Probably not. The point of a cryptographic hash function is to be resistant to analysis.
Can we prove it has the much simpler property that toggling one bit of the input will, on average, toggle half of the bits in the output? (Probably not.)
If you calculate a billion sha256 hashes and look at the results you'll have an even enough distribution to say it's proven, but, it's not "mathematically" proven.
Can we prove it has the much simpler property that toggling one bit of the input will, on average, toggle half of the bits in the output? (Probably not.)