Can it be mathematically proven that a block can be solved? In other words, that there exists a nonce for the block that will result in sha256(sha256(block header + nonce)) hash under the current difficulty barrier number.
The clients make use of the Target to determine if a block hash is valid and will be accepted by the network. The Bitcoin wiki states the following:

The target is a 256-bit number (extremely large) that all Bitcoin
clients share. The SHA-256 hash of a block's header must be lower than
or equal to the current target for the block to be accepted by the
network. The lower the target, the more difficult it is to generate a
block. [...]
Each hash basically gives you a random number between 0 and the
maximum value of a 256-bit number (which is huge). If your hash is
below the target, then you win. If not, you increment the nonce
(completely changing the hash) and try again.

The current target value can be found here: http://blockexplorer.com/q/hextarget