Close. I'm voting for sequential electromagnets, one on each ring. They fire off one after the other, attracting the ring to one on the other end of the arc but repelling from the other end. This would cause the slinky to move. It's basically a biomimietic technology based on how worms move. Deceptively simple, especially with the nanoelectronics to control it and the power source, which is likely some kind of zero-point energy tap.
The problem appears to be that they have hit a "Sorcerer's Apprentice's Broomstick" problem. They can make it go up the stairs. The problem is that there is no way to stop it. Ever.
There are two things that immediately occur to me:
1) The Quinten Quads are clearly smart enough to do anything, no matter how cutting edge and how many boundaries it knocks flat. This is worrying;
2) I have clearly been thinking about how one would make a slinky go up stairs for a long time. This is very worrying.
And that. Both of you have fulfilled so much win! It warms my heart. And don't worry, Ben! It's the perfect qualification to enter the new, secret field of becoming an evil scientist. They pay well! I hear there's an opening at this facility called Arkham!