so the puzzle author could "cheat" and just present a 256-bit number and not know the preimage at all, which would be a fun shortcut.
In a secure hash function, all output bits are without bias. So all combinations exist.
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.