E-Bike Stop told me my S3X has a two year warranty, so I called Sturmey-Archer America, and talked to David, the lone American S/A representative, about my broken axle nut, and how I can’t get the shifting to work with the new, longer nut. I had worried that I broke the hub completely, but it still shifted, even with the broken nut. Turns out it’s just the nut, and David’s mailing me a “short guide nut,” no charge. Super nice.
The long “guide nut”* I had gotten as a replacement (the right-side axle nut with the hole for the indicator chain), doesn’t allow the gear adjustment you need. I’m not entirely sure how that could be; it seems like you’d just slacken off the cable, but it sure doesn’t work that way. I think the length of the nut and the cable angle must act like a pulley, and change the amount of travel in the shifter.
If you need to replace your S3X axle nut, get the Sturmey number and call directly. A shop may not be able to get the short one from the usual channels, and it will most likely cost $5 or so.
I actually called David back and asked if Sturmey Archer had a quick release for the cable. He thought I meant a quick release for the hub, and said Coombs Cycling Components is going to be showing him an axle quick release system for the S3X at Interbike in a couple weeks. That sounds really cool. The Coombs quick releases look dorky with a QR lever on each side of the axle, but I had a flat the other week with no wrench, so I can see the value.
* I had no idea this is what they were called. Good to know.