Ehh... you can often prove that X algorithm is not susceptible to A, B or C attacks, but you cannot usually prove that the algorithm is "secure" in a fundamental sense.
With some exceptions, those proofs make assumptions about their primitives like having block ciphers be "unpredictable permutations" or having hash functions be "random oracles". These proofs also make assumptions that information doesn't leak in other ways (like BEAST, heartbleed, timing attacks, poor entropy sources, etc).
In other words, those proofs don't have as much real-world significance as you'd like.
Yeah, but this is not what most people think of when they think secure, is it? That is, this proves that people have to actually break an algorithm to know what it is protecting. This does nothing to show that an algorithm is unbreakable.
(As an example of my understanding, rot13 does nothing to randomize distribution of characters, so it "leaks" information about what it has modified. This sort of leakage can be proven as absent from your algorithm. Anything else is a bit tougher.)
You can prove that it's secure in that the algorithm itself does not leak information.