This probably really stupid

[RestonRobert]

I have been reading about reducing SHA-256 to a boolean function (see https://bitcointalk.org/index.php?topic=55888.0). The outcome seems to be that this is not very efficient compared to straight forward computation and therefore it's not a worthwhile idea.

However isnt the object of mining to produce a hashed result less than a specific value?

So wouldn't it be useful to calculate probable nonce ranges  based on the other input. - rather than just rerunning the algorithm with an incrementing integer. Even if nothing could be predicted for the first few iterations- wouldn't it be useful to use the output to develop a more directed guess for later nonce values.

Maybe I'm missing something here?

[BlockChainLottery]

The nice thing about a SHA-256 hash is that you cannot predict it. If you hash a string and make another hash from that string but changing just one character, you get a completely different hash. They are not correlated, so you cannot use the outcome of one as an input for another to make an educated guess or something.