As this discussion continues, I thought I would add the following, for reference...
I revisited Greg Rogers review on this projector (Jan 2007), pre-firmware fix. He says that it accepted 1080p24, 1080p24sf, and 1080p48 signals over HDMI, but that they did not display at the input frame rate (or exact multiple), and displayed significant judder and dark flashes.
He also says the XV-Z20000 produces "pixel perfect" images when its DVI or HDMI inputs are driven by 1080i or 1080p digital video signals.
(I wish he had tested the projector again, after the firmware fix - anybody know if he did?)
After reading the other thread I mentioned earlier (see post #232), and looking at the resolution chart on page 60 of the owners manual, it would seem to me that it would be far easier to design a vert. osc. circuit to lock onto the incoming vert rate, and display that rate, than to convert the noted incoming freq. rates [43,50,56,60,70,72 (yes-72),74,75,85,90,96,100], back to 50 or 60hz. So, it would not surprise me, if the Sharp displays at other rates.
I suggest we pursue more testing, to nail down a conclusive answer here, using direct 1080p24 and 1080p48 sources, with both the DVI and HDMI inputs, with autosync and manual experimentation.
Gerry, can you be very specific on the details of your hookup and menu selections?
Thanks to all, for your help!!

Tom