it’s a version of field theory where the rules aren’t all properly explained?

Literally lol, that's the funniest thing I've heard in a few days. I do remember studying physics and getting a bit like "ok but wtf are numbers cause this is a mess?" and studying sets in my own time, then a bit of field theory and going "Oh yes, this actually makes sense now" although I was only ever at the level of amateur dabbler.

I burned out of a physics PhD (funniest thing I've heard that referred to is "I have a post mortem in X") but even by the time I started it I just wished I could go back and choose pure maths, it's not like there are jobs for physics anyway :P (unless you want to make bankers richer, or build weapons which both indelibly mar the soul). Algebraic topology is something I will understand one day. I just need more time, and to move closer to a uni. Even if I'm like 70 and wasting state money I'm gonna study it some day.

Riemann spheres are awesome, I just want to be careful with my language in a space where people don’t even know what a field is, generally.

Yeah it's reasonable, I guess I don't want people just thoughtstopping at "you can't divide by zero" and never thinking there's anything deeper to it instead of maybe going "well, what is division by zero why can you do it sometimes and not other times? why do we sometimes pretend we can do it in systems where we can't?". At the end of the day I think we're on the same side of "maths is cool and people should learn more of it's nuances" but worry about people taking different things from the same trivial remark haha.

Good luck on getting back to it! I hope you do better than me at your studies. It's such a fun little world of order and puzzles.