Turns out there's not a lot of info on the "Byzantine Chain Sheet" (which sounds like a paradox, it's a chain AND a sheet!) I call it Byzantine Interwoven Sheet because there's also a way that you could do something similar without having it interwoven.
At core, this is two strings of byzantine put together, but they share rings down the center, and thus are interwoven. This has a very curious effect, as normally Byzantine is a very tight-AR weave, requiring something around 3-3.5 AR to work correctly, but this is using like 5.3 AR and could actually use greater. I think 5.5 or 5.7 would give slightly different results, due to the way the rings sometimes touch, which would change if the rings were a different size. When I made the starting chain for this, I was using 3 connector rings, but once built up, 1 connector works fine because the whole thing changes. It's less about the ring AR and more about the weave shape. The pure structure of the weave keeps its shape, so larger rings can work more easily due to the different tension that the sheet-style imposes.
End result? Very thin rings, loose design, a bit more delicate appearance, but still fairly strong. This makes for a thoroughly glorious weave and I love it.
OH, and this is a work in progress. It's currently 2 rows wide, but I intend to make it 3.