Bought a new seat, so figured I should actually read the rules for mounting it. Found 9.3.41 Seats, which includes this:

The back of the seat shall be firmly attached to the main roll hoop, or its cross bracing, so as to provide aft and lateral support.

I certainly know about a seat back support requirement, but didn't know it had to be attached to the main hoop cross bracing (my prior seat's back support was to a bar between the rear wheel wells), nor that it was required to supply lateral support. So I guess this new seat gets rigidly bolted to the cross brace right behind it.

I'm a little surprised at this because I thought some lateral "give" was supposed to be a good thing.