After building an entire optical simulation system yesterday which showed me that my theory was wrong, I started to rethink what I was trying to do. Seeing where the rays of light actually went gave me a much better understanding of how such a system really operates.
So today I jumped back in with a new approach to the problem. Fortunately I already had the tools built from yesterday, so the new approach took much less time. And this new approach worked!
Whether it will actually work in the real world is a whole other question. There are issues of materials, manufacturing processes and tolerances to work through, so even if the basic mathematical model itself checks out, the thing might still not be practically buildable.
But now, thanks to the insight I got from trying and failing, I’ve managed to go from “theoretically, this won’t work” to “theoretically this could work just fine.” And that’s progress of a sort.