We need first to agree on the fundamental point that it is VERTICAL lines of resolution which is used to define the resolution of an image, and has been for decades.
We called 480p, “480”, because of its 480 vertical lines of resolution.
Same goes for 720p—720 lines of vertical resolution.
Same even goes for 240, which was the VHS and previous television standard.
We called 1080p, “1080”, because we were counting vertical lines of resolution.
The same thing applies to what you wrote here:
Digital Cinema “4k”: 4096×2160
Ultra HD Home Video: 3840×2160
Flipping the numbers around and placing Vertical resolution at the end (2160), and Horizontal resolution at the beginning (3840 or 4096), does not suddenly, magically, make your resolution 4K. Both of these are STILL 2K, because, again, video resolution has always been defined by lines of Vertical resolution, which in both cases is, 2160. 2K. 2000.
It appears that the consumer market has been brainwashed and reprogrammed with new marketing nomenclature, and manipulated into believing a total lie as the new standard.
Now suddenly, just because we flip V x H, to H x V, we have double the former resolution? Obviously, in widescreen format, the horizontal resolution is almost double that of the vertical. So if you flip the expression 1080 x 1920, around to 1920 x 1080, you can give the illusion of having “2K” resolution, but that is a cheap cheat, probably snuck in by marketing strategists. I call foul on this, based on decades of precedent. 1080 was always 1k, and doubling it to 2160 will only give you 2k, which is what UHD actually is.
Regarding 1:1 pixel mapping.
I was referring to significant pixel loss during actual playback between a UHD source device and a UHD television. Joe Kane talked about this at length on an AVS interview a few years ago. Because of the bandwidth limitations of HDMI, and fear that consumers would balk heavily at AGAIN having to repurchase all their equipment to get the video bandwidth needed to transmit the full video signal of UHD along with full audio, the decision was made to trim the source material by 10 or 12% to allow for passage through the insufficiently narrow HDMI 2 bandwidth. The content on UHD discs will actually be trimmed before passing to your television. Then either your player or TV has to rescale (read STREEETCH) the downgraded image to fit its full array of pixels. There will certainly be a loss of clarity due to this process, what to speak of the boarder information in your picture.
This kind of crap is exactly why videophiles jubilantly switched from DVD to Blu Ray. No more 3:2 pull down, no more motion blur, jaggies and scaling artifacts. A true, 1:1 pixel presentation!
Now, shot right back to hell by the new format.