Best of luck with this, botnet, but I feel it's a wild goose chase. I first heard about bitcoins in autumn 2011 and immediately set about trying to 'simplify' sha256(sha256(x)) in the same way as you, by trying to find boolean expressions for the first bits of the output in terms of the 640 bits of input. As you've found out, it's the seemingly simple addition of two 32-bit numbers that's the big problem. It simply doesn't lend itself to being simplfied, as it's recursive. As you crawl along the bits from right to left the expression balloons exponentially. All the other operations in SHA-256, the logical connectives and the rotations/shifts are trivial by comparison.

You might want to look into the structure of the block header to see how the 640 bits change at different rates. Fr'instance I believe the version number (first 32 bits of the header, but in stupid-endian) has been 1 for most of bitcoin's history but may be 2 now or soon. That's 30 zero bits right there. Then you have the first (or, goddammit, probably last) 32 bits of the previous hash guaranteed to be zero. The difficulty changes every two weeks, the merkle hash every block, and so on.

But I still think you haven't quite grasped the magnitude of the P/NP problem yet! Then again, if the impossible ever gets done, it'll be as always, by someone who didn't know it was impossible.