It would be easy to construct a proof that showed there was some qualitative difference between any two addresses or groups of addresses that corresponded with differences in private keys. It might be an extremely tiny difference but that is all that is needed.
Easy? I doubt it. You seem to think that a hash is calculated from a mathematical formula whether in fact the process is more akin to shuffling and combining a list of bits.
Many people believe that bitcoin's cryptographic strength comes from the high number of possible keys.
Right, it is a common misconception carried on by popular culture.
That does make brute forcing near impossible, but it also makes a more sophisticated attack much easier.
Easier? Why? Just because people don't know about the iron door doesn't make it less sturdy.
I'll research it a bit more and decide whether to admit defeat or not.
I recommend reading the papers on MD5. It was successfully cracked and its construction is the same as SHA-256.