Obviously we all have strengths and weaknesses in our skiing, and I certainly have weaknesses. I certainly can't do the Ted Ligety-on-acid-with-a-clown-wig Charleston turns that I describe above, and probably never will be able to. Huge turns at cheek-flapping speeds aren't particularly my forte either. However, I don't think it's too conceited to say that on a good day I have some ability with regard to carrying off the kind of short-radius clean turns being discussed in a bunch of the posts. Some of my ski friends have asked about this too. They seem to suspect I have some kind of joint mutation or something that makes it unfairly easy for me. ;) Then I point out that most of the local high school slalom racers can do it too (much better than I), so it has to be something you can learn. Some of it is skis - it helps not to be on GS boards - but not all of it. I'm not an educator, so I'm likely to make a hash of this, but I'll try to articulate some of what FEELS like truth to me about this kind of turn.
1) Yes, I am aiming for a carved turn here, but there is something about practicing solid bump turns that helps remind me what I'm supposed to be doing with my feet / ankles / shins. This remains true even though my bump turns cover a spectrum from semi-carved to entirely skidded, and are essentially never arc-to-arc. I imagine that this is about becoming familiar with the fact - at least, I think it's a fact - that if you are going to be making a lot of turns in a short amount of time, it's mostly going to be happening from the knees down. You just don't have time to be cycling the entire length and mass of your legs back and forth from left-hip-on-the-snow to right-hip-on-the-snow when you're making turns this short and quick.
2) It starts with the feet and is followed by two knee dips. I have some friends who are pretty good at GS style arcs. They are patient. They wait for the ski to drift out from under them, and let it gradually carry their whole (upper and lower) leg out away from their bodies. Then they are very patient again and wait for the ski gradually to come around underneath them for the transition. It seems like they are using mostly their hips to get the angle. By contrast, when I am feeling good about my high-cadence short turns, I think I'm starting the turn using mostly some pretty aggressive foot and ankle flexing. The sensation is similar to one you can get pretty easily on ice skates, even if you're not an expert skater; you're not rotating, but you're definitely tipping and guiding forcefully with a lot of forward and lateral pressure on the cuff of the boot (skate). In the middle of the turn, I'm doing a pronounced little dip with my inside knee, but not sinking inside much with the hip or following the turn with my shoulders. This first knee dip is the catalyst or accelerator that really gets the ski coming around quickly. Then there is a second knee dip - outside knee this time - at the end, that finishes the turn and gets me to the transition quickly. I think this is what some of the more well-schooled posters are calling retraction. Making these two knee dips really assertive and meaningful, without a lot of associated leaning or hip action are, I think the pieces that a lot of people miss out on.
My two cents. Ignore if it seems like hogwash in your body's terms.