There are very few non-trivial proven lower bounds in computational complexity, and none whatsoever in cryptography, as far as I know.
Almost all results are of the form "If X is infeasible, then so is Y." For example, "If AES is a secure pseudo-random function, then so is CBC-AES given parameters such-and-such."
RSA and traditional Diffie-Hellman use integers and finite fields (integers modulo a prime). Those mathematical objects have, in some ways, "too much structure"; there are short cuts to factoring and to discrete log, so you have to use much larger keys than you would expect if you did not know about those short cuts.
As far as anyone knows, elliptic curves have "just enough" structure to do cryptography, but no additional structure that can be exploited to extract discrete logarithms via some short cut. That is why the number of bits for EC algorithms is generally chosen to be something like twice the symmetric key length, instead of something like 2048+ bits for RSA.
If you want some mathematical details, this thesis has a good overview:
But again, none of these problems have been proven to be intractable. Maybe there is some linear-time factoring algorithm out there. Maybe elliptic curves -- or the particular curves approved by NIST -- have some unknown structure that could allow discrete log approaches with comparable complexity to those for finite fields. Maybe the NSA's mathematicians have found that structure ahead of everybody else...