THIS FILTER IS WORKING!
It is still very buggy and I hear a lot of cracks and short bursts of really bad distortion in the sound, but the concept seems to work very well!
Not only that, but I think I can prove that the output (when the bugs are resolved, assuming that I can resolve them) is the BEST POSSIBLE APPROXIMATION of the original signal, based on all the present data. This is not an approximation by making estimations from 2 different sides of a clipped region as other applications apparently use - this thing really generates a formula that describes the simplest function that goes through all the known samples - and then it uses that formula to calculate the missing (clipped) samples.
I've run a test with some of the most difficult tracks from the Torture Test (original_wave.flac), clipped at 1/4th of its peak level, and - except for a really really HORRIBLE amount of clipping and static, which really needs to be fixed to make this usable - except for that, it reproduces the original sound perfectly - with apparently NO DISTORTION at all. (I say 'apparently' because it's sometimes hard to determine due to all the static).
Short teaser (first clipped sound, then repaired sound):
http://www.stereotool.com/temp/TEASER.mp3