1800p is 3200x1800, or four times 900p, right?

And checkerboard rendering draws half as many pixels per frame as the native equivalent, so that's 2,880,000 pixels per frame, or about 40% more pixels than 1080p?

That is correct.

In that case, if the base PS4 can hit a stable 1080/60fps, you'd think the Pro could do checkerboard 2160p as that would actually be 2K in terms of pixel fill, not 4K.

It should, but it is still in beta, and we don't know if anything else is going on as far as processing goes. Oh, and I believe checkerboarding still requires some power.

But I would think that by launch it would be checkerboard 2160p.

Checkerboard still has some overheads

http://www.eurogamer.net/articles/digitalfoundry-2016-inside-playstation-4-pro-how-sony-made-a-4k-games-machine

Checkerboarding up to full 4K is more demanding and requires half the basic resolution - a 1920x2160 buffer - but with access to the triangle and object data in the ID buffer, beautiful things can happen as technique upon technique layers over the base checkerboard output.

"First, we can do the same ID-based colour propagation that we did for geometry rendering, so we can get some excellent spatial anti-aliasing before we even get into temporal, even without paying attention to the previous frame, we can create images of a higher quality than if our 4m colour samples were arranged in a rectangular grid... In other words, image quality is immediately better than 1530p," Cerny explains earnestly.

"Second, we can use the colours and the IDs from the previous frame, which is to say that we can do some pretty darn good temporal anti-aliasing. Clearly if the camera isn't moving we can insert the previous frame's colours and essentially get perfect 4K imagery. But even if the camera is moving or parts of the scene are moving, we can use the IDs - both object ID and triangle ID to hunt for an appropriate part of the previous frame and use that. So the IDs give us some certainty about how to use the previous frame. "

So how does geometry rendering vs checkboarding work out in terms of their plus and minus points?

"[With] checkerboard rendering, the first two pluses are the same: crisp edges, detailed foliage, storm fences, but also, increased detail in textures, increased detail in specular effects," Mark Cerny explains. "But we're doubling pixel shader workload, there are other overheads as well and it may not be possible to from 1080p native all the way up to 2160p checkerboard."

So it might indeed not be possible.

If anyone likes watching cars go no track this game is an treat to eyes. By far the best looking racing game and it is on beta stage. GT6 probably has best skid marks and smoke affect even today
https://www.youtube.com/watch?v=PrDI3CMthqQ
https://www.youtube.com/watch?v=HxKKptCZrrE&t

and that is not on the beta version. I expect retail version to be even better. It is a shame though that no daynamic TOD, weather is there. Hopefully like DC they can patch it ? May be for PS4P but I do not think it will happen