Would it be possible to program your gpu, or the mining program to only hash under a certain point, i.e. the target? Obviously I am guessing that the answer is no, but why is this?
I think it is because the hash we use (SHA256) was specifically designed to make it hard to demand certain properties of the output from wise choices of input. So that one could trust a file signed with an SHA256 hash was not modified since if one could generate hashes with certain properties (equal to an observed hash) this would not be possible.