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Just change the display. If you try to send UHD to an HD display, the UHD source gets handshake info from the display that includes the maximum resolution of the display. If the display says max resolution is 1080p, the source device sends 1080p regardless of what you may TRY to send to the 1080p display.

What do you mean by "short throw projector"? Do you mean a laser projector that sits below a wall-mounted screen and shoots up and at a diagonal to the screen? Or a conventional projector with a lens that allows it to focus and fill the projection screen from 6 to 8 feet from the screen? I can tell you this: in ANY home theater, having the projector right in front of you, in your line of sight is a very very very bad idea---it is MASSIVELY distracting. And there is no projector with a conventional high-pressure projection lamp that is quiet in operation --- therefore, the closer the projector is to your ears, the more noise you will hear.

I have 5 projectors here that range in price from $2000 to $35,000 and 3 of them are 4K or UHD resolution while the others have DLP or LCD imagers with pixel shift and will display UHD sources, though with over-sized and overlapping pixels. And I have a 75 inch flat panel TV that can produce 2700 nits with more than 400 local dimming zones and costs about $1900 right now (Vizio PQX series); a $900 Hisense 65-inch model with 150 local dimming zones, and edge lit LG, Samsung Q80R 65-inch. When the next big movie gets here that I haven't seen before, I have all those displays to choose from to watch the movie the first time. The projection screen I have is 110 inches wide (that's something like 125 diagonal maybe) and has motorized masking and 2.40:1 aspect ratio so I can use it for constant image height projection, just like movie theaters. My choice for watching the next big movie would be in this order:
1) Vizio 75-inch $1900 TV (2700 nits, over 400 dimming zones
2) Samsung 65 inch Q80R (1300-1400 nits, forget how many dimming zones)
3) Hisense H9F $900 flat panel TV (1100 nits, 150 dimming zones)
4) $35k projector
6) LG edge-lit TV
7) any other projector here (which range in price from $12,000 down to $2000

Note that a $1900 TV would be my choice over a $35K projector with native UHD/4K resolution (no pixel shifting). People always (almost always, anyway) think "cinematic" means big. That's not correct. What makes something cinematic is the VIEWING ANGLE and understanding that changes everything. A 75 inch TV viewed from 7 feet away has a slightly larger viewing angle that a 110 inch screen viewed from 12 feet. In this example, the 75-inch TV produces a cinematic experience that is SUPERIOR in every possible way to a projector costing as much as 18 of those TVs. The TV has a much larger color volume. HDR looks 1/4 baked on the projector and fully baked on the TV and remember, both are producing images with 3840x2160 pixels in 3 colors. But you are thinking... 7 feet from a 75" TV is too close. But it is NOT too close, it is simply cinematic with a 42.5 degree viewing angle while the projection setup in this example has a 41.8 degree viewing angle. To see all the detail visible in UHD images, you can sit as close as 1.5 times the height of the projection screen. The 75 inch TV is just about 36.5 inches high (picture area), that means you won't see individual pixels on the 75-inch TV screen unless you sit CLOSER than 1.5 times the 36.5 inch height of the screen... so you CAN watch the 75-inch TV as closely as 55 inches (rounded) or 5 inches less than 5 feet. So sitting 7 feet away is really farther than you could be, so you COULD make a 75 inch TV more cinematic than a 110 inch screen viewed at 12 feet.

The expensive projector cannot produce luminance above about 530 nits on a 1.0 gain screen. That means the projector will never display all the colors available in UHD video.. so UHD content will be "dulled down" compared to how it looks on a 2700 nit flat screen TV. Not because peak white is lower, but because you need lots of nits above 1000 to produce the highly saturated colors you can get in UHD video.

I believe projection as a video display technology is on life support. Movie theaters are beginning to install LED screens and will do away with projection completely in the next 10 years or so as cost of large theatrical LED screens comes down. Current movie screens are almost always perforated to allow sound from behind the screen to get to the audience... that causes a loss of about 10% of the light in the images while an LCD screen will get substantially brighter than any cinema projector available at reasonable cost. Furthermore, the light engine and lens in a projector degrade the image--no way to avoid that. TVs have none of those components in the path of the light from the pixels to your eyes so the images you see from a TV are precisely placed and perfectly focused at all times.

My advice to anybody playing around with the idea of getting a projection system... forget about it. Waste of money at this point in time. Put the money you would spend on a projector and screen on a "big"ish TV and use the leftover money to help pay for UHD/HDR content for years. Sit a little closer to the TV screen than to a larger projection screen. Projectors used to be kind of a home theater status symbol, but after so many direct comparisons of content on good TVs vs. on good projectors... TVs make UHD/HDR look incredible and projectors NEVER do, not even a $60,000 projector is wildly outperformed re. image quality by the $1900 Vizio 75 inch TV (or the 65 inch Vizio PQX model that's only $1200 or so right now, just sit 6 feet from it to get the same cinematic viewing angle).

There is an audio element to this also... when you are 12 feet from the screen, and speakers are around 12 feet from you also, the reflected sound in the room overwhelms the directly radiated sound from the speakers. To avoid that, you need to sit 7 to 8 feet from most home theater speakers. That puts you close enough that the direct sound from the speakers will be louder than the reflected sound from every surface in the room. So sitting closer gets you the cinematic viewing angle you want AND the best sound quality in your room.

DISCLAIMERS: I am retired. I was an imaging systems engineer for decades. I did professional video display calibration as a side gig for over 10 years. The TVs mentioned here aren't any kind of "best" TVs list, but each one mentioned is certainly worthy of being owned by someone interested in images that are BETTER than what you can see in ANY current movie theater on earth--provided you use a UHD/HDR disc as the source. You may find an IMAX with a pretty huge viewing angle, but you can just move closer to the TV to replicate that at home if that is your goal. Oh, wait, the edge-lit LG... I would skip over anything edge lit... they just don't get very bright and the vertical banding when they use local dimming is distracting.
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