@Robin When you say that it is "obvious" that visual environments and proofs have a tension, I guess (?) you mean that visual proofs are criticized as being not as rigorous as algebraic, because it seems to be more about intuition. This is an interesting philosophical issue and may end up being a problem for me, but I'm willing to bet that it will be sufficiently non-small
The kinds of proofs I am interested to give are (as you'd expect) fundamentally GA-related, so for example, proving that rxr* is a rotation when r is a rotor and r* is its reverse and x is a vector. Or, expanding a bit, classical mechanics, so proving that ellipses are solutions to the two-body problem. And yes, if I get to it, quantum computing! Subalgebra closure is probably what you were thinking of. It wouldn't be super surprising if the system ends up somewhere in between "strong enough to prove pythagorean theorem" and "strong enough to prove QM subalgebra closure"
"How would you prove the pythagorean theorem?" Thanks a lot for asking me that question, it's a great example to start with and I'd not thought about it :D and now I've thought about it and I know how to prove it. It's a combination of defining a "right angled triangle" by the constraints that are on it (which are visual) and then applying the rules of geometric algebra, which are all visual, to that constraint. I can say more but it might just sound weird/spoil the fun when you eventually see it 😛