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I think it is because you are inputting that into the resolution box so you know what it is. that info box/cell in the upper left displays it so you can see what it is.
Yes, that's a possible reason, but it doesn't seem like a particularly good one. My experience with Lightroom suggests that things are done for good, solid reasons.
Here's an example illustrating the problem caused by this behavior. In order to enter a printer-friendly resolution in the Print Job panel, you check Print Resolution. Then you look to your left to see the printer-unfriendly resolution in the cell, which you need to select the appropriate printer-friendly resolution. But it has disappeared. So you need to toggle Print Resolution off to see it again. That's awkward and error-prone. It would be particularly confusing to new users.
The value displayed in the cell is read-only, so what's the harm in simply displaying the value you enter in the Print Job?
I think JSM is right. It is showing the resolution that is used when printing. So, if you enter your own value the calculated from the image is hidden. It may be a good idea to still show the calculated, and the used in addition. Something like: 816 or with own value 816 > 360 .. or whatever
You may want to post this idea here: Photoshop Lightroom | Photoshop Family Customer Community
IMO the better (and more comprehensible) metric to show here, in this circumstance, would be the resampling FACTOR.
Your original pixels when cropped a certain amount and placed at a certain scaling on the page, lead to an effective native PPI (pixels per inch) for that image in that circumstance. That's useful to display while you are still working out what the images are capable of; how far they will "stretch" (which is an art as well as a science). This needs to, and does, change whenever you change the size of that image.
Displaying on an image that it is going to be resampled to print to 360dpi, say, does not help you in sizing a given image - if this resampling is going to get applied to the whole page anyway, regardless of the sizing of that image. It also doesn't help you decide which is the most appropriate "page" DPI to set.
But displaying how an individual image's current "effective native PPI" relates to the current "imposed page DPI", helps you in making both of those decisions: perhaps displayed as a percentage, or as a ratio.
Experience tells us empirically (for each physical printing method, perhaps different for matt and gloss paper say) what degree of upsizing for printing, is found to work acceptably in practice. Maybe we can go to 150% size factor, but get diminishing returns above 200%, or whatever.
So IMO that's what it might be useful to see against each image, after we've checked this "resolution" box and entered a figure.
As I understand printing in LR, you set the 'printer-friendly' resolution, usually 360 (the default value for Epson printers), in the Print Job panel - printer resolution. The value shown in the image with the dimensions is simply the pixels available at the size you have chosen to print. If this is very low it gives you the chance to reduce the size of the print. Then LR resamples the image from the size shown in the image to the 'printer-friendly' value you have set in the Print Job panel, and then sends that to the printer.
AFAIK you cannot change the resolution value shown in the image, other than by changing the print size. I find the resampling in LR is good enough to allow me to print with values as low as 180 or thereabouts, although I prefer values above 240. But you don't resize just to get a value of 360 shown in the image, unless you are not bothered about the size of your prints.
It's much simpler than PS printing!
Thanks @richardplandon. You gave me the good, solid explanation I was looking for!