Does the Baby Jogger City Select seat frame fit on the Baby Jogger City Premier chassis?
Based on the attachment system, it should, YES.
However, it's a smaller seat frame, and the center of gravity doesn't count on a differently sized seat to be fitted, so it might happen that a different pushchair model's seat might affect the maneuverability of the stroller, ranging from just a little to tipping over in some cases (which means endangering your baby).
In general, then, the Baby Jogger City Select seat should fit on the Premier chassis just fine, but it might be a worse stroller experience. That is why I don't recommend fitting a seat that is not meant for the frame.