duplicating a pixel layer in Photoshop gradually decreases its quality

I never thought it baked in the new pixel locations when I duplicated.  I imagined Photoshop kept the original pixels in memory and just described and rendered the translation on the fly... but I guess that would be too cumbersome

yeah, I think you've solved my mystery.  that makes perfect sense.  thank you 

I didn't save this file.  I reworked it, duplicating from the same original layer each time individually, and that fixed it

Picture this: you have a graphic three pixels wide by 1 pixel tall. The first pixel is black, then white, then transparent. If you move the black and white pixels one half pixel to the right, Photoshop has to calculate what goes into each pixel space. The pixel grid stayed put, but the colors moved. That's where the degradation comes from.

Hard to tell what's going on without seeing the layers panel or the actual file to troubleshoot. Can you share a link to the PSD?

Your copied pieces look like they had an excessive anount of Unsharp Masking appled to it.

I'm wondering if you have a Smart layer with a sharpening effect applied that is being reapplied over an over again as you copy?

I have been working on a digital composition of - one diamond - hundreds of times in a single document.

 I took a real diamond photo and imported it to the new document, then rotated its layer and pressed Command J, dragged and rotated again (but never transformed its size). Then I would press Command J again when I liked the new placement, so I could place the next iteration around the composition.  

After about 100 of these iterations the file was very noticably bad... and in my screenshot you can see the original few generations looked ok (on the left), but the ~180th iteration (on the right side) looks like a marble notebook cover.

That is SERIOUSLY odd - Photoshop DOES perfectly and exactly duplicate a layer, pixel to pixel. Can you walk us through what you're precisely doing (how you're doing it) that makes this happen?