.
This isn’t my area of expertise, but I think I have a sketch for a very simple weak proof:
The conjecture states that V runtime and π length are polynomial in C size, but leaves the constant open. Therefore a counterexample would have to be an infinite family of circuits satisfying P(C), with their corresponding π growing faster than polynomial. To prove the existence of such a counterexample, you would need a proof that each member of the family satisfies P(C). But that proof has finite length, and can be used as the π for any member of the family with minor modification. Therefore there can never be a proven counterexample.
Or am I misunderstanding something?
That’s an interesting point! I think it only applies to constructive proofs, though: you could imagine disproving the counterexample by showing that for every V, there is some circuit that satisfies P(C) but that V doesn’t flag, without exhibiting a particular such circuit.